Quasiparticle states driven by a scattering on the preformed electron pairs

0.38 MB·

by  T. Domanski

CondensedMatterPhysics,2016,Vol.19,No1,13701:1–11
DOI:10.5488/CMP.19.13701
http://www.icmp.lviv.ua/journal
Quasiparticle states driven by a scattering
on the preformed electron pairs
∗
6
1 T.Domański
0
2 InstituteofPhysics,M.Curie-SkłodowskaUniversity,20-031Lublin,Poland

r ReceivedNovember13,2015,infinalformNovember27,2015
a
M
Weanalyzeevolutionofthesingleparticleexcitationspectrumoftheunderdopedcupratesuperconductors
neartheanti-nodalregion,consideringtemperaturesbelowandandabovethephasetransition.Weinspectthe
4
1 phenomenologicalself-energy that reproducesthe angle-resolved-photoemission-spectroscopy(ARPES) data
and we show that above thecriticaltemperature,such procedureimpliesa transfer of thespectral weight

] from the Bogoliubov-typequasiparticlestowards the in-gap damped states. We also discuss some possible
n
microscopicargumentsexplainingthisprocess.
o
c Keywords:superconductingfluctutations,Bogoliubovquasiparticles,pseudogap
-
r
p PACS:74.20.-z,74.20.Mn,74.40.+k
u
s
. 1. Introduction
t
a
m
Superconductivity(i.e.,dissipationlessmotionofthechargecarriers)isobservedatsufficientlylow
- temperatures,whenelectronsfromthevicinityoftheFermisurfaceareboundinthepairsandrespond
d
n collectively (ratherthanindividually) toanyexternalperturbationsuchaselectromagnetic field, pres-
o sure,temperaturegradient,etc.Dependingonspecificmaterials,thepairingmechanismcanbedriven
c by phonons (in classical superconductors), magnons (in heavy fermion compounds) or by the antifer-
[
romagneticexchangeinteractionsoriginatingfromtheCoulomb repulsion(incuprateoxides).Inmost

2 cases,theelectronpairsareformedatthecriticalvalueTc,markingtheonsetofsuperconductivity.There
v are,however,numerousexceptionstothisrule.Forinstance,inthecupratesuperconductors[1]orinthe
2 ultracoldfermionicgasses[2],suchpairspre-existwellaboveT .Tosomeextent,theirpresencecauses
c
9
thepropertiesreminiscentofthesuperconductingstate.
5
1 EarlyevidenceforthepreformedpairsexistingaboveTc hasbeenindicatedinthemuonscattering
0 experiments[3].Lateron,theirexistencewassupportedbytheultrafast(tera-Hertz)opticalspectroscopy
. [4,5]andthelargeNernsteffect[6,7].Spectroscopicsignaturesofthepreformedpairshavebeenalso
1
0 detecteddirectlyintheARPESmeasurementsonyttrium[8]andlanthanum[9]cuprateoxides,revealing
6 the Bogoliubov-type quasiparticle dispersion above T . Furthermore, the STM imaging provided clear
c
1 fingerprintsofsuchdispersiveBogoliubovquasiparticles(bytheuniqueoctetpatterns)beingunchanged
v: from temperatures deep in the superconducting region up to 1.5Tc [10].Superconducting fluctuations
i aboveT havebeenalsoreportedbytheJosephson-liketunneling[11]andtheproximityeffectinduced
X c
inthenanosizemetallicslabsdepositedonLa2 xSrxCuO4[12].Morerecently,theresidualMeissnereffect
r −
a hasbeenexperimentallyobservedabovethetransitiontemperatureTcbythetorquemagnetometry[13]
andothermeasurements[14,15].Additionalevidenceforthesuperconducting-likebehaviouraboveT
c
hasbeenseeninthehigh-resolutionARPESmeasurements[16],thesuperfulidfractionobservedinthe
c-axis optical measurements Re{σ (ω)} [17], the Josephson spectroscopy for YBaCuO-LaSrCuO-YBaCuO
c
junctionusingLaSrCuOinthepseudogapstatewellaboveT [18],opticalconductanceinthepseudogap
c
stateofYBaCuOsuperconductor[19]andthephoto-enhancedantinodalconductivityinpseudogapstate
ofthehighT superconductors[20].
c
∗ThisworkisdedicatedtoprofessorStefanSokołowskiontheoccasionofhis65-thbirthday.
©T.Domański,2016 13701-1
T.Domański
PreformedpairsarecorrelatedaboveTc onlyonsomeshorttemporalτφ andspatiallφ scales[21–
23]. For this reason, the superconducting fluctuations are manifested in very peculiar way [24]. Their
influenceonthesingleparticlespectrumismanifestedby:a)twoBogoliubov-typebranchesandb)addi-
tionalin-gapstatesthatareover-damped(haveashortlife-time).Temperaturehasastrongeffectonthe
transferofthespectralweightbetweentheseentities.Intheunderdopedcupratesuperconductors,such
atransferisresponsibleforfilling-intheenergygap[16,25],insteadofclosingit(asintheclassicalsu-
perconductors).Someearlyresultsconcerningsuperconductingfluctuationshavebeenknownforalong
time[26,27],buttheyattracted muchmoreinterestinthecontextofcupratesuperconductors[26–36]
andultracoldfermionsuperfluids[37,38].
Inthiswork,westudyqualitativechangeoverofthesingleparticleelectronicspectrumoftheunder-
dopedcuprateoxidesfor temperaturesvarying frombelowT (inthesuperconducting state) toabove
c
T , where the preformed pairs are not long-range coherent. In the superconducting state, the usual
c
Bogoliubov-type quasiparticles are driven by the Bose-Einstein condensate of the (zero-momentum)
Cooper pairs. Wefind that above T , the Bogoliubov quasiparticles are still preserved, but the scatter-
c
ingprocessesdrivenbythefinitemomentumpairscontributethein-gapstateswhoselife-timesubstan-
tiallyincreaseswithincreasingtemperature.Wediscussthisprocessonthephenomenological aswell
asmicroscopicarguments.Roughlyspeaking,afeedbackoftheelectronpairsontheunpairedelectrons
resembles thelong-range translational and orientational orderthat develops between the amphiphilic
particlesinpresenceoftheionsatsolidstatesurfacesstudiedbyS.Sokołowskiwithcoworkers[39].
2. Microscopic formulation of the problem
Toaccountforthecoherent/incoherentpairingweconsidertheHamiltonian
1
Hˆ =kX,σ¡εk−µ¢cˆk†σcˆkσ+N kX,k′,qVk,k′(q)cˆk†′↑cˆq†−k′↓cˆq−k↓cˆk↑ (2.1)
describingthemobileelectronsofkineticenergyεk(whereµisthechemicalpotential)interactingviathe
two-bodypotentialVk,k′(q).WeassumeaseparableformVk,k′=−g ηkηk′ ofthispairingpotential(with
g 0). In the nearly two-dimensional cuprate superconductors with the prefactor
ηk=>21 cos(akx)+cos(aky) (whereaistheunitlengthinCuO2planarstructure),suchpairingpotential
inducesthed-wavesymmetryorderparameter[40,41].
£ ¤
TheHamiltonian(2.1)canberecastinamorecompactform,byintroducingthepairoperators
1
bˆq ηkcˆq k cˆk (2.2)
= pN − ↓ ↑
k
X
andbˆq†=(bˆq)†,whenthetwo-bodyinteractionssimplifyto
1
N kX,k′,qVk,k′(q)cˆk†↑cˆq†−k↓cˆq−k′↓cˆk′↑=−Xq gbˆq†bˆq. (2.3)
UsingtheHeisenbergequationofmotion( 1)
ħ=
d 1
idtcˆk↑= εk−µ cˆk↑−gηk pN q cˆq†−k↓bˆq (2.4)
¡ ¢ X
weimmediatelynoticethatthesingle-particlepropertiesofthismodel(2.1):
a) are characterized by the mixed particle and hole degrees of freedom (because the annihilation
operatorscˆk↑coupletothecreationoperatorscˆq†−k↓),
b) dependonthepairingfieldbˆq(appearingintheequationofmotion ddtcˆk ).
↑
Boththesefeaturesmanifestthemselvesinthesuperconductingstate,whenthereexiststheBose-Einstein
(BE) condensate bˆq 0 ,0 of the Cooper pairs. They also survive in the normal state, as long as the
〈 = 〉
preformed(finite-momentum)pairsarepresentbelowthesamecharacteristictemperatureT∗marking
anonsetoftheelectronpairing.Inthenextsectionweexploretheirroleinthesuperconductingstate
(T Tc)andinthepseudogapregion(Tc T T∗).
É < <
13701-2
Quasiparticlestatesduetopreformedpairs
3. Pairs as the scattering centers
TheHeisenbergequationofmotion(2.4)indicatesthattheelectronicstatesareaffectedbythepairing
ˆ
fieldbq.LetusconsiderthegenericconsequencesofsuchAndreev-typescattering,separatelyconsider-
ing:theBEcondensedq 0andthefinite-momentumq,0pairs.
=
3.1. TheeffectoftheBose-Einsteincondensedpairs
WestartbyconsideringtheusualBCSapproach,whenonlythezeromomentumpairsaretakeninto
account.Thissituationhasaparticularlyclearinterpretationwithinthepathintegralformalism,treating
thepairingfieldviatheHubbard-Stratonovichtransformationanddeterminingitfromtheminimization
ofaction(thesaddlepointsolution).Thesameresultcanbeobtainedusingtheequationofmotion(2.4),
focusingontheeffectofq 0pairs
=
d bˆ
idtcˆk↑ ≃ εk−µ cˆk↑−gηkcˆ−†k↓ p0N, (3.1)
¡ ¢ ˆ†
d b
idtcˆk†↓ ≃ − εk−µ cˆk†↓−gηk p0N cˆk↑. (3.2)
¡ ¢
ˆ(†)
Macroscopicoccupancyoftheq 0stateimpliesthatthebosonicoperatorsb canbetreatedascomplex
= 0
( )
numbersb0∗.Byintroducingtheorderparameter
bˆ 1
∆k=gηk〈p0N〉 =gηkN ηk′〈cˆ−k′↓cˆk′↑〉 (3.3)
Xk′
theequations(3.1),(3.2)canbesolvedexactlyusingthestandardBogoliubov-Valatintransformation.In
such BCS approach, the classical superconductivity has close analogy with the superfluidity of weakly
interactingbosons,whosecollectivesound-likemodeoriginatesfromtheinteractionbetweenthefinite-
momentumbosonsandtheBEcondensate.
Inthepresentcontext, thezero-momentum Copperpairssubstantially affectthesingleparticleex-
citationspectrum(andthetwo-bodycorrelationsaswell).Thesingle-particleGreen’sfunctionG(k,τ)
−Tˆτ〈cˆk↑(τ)cˆk†↑〉,whereTˆτisthetimeorderingoperator,obeystheDysonequation =
[G(k,ω)]−1 ω εk µ Σ(k,ω), (3.4)
= − + −
withtheBCSself-energy
∆ 2
Σ(k,ω) | k| . (3.5)
=ω (ε µ)
k
+ −
The self-energy (3.5), accounting for the Andreev-type scattering of the k-momentum electrons on the
Cooper pairs, can be alternatively obtained from the bubble diagram. The related spectral function
A(k,ω) π−1ImG(k,ω i0+)isthuscharacterizedbythetwo-polestructure
=− +
A(k,ω)=uk2δ(ω−Ek)+vk2δ(ω+Ek) (3.6)
with the Bogoliubov-type quasiparticle energies Ek =± εk−µ 2+∆2k and the spectral weights uk2 =
21 1+(εk−µ)/Ek and vk2 =1−uk2. Let us remark that qthe¡se qua¢siparticle branches are separated by
th£e (true) energy¤gap |∆k|. In classical superconductors, ∆k implies the off-diagonal-long-range-order
(ODLRO)thatquantitativelydependsonconcentrationoftheBEcondensedCooperpairs.ODLROisre-
sponsible for a dissipationless motion of the charge carriers and simultaneously causes the Meissner
effectviathespontaneousgaugesymmetrybreaking.
13701-3
T.Domański
3.2. Theeffectofthenon-condensedpairs
Inthissectionweshallstudyeffectofthefinite-momentumpairsexistingaboveT ,whichnolonger
c
developanyODLRObecausethereisnoBEcondensate.Nevertheless, accordingto(2.4),thesingleand
pairedfermionsarestillmutuallydependent.ThisfactsuggeststhatthepreviousBCSform(3.5)should
be replaced by some corrections originating from the finite momentum pairs. Let us denote the pair
propagatorbyL(q,τ) Tˆτ bˆ(r,τ)bˆ†(0,0) andassumeitsFouriertransforminthefollowingform
=− 〈 〉
1
L(q,ω) , (3.7)
=ω E iΓ(q,ω)
q
− −
whereEq standsfortheeffectivedispersionofpairsandΓ(q,ω)describestheinverselife-time.Taking
intoaccounttheequation(2.4),weexpresstheself-energyviathebubblediagram
1
Σ(k,ω)=−T ω ξ iν L(q,iνn), (3.8)
iνXn,q − q−k− n
whereξq k εq k µandiνn isthebosonicMatsubarafrequency.SinceaboveTc thepreformedpairs
− = − −
areonlyshort-rangecorrelated[21–23],weimpose
t r
Lˆ†(r,t)Lˆ(0,0) exp | | | | . (3.9)
〈 〉∝ −τ −ξ
µ φ φ¶
Following T.Senthil andP.A. Lee [22, 23], onecanestimate thesingle particleGreen’sfunction G(k,ω)
usingthefollowinginterpolation
ω ξ
Σ(k,ω) ∆2 − k , (3.10)
= ω2 ξ2 πΓ2
− k+
where∆istheenergygapduetopairingandtheoth¡erparam¢eterΓisrelatedtodampingofthesubgap
states.Inthelowenergylimit(i.e.,for ω ∆)thedominantcontributioncomesfromthein-gapquasi-
particlewhoseresidueis Z 1 ∆2/(|πΓ|≪2) −1,whereas athigher energiestheBCS-typequasiparticles
≡ +
arerecovered.Thisselfenergy(3.10)canbederivedfromthemicroscopicconsiderations[42]withinthe
£ ¤
two-componentmodel,describingitinerantfermionscoupledtothehard-corebosons[43–50].
Theother(closelyrelative)phenomenologicalansatz[31,32]
∆2
Σ(k,ω)=ω ξk iΓ0−iΓ1 (3.11)
+ +
hasbeeninferredconsideringthe“smallfluctuations”regime[26].Experimentallineshapesoftheangle
resolvedphotoemissionspectroscopyobtainedforthecupratesuperconductorsatvariousdopinglevels
and temperatures(including the pseudogap regime) amazingly well coincide withthis simple formula
(3.11).ThegapandthephenomenologicalparametersΓ0,Γ1 areingeneralmomentum-dependent, but
foragivendirectionintheBrillouinzoneonecanrestrictonlytotheirtemperatureanddopingvaria-
tions.Fromnowonwardsweshallfocusonsuchantinodalregion.
Intheoverdopedsamples,Γ0canbepracticallydiscardedfrom(3.11)andtheremainingparameter
Γ1simplyaccountsforT-dependentbroadeningoftheBogoliubovpeaksuntiltheydisappearjustabove
Tc.PhysicaloriginofΓ1ishencerelatedtotheparticle-particlescattering.Onthecontrary,intheunder-
dopedregime,thereexistsapseudogapuptotemperaturesT∗whichbyfarexceedTc.Toreproducethe
experimentallineshapes,onemustthenincorporatetheotherparameterΓ0 (nonvanishingonlyabove
Tc)whichisscaledbyT Tc asshowninfigure1reproducedfromreferences[31,32].SinceΓ0 enters
−
theself-energy(3.11)throughtheBCS-typestructure,itsoriginisrelatedtotheparticle-holescatterings.
Wenowinspectsomeconsequencesoftheparametrization(3.11)applicableforthepseudogapregime
T Tc intheunderdopedcuprates.SinceneitherthemagnitudeofΓ1 nor∆seemtovaryoveralarge
>
temperatureregionaboveT ,itisobviousthatthequalitativechangesaretheredominatedbyscatter-
c
ingsinthe particle-hole channel, i.e., duetoΓ0. Roughly speaking, these processes areresponsible for
filling-in the low energy states upon increasing T as has been evidenced by ARPES [16] and STM [25]
13701-4
Quasiparticlestatesduetopreformedpairs
200
150
Γ
1
eV) 100
m
(
∆
50
T Γ
c 0
0
0 50 K 100 K 150 K
Temperature
Figure 1. (Color online) Temperature dependence of the phenomenological parameters ∆, Γ0 and Γ1
which, through the self-energy (3.11), reproduce the experimental profiles of the underdoped Bi2212
(Tc 83K)sample.Thisfittingisadoptedfromreferences[31,32].
=
measurements. On a microscopic level, such changes can be assigned to scattering on the preformed
pairs.
Foranalyticalconsiderations,letusrewritethecomplexself-energy(3.11)as
∆2
Σ(k,ω)=(ω+ξk)(ω ξ )2 Γ2−iΓk(ω), (3.12)
+ k + 0
wheretheimaginarypartis
∆2
Γk(ω)=Γ1+Γ0(ω ξ )2 Γ2. (3.13)
+ k + 0
InwhatfollowsweindicatethataboveT theexcitationspectrumcanconsistofaltogetherthreedifferent
c
states,twoofthemcorrespondingtotheBogoliubovmodes(signifyingparticle-holemixingcharacteris-
tic for the superconducting state) and another one corresponding to the single particle fermion states
whichforminsidethepseudogap.ThesestatesstarttoappearatT =Tc+ andinitiallyrepresentheavily
overdampedmodescontaininginfinitesimalspectralweight(seereference[22,23]foramoredetailed
discussion).Uponincreasingtemperature,theirlife-timegraduallyincreasesandsimultaneouslythein-
gapstatesabsorbmoreandmorespectralweightattheexpenseoftheBogoliubovquasiparticles.Finally
(intheparticularcaseconsideredhere,thishappensroughlynear2T )thesingleparticlefermionstates
c
becomedominant.
Anticipatingtherelevanceof(3.11)tothestronglycorrelatedcupratematerials,onecandetermine
thesingleparticleGreen’sfunctionG(k,ω)andthecorrespondingspectralfunctionA(k,ω).Quasiparticle
energiesaredeterminedbypolesoftheGreen’sfunction,i.e.,
ω ξk Re{Σ(k,ω)} 0 (3.14)
− − =
providedthattheimaginarypartΓk(ω)disappears.Weclearlyseethatthelatterrequirementcannotbe
satisfiedforΓ1,0regardlessofΓ0.Formallythismeansthatthelife-timeofhereindiscussedquasipar-
ticlesisnotinfinite.Letuschecktheseeventual(finitelife-time)quasiparticlestatesdeterminedthrough
(3.14).Usingtheself-energy(3.11),thecondition(3.14)isequivalentto
∆2
(ω−ξk)−(ω+ξk)(ω ξ )2 Γ2 =0. (3.15)
+ k + 0
Ingeneral,thereappearthreesolutions(figure2)dependingontemperatureviatheparameterΓ0.
Superconductingregion.Thefittingprocedure[31,32]hasestimatedthattheparameterΓ0 vanishes
in the superconducting state T T . Under such conditions, (3.15) yields the standard BCS poles at
c
É
Ek=± ξ2k+∆2.DuetoΓ1,0,theyshowupinthespectralfunctionA(k,ω)asLorentzianswhosebroad-
eningcqorrespondstotheinverselife-timeoftheBogoliubovmodes.OwingtoT-dependenceofΓ1 (see
13701-5
T.Domański
Figure2.(Coloronline)Dispersionofω=Ekrepresentingthesolutionsofequation(3.14)forT/Tc=0.95
(dottedline),1.1,1.5and2(solidcurvesasdescribed)obtainedfortheparametersusedinreferences
[31,32]. We notice three different crossings (3.15), two of themrelated totheBogoliubov modes and
additionaloneappearinginbetween.
figure1),thebroadeningofthesepeaksincreasesuponapproachingT frombelow,albeitA(k,ω 0) 0.
c
= =
Experimentallythisprocesscanbeobservedasthesmearingofthecoherencepeaks[1].
Pseudogapregime.WiththeappearanceofΓ0,0aboveTc,therealpartoftheself-energybecomes
acontinuous functionofω(seefigure3).Consequently, besidestheBogoliubovmodes, wenowobtain
anadditionalcrossinglocatedin-between.Figure2showstherepresentativedispersioncurvesobtained
for1.1T ,1.5T ,2T andcomparedwiththesuperconductingstate(dottedline).Weobserveeitherthe
c c c
threebranchesorjustthesingleoneatsufficientlyhightemperatureswhenthespectralfunctionA(k,ω)
evolvestoasinglepeakstructure.
Assomeusefulexample,letusstudytheFermimomentumk ,when(3.15)simplifiesto
F
∆2
ω 1 0. (3.16)
à −ω2+Γ20!=
Inthiscase,weobtain:a)twosymmetricquasiparticleenergiesatω ∆˜,where∆˜ ∆2 Γ2,andb)
thein-gapstateatω0 0.ThecorrespondingimaginarypartsΓk(ω)±ar=e± ≡ q − 0
=
Γk(ω ) Γ1 Γ0, (3.17)
± = +
∆2
Γk(ω0) = Γ1+Γ0 . (3.18)
Figure3.(Coloronline)Therealpartoftheself-energyΣ(k,ω)forεk=µatseveralrepresentativetem-
peraturesT/Tc=0.95(dottedline)and1.1,1.5,2.0(asdenoted).BelowTcthereexisttwopolesatω ∆
=±
whereasforT Tc,weobtainaltogetherthreecrossingswhichathighertemperaturemergeintoasingle
>
one.
13701-6
Quasiparticlestatesduetopreformedpairs
Figure4.(Coloronline)TheimaginarypartΓk(ω)forthesamesetofparametersasusedinfigure3.The
filledsymbolsindicatethevalueofΓk(ω)andpositionofthecrossingsω=EkofthelowerBogoliubov
mode(squares),in-gapstate(circles)andtheupperBogoliubovbranch(triangles).
SinceΓ1doesnotvaryaboveTc,thetemperaturedependenceofΓ−k1(ωi)iscontrolledbyΓ0.Usingtheex-
perimentalestimations[31,32],wethusfindthequalitativelyoppositetemperaturevariationsofΓk(ω )
andΓk(ω0)showninfigure5.Thesequantitiescorrespondtothelife-timesofquasiparticlesand,ther±e-
fore,weconcludethat:
a) in-gapquasiparticlesareforbiddenforthesuperconductingstateduetovanishingΓ−k1(ω0)=0(in
otherwords,spectrumconsistsofjusttheBogoliubovmodestypicaloftheBCStheories),
b) inthepseudogapstateaboveTc,where∆,0andΓ0,0,besidestheBogoliubovbranchesthere
emergein-gapstateswhichinitiallyatTc+representtheheavilyoverdampedmodes.
Atafirstglance,ourconclusionsseemtobeinconflictwiththeARPESdata,whichhavenotreported
anypronouncedin-gapfeatures.Nevertheless,variousstudiesofthepseudogapclearlyrevealedarather
negligibletemperaturedependenceof∆(T)uponpassingT (atleastfortheanti-nodalareas).Insteadof
c
closingthisgap,thelowenergystatesaregraduallyfilled-in[25].Suchabehaviorcanbethoughtasan
indirectsignatureofthein-gapstates,whichforincreasingtemperaturesabsorbmoreandmorespectral
weight.Tosupportthisconjecture, weillustrateinfigure6anongoingtransferofthespectralweights
between the Bogoliubov quasiparticles and in-gap states. Using (3.11), we show the spectral function
A(k,ω)subtractingitsvalueatT inanalogytothedetailedexperimentaldiscussionbyT.Kondoetal.
c
[16]. In-gap states emerge around ω0 and gradually gain the spectral weight (figure 7) simultaneously
increasingtheirlife-time.
Intrinsicbroadeningofthein-gapstates[53]unfortunatelyobscurestheirobservationbythespectro-
scopictoolsattemperaturesclosetoT .Thesestatesmightbe,however,probedindirectly.T.Senthiland
c
P.A.Lee[22,23]suggestedthatsuchstatescouldberesponsibleforthemagnetooscillationsobservedex-
0.02
ε = µ
k
-1V) 0.015
e
m
-1Γ (k 0.01 Bogoliubov peaks
0.005 In-gap states
0
0 50 K 100 K 150 K 200 K
Temperature
Figure5.(Coloronline)TemperaturedependenceoftheinversebroadeningΓ−k1 whichcorrespondsto
theeffectivelife-timeoftheBogoliubovmodes(thinline)andthein-gapstates(thickcurve)obtainedfor
εk=µ.
13701-7
T.Domański
ε = µ
k
T = 2.5 T
Tc c
=
T
ω)|
k,
A( T = 2.0 Tc
ω) -
k,
A( T = 1.5 T
c
T = 1.1 T
c
-200 -100 0 100 200
ω (meV)
Figure6.(Coloronline)TransferofthespectralweightfromtheBogoliubovquasiparticlepeakstowards
thein-gapstatesobtainedusing(3.11)forεk=µ.Temperaturedependenceofthetotaltransferredspec-
tralweightisshowninfigure7.
perimentallybyN.Doiron-Leyraudetal.[51,52].Theyindicatedthatpair-coherenceextendingonlyover
shortspatial-andtemporallength naturallyimpliesthepairdecay(scattering)intothein-gapfermion
states.Thislineofreasoninghasbeenalsofollowedbysomeothergroups[53,54].
4. Microscopic toy model
Pairingofthecupratesuperconductorsoccursonalocalscale,practicallybetweenthenearestneigh-
borlatticesites.Toaccountforaninterplaybetweenthepairedandunpairedchargecarrierstakingplace
inthepseudogapregimeweconsiderherethefollowingsimplifiedpicture
Hˆloc=ε0 σ cˆσ†cˆσ+E0bˆ†bˆ+ ∆bˆ†cˆ↓cˆ↑+h.c. , (4.1)
X ³ ´
wherecˆ(†)correspondtotheunpairedfermionsandbˆ(†)tothepairs(hard-corebosons).Weassumethat
σ
in the pseudogap state, neither the fermions nor the hard-core boson pairs are long-living because of
theirmutualscatteringbytheAndreevchargeexchangeterm.Thesametypeofscattering,althoughin
4 Result from eqn (3.11)
Andreev scattering
% )
s ( 3
e
at
st 2
p
a
g
n- 1
I
0
0.1 1 10
T / T
c
Figure7.(Coloronline)Spectralweightcorrespondingtothein-gapstatesobtainedfromthephenomeno-
logicalansatz(3.11)forεk=µ(dottedcurve)andsolutionofthetoymodel(4.1)forε0=0=E0(solidline).
13701-8
Quasiparticlestatesduetopreformedpairs
themomentumspace,hasbeenconsideredinreference[22,23]withinthelowestorderdiagrammatic
treatment.Onamicroscopicfooting,theHamiltonian(4.1)canberegardedastheeffectivelowenergy
descriptionoftheplaquettizedHubbardmodel[47,48].
Neglecting theitinerancyofthechargecarriers,wecanobtainarigoroussolutionforagivenlocal
cluster(nottobeconfusedwiththeindividualcoppersitesinCuO2planes[50]).Exactdiagonalizationof
theHilbertspaceyieldsthefollowingsingleparticleGreen’sfunction[42]
Z 1 Z
G(ω) QP − QP . (4.2)
= ω−ε0+ω−ε0−ω |ε∆0|2E0
+ −
Letusnoticethatthesecondtermonrhsof(4.2)acquiresthesamestructureasimposedby(3.11).Inthe
presentcase,noimaginarytermsappearbutthestructureoftheGreen’sfunction(4.2)mimicstheroleof
Γ0.Formally,itdescribesthebondingandantibondingstatesoriginatingfromtheAndreevscatteringand
besidesthatwealsohavearemnantofthenon-interactingpropagator[ω ε0]−1whosespectralweight
−
isgivenbyZ .
QP
The quasiparticle weight Z depends on occupancies of the fermion and boson levels. As an ex-
QP
ample,weexploreherethesymmetric(i.e.,half-filled) casewithε0 0andE0 0whenZQP 2/[3
= = = +
cosh(∆/k T)].Assumingthetypicalratio ∆/k T 4,weplotinfigure7thetemperaturedependence
B B c
| | | | =
oftheunpairedstates contributionZ tothespectrum. Wefind averygood agreement between our
QP
simpletreatmentandtheestimationsusingtheself-energy(3.11).ItmeansthattheparameterΓ0 intro-
ducedinreferences[31,32]andthelocalAndreev-typescatteringconsideredhereaccountforthevery
sameparticle-holeprocessesinducingthein-gapstates.Transferofthespectralweightfromthepaired
tounpairedstates(figure7)confirmsthequalitativeagreementoverabroadtemperatureregionandthe
indicationforthesamecriticalpoint.
Formorerealisticcomparisonofthepresentstudywiththeexperimentaldata[16],oneobviouslyhas
toconsidertheitinerantchargecarriers.Asanaturalimprovementofthelocalsolution(4.2)wewould
expectthefollowingtypeofGreen’sfunction
Z (k) 1 Z (k) f(q,k)
G(k,ω) QP − QP , (4.3)
= ω−εk +Xq £ω−εk−ω+¤ε|∆q−kk|2−Eq
whereT-dependentcoefficients f(q,k)shouldbedeterminedviathemany-bodytechniques.Approach-
ingT from above the predominant influencecomes from q 0 bosons and then we noticethat (4.3)
c
→
reduces to the ansatz (3.11). Such results have been recently reported from the dynamical mean field
calculationsfortheHubbardmodel[55,56].
5. Conclusions and outlook
Wehaveshownthatthepairingansatz(3.11),widelyusedforfittingtheexperimentalARPESprofiles,
aboveT correspondstothepairscatteringinducingthesingleparticlefermionstatesinsidethepseu-
c
dogap.Temperaturedependentphenomenological parameterΓ0 isfound tocontrolthetransferofthe
spectralweightfromtheBogoliubovquasiparticlestotheunpairedin-gapstates.Tomodelsuchapro-
cessonamicroscopiclevel,wehaveconsideredthescenarioinwhichthelocalpairsarescatteredinto
singlefermionsviatheAndreevconversion[22,23,47,48,50].Wehavefoundauniquerelationbetween
thetransferredspectralweight(fromthepairedtounpairedquasiparticles)withthenon-bondingstate
Z .Itwouldbeinstructivetoextendthepresentanalysisontothecaseofk-dependentenergygap.Such
QP
aproblemwould becloselyrelatedtotheissueofFermiarcs,i.e.,partiallyreconstructedpiecesofthe
Fermisurface,andtonontrivialangulardependenceofthepseudogap[1].
Acknowledgements
AuthorisindebtedforthefruitfuldiscussionswithAdamKamiński,RomanMicnas,JuliusRanninger,
andKarolI.Wysokiński.ThisworkissupportedbytheNationalScienceCentreinPoland throughthe
projectsDEC-2014/13/B/ST3/04451.
13701-9
T.Domański
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