In-plane optical conductivity of La$_{2-x}$Sr$_x$CuO$_4$: Reduced superconducting condensate and residual Drude-like response

0.3 MB·English

by  S. Tajima

In-plane optical conductivity of La Sr CuO : Reduced superconducting condensate
2−x x 4
and residual Drude-like response
S. Tajima1)∗ and Y. Fudamoto1)*, T. Kakeshita1,2), B. Gorshunov3)**,
V. Zˇelezny´1)***, K. M. Kojima2), M. Dressel3), and S. Uchida2)
4 1)Superconductivity Research Laboratory, ISTEC, 1-10-13 Shinonome, Tokyo 135-0062, Japan
0 2)Dept. of Physics, The University of Tokyo, Tokyo 113-8656, Japan
0 3)I. Physikalisches Institut, Universit˝at Stuttgart, D-70550 Stuttgart, Germany
2 (Dated: February 2, 2008)

n Temperature dependences of the optical spectra of La2−xSrxCuO4 with x=0.12 and 0.15 were
a carefully examined for a polarization parallel to theCuO -planeovera wide frequency rangedown
2
J to 8 cm−1. Selection of well-characterized crystals enabled usto measure purely in-planepolarized

3 spectra without any additional peak. The weight of superconducting (SC) condensate estimated
2 from the missing area in σ (ω) well agrees with the estimate from the slope of σ (ω) vs 1/ω plot,
1 2
showing no evidence that theFerrell-Glover-Tinkham sum-rule is violated in the optical spectrum.

] WedemonstratethattheopticallyestimatedSCcondensateismuchsmallerthanthevalueobtained
n
from the µSR measurement of magnetic penetration depth. We also find an anomalous increase
o
c of conductivity in sub-millimeter region towards ω=0 below Tc, which suggests the microscopic
- inhomogeneity in the superconducting state. Both observations are discussed in relation with the
pr inhomogeneous electronic state that might be inherent to high-Tc cuprates.
u
s
. I. INTRODUCTION of this remaining conductivity is an important problem
t
a to be solved.
m
The superconducting (SC) condensate, one of the The other reason for the difficulty in determining SC
- essentail parameters for a superconductor, has been condensate of LSCO is linked to the difficulty in obtain-
d
a subject of discussion in high-T superconducting ing a pure in-plane spectrum. Since the early stage of
n c
cuprates(HTSC).Therecentprogressinscanningtunnel- HTSC research, various different in-plane spectra have
o
c ing spectroscopy has revealed a strange inhomogeneous been reported for LSCO10,11,12,13,14 and La2CuO4+δ15.
[ distribution of the gap in Bi Sr CaCu O (BSCCO)1. Some of the reportedin-plane spectra show rich features
The effect of this real-space ga2p i2nhomo2gen8eity on a su- in a far-infrared (FIR) region13,14,15, which are ascribed
1
perconducting (SC) condensate is unclear. On the other eithertotheLO-phonons13ortotheexcitationrelatedto
v
7 hand, the angle-resolved photoemission spectroscopy the polarons14. According to the variety of spectra, dif-
4 (ARPES) has demonstrated a distinct k-dependence of ferentvalues of the SC condensate werederived, ranging
4 the gap at the Fermi surface in the pseudo-gap state or from250to430nm10,11,12,13,14,15 intermsoftheLondon
1 the stripe ordered state2,3. For the system with such a penetration depth λFLIR(=1/2πωps), which do not cor-
0 partiallygappedFermisurface,the sourceofSCconden- relate with the Tc -value of the studied sample. Some
4
sate is not obvious. of them are considerably different from the value deter-
0
/ As one of the direct methods to detect a SC gap and mined by µSR measurements16,17.
t
a the weight of SC condensate, optical studies have been The purpose of this work is two-fold. The first one
m extensivelyperformedforHTSC.Aclearconductivityde- is to clarify the origin of such a confusing situation in
pression was observed below ∼500 cm−1 in YBa Cu O optical study of LSCO. We report a purely in-plane po-
- 2 3 y
d (YBCO)4,5,6,7 and BSCCO8. Then, the weight of SC larized spectrum of LSCO and demonstrate the effect
n condensate (ρ ∼ ω2 ) was estimated from the missing of the c-axis component mixing, by comparing with the
o s ps
area (A) of the realpart σ (ω) of conductivity spectrum ”dirty” spectra contaminated by the c-axis component.
c 1
or fromthe low-frequencybehaviorofits imaginarypart Possiblereasonsforthec-axiscomponentmixingaredis-
:
v σ (ω). cussed. The second and main purpose is to examine the
2
i
X The estimate of SC condensate from optical spectra, electrodynamic response in the low energy regionand to
however, has been subject to uncertainty. One of the estimatetheSCcondensate,basedonthepurelyin-plane
r
a reasonsis that,unlike the caseofconventionalsupercon- polarized optical spectra of La2−xSrxCuO4 (x=0.12 and
ductors,thereoftenremainssomeconductivitybelowthe 0.15). The spectra were measured over a wide range of
gapenergy,formingaDrude-likeincreasetowardsω=09. frequency 8 - 30000 cm−1, down to the sub-millimeter
Since the missing area of σ (ω) is large in YBCO and region. At T ≪ T , we have found a depression of con-
1 c
BSCCO, the remaining conductivity effect is relatively ductivitybelow200cm−1 (70cm−1)forx=0.15(x=0.12)
smallinestimationofSCcondensate. Bycontrast,inthe butasteepincreaseoftheconductivitybelow40cm−1(10
caseoflowerTc-HTSCsuchasLa2−xSrxCuO4(LSCO),it cm−1), which leads to a significantly small missing area
must givemore serious effect. Nevertheless,this residual comparedtothevalueestimatedfromtheLondonlength
conductivitywasignoredinmostofthepreviousestimate λ determined by µSR measurement. We conclude that
L
ofSC condensateexceptfor a few works10,11. The origin a discrepancy of the SC condensate between the opti-
2
cal and µSR-estimations is commonly observed in many 1.0 1.0
HTSC. Possible scenarios for these anomalous features
0.8
are discussed.
y 0.8
vit 0.6
cti
II. EXPERIMENTS efle LLSSCC##21 EE////aa((bb)) 0.4
R 0.6 Lθ=N1S5Cº#1 E//ab a E//c 0.2
SinglecrystalsofLa2−xSrxCuO4 withx=0.12and0.15 La1.85Sr0.15CuO4 R.T.
were grown by a traveling solvent-floating zone (TSFZ) 0.4 0.0
100 200 300 400 500 600 700
method. ThesuperconductingtransitiontemperatureT
c
is 36K for x=0.15 and 30K for x=0.12. Fixing crys- Wave Number (cm-1)
tal axes by observing X-ray Laue patterns at the sur-
face of as-grown crystal rods, we cut the crystals along
FIG. 1: Comparison of the in-plane reflectivity spectra of
the c-axis and took several pieces of samples with a
thickness of about 2 mm. We call them ”ac-face” sam- three La(Nd)1.85Sr0.15CuO4 crystals at 300K. LSC#1 and
LSC#2weremeasured ontheac-faces,whileLNSC#1isthe
ple hereafter. For comparison, we also cut some crys-
spectrum for the ab-face measured without polarizer. The
tal pieces with the c-axis perpendicular to the measure- spectraofLSC#2andLNSC#1showasubstantialcontribu-
ment surfaces called ”ab-face” sample. In both cases, tionfrom thec-axisspectrum. Thec-axisspectrumofLSCO
a- and b-axes are not distinguished although there is a is indicated by a dashed curve as a reference. We can re-
small difference between them. After a number of trial produce a similar profile to LSC#2 and LNSC#1, using the
measurements of far-infrared spectra, we realized that purein-planespectrum(LSC#1)andthec-axisspectrum,as
even Laue-characterized crystals were sometimes multi- described in the text. The calculated reflectivity assuming
a 15 degree rotation around the ac-/bc-plane or equivalently
domains, showing the c-axis phonon admixture.
the 6.7%-admixture of the c-axis component is indicated by
The second characterization method to detect multi-
a dash-and-dotcurve.
domain is to observe the surface using a polarized op-
tical microscope. Prior to this observation, the sample
surfaces were polished with Al O powder of different
2 3 ing of the field distribution, suchas randompinning, are
particle sizes at several steps. The particle diameter for
correctedbyassuminganadditionalGaussianrelaxation
the final polishing was 0.3 µm. It was found that some
with the relaxation rate 0.39-0.43MHz.
pieces of crystals consisted of multi-domains. We care-
fullyselectedthesingledomaincrystalsformeasurement
of purely in-plane polarized spectrum. A typical diam-
III. RESULTS AND ANALYSIS
eter of the sample disks is about 5 mm. A spot size of
the incident light is about 3 mm, well smaller than the
A. Effect of c-axis component admixture
sample size.
Optical reflectivity spectra were measured using a
coherent-source spectrometer18 in a millimeter and sub- Figure 1 shows the far-infrared reflectivity spectra at
millimeter wavelength region (8 - 33 cm−1), a Fourier room temperature for several samples. LSC#1 is the
transformationtype spectrometer for infrared region(30 almost perfectly in-plane polarized spectrum for the ac-
- 8000 cm−1) and a grating type one for higher energy faceofLa1.85Sr0.15CuO4 crystal,whileLSC#2,thespec-
region above near-infrared (4000 - 30000 cm-1). Both a trumforanotherac-faceLa1.85Sr0.15CuO4 crystal,shows
sampleandagoldmirrorweremountedonacopperplate the bumps centered at around 300 cm−1 and 520 cm−1.
inaHe-gasflowcryostatfortemperaturecontrol. Before LNSC#1, measured without polarizer for the ab-face
each measurement of a sample spectrum, a spectrum of of La1.45Nd0.4Sr0.15CuO4, also shows a similar bump-
gold mirror was measured as a reference. Reflectance of like feature at the same position. Compared to the c-
polarized light was measured with the s-polarized geom- polarized spectrum (E k c) in Fig.1, these bumps can
etry at the incident angle of about 5 degree. be assigned to the two A2u mode phonons with the TO-
InordertocomparetheSCcondensateestimatedfrom frequenciesof240cm−1 and500cm−1 thatarethec-axis
the other experimental methods, we measured the Lon- vibrationmodes ofoxygens21. Namely, the bumps result
don penetration depth using muon spin rotation (µSR) from mixing of the c-polarized component into the in-
technique. To improve the reliability of the missing area plane spectrum.
analysis and discussion, the µSR measurement has been We can reproduce a similar spectrum to LSC#2 and
carried out on the same single crsytal as used in the LNSC#1, using LSC#1 as a pure in-plane reflectivity
optical study. The analysis of the field distribution of data (Ra/b) and the measured c-polarized reflectivity
a vortex lattice in the applied transverse field H =0.2 data (R ), with R = R cos2(θ) + R sin2(θ). The
t c θ a/b c
Tesla has been performed using a London model19 with dashedcurve in Fig.1(a)representsthe calculatedreflec-
a Lorentzian cut-off20 of 4.5-6.5 nm as a non-local cor- tivityspectrumforθ=15degreedeviationofthepolariza-
rection. Other possible contributions for the broaden- tiondirectionfromthe purea(b)-axisorequivalently the
3
1600 YBCOcrystalsgrownbyafluxmethod,naturalsurfaces
La1.85Sr0.15CuO4 can be used for optical measurement, and the BSCCO
-1m) 1200 E//a(b) cthryasttcarlyssgtraolwcunttbiyngaisTnSoFtZnemceestshaordytaoreexstoraecatstilhyecalbe-afvaecde
-1Ωc 800 samples.
σ (1 400 300K Themulti-domainproblemisseriousintheTSFZcrys-
(a) 300K(θ=15º) tals. Itoftenappearsintheearlystageofcrystalgrowth,
namely, in the crystal portion close to a seed. It is hard
0
to confirm that a sample is of single domain by a usual
5000
(b) X-ray Laue pattern measurement, because a typical X-
4000 ray spot size is much smaller than the sample surface
area of about 3x3 mm2. This was the case for LSC#2.
-1m) 3000 In order to scan the whole area of sample surface, we
c
τ1/ ( 2000 300K(θ=15º) need an X-ray topography measurement. An easier way
300K to examine the existence of multi-domain feature is to
1000 40K
observe a polished sample surface using a polarized op-
5K
0 tical microscope. In the worst case, even the sample se-
0 500 1000 1500 2000 lectedby this method shows some admixture ofdifferent
Wave Number (cm-1) direction component in its far-infrared spectrum. The
mis-oriented domain behind the surface within a pene-
tration depth (∼ 1 µm) of far-infraredlight is a possible
FIG.2: (a)Comparison ofthein-planeconductivityσ1(ω)of origin. The dielectric response of the light penetrating
La1.85Sr0.15CuO4 (LSC#1) and the c-axis component mixed through multi-domains must be complicated, giving an
data for θ=15 degree. (b) Comparison of the scattering rate
unusual structure in far-infrared spectra. The obtained
1/τ(ω)forLSC#1andthec-axiscomponentadmixturecase.
reflectivity cannot merely be interpreted as a weighted
Itisclearthatthedoublepeakisaresultofthec-component
summation of the in-plane component R and the c-
admixture. It is also demonstrated that the kink at ∼700 a/b
cm−1evenat300Kandisindependentofthesuperconducting axis component Rc in this case.
response below 200 cm−1. The final problem related to the p-polarization must
be considered when we measure the in-plane spectrum
onthe ab-faceusingnon-polarizedlight. Inordertogain
6.7%-admixture of mis-oriented volume. Corresponding asignal,wesometimesdidnotuseapolarizerforthe ab-
to the rapid dropof reflectivity above 600 cm−1, the dip face measurement because the light intensity is reduced
at ω ∼ 470 cm−1 and the suppression below 250 cm−1 toaboutahalfwhengoingthroughapolarizer. However,
inthec-axisspectrum,theweakreflectivitysuppressions this geometry measurement always gave a weak bump-
giving the bump features in LSC#2 and LNSC#1 are likefeaturethatroughlycorrespondstothec-axisphonon
well reproduced in the calculated spectrum. peak or high-frequency reflectivity suppression. An ex-
The admixture of the c-axis spectral component into ample is shownin Fig.1(a)as LNSC#1 for theab-faceof
thein-planespectrumseriouslyinfluencestheconductiv- (La,Nd,Sr)2CuO4 crystal which was well characterized
ity and scattering rate spectra calculated from the re- by an X-ray diffraction. A similar problem can be seen
flectivity spectrum via Kramers-Kronig (KK) transfor- in the set of data by Lucarelli et al.14 About a half of
mation. As shown in Fig.2(a), the c-component admix- the seven spectra were measured on the ab-face with-
ture creates the conductivity dip between 300 and 1000 out a polarizer, and they show a clear peak correspond-
cm−1 and the peak around 500 cm −1. These additional ing to the c-axis phonon22. The most plausible origin of
featureswereobservedinbothofconductivityandreflec- this c-axis component mixing is the p-polarization com-
tivity spectra for the previous studies of LSCO13,14 and ponent in the non-polarized light, as van der Marel et
La CuO 15, which clearly indicates the admixture of al. pointedout23. Althoughtheintroducedc-component
2 4+δ
thec-axiscomponent. Figure2(b)demonstratesthatthe is very small if the surface is perfectly ab-oriented, the
double peak in 1/τ(ω) reportedby Startseva et al.13 can effect possibly becomes evident in extremely anisotropic
also be considered as the same admixture effect. materials like HTSC.
There are several possible reasons for the c-axis com- All the problems due to the c-axis component mix-
ponent mixing, such as (i)mis-cut of crystals, (ii)multi- ing become serious only in strongly anisotropic systems,
domains in as-grown crystals, (iii)p-polarization admix- namely, when the reflectivity values are remarkably dif-
ture in non-polarizedmeasurement. Mis-cut of as-grown ferent in the ab- and c-polarized spectra. This is well
crystal is the most primitive mistake that gives a mis- demonstratedin the optical spectra of intentionally mis-
orientedsurfaceforreflectivitymeasurement. Thisprob- alighed films of (La,Ce) CuO 24, where only 1 degree
2 4
lem is peculiar to the LSCO crystals, because the cut- mis-orientation results in a clear feature of the c-axis
tingprocedureisinevitableforextractinganappropriate phonons in the spectra. This remarkable effect is due
size sample from an as-growncrystal rod. In the case of tothe completely insulatingspectralprofileforE kc. In
4
1.0 10000
La2-xSrxCuO4 5K 15
Reflectivity 0.9 1.00 x=0.15 1234 200000000KKKKK -1-1σ (Ωcm)1 8642000000000000 20K 4−1-1σ(10 Ωcm)1 15000 ω (5c0m-1) 100
0.8 0.98 x=0.15
R SMM
0.96 0
FIR 6 8 2 4 6 8 2 4 6 8 2
5000 10 100 1000
0.94
10..070 0x=2W02000.1ωa v2(4ce0m -N1)6u401m0.00b8e0r (cm6-10)00.15 800 −1-1cm) 43000000 123 0005W50000KKKKKave N3-1-1uσ(10 Ωcm)m154321ber (cm-1)
R 0.5 0.12 σ(Ω1 2000 00 ω(c5m0-1) 100
0.9
ectivity 0.00.0 0ω.5 (1014. 0cm-11). 5 2.0 10000 x=0.12
Refl 5K 6 810 2 4 6 8100 2 4 618000 2
0.8 50K Wave Number (cm-1)
100K
200K
300K
FIG. 4: Temperature dependence of in-plane conductivity
0.7
0 200 400 600 800 spectraofLa2−xSrxCuO4withx=0.15andx=0.12withE ka
orb. Insetfiguresshowthelow-ω spectraandthedcconduc-
-1
Wave Number (cm ) tivity. The optical data can besmoothly extrapolated tothe
dcvalues.
FIG.3: Temperaturedependenceofin-planereflectivityspec-
tra of La2−xSrxCuO4 with x=0.15 and x=0.12 with E ka or the in-plane oxygen bending mode phonon. No admix-
b. Theinset oftheupperpanel showsthespectra for x=0.15
ture of the c-axis component is observed in both sample
in sub-millimeter (SMM) and far-infrared (FIR) regions sep-
spectra. The FIR reflectance is smoothly connected to
arately. Theinsetofthelowerpanelshowstheroom temper-
ature spectra in a wider range of frequency. the sub-millimeter reflectance as shown in the inset of
Fig.3(a). When the temperature is lowered below T ,
c
the reflectance for x=0.15 exhibits a rapid increase be-
fact, less anisotropic materials such as well-oxygenated low150cm−1 andaplateaubelow40cm−1,whichcould
YBCO have never shown the c-axis component in the be originating from the condensation of low frequency
spectrum measured with the in-plane polarized light. spectralweighttransferredtoδ(ω)atω=0. Asimilarfea-
Therefore, the measurement of in-plane spectrum for tureisobservedinthein-planeSCspectraofYBCOand
LSCOandprobably(Nd,Ce) CuO requiresparticularly BSCCO,butathigherfrequencyof∼500cm−1,probably
2 4
high quality samples without any multi-domain feature, owing to the larger energy scale of SC gaps.
high accuracy in cutting angle and careful polarization Thereflectanceis transformedto the complexconduc-
measurements. Theserequirementsarequite sever,com- tivityσ(ω)byutilizingKKrelations. Figure4showsthe
pared to that for other measurement techniques such as real part conductivity σ (ω) that can be smoothly ex-
1
neutron scattering. trapolatedtoadcvalueateachtemperatureaboveT as
c
shownintheinset,justifyingthevalidityofourKKanal-
ysis. The low-ω σ (ω) for x=0.15 increases with reduc-
1
B. In-plane optical spectra ing the temperature, forming a Drude-like peak at ω=0,
as is seen in Fig.4(a). When the temperature decreases
Hereafter we focus on the results of the sample for below T , σ starts to be suppressed below 250 cm−1,
c 1
LSC#1 which we believe the purely in-plane polarized corresponding to the SC gap opening. This SC response
spectrum, and for comparison on the results of the is, however, followed by an unusual Drude-like increase
x=0.12 sample with ac-face which was also well char- below 40 cm−1 even at 5K≪ T . A similar Drude-like
c
acterized. Figure 3 shows the in-plane reflectivity spec- response is also seen in the low-ω upturn of the spec-
tra for x=0.15 (a) and 0.12 (b) at various temperatures. trum for x=0.12 (Fig.4(b)). The Drude-like peak width
Phonon peaks are almost screened by free carriers ex- isnarrowerinthex=0.12samplethanthatforx=0.15. A
cept for the small structure at ∼370 cm−1 ascribed to hugeamountofresidualconductivityatT ≪T hasalso
c
5
beenreportedinadirectmeasurementofconductivityof
La Sr CuO E//a (a)
Lreag1i.o8n4S10r.0.1M6CorueOov4erfi,lmthiisnfetahteurseubsememillsimtoetbere cwoamvemleonngtinh -1m) 12 1.85 0.15 4
c 5K
many HTSC such as YBCO and BSCCO9, which should -1Ω 8 d-wave approx.
affect the estimation of SC condensate. 310 4A0µKSR
To compare with the previous results of LSCO13, σ (1 4
YBCO25 and BSCCO26, we derive an ω-dependent scat-
tering rate 1/τ(ω) = (ω2/4π)Re[1/σ(ω)] for x=0.15 as 0
p 0 50 100 150 200 250
plotted in Fig.2(b). Here ω is obtained from the in-
tegral ω0σ (ω)dω=ω2/8. Wpe take ω =10000 cm−1 so Wave Number (cm-1)
0 1 p 0
that thRe integral range includes the reflectance plasma
1.0 (b)
edge, but is well below the charge transfer excitation
MA#1
energy in La2CuO427. The spectrum of 1/τ(ω) clearly µSR MA#2
sinho1w/sτ(aωk)iniskcaotm6m50onclmy−o1bsaetrvaelldtaemt npeeararltyurtehse. sTahmeekfirnek- ω) / A0 0.5
quency for all HTSC studied so far26. This has been A(
0.0
ascribed to the opening of a pseudogap aboveT and/or
c
the coupling of carriers with a magnetic collective mode 0 100 200 5000 10000
observedasaresonancepeakinneutronscattering28. Re- ω (cm-1)
0
cently anothercandidateofsourceforthis kink hasbeen
proposed by the study of angle-resolved photoemission
spectroscopy (ARPES)29. It was demonstrated that the
kink inenergy-dispersioncurveofARPESis observedat FIG. 5: (a) Optical conductivity of La1.85Sr0.15CuO4
nearlythe sameenergyinallHSTC,irrespectiveofpres- (Tc=36K)withE kaorbatT=40Kand5K.Themissingarea
ence of the magnetic resonance peak. A phonon mode AµSR expected from λL=280 nm obtained in the µSR study
is, then, proposedas the mostlikely originofthe kink in is shown by a hatched area. The solid circle indicates the dc
energy-dispersioncurve of ARPES and 1/τ(ω). conductivity at 40K. (b) The ratio of the missing area ob-
The present result confirms that the kink in 1/τ(ω) tainedfromthepresentopticalstudyA(ω0)toAµSR. MA#1
existsat700-800cm−1 commonlyinHTSC.Adifference roefpirnetseegnrtasltohfe8racmtio−1fo.rMAA(ω#02) cisalfcourlattheeddwaittahotfhAe(loωw)ercalilmcuit-
ofourresultforLSCOfromthoseofYBCOandBSCCO 0
lated bythe dashed curveextrapolation in Fig.5(a).
is that the kink is observed even at room temperature
in the optimally doped sample with x=0.15. The pseu-
dogap temperature at this composition is expected to
In the case of HTSC, it is not a trivial issue that these
be70-80KfromtheT-dependencesofin-planeresistivity
twomethodsgiveanidenticalvalueofω ,since apossi-
andmagnetization30. Itmeansthatthe kinkstructureis ps
bility of kinetic energy contribution has been suggested
not related to the pseudogap opening. Another impor- by both theory31 and experiment32,33 We compare the
tant difference is the SC-gap related suppression which
twoopticallyobtainedvaluestochecktheFerrell-Glover-
is observed at ω ∼100 cm−1 that is far below the kink
Tinkham (FGT) sum rule, adn also compare them with
frequency. Thisisinsharpcontrasttothecontinuousde-
thoseestimatedfromtheLondonpenetrationdepthsthat
velopmentfromapseudogaptoaSCgapobservedinthe
were determined by µSR measurement.
other HTSC such as YBCO and BSCCO25. The present
Below,wefocusonthecondensateweightatthelowest
observation indicates that the kink feature is irrelevant
temprerature (∼5K) well below T . The conductivity
c
to a SC gap.
missing area for x=0.15 is defined by the integral
ω0
IV. DISCUSSIONS A(ω0)=Z [σ1(ω,40K)−σ1(ω,5K)]dω, (1)
0+
A. SC condensate where we set ω = 10000 cm−1. We first put the lower
0
limit of the integral at 8 cm−1 which is the limit of our
The observed SC response is most naturally inter- optical measurement (labeled as MA#1). In Fig.5(a),
preted as the signature of formation of SC condensate the missing area A =1/8(2πλµSR)2 expected from
µSR L
associatedwithaSCgapopening. Whenadelta-function λµSR=280 nm obtained by µSR study34 is shown by the
L
peak develops atω=0in σ (ω) at the expense ofits sup-
1 hatchedarea. ItisobviousthatA ismuchlargerthan
µSR
pression in the low-ω region, σ2(ω) behaves as propor- themissingareaintheopticalconductivityspectrumbe-
tional to ω2 /ω. There are two ways to estimate the SC
ps tween 40 and 5K. The ratio of the optical missing area
condensateω2 fromthe opticalspectrum. Oneistocal- A(ω )andtheµSRvalueA isplottedinFig.5(b)asa
ps 0 µSR
culate the missing area A(= ω2 /8) in σ (ω), and the functionofω . A(ω )/A increasesandthensaturates
ps 1 0 0 µSR
other is to estimate ω from the slope of σ (ω) vs 1/ω. to0.25±0.05,showingnoappreciablechangeupto10000
ps 2
6
cm−1. Thepenetrationdepthcorrespondingtothisvalue 2.5 0.6
x=0.15 x=0.12
oostffrAathi(geωh0Dt)lriiusnd5ee3(-0slihnkoemwc.noEmbvypenoaniedfnawtsehaeetldim5lKiinneaatniendtFhrieegpc.l5oa(ncate)r)iibtwubhtyiiochna -1-1Ωcm) 21..05 ωps~ 2700 cm-1 σ2 -1-1Ωcm) 0.4 ωps~ 1600 cσm2-1
mtoim0.5ic.sTahdi-swisavpeloStCtedbeahsaavdioars,hAed(ωc0u)r/vAeµlSaRbealemdoausnMtsAon#ly2 4σ (102 10..05 σ2−σ2res4σ (102 0.2 σ2−σ2res
in Fig.5(b), giving an upper limit of the missing area(or
0.0 0.0
the lower limit of λFIR=400 nm) in our experiment. 0.00 0.05 0.10 0.15 0.00 0.05 0.10
L
The anomalously small missing area is also observed 1/ω (cm) 1/ω (cm)
for x=0.12. Compared to the SC condensate estimated
from the µSR λµSR=310 nm, the optical conductivity
L
missing area A(ω ) is substantially small with the ra- FIG. 6: Imaginary part of condutivity σ (ω) of
0 2
tio A(ω0=10000cm−1)/AµSR ∼0.1. The opticallydeter- La1−xSrxCuO4 for x=0.12 and 0.15 at low frequencies.
mined penetration depth turns out to be ∼ 1µm in this E k a(b) and T=5K. σ2res is calculated from the residual
case. σ1(ω) at 5K using KK-relation. When σ2res is substracted,
σ well scales with 1/ω, which is the signature of supercon-
Strictly speaking, the missing area which is connected 2
ductingcarrier response.
withtheSCcondensateshouldbecalculatedfromthedif-
ferencebetweenthenormalandthesuperconductingcon-
ductivity at T=5K. In the above estimation of A (ω ),
we assume that σ (ω,40K)dω= σn(ω,5K)dω, 0whe0re below 50 cm−1 strongly influencs σ2(ω). In order to re-
1 1
σn(ω,5K)istheRconductivityinthRecasethatthesystem move the contribution of this residual conductivity, we
1 adopt the method proposed by Dordevic et al.36. First,
keepsthenormalstatedowntoT=5K.Inaconventional
we assume σ (ω) at 5K as the conductivity of unpaired
metal(superconductor), this assumption holds well, be- 1
carriers σres(ω) which do not condense at ω=0. Next,
cause the low-T resistivity is almost constant (the resid- 1
we calculate σres(ω) corresponding to σres(ω) using a
ualresistivityregime). Inthe case ofLSCO withx=0.12 2 1
KK-relation. Then, the σs(ω) which should represent
and 0.15, the ’normal state’ resistivity realized by the 2
the ”true” superfluid response is obtained as σs(ω) =
application of intense magnetic fields increases weakly 2
withloweringT belowTc35,suggestiveofa’normal’state σ2(ω)−σ2res(ω).
rathersimilartotheconventionalone. Asregardstheob- The obtained σs(ω) is proportional to 1/ω both for
2
servedT-dependenceofthenormal-statespectrum,Fig.4 x=0.12 and 0.15, indicating a SC response in this fre-
(and its inset) indicates that the frequency range where quencyrange. TheSCcondensatesgivingλFIR=990nm
L
σ showsanappreciableT-variationshrinkswithdecreas- for x=0.12 and 590 nm for x=0.15 are estimated from
1
ing T. For the spectrum at T=50K, a difference from the slopes, respectively. These are in agreement with
that at T=100K is seen restricted below 30 cm−1. The the values (1 µm for x=0.12and 530nm for x=0.15)ob-
trend suggests that the T-dependence of σ1n arises pre- tainedfromthemissingareainσ1(ω). Itmeansthatthis
dominantly from the Drude term. From this, and also methodforestimationofSCcondensateisself-consistent
from the result of dc resisitivity measured under intense within the optical spectra, and thus it is concluded that
fields, we guess that, if the normal state persists below the FGT sum rule almost holds within an infrared fre-
T , the Drude term would not change significantly from quency range. The present result does not completely
c
thatatT ,andhencethedifferencebetweenσ (40K)and excludeapossibilityofsmallcontributionfromthe high-
c 1
σn(5K) would be small. ω region, namely, the kinetic energy contribution to the
1
Anotherprobleminestimationusingeq.(1)isthe pos- condensate weight, as was observed in BSCCO32,33, but
sible contribution from the higher frequency spectrum. it is beyond the accuracy of our measurement.
Recently the spectral weight reduction in the high en- The previously reported values of λFIR for the opti-
L
ergy region above 10000 cm−1 has been observed in mally doped LSCO range from 250 nm to 430 nm, de-
BSCCO by two independent methods and the change pending on the sample and the research group but not
in kinetic energy at superconducting transition has been on the T -value. The value estimated from the spectrum
c
reported32,33. WehavealsofoundsmallT-dependenceof showingthec-axiscomponent13,15isnotreliable,because
thepresentspectrainthevisiblefrequencyregion,which the c-axisadmixture modifies the conductivity spectrum
gives additional weight to A(ω =10000cm−1). However, as demonstrated in Fig.1. In fact, the estimated λFIRs’
0 L
the integral up to ω =30000 cm−1 increases A(ω ) only from their data are shorter than ours.
0 0
by 20% at most. The kinetic energy change, if it ex- Anotherpointtojudgethedataqualityiswhetherthe
ists,cannotexplainthe smallmissingareainthe present low-ω Drude-like component is observed and taken into
results. account or not. If the remaining conductivity in the SC
The second method to estimate the SC condensate is stateisignored,the missingareamustbe overestimated.
to use σ . In Fig.6, σ is plotted as a function of 1/ω. The relatively larger missing area (or equivalently the
2 2
It is substantially deviated from the linear-relationship, shorter λFIR=250nm), reported in the paper by Gao12
L
because large amount of remaining conductivity σ (ω) may partly result from this over-estimation of the miss-
1
7
TABLE I: Comparison of penetration depthsestimated from FIR and µSR measurements.
(The FIR data of YBa2Cu3Oy are for E ka, while the µSR data are theaverage within the ab-plane.)
La2−xSrxCuO4 YBa2Cu3Oy Bi2Sr2CaCu2Oz
x=0.15 x=0.12 y≈6.9 y≈6.6 Tc ≈70K
λFIR(nm) 400-530 630a-990 160b ∼300c 680d
L
λµSR(nm) 280e 310f 112g 170g 190f
L
afromRef.39
bfromRef.7
cfromRef.42
dfromRef.43
efromRef.34
ffromRef.17
gfromRef.41
ing area, because the 5K-σ (ω) was set to zero below mationofλFIRbysubtractingthelow-ωresidualconduc-
1 L
70 cm−1. The measurement by Gorshunov et al.10 on tivity is about250nm for YBa Cu O 42. This is much
2 3 6.6
tthhee fLraeq1.u84eSnrc0y.1r6aCnugOe4dofiwlmn t(oTc5=c3m9.−5K1,)c3l7e,awrlhyicdhetceocvteedreda lFoInRgepretnheatnratthioenµSdRepvthaluisea(λlsµLoSRrep=or1t7e0dnfmor)4u1n.dAerldoonpgeedr
narrow Drude-like band with the large weight ωp ∼7800 BSCCO with Tc=70K43. Compared with the µSR pen-
cm−1 at 5K. Subtracting the Drude-like spectral con- etration depth (∼190 nm) for BSCCO with T =75K17,
c
tribution by fitting, they obtained a long penetration the reported value of λFIR (=680 nm) for the T =70K
L c
depth λFIR =400 nm. A little longer penetration depth sample43 is extremely long. We summarize the compari-
L
λFIR=430 nm was reported by Somal et al.11 for a sin- son of the FIR- and µSR-penetration depths for various
L
gle crystal. These values are in fairly good agreement HTSC in Table I. As is clearly seen in the table, the dis-
with our estimate of the shorter limit of λFIR. There- crepancy between the FIR and µSR data is quite robust
L
fore, we conclude that the spectra successfully measured for HTSC.
without contamination of the c-axis component give a One may consider the possibility that a FIR spectrum
similar value of λL (400-500 nm) that is longer than the is not correctly measured by some reason, for example,
estimate (250-310nm) from the contaminated spectra. by the [110] surface giometry problem arising from the
Themicrowave(MW)conductivitymeasurementgives d-wave symmetric gap, as recently pointed out by Tu et
a similar large value (=λMW=400±100 nm for x=0.15) al.44. It is, however, unlikely because a small estimate
L
to our FIR result, although the error bar is quite large ofSC condensatewasobtainedbyvariousmeasurements
in this technique38. By contrast, the µSR penetration with various surface giometries.
depth16,17 isclearlyshorterthantheFIRandmicrowave Itisalsoageneraltrendthatthediscrepancybecomes
values. For x=0.12, our estimate from the FIR data larger as one goes to more underdoped regime. This is
is λFIR=990 nm, the recent FIR data for x=0.125 by suggestive of electronic inhomogeneity as a possible ori-
L
Dumm et al.39 shows λFIR=630 nm, and the microwave ginofthe discrepancy. The SCorderparametermaynot
L
dataforx=0.12isλMW=500±200nm38. NotethatλFIR be uniform in real sapce and vanish in some parts, ac-
L L
as well as Tc is very sensitive to the Sr-content x around cording to the recent observation of STM on BSCCO1.
x=0.12 owing to the so-called 1/8-anomaly (Tc is most InLSCO,the stripefluctuationmayintroduceelectronic
suppressedwhenxistunedto0.115). Eveniftakinginto inhomogeneity. Aswediscussbelow,thepresenceofnon-
accountthissituation,wecanconcludethatthevaluesof superconducting area appears to be correlated with the
λFIR are much longerthan the µSR estimate λµSR=310 residualconductivityintheSCstate,whichisresponsible
L L
nm. partly for the reduced missing area in σ (ω).
1
A similar discrepancy between the FIR- and the µSR- A distinct difference from the µSR results was recog-
penetrationdepths is also seenin the caseof YBCO and naizedin the impurity-substituted YBCO7, YBa2Cu4O8
BSCCO particularly in the underdoped regime. In the and many other disordered HTSC45. Disorder, impurity
optimally doped YBCO, the average value of a- and b- or defect, decreases the FIR SC condensate (λFIR)−2 at
L
polarized data (λFIR=160 nm for E ka and 117 nm for much steeper rate than that expected from the linear
L
E k b)6,7 is larger than the µSR value [λµLSR(ab)=112 Tc-(λµLSR)−2 relationshipobservedbyµSR.Forexample,
nm41]. Comparing the conductivity spectra of optimally 0.4% Zn-substitution for optimally doped YBCO sup-
and under-doped YBCO25,40, we find that the missing presses Tc from 93K to 85K and duplicates λFLIR (∼340
area of σ in the underdoped YBCO with T =56-59K nm for E k a), while λµSR is expected to increase only
1 c L
is about 20% of that for the optimally doped one with 20%atmostbythesameZn-substitution46. Thediscrep-
T =93K. This suggests the penetration depth of about ancybetweenthe FIRandµSRdataseemstobe univer-
c
300-400 nm for the underdoped YBCO. The recent esti- sal in the disorderedor inhomogeneous cuprates, includ-
8
ing the underdoped cuprates for which STM suggests an Josephson coupling strength is increased and strongly
inherent spatial modulation of the SC order parameter modulatedspatiallybydecreasinganaveragespacingbe-
over a lenght scales of a few nm. It may follow that the tween stripes and/or by enhanced stripe fluctuation, the
FIR measurement is more sensitive to microscopic inho- weight of residual conductivity would increase and be
mogeneity than µSR, and that the ordinary relationship distributed over a fairly wide frequency range.
between ρ and λ (ρ ∼ λ−2) no more holds for such V. CONCLUSION
s L s L
microscopically inhomogeneous SC-state.
The in-plane polarized spectra of LSCO with x=0.12
and 0.15 were carefully measured. The peak at ∼500
B. Low-ω residual conductivity at T ≪Tc cm−1 that was observed in the previous reports and/or
insomeofoursampleswasassignedtothec-axisphonon
Finally we discuss the residual conductivity in the SC mode (A ). Various possible sources for this admixture
2u
state. The Drude-like up-turn of conductivity towards of c-axis spectral component were examined. Inaccu-
ω=0isobservedalmostinallHTSCevenatT wellbelow rateangleofcrystalcuttingand/ormulti-domainsinthe
Tc9. In the present study, the low-ω Drude-like spectral TSFZ-crystals possibly introduces some amount of the
weigh at T ≪ Tc is smaller for x=0.12 than for x=0.15, dielectric response for E k c. Measurement on ab-face
the Drude-like peak width being narrowerin the former. with non-polarizedlight or with p-polarizationgeometry
This is consistent with the general trend that the spec- alsoresults inthe c-axiscomponentmixing. Inanycase,
tral weight of the residual conductivity increases with the problem originates from the large anisotropy in the
doping47. electronicsystemofLSCOwhereonlyasmallamountof
The low-ω conductivity remaining at T ≪Tc is an in- c-axis component makes a serious effect on the in-plane
dication of the presence of non-superconducting region. spectrum, and therefore, this is the problem peculiar to
So far known are three types of inhomogeneous state in a strongly anisotropic material like LSCO.
HTSC:(i)Presenceofnon-SCregionwithintheSCarea,
The obtained pure a/b-axis spectrum showed a clear
either metallic as speculated for overdoped cuprates by
superconducting response, the suppression of σ (ω) and
FIR48 and µSR measurements49,50 or pseudogapped as 1
1/ω-behaviorofσ (ω). Theestimationbothfromσ and
2 1
observed by STM for underdoped BSCCO1, (ii) Alter-
σ give a consistent value of SC condensate, which indi-
natingarrayofanti-ferromagnetic(AF)andSCstripes51, 2
cates that the kinetic energy contribution is not appre-
(iii) Local suppresion of the SC order around impurities
ciable. We find that the SC condesate is much smaller
such as Zn, as was directly observed by STM52.
than that determined by µSR. This discrepancy is pos-
The impurity-induced inhomogeneity [the case (iii)]
sibly caused by microscopic inhomogeneity in the elec-
has been investigated by many experimental techniques.
tronicstateofsupercnductingCuO -planes,probablyre-
2
In the optical spectra, Zn-substitution creates a huge
lated to the stripe fluctuation in the case of LSCO. It is
residual conductivity, which dramatically depresses the
guessed that FIR-measurement is a probe sensitive to
missingarea7. Asimilarobservationwasreportedforthe
disorder which suppresses the SC order parameters over
Zn-doped YBa Cu O 45 and in the irradiated YBCO53,
2 4 8 a length scale of nano-meter. The inhomogeneous elec-
givingtheevidenceforthesensitivenessoftheFIRprobe
tronic state also seems to manifest in the spectrum as a
to microscopic inhomogeneity.
residual Drude-like response at very low frequencies be-
The case (ii), the spin and charge stripe order can low 50 cm−1 in the SC state.
be considered as an ”ordered” inhomogeneous state,
which has been detected by neutron scattering experi-
mentasastaticorderin(La,Nd,Sr) CuO 51,andproba-
2 4
blyasadynamicalforminLSCO54,wheretheSr-content Acknowledgement
x=0.12(∼1/8) is expected to be closer to the static or-
der. In this case, the charge stripes on which SC oreder This work was supported by the New Energy and In-
mightdevelopareseparatedbytheAFspindomains,and dustrailTechnologyDevelopmentOrganization(NEDO)
would form a periodic Josephson coupled array. A weak as Collaborative Research and Development of Funda-
spatial modulation of the Josephson coupling strength mentalTechnologiesfor Superconductivity Applications,
between the stripes, due to e.g. stripe meandering, is a and by a COE Grant and a Grant-in-Aid For Scientific
possible source of the low-ω conductivity peak, as was Research on Priority Area from the Ministry of Educa-
observedin the c-axis optical response55,56,57. When the tion, Japan.
∗
Electronic address: [email protected] Academy of Sciences, 117942 Moscow, Russia.
* Present address:National Institute for Material Science, *** Present address:Institute of Physics, ASCR,Na Solvance
Tsukuba, Ibaraki 305-0044, Japan. 2, 182 22 Praha, Czech Republic.
** Permanent address: General Physics Institute,Russian 1 S. H. Pan, J. P. O’Neal, R. L. Badzey, C. Chamon, H.
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