Novel magnetic order in pseudogap state of high Tc copper oxides superconductors

1.3 MB·English

by  Philippe Bourges

Physics
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0 Novel magnetic order in pseudogap state of high Tc copper oxides
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superconductors
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J Philippe Bourgesa, Yvan Sidisa

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aLaboratoireL´eonBrillouin-Orph´ee,CEA-Saclay,91191Gif-sur-Yvette,FRANCE
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Received*****;acceptedafterrevision+++++
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o
c
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r Abstract
p
u One of the leading issues in high-Tc copper oxide superconductors is the origin of the pseudogap phase in the
s underdoped regime of their phase diagram. Using polarized neutron diffraction, a novel magnetic order has been
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t identified as an hidden order parameter of the pseudogap as the transition temperature corresponds to what is
a
expectedforthepseudogap.Theobservedmagneticorderpreservestranslationalsymmetryaspredictedfororbital
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momentsinthecirculatingcurrenttheory.Beingnowreportedinthreedifferentcupratesfamilles,itappearsasa
- universalphenomenonwhateverthecrystalstructure(withsingleCuO layerorbilayerperunitcell).Todate,it
d 2
isthefirstdirectevidenceofanhiddenorderparametercharacterizingthepseudogapphaseofhigh-T cuprates.
n c
To cite this article: P. Bourges and Y. Sidis, C. R. Physique xx (2010).
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R´esum´e
[

Ordremagn´etiquedel’´etatpseudogapdesoxydesdecuivresupraconducteurs`ahautetemp´erature
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critique Un des probl`emes majeurs des oxydes de cuivre supraconducteurs a` haute temp´erature critique est
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l’originedelaphasepseudogapdansl’´etatsous-dop´e.Enutilisantdesmesuresdediffractiondeneutronspolaris´es,
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nousavonsidentifi´eunordremagn´etiquecach´equiapparaˆıta`latemp´eratureattenduedecet´etatdepseudogap.
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7 Cet ordre magn´etique respecte la sym´etrie de translation du r´eseau comme cela a ´et´e pr´edit dans la th´eorie des
1 boucles de courants circulants. Nous avons g´en´eralis´e cette d´ecouverte dans trois familles distinctes de cuprates.
. Cettemesureestlapremi`erepreuveexp´erimentaledirected’unparam`etred’ordreuniverseldel’´etatdepseudogap
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des oxydes de cuivre supraconducteurs.
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Pour citer cet article : P. Bourges et Y. Sidis, C. R. Physique xx (2010).
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1
:
v Keywords: Neutrondiffraction,high-Tc superconductors,pseudogapphase
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X Mots-cl´es:diffractiondeNeutron,supraconducteurs hautetemp´eraturecritique,phasedepseudogap
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a
Emailaddresses: [email protected](PhilippeBourges),[email protected](YvanSidis).
PreprintsubmittedtoElsevierScience January20,2011
1. Introduction
Theoriginofhigh-T superconductivityincopperoxidematerialsisstillhotlydebatedmorethantwentyyears
c
after its discovery. On the one hand, conventional phonon-mediated superconductivity has been advocated to
explainanomaliesinelectronicspectroscopiesalthoughtheelectron-phononcouplingseemsinsufficienttoexplain
the high value of the critical temperature. On the other hand, the antiferromagnetic (AF) spin fluctuations
observed in these strongly electronic correlated materials could also lead to an unconventional superconducting
(SC) pairing mechanism, where the conventional electron-phonon coupling would be replaced by a spin-fermion
coupling. For instance, unconventional AF excitations, the so-called resonance modes, have been reported in the
SCstateofmostofcupratesfamilies[1].Assumingaratherlargespin-fermioncoupling(∼1eV),thesecollective
AF excitations may account for several anomalies in charge excitation spectra and for the angular momentum
dependenceoftheSCgap(seeforinstanceRef.[2]).However,thespinfluctuationmediatedpairingscenariohas
to face a serious problem: high-T superconductivity survives even when the AF spin fluctuation spectral weight
c
becomes strongly reduced. If certain aspects of the physics of cuprates can be understood using either phonons
or AF spin fluctuations, the mystery of high-T superconductivity in cuprates remains unsolved.
c
Allthehigh-T SCcupratesshareacommoncrystallographicstructure:Theyarelayeredmaterialscharacterized
c
by the stacking of CuO planes. These planes are paved with squared CuO plaquettes. The charge density in
2 2
CuO planes can be tuned using either electron or hole doping. All the hole doped high-T SC cuprates exhibit
2 c
the same remarkable phase diagram (Fig. 1). These compounds are AF Mott insulators at zero doping. The AF
state is quickly destroyed once a small amount of doped holes is introduced in the CuO planes. Increasing the
2
hole doping, the system becomes metallic and superconducting below the critical temperature T . At optimal
c
doping (p∼0.16 holes /Cu), T reaches its maximum value. Two distinct regimes develop on both sides of the
c
optimaldoping.Intheoverdopedregimewhenincreasingtheholedoping,theelectronicpropertiesinthenormal
statecanbedescribedusingamoderatelycorrelatedFermiliquidpicture.Atvariance,intheunderdopedregime
when reducing the hole doping, the materials behave as a strongly correlated metals and the standard Fermi
liquidpicturefailstoaccountfortheirunconventionalelectronicproperties.Inparticular,theunderdopedregime
is dominated by a phase with highly unusual physical properties where magnetic, transport and thermodynamic
measurements point towards a diminution of the electronic density of states below a temperature T* [3,4,5]
although the system remains metallic. As the density of states remains non-zero at the lowest temperature, this
phasehasbeennamedasapseudogapphasesinceitsfirstevidence[6].However,belowT*,aq-dependentgapopens
500 Underdoped Overdoped
400 T
N
e T*
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t 300 strange metal
Tempera 120000 AF psTecudogap SC 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id
0
0 0.05 0.1 0.15 0.2 0.25 0.3
hole doping p
Figure1.Genericphasediagramofhightemperaturesuperconductingcopperoxidesasafunctionofholedoping(p).IntheMott
insulationstate,theantiferromagneticphasedevelopsbelowtheN´eeltemperature(TN).Upondoping,thesystembecomesmetallic
andsuperconducting(SC)belowthesuperconductingcriticaltemperature(Tc).Inthenormalstate,thelostofspectralweighton
certainportionsoftheFermisurfacehighlightstheopeningofthemysteriouspseudogapphasebelowthetemperatureT*.
2
inthesingle-particleexcitationspectrummeasuredinangle-resolvedphotoemission(ARPES)experimentswitha
maximumatthewavevectorM=(π,0)andsymmetryrelatedpointsoftheBrillouinzone.Otherspectroscopies[4]
spanning from scanning tunneling microscopy (STM), electronic Raman spectroscopy and optical conductivity
reportedanomaliesatsimilarenergy.Theunderstandingofthemicroscopicoriginofthepseudogapphaseandof
its interplay with unconventional d-wave superconductivity has become one of the major challenges to overcome
in order to crack the mystery of superconducting cuprates.
Two major classes of theoretical models attempt to explain the pseudogap state. It has been at first proposed
thatthepseudogapphenomenoncanbedescribedasaprecursorofthesuperconductingd-wavegap,butwithno
phase coherence, which would occur only below T [7]. This preformed pairs scenario has largely been favoured
c
over the years in relation to the doping dependence of the SC gap which does not follow T (as it is expected
c
in the conventional BCS theory of superconductivity) but rather T*. This has been discussed in conjunction of
manyexperimentaltechniquessuchasARPES,STM,electronicRamanscatteringexperiments.However,dueto
lacking experimental evidence of such preformed pairs[8], this approach is questioned.
Many other theories attribute the pseudogap origin to the proximity of a competing state, but there is a
wide disagreement about the nature of this state. In certain theoretical models, the pseudogap state is a long
range ordered phase, with a well defined order parameter and a related broken symmetry [9,10,11,12]. In this
scenario,theorderingtemperatureT*andtheorderparametershouldvanishataquantumcriticalpoint(QCP)
locatedbehindtheSCdome,closetooptimaldoping(seedashedlineinFig.1).Alargenumberofexperimental
properties ranging between transport, thermodynamic and magnetic [5] point towards the existence of a QCP
for a doping level of p=0.19. The order parameter may involve charge and spin density waves or charge currents
flowingaroundorinsideCuO plaquettes[9,10,11,12].Interestingly,whilethelow-dopingphasemaycompetewith
2
superconductivity,thefluctuationsassociatedwiththebrokensymmetryareexpectedtoberatherstrongaround
the QCP and could play the role of pairing glue leading to high temperature superconductivity. This appealing
scenario has nevertheless to face two major experimental facts: there is no clear jump of the specific heat at T*,
as expected for many phase transitions (but not all) and there is no indication that the translation symmetry of
the lattice (TSL) is broken in the pseudogap state.
Alternatively, other theoretical approaches have been developed still on the basis of a competing electronic
instabilityresultingfromstrongelectronicinteraction,butintheseapproachesthepseudogapstateisnolongeran
orderedphase[13,14,15,16,17].Itisadisorderedphasedominatedbythefluctuationsassociatedwithacompeting
electronic instability and T* is only a crossover of dynamical properties. Nevertheless, a true ordered state could
be stabilized at low temperature at the expense of superconductivity when applying an external perturbation
(externalmagneticfield,disorder,structuraldistortion,etc...).Suchanapproachiswellillustratedbythestripes
modelwherechargesself-organizetoformlines,separatedAFdomainsinanti-phase[13].When1Dchargestripes
appear,butstillfluctuate,theC4rotationalsymmetryisfirstbroken.Whenfluctuatingstripesarepinneddown
onthelatticeordefects,theybecomestaticandtheresultingchargeandspinorderultimatelybreakstheTSL.In
La Sr CuO (LSCO),incommensuratespinfluctuationsareusuallyinterpretedintermsoffluctuatingstripes,
2−x x 4
that can be pinned down at low temperature by disorder or stabilized under an external magnetic field [18]. A
static stripe phase has been further reported in La Ba CuO or in (La,Nd) Sr CuO around the critical
2−x x 4 2−x x 4
hole doping p=1/8. In addition, in strongly underdoped YBa Cu O (p=0.085) [19] as well as for similar
2 3 6.45
dopings [20], anisotropic incommensurate spin excitations are observed at low temperature, suggesting that the
C4rotationalinvarianceofthesystemisspontaneouslybrokenbelow∼150K.Thesefluctuationsfurtherfreezeat
verylowtemperature,yieldingaglassyshortrangespindensitywave(SDW)state.ThisSDWstateisenhancedby
applyingaweakexternalmagneticfield[21],butstillremainsatshortrange.Meanwhile,smallFermipocketshave
beendetectedinquantumoscillationmeasurementscarriedoutathighmagneticfield[22,23,24]inYBa Cu O
2 3 6+x
system (only for x≥0.5, i.e, p≥0.09). The most simple explanation for these Fermi pockets is a Fermi surface
reconstruction[23,25],triggeredbyaspinand/orchargedensitywaveorderbreakingtheTSL.Thereconstruction
oftheFermisurfaceinferredfromquantumoscillationmeasurementscouldbelinkedtotheSDWorderingreported
in YBCO with and without magnetic field [19,21,20], but for a lower hole doping (x<0.5, i.e, p<0.09). However,
one should be rather careful since the correlation length associated with the glassy SDW state are too short to
account for the observed quantum oscillation and no quantum oscillation have been reported so far at the small
hole doping at which the SDW is observed.
Infollowingsections,wewillpresentcompellingevidenceofanorderedmagneticphaseinthepseudogapstate.
We will show that this order phase displays strong similarities with the circulating current (CC) phase proposed
byC.M.Varma[10,11].TheCC-phaseisassociatedwithaQ=0electronicinstabilityandthereforepreservesthe
TSL.Furthermore,itsgeneralizedIsing(Ashkin-Teller)phasetransitiondoesnotproduceanystrongjumpinthe
specific heat for a large range of parameters. While a 3D long range Q=0 magnetic order is clearly visible in the
3
pseudogapstateofsystemsuchYBa Cu O (YBCO)orHgBa CuO (Hg1201),wewillshowthatthisorder
2 3 6+x 2 4+δ
is frustrated in LSCO and is likely to be in competition with a stripe instability.
2. Evidencing circulating currents
2.1. Loop current order
Beyondusualchargeorspininstabilities,moreexoticelectronicphases,spanningvariousCCstates[9,10,11,12],
havebeenproposedtoaccountforahiddenorderparameterassociatedwiththepseudogap.Onemayhaveasingle
chargecurrentperCuO plaquettestaggeredintheneighbouringcell,referredtoasaD-densitywave(DDW)[12].
2
TheDDWstateimpliesadoublingoftheunitcell.Theinterestforsuchastatehasbeenrecentlyrevived[26,27]
owing to the observation of small Fermi pockets in quantum oscillation measurements at high field. In the DDW
state,chargeloopsgiverisetoorbital-likemagneticmomentperpendiculartoCuO planes.TheAFarrangement
2
of such orbital-like moments should be directly probed by neutron scattering diffraction. However, to date, the
neutron detection a DDW state is rather scarce [28].
OtherCCstateshavebeenpredicted[9,10,11]fromthe3-bandHubbardmodelinvolvingbothcopperd-orbitals
and in-plane oxygen p-orbitals. As a result of the Cu-O repulsion term, a loop-current state is stabilized yielding
patterns with two (CC−Θ phase), or four (CC−Θ phase) current loops per unit cell. The phase CC−Θ
II I II
correspondingtotwooppositecurrentloopsisdepictedinFig.2.A,itbelongstotheE irreductiblerepresentation
u
of the D point group [30,31]. A current flows from the Cu atom through the nearest oxygen atoms, then back
4h
to Cu. By principle, these phases break the time-reversal symmetry (TRS). The broken TRS has been indeed
observed byARPESin Bi Sr CaCu O (Bi2212)througha spontaneous dichroismbelowT* [29].Assketched
2 2 2 8+δ
inFig.2.B,eachclosedcurrentloopproducesanorbitalmagneticmomentwhichshouldbemeasurablebyneutron
diffraction. A sizeable orbital magnetic moment of 0.1 µ is typically expected [9], pointing perpendicularly to
B
the CuO planes. In contrast to the DDW state, these phases preserve the TSL, as the same pattern is exactly
2
reproducedineveryneighbouringcells.Becausetheloopsarestaggeredwithineachunitcell(fig.2.B),thereisno
netmagnetizationcontrarytoaferromagneticphase.Therefore,themagneticstatecorrespondstoaQ=0orbital
AF order(Q=0, AFO).
Motivatedbythisapproach,wedevelopedaneutrondiffractionexperimenttoevidencesuchamagneticorder.
Fig.2.CrepresentsthemagneticintensityinthereciprocalspaceofaquadraticCuO planetakingthemagnetic
2
structure given in Fig. 2.B. As the state preserves the TSL, the magnetic scattering is only expected on top of
nuclear Bragg peaks for integer values of H and K. Importantly, due to the existence of two staggered moments
in each unit cell, no intensity is predicted for H=K=0. In CC−Θ state, 4 classical domains can exist [11].
II
One CC state is shown in Fig. 2.A. The three other states can be obtained by reversing the current flow or by
rotating Fig. 2.A by 90o [32]. Two domains should produce the neutron scattering pattern given in Fig. 2.C and
thetwoothersshouldproduceasimilarscatteringpatternbutrotatedat90o.ThescatteringpatterninFig.2.Dis
obtainedbyassumingthesamepopulationdistributionforeachdomain,restoringthequadraticC symmetryin
4
thetotalscatteringpattern.Oneseesthatthemostintensemagneticpeakwouldbefor(H,K)=(±1,0)≡(0,±1).
2.2. Polarized neutron technique
Toseparatethenuclearandmagneticscatterings-superimposedatthesameBraggposition-requirespolarized
neutron diffraction experiments. As shown in Fig. 2.E, the nuclear scattering interaction conserves the neutron
spin whereas the magnetic scattering interaction can flip the neutron spin. In a polarized neutron diffraction
measurement, the incoming neutrons are in the same spin state and one analyses the spin state of the scattered
neutrons. On can then distinguish the spin-flip (SF) scattered intensity from the non-spin-flip (NSF) one. This
techniqueisextremelypowerful.Unfortunately,the(NSF)nuclearintensityismuchlargerthanthe(SF)magnetic
intensity, typically one thousand time larger. The difficulty of the experiment then resides in the capability of
producingahighpolarizedneutronbeam,reliableandstableintimeandinposition.Inthecasewherethemagnetic
orderisshortrange,themagneticintensityisredistributedinmomentumspace.Inthislimit,thetechnicalissue
is related to the weakness of the magnetic signal/background ratio, but the polarization stability is no more
relevant. These experimental constraints are the best satisfied using the polarized neutron diffraction on the 4F1
triple-axis spectrometer around the reactor Orph´ee at the Laboratoire L´eon Brillouin (LLB), Saclay (France)
4
Cu O
A B
Phase
ordonnée
supra
C
mmaaggnneettiicc
SF
NSF
y
t
si
D tén
nsinte nuclear
e
ti
n
I
E
Wvecatveuer vde'ocndtoer
Figure2.A)SchematicdescriptionofthecirculationcurrentsflowinginsideaCuO2 plaquettes.IntheCC−ΘII phase,thereare
twocurrentsloopsturningclockwiseandanti-clockwise.Reversingthecurrentloopsand/orrotatingthefigureby90o allowsone
to build the 4 classical domains of the CC−ΘII phase. B) The current loops generate staggered orbital-like magnetic moments,
perpendicular to the CuO2 planes. C-D) Neutron scattering patterns for a C) single- and D) a multi-domain CC−ΘII phase,
respectively. In these two figures, a typical magnetic form factor has been used to compute the magnetic structure factor. E)
TheCC−ΘII phaseis aQ=0 AFphase (whichbelongsto theEu irreductible representationof theD4h pointgroup [30,31]):it
breakstimereversalsymmetry,butpreserveslatticetranslationinvariance.Asaconsequence,thenuclearandmagneticintensity
aresuperimposedonBraggreflections.Thepolarizedneutronscatteringtechniqueallowsonetodisentanglenuclearandmagnetic
scatterings,whichappearselectivelyinthenon-spin-flip(NSF)orspin-flip(SF)channels.
[33,34,35,36]. The observation of a magnetic scattering whatever the experimental difficulties is a significant sign
of the universality of the phenomenon.
As discussed in [33,34,35,36], the polarized neutron diffraction setup is similar to that originally described in
[37] with longitudinal polarization analysis (see also refs. [38,39] in the context of high-T cuprates). A polarized
c
incidentneutronbeamisobtainedwithapolarizingmirror(bender),theneutronenergyisfixedatE =13.7meV.
i
The polarization analysis is performed with an Heusler analyser. A radio-frequency Mezei flipper is put between
the bender and the sample in order to reverse the incident neutron polarisation. A pyrolythic graphite filter has
been put before the bender to remove higher order neutrons. The direction of the neutron spin polarization, P,
at the sample position is selected by a small guide field H of the order of 10 G.
In polarized neutron scattering technique, one can define the inverse of the flipping ratio (R−1) as R−1 =
I /I where I and I stand for the SF and NSF neutron intensities, respectively. For a perfectly
SF NSF SF NSF
polarized neutron beam, R−1 should be equal to 0 in absence of any magnetic scattering. In real experiments,
this is not the case. As a results of the unperfect neutron beam polarization, there is a leakage of the NSF
intensity into the SF channel. At the spectrometer 4F1, a highly polarized neutron beam is routinely obtained
withR−1rangingbetween1/40and1/100.Inabsenceofamagneticorder,R−1 exhibitsasmoothbehaviourwith
temperature,ideallyconstantbutinrealityonetypicallyobservesaslightdriftofR−1 uponcooling.Theoriginof
5
thistemperaturedependenceisunclearandvariesfromoneexperimenttoanother.Itislikelyduetoimperfections
intheneutronpolarizationandthedisplacementofthesampleintheneutronbeamwhenchangingtemperature.
For a better data analysis, one needs to care about this change of the flipping ratio with temperature. R(T) is
measured from the evolution of flipping ratios at Bragg peaks where the magnetic order is not present. In the
presence of an additional magnetic scattering, an upturn at low temperature in R−1 shows up. We prove that
method to be efficient enough to see weak magnetic moments (∼0.05µ ) on top of nuclear Bragg peaks, see e.g.
B
the determination of the A-type antiferromagnetism in Na cobaltate systems [40].
In an unpolarized neutron diffraction measurement and for a magnetism associated with unpaired electrons,
the interaction between the neutron spin and the magnetic moments in the sample is of dipolar type [41]. As
a consequence, only the components of the order moment M which are perpendicular to Q the transferred
momentum contribute to the magnetic scattering |F |2 [41]: |F |2 ∝|M |2 with M =M−(M.Qˆ)Qˆ with Qˆ
M M ⊥ ⊥
theunitvectoralongQ(|Qˆ|=1).Inapolarizedneutrondiffraction,theamountofmagneticscatteringintheNSF
channel is proportional to |M .P|2, theneutron polarization vectorbeing normalized (|P|2 =1). The restof the
⊥
magnetic scattering (i.e |M |2−|M .P|2) appears in the SF channel. Thus, for the neutron spin polarization
⊥ ⊥
P//Q, the full magnetic scattering appears in the SF channel.
2.3. Samples
Experiments were performed on about a dozen samples of either the YBCO system [33,34], or the Hg1201
system[35].Thesesystems,whosecrystalstructuresaregiveninFig.4,allowustocoveralargerangeofdoping,
especially in the underdoped regime where the pseudogap occurs. In both systems, it is worth pointing out that
the experimental results are very consistent with each other and with the expected phase diagram, whatever the
origin of samples, excluding extrinsic effects caused by impurity phases as it has recently argued from a single
sample[42].InHg1201system,thisstudyhasbeenmadepossibleduetotherecentbreakthroughinsinglecrystal
synthesis of samples with mass ∼ 1 g [43].
ThescatteringBraggwavevectorQ=(H,K,L)isgiveninunitsofthereciprocallatticevectors,a∗ ∼b∗ =2π/a
and c∗ =2π/c. Most of the data have been obtained in a scattering plane where all Bragg peaks like Q=(H,0,L)
were accessible. In the YBCO, the crystal structure is orthorhombic and plain samples are twinned such that
(H,0,L)≡(0,K,L). However, by applying an uniaxial pressure at high temperature (see e.g. [19]), it is possible to
detwin YBCO single crystals. In such a case, one can distinguish scattering along a* and b*.
3. Magnetic order in the pseudogap phase
3.1. Magnetic order in YBCO and Hg1201
As discussed above, the magnetic scattering is expected to be the largest at in-plane Bragg indices (H,K)=
(±1,0) ≡ (0,±1) for any integer L value along c*. To evidence the magnetic signal, we need to look for Bragg
peakswherethestructuralnuclearintensityisreduced.Inallcuprates,thissituationisbetterrealizedforL=1.The
BraggpeakintensitiesinbothNSFandSFchannelsareshowninFig.3.AforadetwinnedYBCO sample[33].
6.6
TheNSFintensityslightlyincreaseswithdecreasingtemperatureasitisexpectedatsmall|Q|duetotheDebye-
Wallerfactor.Incontrast,theSFintensitydepartsfromthisbehaviourbelowT =220Kevidencingamagnetic
mag
scattering |F |2 developing on top of a smooth background in the SF channel, given by the polarization leakage
M
fromthenuclearintensity|F |2/R.From bothintensities,onecanestimatearatio|F |2/|F |2 of∼400forthis
N N M
detwinnedYBCO sample.Typically,thisratioislargerintwinnedYBCOandHg1201samples.Convertingthe
6.6
relationsgiveninFig.3,onecandeducethemagneticscatteringcross-sectionasithasbeenperformedformany
samples (Fig. 3.B) corresponding to different doping levels ranging from an underdoped YBCO sample to an
6.5
overdoped (Y,Ca)BCO sample [33]. These data were calibrated in absolute units by scaling the Bragg intensity
to the strong nuclear Bragg peak Q=(0,0,4). One sees in Fig. 3.B that the magnetic scattering appears below
a temperature, T . Both T and the magnitude of the magnetic scattering increase with decreasing hole
mag mag
doping as expected for the pseudogap phenomenon. At variance, similar analysis at Q=(0,0,2) (empty symbols
on Fig. 3.B) show no additional magnetic scattering below T in agreement with the expected null magnetic
mag
structure factor for orbital moments (Fig. 2.D) related to a CC order[9,10,11]. At large |Q|, the magnetic form
6
Figure3.A)YBCO6.6:TemperaturedependenceoftheSFintensity(red)andtheNSFintensity(blue)measuredatQ=(0,1,1)for
apolarizationP//Q.Theformulagivetheexpectedscatteringinbothchannelswhere|FN|2 and|FM|2 arethenuclearandthe
magneticscatteringrespectively.BothcurvesinAwererescaledathightemperaturebydividingtheNSFintensitybyR=40(the
flippingratioforthatexperiment).B)(Y,Ca)BCO6+x:TemperaturedependenceofthemagneticintensityforP//QatQ=(0,1,1)
(≡(1,0,1))(fullsymbols)andQ=(0,0,2)(opensymbols).Fromtoptobottom,theholedopingincreasesthroughthegradualchange
oftheoxygenstoichiometryorextrasubstitutionofCaforY.(adaptedfrom[33]).
factorisexpectedtoconsiderablyreducethesignal.Accordingly,measurementsattheQ=(2,0,1)reflectionshow
no magnetic scattering [34].
3.1.1. Magnetic moment
Next,from|F |2(Fig.3.B),onecanestimatethemagnitudeoftheorderedmagneticmoment|M|definedusing
M
the magnetic neutron cross section for the CC−Θ phase. Keeping in mind that there are 2 opposite orbital
II
moments per CuO plaquette, the neutron cross section of the magnetic order given in Fig. 2.B and for a Bragg
2
position Q=(H,0,L) simply reads
|F |2 =r2f(Q)2β(L)24sin2(2πx H)|M |2 (1)
M 0 0 ⊥
where r =-0.54 10−12 cm (i.e. r2 = 290 mbarns) is the neutron magnetic scattering length. x = 0.146 is the
0 0 0
position of the magnetic moment within the unit cell (i.e. the center of the triangle in Fig. 2.A).
f(Q) stands for the magnetic form factor comprised by principle between 0 and 1. The calculation of the
magnetic form factor necessitates in principle a detailed knowledge of the real space extension of the orbital
moments, involving the Cu-d and O-p orbitals. So far, such a calculation has not been performed for the
x2−y2 x,y
observedmagneticorder.Wetakeanarbitraryestimateoff(Q)2 =0.5aroundQ=(1,0,1).Thiscrudeassumption
has been previously made for YBCO and Hg1201 [33,34,35] to be able to make a first estimate of the magnetic
moment.
β(L)isthemagneticstructurefactoralongc*relatedtothemagneticstructurewithinthe(CuO ) bilayer(in
2 2
thecaseoftheYBCOsystem).AsshownfromtheL-dependenceonthreedifferentpeaks[28,33,34],thecoupling
alongc*betweenthemagneticmomentshastobeferromagneticasthelargestmagneticintensityisobservedfor
L=0.Accordingly,anaturalformforβ(L)isthen2cos(πzL)withz=0.29isthereduceddistancebetweenCuO
2
layers in YBCO[28].
Considering, on the one hand, a magnetic moment M with the 3 components (M ,M ,M ) along the main
a b c
crystallographic directions and, on the other hand, the existence of 4 domains in the CC−Θ phase (built from
II
Fig. 2.A), the square of the magnetic components perpendicular to Q reads:
7
YBa Cu O HgBa CuO
2 3 6+x 2 4+x
C
A B
Figure4.A)CrystalstructureofYBa2Cu3O6+x(a=3.85˚A,c=11.7˚A).Itcontains2CuO2planesperunitcellandisorthorhombic.
CusitesareembeddedintooxygenpyramidsandthereisadimplingofCuO2planes.B)CrystalstructureofHgBa2CuO4+δ(a=3.87
˚A,c=9.5˚A).ThereisonlyoneCuO2planeperunitcellandthesystemistetragonal.Cusitesarelocatedinsideoxygenoctahedra.
Copperandplanaroxygenslieexactlyinthesameplane.C)Holedopingdependenceofthesuperconductingcriticaltemperature
Tc (stars)andtheorderingtemperatureTmag ofthe(Q=0,AFO)magneticstate(bullet).Redandlightbluesymbolscorrespond
to YBa2Cu3O6+x (YBCO) and HgBa2CuO4+δ (Hg1201) systems. The relationship between p and Tc has been established by
a systematic study of the c lattice parameter in YBCO where the true doping can be simply estimated in the oxygen-ordered
orthorhombic phases[44]. Unfortunately, the exact doping level in Hg1201 cannot be determined with accuracy. However, the Tc
curveexhibitsthesameshapeindopingasYBCOgivingconfidencethatthesamerelationshipbetweenpandTccanbeused.The
sizeofsymbolsisproportionaltothemagnitudeoftheorderedmagneticmoment.(adaptedfrom[35]).
1 c∗L c∗L
|M |2 = [1+( )2]M2 +[1−( )2]M2 (2)
⊥ 2 Q a,b Q c
with M = (cid:112)M2+M2 the magnitude of the in-plane magnetic component. At Q=(1,0,1), (c∗L)2 is equal to
a,b a b Q
0.1. If M were perpendicular to the CuO plane, then |M |2 would be ∼ |M|2. As will be shown in the next
2 ⊥
section,Misnotsimplyalongthecaxis,butrathertiltedat∼45o.Onegetsaroughestimate|M |2 ∼ 3|M|2.
⊥ 4
Finally,usingthesevariousapproximations,oneobtainsamagneticmomentof∼0.1µ when|F |2 ∼1mbarn
B M
inFig.3.B.Thisvaluedecreaseswhenincreasingholedoping.Itshouldbeemphasizedthatthismomentisgiven
pertriangularcurrentloopintheCC−Θ modelofFig.2.Aandcorrespondstotheorderofmagnitudeexpected
II
theoretically[9]. Of course, the precise value M would change depending on both the model and the accuracy of
the different parameters appearing in Eq. 1. However, the order of magnitude of the ordered moment is actually
quite robust whenever other assumptions are made. In Hg1201[35], the same expression Eq. 1 can be used with
β(L)=1asthereisasinglelayerperunitcell.Usingsimilarassumptions,themagneticmomentpertrianglecan
reach ∼0.2 µ in the strongly underdoped state[35].
B
3.1.2. A true phase transition
We then identify this magnetic order to the hidden order parameter of the pseudogap phase in YBCO[33,34].
Similar measurements in Hg1201[35] display the same doping dependence which can be summarized in Fig. 4.C
where the doping level is determined for both YBCO and Hg1201 using the same updated relationship between
thedopingandT [44].ItshouldbeemphasizedthatthetemperatureT matchesthepseudogaptemperature
c mag
T* deduced from resistivity measurements[33,35]. For both systems, the magnetic ordering temperature T
mag
extrapolates to zero at p=0.19, which matches the expected end point of the pseudogap phase as deduced from
various physical properties such as magnetic susceptibility, entropy or resistivity [5]. This points towards the
existence of a quantum critical point at p=0.19. Clearly, the neutron data provides a strong support in favor
of a true phase transition at T* [3,45]. In addition, high resolution magneto-optic (Kerr effect) measurements
consolidate this conclusion as it also evidences a time reversal breaking symmetry in the pseudogap phase of
YBCO [46].
The pseudogap affects all physical properties. However, the value of T* for a given doping varies through the
literaturedependingonthedataanalysisoroneachexperimentaltechnique.ThatisactuallythereasonwhyT*
8
Polarization analysis
A
B C
AAnnggllee ((MM,,cc**)) ~~ 4455°°
D E
F
Figure5.A)Schematicdescriptionofthethreeorthogonalneutronpolarizations(seetext).Theshadedarearepresentsthescattering
planedefinedbythedirections(100)and(001).B-E)TemperaturedependenciesoftheintensitiesintheSFchannel(magenta):B)
forP//QatQ=(1,0,1)and(C)atQ=(2,0,1),(D)forP//zand(E)P⊥QatQ=(1,0,1).Forallpanels,bluesymbolsindicate
the polarization leakage, given by the NSF intensity divided by the temperature dependence of the flipping ratio, R(T). R(T) is
foundbyafitinpanelCoftheratioNSF/SFoftheBraggpeakQ=(2,0,1)asR(T)=R(300K){1+0.02(1-T/300)F)Orientationof
themagneticmomentwithrespecttothecaxis(greenarrow),asdeducedfromthepolarizationanalysis(adaptedfrom[34]).
hasbeenlargelyconsideredasacrossoverphenomenon.Inordertocomparetheeffectofthepseudogaponagiven
physical properties, one needs to know how this quantity can couple to the order parameter of the pseudogap
phase. For instance, it has been argued[47] that the CC−Θ theory could be mapped onto the Ashkin-Teller
II
model(amodelwithapairofIsingspinsateachsitewhichinteractwithneighboringspinsthroughpair-wiseand
four-spin interactions). Monte Carlo simulations[45] of the Ashkin-Teller model show no sharp thermodynamic
anomalyatthephasetransitionforalargesetofparameters.Thatmaycorrespondstothatobservedincuprates
at T*[5]. Furthermore, the order parameter exponent β =0.18 [34] that one can extract from the neutron data
falls into the proper range for such an Ising type model. Another example of a thermodynamic signature of the
existence of a phase transition in the pseudogap state of underdoped YBCO is given by recent high-precision
magnetization measurements[48]. The temperature derivative of the uniform susceptibility indicates a singular
point at a temperature corresponding to T . This can be understood[45,48] as a biquadratic coupling of the
mag
magnetic order parameter with the uniform magnetization in a way similar to what happens in antiferromagnet.
3.1.3. Polarization analysis
Polarized neutron diffraction technique allows us to determine the direction of the magnetic moment through
apolarizationanalysis[37].Astheinteractionoftheneutronspinwithmagneticmomentsisofdipolartype,only
the magnetic components perpendicular to the wavevector Q are measurable[41]. Furthermore, only a magnetic
component perpendicular to the neutron spin polarization P is spin-flip whereas a component along P is non-
9
Figure6.YBCO6.6:A)[100]/[001]scatteringplaneandschematicdescriptionofascanalongthe[001]directionacrossQ=(1,0,1).
B)DifferencebetweenL-scansat70Kand300KintheSFchannelandforpolarisationP//Q(magenta).Themagneticsignal,
centered at Q=(1,0,1), is resolution limited. The blue symbols stand for the resolution limited nuclear Bragg scattering at Q
=(1,0,1)intheNSFchannel(adaptedfrom[34]).
spin-flip[41]. The amount of magnetic scattering in the SF channel therefore depends on the choice of P.
For a polarized neutron analysis, one usually introduces a set of three orthogonal directions: The xˆ axis is
parallel to Q, the yˆ axis is perpendicular to Q in the scattering plane and the ˆz axis is perpendicular to the
scattering plane. For a neutron spin polarization along xˆ (P//Q ≡ H ), the full magnetic intensity appears in
x
theSFchannel:I ∝M2 +M2 ,M andM arethecomponentsofM alongyˆ andˆzrespectively.For
P//Q ⊥y ⊥z ⊥y ⊥z ⊥
aneutronspinpolarizationappliedalongyˆ (P⊥Q≡H ),onlyonepartofthemagneticintensityappearsinthe
y
SFchannel:I ∝M2 .ThecomplementarymagneticintensityI ∝M2 showsupintheSFchannelwhen
P⊥Q ⊥z P//z ⊥y
the neutron spin polarization is applied along ˆz (P//z ≡ H ). As a result, in the SF channel, one can deduce
z
a specific polarization sum rule for the neutron intensity: I = I +I . It is worth noticing that this
P//Q P⊥Q P//z
polarization sum rule is only fulfilled for a magnetic scattering in the absence of chirality in the system.
ThepolarizationanalysishasbeenperformedinYBCO[33,34]fortheBraggpeakQ=(1,0,1)intwinnedsamples
as shown in Fig. 5. In first approximation, the flipping ratio is constant in temperature[33,35]. To improve the
data analysis, one needs further to fit it empirically by a straight line [34,49] as it has been done in Fig. 5.C.
First,themagneticscatteringislargerforP//Qasitshouldbe.Next,thepolarizationsum-ruleaboveisfulfilled
provingthemagneticnatureofthesignalobservedbelowT .Thesameconclusionhasbeenalsodemonstrated
mag
in another YBCO sample [33] as well in recent data obtained in Hg1201 [49].
6.6
The data reported in Fig. 5 were measured in the scattering plane given by the directions (100) and (001):
the ˆz axis therefore corresponds to the (010) direction. The orbital magnetic moment in a CC-phase is then
expected to be parallel to c (Fig. 2.B). Thus, one should expect I = I and I =0. At variance, one
P//Q P//z P⊥Q
observesamagneticcontributionforallneutronpolarizations(Fig.5.B,D,E).Morespecifically,I =1.4mbarn
P//Q
and I = I =0.7 mbarn in the YBCO sample[34] shown in Fig. 5. Qualitatively, the fact we observe a
P⊥Q P//z 6.6
non-zeromagneticintensityforP//z(Fig.5.D)provesthatalargepartofthemagneticmomentispointingalong
c.However,theintensityforP//zissmallerthantheoneforP//Qindicatingthatanin-planecomponentisalso
present.
Next, to discuss the magnetic moment direction, we use the same simple model used in previous sections to
evaluate the order of magnitude of the ordered magnetic moment. We assume a model with collinear moments
pointingalongagenericdirectiongivenby,M=(M ,M ,M )withnopreferentialin-planedirection(Fig.5.F).One
a b c
thenneedstocalculatetheneutronintensityfromEq.1andEq.2forQ=(1,0,1)andforeachneutronpolarization
[41].ThatgivesI ∝[1M2 ],I ∝[0.05M2 +0.9M2]andthesumofbothtermsforP//Q.Onecandefine
P⊥Q 2 a,b P//z a,b c (cid:113)
the angle, φ between the magnetic moment and c (see Fig. 5.F) which can be written as tan(φ) = M2 /M2.
a,b c
10