Charge-fluctuation contribution to the Raman response in superconducting cuprates

0.22 MB·English

by  S. Caprara

Charge-fluctuation contribution to the Raman response in superconducting cuprates
S. Caprara, C. Di Castro, M. Grilli, and D. Suppa
1Istituto Nazionale di Fisica della Materia, Unit`a Roma 1 and SMC Center, and Dipartimento di Fisica
Universit`a di Roma ”La Sapienza” piazzale Aldo Moro 5, I-00185 Roma, Italy
(Dated: February 2, 2008)
5
0 We calculate the Raman response contribution due to collective modes, finding a strong depen-
0 denceonthephotonpolarizationsandonthecharacteristicwavevectorsofthemodes. Wecompare
2 our results with recent Raman spectroscopy experiments in underdoped cuprates, La2−xSrxCuO4
n and (Y1.97Ca0.3)Ba2CuO6.05, where anomalous low-energy peaks are observed, which soften upon
a lowering the temperature. We show that the specific dependence on doping and on photon polar-
J izations of these peaks is only compatible with charge collective excitations at finite wavelength.

7
PACSnumbers: 74.20.Mn,78.30.-j,74.72.-h,71.45.Lr
2

]
n There is an increasing experimental evidence that in the Raman response with distinct doping and polariza-
o the high T superconducting cuprates there are finite- tiondependencies[12,13],whichinturnallowtoextract
c
c
energy excitations, which are distinct from the usual valuable informations about the momentum dependence
-
r single-particle excitations characterizing the spectra of of the excitations. This is the main issue of this work.
p ∗
standard metals. In optical conductivity there are We consider the doping-dependent line T (x) ∼ T (x)
u CO
s many examples of absorption peaks at finite frequen- [15]. Intheunderdopedregimebelowthislinethesystem
. cies,whicharequitedistinctfromthezero-energyDrude would order in the absence of competing effects. There-
t
a peak, occurring in hole-doped [La2−xSrxCuO4 (LSCO) forestrongCOfluctuationsarepresentinalargeportion
m
[1], La1.6−xNd0.4SrxCuO4 (LNSCO) [2], Bi2Sr2CuO6 of the phase diagram and provide an additional chan-
- (Bi2201) [3]] as well as in electron-doped materials nel for the Raman response, besides the usual one ob-
d
n [Nd2−xCexCuO4 (NCCO) [4]]. In all these cases absorp- tained from Fermi-liquid quasiparticles (QP). The role
o tion peaks are observed below 200 cm−1 which substan- of fluctuations in the response functions was established
c tially soften upon decreasing the temperature T. It is by Aslamazov and Larkin for the paraconductivity in
[
natural to assign to these excitations a collective char- superconductors [16]. This theory was extended to the

1 acter. Moreover in the case of Bi2201 there is a clear particle-holechannelforone-dimensionalcharge-density-
v evidenceofscalingpropertiesforthe finite-frequencyab- wave systems in Refs.17, 18. In this latter case a partial
1
sorption [5]. This suggests that these excitations are cancellation leads to a less singular paraconductivity as
7
nearly critical fluctuations associated to some critical- compared to the particle-particle case. Here we recon-
6
1 ity occurring in the cuprates below a critical line T∗(x), sider this scheme in two dimensions for the Raman re-
0 ending near optimal doping at T = 0. The proposal of sponse diagrammatically represented in Figs. 1(a) and
5 a quantum critical point in the strongly underdoped [6] 1(b) involving the incommensurate charge CM. We find
0
or near optimal doping [7, 8, 9, 10, 11] is by now acquir- that the above cancellation does not generically occur
/
t ing consensus in the community, but the nature of the for the corresponding Raman diagrams, generating im-
a
m ordered phase still remains debated. In this regard it is portant contributions, which in turn depend on the se-
quiteimportanttoestablishwhetherthecollectivemodes lected symmetry. We will argue that only these contri-
-
d (CM)responsibleforthefinite-frequencyabsorptionsoc- butions account for the strong symmetry dependence of
n cur at finite wavevectors and are therefore associated to the Raman response. Other diagrams with self-energy
o
some form of spin and/or charge spatial ordering [6, 8] and vertex corrections are weakly anisotropic and pro-
c
or have infinite wavelength [7, 9]. vide the usual weakly temperature-dependent QP con-
:
v tribution. Specifically each dashed line appearing in the
i In order to clarify this issue, in this letter we calcu-
X Raman response in Fig. 1 (a,b) is the gaussian diffusive
late the Raman response contribution due to collective
CM propagatorin Matsubara frequencies
r
charge-order(CO)excitationsassociatedtostripefluctu-
a
ations and eventually use our results to interpret recent
D(q,ω )=(|ω |+Ω )−1, (1)
findings of Raman-scattering experiments [12, 13]. Ra- m m q
manscatteringisanimportanttoolintheaboveissuefor
several reasons. First of all it is a probe directly access- where Ω =ν|q−q |2+m(x,T) with ν a constant elec-
q c
ing the bulk properties in the relevant frequency range. tronic energy scale (we consider a unit lattice spacing).
Mostimportantly,asuitablechoiceofthepolarizationof m(x,T) is the mass of the CM and encodes the distance
the incomingandoutgoingphotonsselectstheregionsin from the critical line T (x). This propagator is domi-
CO
momentum space from where excitations are originated nant at zero frequency and q = q , the wavevector set-
c
[14]. As a consequence, characteristic features arise in ting the modulation of the most singular charge fluctua-
2
vertices for the A1g symmetry only. The same holds for
spin CM with diagonal incommmensurate wavevectors
q ≈(±(π−δ),±(π−δ)), which is the case of LSCO at
c
lowdoping(x≤0.04)[23]. Onthe otherhand,athigher
doping, for small vertical and horizontal incommensura-
tion q = (±(π − δ),±π),(±π,±(π − δ)), there is no
c
complete cancellation in the B1g symmetry. This small
contribution could add to the CO contribution that we
are going to evaluate from the diagrams of Fig. 1(a,b),
FIG. 1: Direct (a) and crossed (b) diagrams for the fluctu-
butcannotexplaintheanomalousRamanresponseinthe
ation contributions to Raman spectra. The dots represent
deeply underdoped regime.
Raman vertices. Solid lines represent fermionic QP propaga-
torsanddashedlinesrepresentCMpropagators. (c)and(d): The resulting CM contribution to B = B1g,B2g Ra-
Hotspotsjoinedbythecriticalwavevectorqc =(0,−qc)rep- man scattering is given by
resented by arrows. Equivalent hot spots have arrows of the ∞
sametype(solidordashed). The+(−)signmarktheregions ∆χ′′ = Λ2 dz[b(z−ω/2)−b(z+ω/2)]
B B
where theB1g and B2g vertices are positive (negative). Z0
z+z−
× z2 −z2 [F(z−)−F(z+)] (3)
+ −
tions. The fermionic loop in Fig. 1(a,b) has the form
where b(z) is the Bose function,
Λ (Ω ;q,ω ) = CT γ (k)G(k,ε +Ω )
α,β l m Xn Xk α,β n l F(z)≡ 1 arctan ω0 −arctan m , (4)
z z z
× G(k−q,ε −ω )G(k,ε ), (2) h (cid:16) (cid:17) (cid:16) (cid:17)i
n m n
where γα,β(k) ≡ ∂2Ek/∂kα∂kβ and C is a constant de- a1n00d−z5±00≡cm(z−1±isωa/n2)u(l1tr+av(iozle±tcωu/t2o)ff2/oωf02t)h.eoHrdereeroωf0th∼e
termined by the coupling of the Raman vertex with the
frequency of the phonons most strongly coupled to the
incoming and outgoing photons and the coupling g of
electrons and driving the systems near the CO instabil-
the CM with the fermions. The choice of the α and β ity [15]. The above expression of ∆χ′′ is actually calcu-
componentsdepends onthe polarizationofthe incoming
latedbyconsideringCOcollectivemodeswithasemiphe-
and outgoing photons [14]. A suitable choice of these
nomenologicalspectral density of the form
polarizations corresponds to specific projections of the
γ(k) vertex on cubic harmonics of the square lattice. 2
ω 1+ ω
To start our discussion we chose the polarization corre- (cid:20) (cid:16)ω0(cid:17) (cid:21)
A(ω,Ω )= (5)
scposonkxdi−ngctoosakyRvaamnaisnhvinegrteaxlowngiththBe1g(0s,y0m)m→etr(y±γπB,1±gπ=) q ω2(cid:20)1+(cid:16)ωω0(cid:17)2(cid:21)2+Ω2q
directions. We are interested in the dominant contribu-
tions of the diagrams in Fig. 1(a,b), which occur when In the limit of infinite ω0, one recovers the spectral den-
the CM are around q . Therefore we set q = q in sity of the above critical diffusive propagator D(q,ω ).
c c m
Eq. (2). In the k summation the largest contribution Experimentally [12] in La1.9Sr0.1CuO4 an anomalous
is obtained when the three fermions are around the so- Raman absorption is observed in the B1g symmetry,
called hot spots, that is those regions on the Fermi sur- while the behavior of the B2g spectra can be accounted
face which can be connected by qc. For each given qc in for by QP only. Moreover the anomaly in B1g softens
Eq. (2), to avoid cancellations, the sum over k must en- upon lowering temperature and tends to saturate sug-
counter equivalent hot spots where the vertex γ (k) gesting a quasi-critical behavior. In Fig. 2(a) we report
B1g
does not change sign. Since γ changes sign under a comparison between our theoretical calculations and
B1g
the ±x vs. ±y interchange, this condition is fulfilled by experimental data. According to the neutron-scattering
stripes or eggbox CO fluctuations at not too small dop- experiments in this material [19, 21] we choose q along
c
ing [8, 19, 20, 21], where q ≈ 2π(±0.2,0),2π(0,±0.2) the(1,0)or(0,1)directions,givinganon-vanishingvertex
c
[see Fig. 1(c)]. On the other hand, when for the same for the B1g symmetry only. We adjust at one tempera-
qc the B2g vertex γB2g = sinkxsinky is considered in turetheoverallintensitybychoosingthe vertexstrength
Eq. (2),theleadingcontributionfromhotspotsvanishes ΛB1g ofEq. (2)andtheultravioletcutoffω0toreproduce
since the equivalent “available” hot spots [see Fig. (1d)] the lineshape. Then, keeping fixed these parameters at
give contributions from regions where γ is opposite all temperatures, we tune the mass m(T) to reproduce
B2g
in sign. With similar arguments one can show both in all the curves. The agreement is manifestly satisfactory,
the superconducting [22] and in the normal phase, that indicatingthatthestrongtemperaturedependenceofthe
spin CM at q = (±π,±π), although having the simi- Raman absorption peaks is basically ruled by the tem-
c
lar hot spots as the CO ones, give rise to non-vanishing peraturedependenceofthelow-energyscalem(T)andby
3
6 order along the diagonal directions [23], do show that
B 150
mW) 5 1g 11 3692865448 KKKKK -1m)100 M(x,T) tshyme maneotmryalaonudsiRsaambsaenntabinsotrhpetiBon1gi.sFpirge.se2n(tb)inretphoerBts2ag
(cps/ 4 (c 50 caocmaplcaurliastoendbCetMweceonnterxipbeurtiimonenftoarltdhaistaBo2fgRcaefs.e.[1I2n] tahnids
T) 3 00 50 100 150 200 case,consistenly with the expected high value of T∗, the
ω, Temperature ( K ) temperature dependence of the mass no longer displays
( 2 a clearlinear behaviorand thereforethe estrapolationof
χ ’’ (a) the high-temperature “linear” part to identify T∗ is not
∆ 1
welljustified. Nevertheless,as acrude estimate fromthe
last three points we obtain T∗ of the order of 150 K,
0
0 100 200 300 400 which is again consistent with the T∗ values reported in
-1
Raman Shift (cm ) the literature (see, e.g., Ref. 15). Notice also that the
tail of the data at the lowest temperature (T = 88K),
substantially lower than T∗, is not well reproduced by
6
our calculations based on a diffusive form of the CM.
88 K 100
W) 5 128525 KK M(x,T) AsimilarbehavioroftheRamanspectraisobservedin
m B2g 305 K -1m) 50 (Y1.97Ca0.3)Ba2CuO6.05 sample [13] at filling p = 0.03,
(cps/ 4 (c wonitlyh.aAtegmaipneirtatisurqeu-diteepennadtuenratlpteoakatitnritbhuetBe2tghissympemaketrtyo
T) 3 0100 200 300 stripe fluctuations along the (1,±1) directions and long
ω, Temperature ( K ) wavelength [q ≈ (p,p)]. Fig. 3 reports the comparison
c
’( 2 (b) betweenexperimentsandourtheoreticalanalysis. Again
’
χ avalueofT∗ ∼120K maybeestrapolatedfromthehigh
∆ 1
temperature mass behavior.
ThedataforYCBCOalsodisplayaremarkablechange
0
0 100 200 300 400
-1 in the lineshape at the lowest temperatures (more pro-
Raman Shift (cm )
nounced than in La1.98Sr0.02CuO4 at T =88K. In par-
ticular the peak at T = 11.0K is much narrower and
FIG. 2: (color online) (a) Comparison between experimen-
decreases more rapidly at high frequencies. This change
tal and calculated Raman spectra on La1.90Sr0.10CuO4. The
intensity is chosen to reproduce the data with Λ2 = 1.7, in the lineshape naturally marks a change in the nature
the high-frequency cutoff is ω0 = 250cm−1, and tBh1egmass is ofthe chargefluctuations fromthe the high-temperature
reported in the inset. (b) Same as (a) for La1.98Sr0.02CuO4 diffusive behavior in the quantum critical regime to a
with Λ2 =0.85. sharper mode behavior typical of the low-temperature
B2g
underdoped regime. In this case the diffusive form of
the propagator fails in reproducing the lineshape, while
the temperaturedependenceoftheBosefactors. Thein- agooddescriptionmaybeobtainedbyafactorizedform,
setreportstheCMmassm(x=0.10,T)neededtofitthe according to previous treatments of the nearly ordered
experimentalcurves: At largeto moderate temperatures underdoped regime [24]. Therefore we also carried out
m(T) has a linear part, which has an intercept on the T the calculations using
axis at a temperature T∗ of the order of 70K. At lower D˜(q,ω )=W(iω )J (q)J (q), (6)
n n x y
temperaturesmtendstosaturateandthesystemcrosses
overto a nearly orderedregime with a finite m(T). This where J (q) = Nγ γ2+(q −q )2 −1. Here N is
x,y x,y cx,y
behavioris clearlyconsistentwiththe behaviorexpected a normalizationfacto(cid:2)r andthe inversecor(cid:3)relationlength
for the mass of critical modes in the underdoped region: is γ. The (real-frequency)-dependent part W(ω) =
Atlargetemperaturesthesystemisinthe quantumcrit- dνA˜(ν)2ν/(ω2−ν2) withA˜(ω)∼Γ/[(ω−ω )2+Γ2]
CM
ical regime with m ∝ (T −T∗), while below a crossover Ris a normalized lorentzian centered around the typi-
temperature T∗ [which for LSCO at x = 0.10 is indeed cal CM energy ω with halfwidth Γ. ∆χ′′ now
CM B
about 60−80K, see, e.g., Fig. (1a) of Ref.15] the CO reads ∆χ′′ = B(T)f(ω/Γ,T/Γ,ω /Γ) with B(T) ∝
B CM
transition is quenched by other effects. g˜(T)4/[γ(T)2Γ(T)],g˜beingthecouplingbetweentheCM
IftheCMhadwavevectorsalongthediagonal(1,1)and and the QP [25]. As shown by the dashed curves in Fig.
(1,-1) directions and small modulus, the role of the B1g 3, the factorized “propagator” provides a good descrip-
and B2g would be interchanged,with this latter display- tionatthelowesttemperature,whileintheintermediate-
ingtheanomalousRamanabsorption. Noticeably,exper- temperature regime (T = 86.2K, T = 127.3K) the
iments in La1.98Sr0.02CuO4, [12], where neutron scatter- diffusive propagator becomes progressively a better de-
ing detect spin incommensurate (but likely also stripe) scription of the charge fluctuations. We also notice that
4
6 11 K 100 is also supported by recent experiments in ladder com-
) 5 B 1 582567...223 KKK 80 ωCΓM ((xx,,TT)) pounds [26]. Here we showed that the CM contribu-
s/mW 4 2g 129504 . 1 KK -1(cm) 4600 m (x,T) tpieoanksquinanRtiatamtaivnelsypaecctcroaunotfsLfoSrCtOheanobdseYrCveBdCaOn.omInaltohuiss
p 20 explanation a crucial role is played by the symmetry of
c
(
T) 3 00 60 120 180 240 300 the Raman vertices: We demonstrate the stringent con-
ω, Temperature (K) nection between the symmetry properties of the Raman
’( 2 spectra and the doping, which implies a dependence of
’
∆χ the CMonfinite wavevectorsspecific of stripe or eggbox
1
fluctuations as inferred from neutron scattering. This
rules out for the interpretation of these Raman exper-
0
0 20 40 60 80 100
iments local (i.e., at all momenta) excitations like po-
-1
Raman Shift (cm )
larons, disorder-localized single particles or resonating-
valence-bond singlets. Also spin CM do not comply
FIG.3: (Coloronline)Comparisonbetweenexperimentaland
withthesesymmetryrequirementsoverthewholedoping
calculated Raman spectra on (Y1.97Ca0.3)Ba2CuO6.05 with
rangeexaminedabove. Forthesamereasonotherexcita-
diffusive propagators [Eq. (1)] (solid lines). The intensity is
chosen to fit the data with Λ2 = 0.41, the high-frequency tionspeakedatq=0(likesuperconductingpairfluctua-
B2g
cutoff is ω0 = 100cm−1, and the mass is reported in the in- tions or time-reversal-breaking current fluctuations) are
set. Thedashed lines are obtained with factorized “propaga- not appropriate. Although we cannot a priori exclude
tors” [Eq. (6)]with overall intensityB(T)=28, 27, 19, 12 thattheproposedCOcollectiveexcitationsareconcomi-
(cps/mW) for T = 11K, 55.2K, 86.2K, 127.3K. Γ(T) tantwith these other more elusive forms of criticality,at
and ωCM are also reported in theinset. the moment the anomalous Raman absorption can only
be interpreted with CM peaked at finite momenta.
WeacknowledgeinterestingdiscussionswithC.Castel-
the mass m(T) of the quantum-critical diffusive prop-
lani, T. Devereaux, and J. Lorenzana. We particularly
agator smoothly connects with the energy of the low-
thank R. Hackl and L. Tassini for discussions and for
temperature modes: m(T)∼ω .
CM
providingus the Ramandata before publication. We ac-
As far as LSCO samples are concerned, the diffusive
knowledgefinancialsupportformtheMIUR-COFIN2003
form of the modes seems to persist down to the lowest
n. 2003020230 006. C.D.C. thanks the Walther Meiss-
temperatures in the available published data. We notice
inpassingthatthetwovaluesforT∗obtainedatx=0.02 ner Institute in Garching and the Humboldt Fundation
and x=0.10 linearly estrapolate to T∗ =0 at x=x ≈ for hospitality and support.
c
0.17, consistent with the theoretical position of the CO
quantum critical point [8, 15] and not inconsistent with
most of the experiments on T∗ in LSCO [11].
In order to emerge clearly from the other electronic [1] A. Lucarelliet al.,Phys.Rev.Lett. 90, 037002 (2003).
excitations, the CM’s need to have rather small masses [2] M. Dumm,et al.,Phys. Rev.Lett. 88, 147003 (2002).
(from the insets of Figs. 2 and 3 a few tens of cm−1). [3] )S. Lupi,et al.,Phys. Rev.B 62, 12418 (2000).
[4] E. J. Singley, et al.,Phys. Rev.B 64, 224503 (2001).
This is the case in underdoped systems where the fi-
nite and sizable values of T∗ carry to finite tempera- [5] S.Caprara,C.DiCastro,S.Fratini,andM.Grilli,Phys.
Rev. Lett.88, 147001 (2002).
turethesmallvaluesofthemass,whichshouldvanishat
[6] Ar.Abanov,A.Chubukov,andJ.Schmalian,Adv.Phys.
TCO(x)∼T∗ if a genuine long-range order would occur. 52, 119 (2003) and references therein.
Around optimal doping, where TCO(xc)= 0, anomalous [7] C. M. Varma, Phys. Rev. Lett. 75, 898 (1995); Phys.
peaksarehardlyvisiblebecausesuperconductivityinter- Rev. B. 55, 14554 (1997) and references therein.
venesathighertemperaturesbeforethemass,beingpro- [8] C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev.
Lett. 75 4650 (1995).
portional to T, becomes sufficiently small. Similarly we
[9] W. Metzner, D. Rohe, and S. Andergassen, Phys. Rev.
argue that the observation of CM peaks in Bi-2212 sys-
Lett. 91, 066402 (2003) and references therein.
tems is unlikely. In these materials, STM experiments
[10] M. Capone, M. Fabrizio, C. Castellani, and E. Tosatti,
suggestthatthe COcoherencelenghtis smallerthan5-6 Phys. Rev.Lett. 93, 047001 (2004).
latticespacings[20],indicatinglargemassvalues,incon- [11] J.L. Tallon, J.W. Loram, Physica C 349, 53 (2001); C.
trast to underdoped LSCO materials, where according Castellani, C. Di Castro, and M. Grilli, J. of Phys. and
to neutronscattering experiments the stripe correlations Chem. of Sol. 59, 1694 (1998).
[12] L. Tassini et al.,cond-mat/0406169 (unpublished).
may extend over tens of lattice spacings [19].
[13] R. Hackland L. Tassini, privatecommunication.
In summary we calculated the CO-CM contribution
[14] See, e.g., T. P. Devereaux, A. Virosztek, and A. Zawad-
to the Raman response function. The idea that charge owski, Phys. Rev.B 51, 505 (1995).
modes are important in the low-energy Raman spectra [15] S.Andergassen,S.Caprara,C.DiCastro, andM.Grilli,
5
Phys.Rev.Lett. 87, 056401 (2001). [23] S. Wakimoto, et al.,Phys.Rev. B 60, R769 (1999).
[16] L. G. Aslamazov and A. I. Larkin, Sov. Phys. Sol. State [24] A. P. Kampf and J. R. Schrieffer, Phys. Rev. B42, 7967
10, 875 (1968). (1990).
[17] B. R. Patton and L. J. Sham, Phys. Rev. Lett. 31, 631 [25] TheintensityofthespectraB(T)hasatemperaturede-
(1973); ibid. 33, 638 (1974). pendence through γ(T), Γ(T), and g˜(T) This latter in
[18] S.TakadaandE.Sakai,Prog.Th.Phys.59,1802(1978). the“factorized-propagator”schemeisproportionaltothe
[19] J. M. Tranquada, et al., 375, 561 (1995). (local)orderparameter[24].Thesedependenciessaturate
[20] C. Howald, et al., Phys. Rev. B 67, 014533 (2003); M. at low T and lead to an increasing and saturating B(T)
Vershihin,et al.,Science 303, 1995 (2004). for T much smaller than T∗.
[21] K.Yamada, et al., Phys.Rev.B 57, 6165 (1998). [26] A. Gozar, et al., Phys.Rev.Lett. 91, 087401 (2003).
[22] F. Venturini,et al.,Phys.Rev. B 66, 060502(R) (2002).