Electronic Inhomogeneity and Breakdown of the Universal Thermal Conductivity in Cuprate Superconductors

0.59 MB·English

by  X. F. Sun

Electronic Inhomogeneity and Breakdown of the Universal Thermal Conductivity in
Cuprate Superconductors
∗
X. F. Sun, S. Ono, Yasushi Abe, Seiki Komiya, Kouji Segawa, and Yoichi Ando
Central Research Institute of Electric Power Industry, Komae, Tokyo 201-8511, Japan
(Dated: February 2, 2008)
6
0 We report systematic, high-precision measurements of the low-T (down to 70 mK) thermal con-
0 ductivity κ of YBa2Cu3Oy, La2−xSrxCuO4 and Bi2Sr2CaCu2O8+δ. Careful examinations of the
2 Zn-andhole-dopingdependencesoftheresidualthermalconductivityκ0/T,aswellasthein-plane
n anisotropyofκ0/T inBi2Sr2CaCu2O8+δ,indicateabreakdownoftheuniversalthermalconductiv-
ity,a notable theoretical prediction for d-wavesuperconductors. Ourresults point to an important
a
role of electronic inhomogeneities, that are not considered in the standard perturbation theory for
J
thermal conductivity,in theunder-to optimally-doped regime.
6

PACSnumbers: 74.25.Fy,74.25.Dw,74.72.Bk,74.72.bk,74.72.Dn
]
n
o
c The “gapless” nature of the d-wave superconductivity suggests a QP localization phenomenon that does not
- inhigh-T cupratesopenedanewpossibilitytostudythe show up in the SCTMA theory. In this regard, an un-
r c
p electronicstateatlowtemperaturethroughquasiparticle usual QP localization in magnetic field was recently ob-
u (QP)heattransport[1,2,3,4,5,6,7,8,9,10,11,12,13]. served in underdoped La2−xSrxCuO4 (LSCO) [12, 13]
s
Inthepresenceofdisorder,QPsarecreatednearthegap and, while it is obviously related to the magnetic-field-
.
t nodes with an “impurity bandwidth” γ [14] and carry inducedspin/chargeorder[12,18],themechanismofthis
a
m heat; intriguingly, those QPs are also scattered by the localization is not yet understood.
- same disorder, and in some cases the effects of creation Theoretically, whereas the SCTMA theory is a per-
d and scattering balance, causing the thermal conductiv- turbation theory, there are other non-perturbative ap-
n
ity κ due to QPs to become independent of the disorder proaches [14, 19, 20] that tend to predict QP localiza-
o
c concentration at low T. This phenomenon is called the tions. Actually, when the superconducting (SC) state is
[ “universal thermal conductivity”, which is best under- inhomogeneous, the SCTMA theory is clearly inappro-

stood in the framework of the self-consistent T-matrix priate and nonperturbative theories becomes necessary.
2
v approximation(SCTMA)theory[14]forBCSd-wavesu- Since the cuprate superconductors have an inherent ten-
2 perconductors. In the SCTMA theory, when γ is in the dency to develop electronic inhomogeneities in the SC
5 universal limit kBT ≪ γ ≪ ∆0 (∆0 is the maximum state because of the short coherence length, and indeed
4 gap),the residualcomponentκ0/T (T →0limitofκ/T) various sorts of electronic inhomogeneities are being dis-
1
becomes independent of the QP scattering rate Γ and is covered in cuprates [21, 22, 23, 24], it is important to
1
5 expressed as [15, 16] sortouttheconsequenceoftheinhomogeneityinthelow-
0 energy physics probed by the QP heat transport.
t/ κ0 = kB2 n vF + v2 ≃ kB2 nvF, (1) InthisLetter,wecriticallyexaminetheuniversalther-
a T 3h¯ c (cid:18)v2 vF(cid:19) 3h¯ c v2
malconductivitybystudyingYBCO,LSCO,andBi2212
m
with n the number of CuO2 planes per unit cell, c the c- for various doping levels; what is crucial to this work
-
d axislatticeconstant,andvF (v2)theQPvelocitynormal is that our samples are among the best single crystals
n (tangential) to the Fermi surface at the node. This is available for these three systems, well characterized by
o useful because, if valid, one can obtain ∆0 by the bulk various transport properties [25, 26, 27, 28, 29, 30], and
c
measurement of κ [7, 8]. the κ measurements are done with a small absolute un-
:
v Experimentally, κ0/T was reported to be approx- certainty of less than 10%. In our experiments, we find
i
X imately independent of the impurity scattering in threefeaturesthatallindicateabreakdownoftheuniver-
r nearly optimally-doped YBa2Cu3Oy (YBCO) [1] and sal thermal conductivity upon a closer look: i) a slight
a Bi2Sr2CaCu2O8+δ (Bi2212) [4], and its magnitude for Zn substitution for Cu induces notable suppression of
Bi2212 was consistent [3] with Eq. (1) using the angle- κ0/T in underdoped and optimally-doped samples of all
resolved photoemission spectroscopy (ARPES) results thethreesystems;ii)inBi2212,boththemagnitudeand
[17]. Hence, it has been considered that the universal the doping dependence of the gap parameter vF/v2 ob-
thermal conductivity is essentially realized in cuprates. tainedfromκ0/T viaEq. (1)differsignificantlyfromthe
However,the experimentalresultssofar[1, 2, 3, 4]carry ARPES results [17, 31]; iii) a large in-plane anisotropy
rather large error bars and thus it has not been clear of ∼2 is observed for κ0/T in Bi2212. We discuss that
to what extent κ0/T is “universal”. Moreover, Hussey the electronic inhomogeneity plays a major role in the
et al. [5] found an absence of κ0/T in an underdoped departure from the universality by causing a partial QP
YBa2Cu4O8, which questions the universal scenario and localizationassuggestedbythenonperturbativetheories.
2
0.08 0.06
YBa(Cu Zn)O YBa(Cu Zn)O (a) 0.10 (b) z = 0.006
2 1-z z3 y 2 1-z z3 y
2k / T (W/Km)a000...000246 (a) zy == 0 z6. 0.=50 060 00..0024 (b) zy == 0z7. 0.=00 006 2k / T (W/Km)ab00..0150 La2-xSrxCu1-zZzn x=z O=0z . 400=0. 2092 2k / T (W/Km)a0.05 Bi2Sr2Ca(Cu1-zZnOzzD) =28O 038+d
0.00 0.00
0.00 0.00 0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03
0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03 T2 (K2) T2 (K2)
(c) z = 0 0.03 (d) z = 0
0.01 FIG. 2: (color online) Zn-substitution effect on the low-T
m) thermal conductivity of LSCO and Bi2212 in the overdoped
2K 0.02 z = 0.017
/ T (W/ab z = 0.006 xz == 00..01100 0.01 x = 0.17 rpsoehgloiidonnoli.nnTedsehcaeordueapltliiannetgahr(astfieetasrteetxoatpe)psatarirmeesnahttoleywκan0ff/beTyct.oepdebnysythmebeollesc.trTohne-
k La SrCu ZnO La SrCu ZnO
2-x x 1-z z 4 2-x x 1-z z 4
0.00 0.00
0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03
0.08
Weemphasizethatourκdataarereproduciblewithtypi-
0.06 (e) (f) z = 0 calvariationsoflessthan10%,thankstotheconsistently
z = 0 0.06
m) highqualityofthecrystalsandagoodcontrolofthedop-
2W/K 0.04 z = 0.006 0.04 z = 0.006 ing level [25, 26, 27, 28, 29].
/ T (a0.02 UD83 0.02 OP94 lowF-iTrstQ, PwehesahtowtratnhsepoerfftecftorodfiffZenr-esnutbsdtiotpuitniognreognimtehse.
k Bi2Sr2Ca(Cu1-zZnz)2O8+d Bi2Sr2Ca(Cu1-zZnz)2O8+d Figure 1 shows the κ/T vs T2 plots for YBCO, LSCO,
0.00 0.00
0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03 and Bi2212 systems in the underdoped and optimally-
T2 (K2) T2 (K2)
doped regimes. The T=0 intercepts of the linear fits to
FIG. 1: (color online) Zn-substitution effect on the low-T the lowest-T data give the residual QP component κ0/T
thermal conductivity of YBCO, LSCO, and Bi2212 in the [1, 2, 3, 4, 5, 6, 10, 11]. Here, the striking feature is
underdoped and optimally-doped regimes. The doping levels a strong decrease of κ0/T upon slight Zn substitution
ofBi2212 samplesareindicatedbytheTc valuesofpureones in all these systems [33], whereas within the SCTMA
(Zn-substitutedonesareannealedtoachievethesameoxygen theory κ0/T is predicted to increase with Γ when the
contentsasthepureones). Thesolidlinesarelinearfitstothe
system is not in the universal limit [15, 34]. We note
lionw(efs)t-(Tseedtaetxat)t.o extract κ0/T, except for the z = 0 sample that in YBCO at y = 7.00, κ0/T decreases by ∼20%
with 0.6% of Zn [Fig. 1(b)]; while this behavior sug-
gests a departure from the universal behavior given our
small error bar (∼10%) [35], it is not inconsistent with
High-quality single crystals of YBa2(Cu1−zZnz)3Oy, the previous results by Taillefer et al. for y = 6.9 [1, 2],
La2−xSrxCu1−zZnzO4 and Bi2Sr2Ca(Cu1−zZnz)2O8+δ where the error bar was as large as ∼40%. A more pro-
are grown by a flux method [25], a traveling-solvent nounced suppression in κ0/T of ∼40% is observed upon
floating-zone method [26] and a floating-zone method 0.6% of Zn-substitution in underdoped YBCO at y =
[28, 29], respectively. The actual Zn concentration z is 6.50 [Fig. 1(a)], for which there has been no previous
determined by the inductively-coupled plasma atomic- report. Apparently, the present higher-precision mea-
emission spectroscopy (ICP-AES). All the YBCO sam- surements indicate that all three systems do not strictly
ples are perfectly detwinned and κ is measured along display the universal behavior in the underdoped and
the a axis to avoid the additional electronic heat trans- optimally-dopedregimes;furthermore,theobservedsup-
port coming from the Cu-O chains. For a doping- pression of κ0/T by Zn substitution is opposite to what
dependence study of Bi2212, Bi2Sr2CaCu2O8+δ crystals is expected from the SCTMA theory. If we remember
aswellasBi2Sr2Ca1−yDyyCu2O8+δ(Dy-Bi2212)crystals that a non-perturbative theory that takes into account
(also produced by the floating-zone method) are care- the gap inhomogeneities induced by impurities predicts
fully annealed at 400–800◦C in suitable atmospheres to a QP localization [19], it seems most likely that the ob-
tunetheoxygencontent. WelabeltheBi2212samplesas servedbehaviorisessentiallyduetothe Zn-inducedelec-
UD70,OP94,OD70,etc.,bytheirdopingregimesandT tronic inhomogeneity in the form of nonsuperconducting
c
values, and the Dy-Bi2212 samples as Dy81, Dy45, etc., droplets [22].
by their T values (all in underdoped regime) [32]. The Figure2 showsthe Zn-substitutioneffect onκ inover-
c
thermal conductivity measurement in millikelvin region dopedLSCOandBi2212. Inoverdopedcuprates,serious
is done by a conventional steady-state “one heater, two electron-phonondecouplingoccursatverylowT [36,37],
thermometer”techniqueinadilutionrefrigerator[6,11]. causing the QP thermal conductivity to be undetectable
3
0.15 0.10
m) 00..0068 Bi2212 00..0068 DDDyyy854022 DDyy306 0.10 (a) k a 0.08 (b) k a
2K +0.01 m) OP94 0.06 UD70
/ T (W/a00..0024 (a) OUDP9843 00..0024 (b) k (W/K0.05 k b 00..0024 k b
k UD70 Dy-Bi2212
0.000.00 0.01 0.02 0.030.000.00 0.01 0.02 0.03 0.000.0 0.2 0.4 0.6 0.000.0 0.2 0.4 0.6
T2 ( K2) T2 ( K2) T ( K) T ( K)
0.06 0.08 0.06
Bi2212 Bi2212 (c) k (d)
Dy-Bi2212 60 Dy-Bi2212 0.06 a k
2k / T (W/Km)000..0024 (c) v/v F22400 (d) Bi2212 (ARPES) 2k / T (W/Km) 00..0024 OPk9b4 00..0024 UDk7ba0
0.00 0 0.00 0.00
0.05 0.10 0.15 40 60 80 100 0.00 0.01 0.02 0.03 0.00 0.01 0.02 0.03
p Tc (K) T2 (K2) T2 (K2)
FIG. 3: (color online) (a),(b) a-axis thermal conductivity of FIG. 4: (color online) (a),(b) T dependencesof a- and b-axis
Bi2212 and Dy-Bi2212 at low T; Dy80 data are shifted up thermal conductivities for optimally-doped and underdoped
by 0.01 W/K2m for clarity. The solid lines are linear fits to Bi2212. (c),(d) κ/T vs T2 plots of the low-T data; the solid
extractκ0/T. (c)Hole-dopingdependenceofκ0/T inBi2212. lines are linear fitsto extract κ0/T.
(d)ThegapparametervF/v2,calculatedfromκ0/T usingEq.
(1),asafunctionofTcintheunderdopedregion;alsoshownis
vF/v2obtainedfromARPES[31]forcomparison. Thedashed
and dotted lines are guides to theeyes. 3(d); also included in this panel are the vF/v2 values
directly obtained from ARPES for several underdoped
Bi2212 crystals [31]. Not only the opposite trend in the
bytheconventionalsteady-statetechnique. Asaresult,a doping dependence, but also the difference in the vF/v2
strongdownturnshowsupinthelowest-T dataofκ/T for values from κ0/T and from ARPES, demonstrate an in-
overdoped samples [36, 37], which is also the case with adequacy of Eq. (1). To be more precise, κ0/T of 0.051
our data in Fig. 2. [Our optimally-doped Bi2212 also W/K2m for the optimally-doped Bi2212 yields vF/v2 =
shows this problem below 100 mK, see Fig. 1(f)]. It is 65and∆0 =10meV(assuminga simple dx2−y2 gap),in
thereforeimpossibletopreciselydetermineκ0/T ofthese contrastto vF/v2 = 20and∆0 = 35meVobtainedfrom
samples from the present data. Nevertheless, for a qual- ARPES [17, 31]. The reason why κ0/T is much larger
itative evaluation of the Zn-substitution effect, one can than what the ARPES-derived vF/v2 would predict via
crudelyestimateκ0/T byalinearfittingtoahighertem- Eq. (1) is notclear,but this shouldbe understoodin re-
perature range, as shown in Fig. 2, and it would be safe lationwiththeelectronicinhomogeneity,giventhatthere
to conclude that upon Zn substitution κ0/T is enhanced islikelyalargevariationintheSCgapmagnitudeinthis
in these overdoped samples. This trend is opposite to material[21,24]. Itmightbe that, depending onthe na-
that observed in underdoped and optimally-doped sam- ture of the inhomogeneity, sometimes the QP creation
ples in Fig 1, and is probably understandable within the effect overwhelms the QP scattering effect (which may
SCTMA theory[15, 34]. Hence, there is a crossovernear be the case near optimum doping) and sometimes the
optimum doping beyond which the nodal QPs behave opposite happens (possibly in the underdoped regime).
more ordinarily. Clearly, more work is called for on the theoretical part.
We next show the doping dependence of κ0/T for Figure 4 shows another novel property observed in
Bi2212, where the gap parameters have been reliably Bi2212; namely, the QPs show very different abilities to
obtained by ARPES studies [17, 31]. Figures 3(a) and conduct heat along the a and b axes that correspond to
3(b) show the low-T thermal conductivity (along the a the two orthogonal nodal directions. Indeed, the data
axis) of Bi2212 and Dy-Bi2212 single crystals, and Fig. in Fig. 4 demonstrate that in the T → 0 limit the
3(c) summarizes the doping dependence of κ0/T. Here, anisotropy in κ0/T is more than a factor of 2. If the
the hole concentration per Cu, p, is determined by the universal scenario holds here and Eq. (1) is valid, the
method employing the Hall coefficient proposed in Ref. in-plane anisotropy would mean that the SC gap is not
[38]. Clearly,κ0/T decreaseswithdecreasingdopingand the standard dx2−y2, a possibility discussed in Ref. [29];
approaches zero upon entering the nonsuperconducting however, given the large anisotropy for T → 0 and the
region — this behavior is essentially the same as that problems in the universal scenario, it seems more natu-
observed in LSCO [6, 7] and YBCO [11]. The gap pa- ral to consider that the source of the anisotropy is not
rameter vF/v2 calculated using Eq. (1) is shown in Fig. anasymmetryintheSCgapbutisthebreakdownofthe
4
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in our optimally-doped Bi2212 are much larger than the [9] R. W. Hill et al.,Phys. Rev.Lett. 92, 027001 (2004).
[10] Y. Andoet al.,Phys. Rev.Lett. 92, 247004 (2004).
values reported previously [3, 4]; such a dependence of
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κ0/T on the sample source suggests that the details of 107001 (2004); Phys.Rev.B 72, 100502(R) (2005).
the electronic inhomogeneities matter in Bi2212.
[12] X. F. Sun, S. Komiya, J. Takeya, and Y. Ando, Phys.
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Bi2212, it is useful to notice that the latest “gap map” [13] D. G. Hawthorn et al., Phys. Rev. Lett. 90, 197004
data obtained by the scanning tunneling microscope ex- (2003).
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(2002).
in the electronic inhomogeneity; in Fig. 2B of Ref. [21],
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the Bogoliubov-deGennes equation suggest [14, 19], one [18] V. P. Gusynin and V. A. Miransky, Eur. Phys. J. B 37,
would expect κ to become smaller than κ in the situ- 363 (2004); M. Takigawa, M. Ichioka, and K. Machida,
b a J. Phys.Soc. Jpn.73, 2049 (2004).
ation as indicated in Ref. [21]. Hence, the unusual in-
[19] W. A. Atkinson and P. J. Hirschfeld, Phys. Rev. Lett.
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to the anisotropic electronic inhomogeneity, that is indi-
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rectly caused by the supermodulation structure peculiar Rev. B 61, 6966 (2000).
to this system. [21] K. McElroy et al.,Science 309, 1048 (2005).
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conductivity is unexpectedly sensitive to the electronic [23] M. Vershinin et al.,Science 303, 1995 (2004).
[24] C. Howald, P. Fournier, and A. Kapitulnik, Phys. Rev.
inhomogeneity, whatever the nature of the inhomogene-
B 64, 100504(R) (2001).
ity is; it can be the local suppression of the SC gap by
[25] K. Segawa and Y. Ando, Phys. Rev. Lett. 86, 4907
Zn impurities [22], or it can be the gap inhomogeneity (2001); Y. Ando and K. Segawa, Phys. Rev. Lett. 88,
produced by dopant atoms [21]. The qualitative change 167005 (2002).
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suggeststhata strongtendency to formanelectronicin- Rev.B65,214535 (2002); S.Komiya,H.-D.Chen,S.-C.
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account the QP localization due to electronic inhomo- [32] Actual Dy contents y of the Dy-Bi2212 samples are de-
termined by ICP-AES: 0.20 (Dy80); 0.27 (Dy52); 0.34
geneities, and the present work demonstrates a crucial
(Dy42); 0.36 (Dy36); 0.51 (Dy0). Dy0is a nonsupercon-
role of the nanoscale inhomogeneity in the low-energy
ducting sample close to theSC boundary.
physics of high-T cuprates.
c [33] Whiletheslopeofthelinearfitisusuallydeterminedby
We greatly thank K. Behnia, P. J. Hirschfeld, N. E. thecrosssectionofthesampleduetotheboundaryscat-
Hussey and J. Takeya for helpful discussions. This work tering of phonons,thelarge slopes in ourZn-freeYBCO
was supported by the Grant-in-Aid for Science provided are indicative of the T3 contribution of the thermally-
by the Japan Society for the Promotion of Science. excitedQPsinveryclean samples[9],whichtestifiesthe
“ultraclean” nature of our YBCO crystals. This addi-
tionalT3termdoesnotaffectthedeterminationofκ0/T.
[34] Y. Sun and K. Maki, Europhys.Lett. 32, 355 (1995).
[35] For those YBCO data, the uncertainty in κ0/T comes
mostlyfromgeometrical-factorerrors(5–10%)andmuch
∗ Presentaddress: CERC,AIST,Tsukuba205-8562,Japan
less from thelinear fitting.
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