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arXiv:cond-mat/0605349v1 [cond-mat.supr-con] 12 May 2006

Search for Cooper-pair Fluctuations in Severely Underdoped Y1−yCayBa2Cu3O6+x

Films (y = 0 and 0.2)

Yuri L. Zuev∗and Iulian Hetel and Thomas R. Lemberger

Department of Physics, The Ohio State University

191 W. Woodruff Ave., Columbus, OH, 43210

The preformed-pairs theory of pseudogap physics in high-TC superconductors predicts a nonan-

alytic T-dependence for the ab-plane superfluid fraction, ρS, at low temperatures in underdoped

cuprates. We report high-precision measurements of ρS(T) on severely underdoped YBa2Cu3O6+x

and Y0.8Ca0.2Ba2Cu3O6+xfilms. At low T, ρS looks more like 1−T2than 1−T3/2, in disagreement

with theory.

PACS numbers: 74.25.Fy, 74.40.+k, 74.78.Bz, 74.72.Bk

Numerous theoretical models have been advanced to

explain the pseudogap in underdoped high-TCsupercon-

ductors. One class of models (boson-fermion models)1,2,3

involves the idea that Cooper pairs form at the pseudo-

gap temperature, T∗, but due to internal fluctuations,

these pairs do not form a condensate until the observed

transition temperature, TC, where superfluid appears. Of

the many results of the model, (4and references therein)

to us the most dramatic and accessible is its prediction

of a nonanalytic T-dependence of the superfluid frac-

tion - a quantity that we can measure with high pre-

cision. In this model, the pairing field ∆pg(T) binds

the carriers into pairs below a pseudogap temperature,

T∗, of the order of several hundred K. The T depen-

dence of ∆pg is assumed to be BCS-like. Below TC, an

energy scale ∆sc responsible for pair condensation ap-

pears. The total gap, ∆(T), for Fermionic quasiparti-

cles is: ∆2= ∆2

sc.

superfluid density is determined by condensation of pre-

formed pairs. The underlying BCS superfluid fraction,

ρBCS

s

(T), which vanishes at T∗, is renormalized by pair

condensation such that the measured superfluid fraction

is: ρs(T) = [∆2

pg+ ∆2

The T dependence of the

sc(T)/∆2(T)]ρBCS

s

(T).

Detailed calculations5,6predict that strong Cooper

pair fluctuations in underdoped materials dominate the

physics and give ∆sc(T), and therefore ρs(T), a nonan-

alytic T-dependence: ∆2

a is near unity. While this is technically a low-T re-

sult, it is expected to hold over a relatively wide tem-

perature range because TC is the only low-energy scale

in the problem.Because T∗is much larger than TC,

∆pg is essentially constant at low T, T ≪ T∗. And, in

moderately disordered d-wave superconductors, ρBCS

is quadratic at low T12.The end result is: ρs(T) ≈

1 − a(T/TC)3/2− b(T/T∗)2, where b is a constant near

unity7. This result is experimentally accessible for two

reasons. First, there is the separation of scales: for

the severe underdoping explored here, T∗is more than

ten times larger than TC, in which case the quadratic

term is negligible. Second, the predicted nonanalytic T-

dependence is not affected by microscopic disorder, which

is present in our films. Numerical results7on disordered

d-wave superconductors find that the limiting behavior,

sc(T) ∝ 1 − a(T/TC)3/2, where

s

(T),

ρs(T) ≈ 1 − a(T/TC)3/2, holds up to about TC/2.

The theory under discussion is physically appealing,

but controversial. Rather than discuss its merits, we pro-

ceed to test its striking prediction for ρS(T) in cuprates.

We note that Carrington et al.8claim T3/2behavior

in organic superconductors.

present the T-dependence of ρS(T) for several heavily

underdoped YBCO films.

in terms of the ab-plane penetration depth, λ, where

ρs(T) = λ−2(T)/λ−2(0).

a clear T3/2dependence over a significant temperature

range. The data are better fit by T2.

We determine the ab-plane superfluid density of thin

films by a two-coil rf technique, described in detail in9,10.

The measured quantity is the sheet conductivity of the

film, σd = σ1d − iσ2d, where d is the film thickness and

σ is the usual conductivity. The ab-plane penetration

depth, λ, is obtained as: λ−2= µ0ωσ2, where µ0is mag-

netic permeability of vacuum and ω/2π is the measure-

ment frequency, 100kHz in this work. σ1is large enough

to be measurable only near the transition, where, in prin-

ciple, it diverges. The width of this peak provides an

upper limit on the inhomogeneity of TC.

In this study we used three films of pure YBCO with

TC= 36, 20 and 10 K, and λab(0) =0.46, 0.75 and 2.43

µm, respectively. These are films A, B, and C in Table

I. They were produced by pulsed laser deposition (PLD)

on (001) polished SrTiO3substrates with subsequent an-

nealing. Films were sandwiched between a buffer and

a cap layer of insulating PrBCO, whose purpose was to

improve homogeneity of YBCO by removing interfaces

with substrate and air (protecting samples from air hu-

midity). In addition, the PrBCO cap helps move the re-

gion of high oxygen concentration gradient (near the free

surface) away from YBCO, thus improving oxygen homo-

geneity. The doping homogeneity is paramount here be-

cause at the doping levels of this work the TCis a strong

function of doping, so that small variations of oxygen con-

centration cause wide transition and potentially much of

the superfluid density curve λ−2(T) is inside the transi-

tion region. The above procedure allows us to achieve

narrow enough σ1 peaks so that no more than about a

quarter of ρs(T) curve is inside the peak. A fourth film

In this brief report, we

The discussion is couched

None of the films exhibits

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FIG. 1: Superfluid density ρs ∝ λ−2(solid curves) and σ1

(dashed curves) vs. T for four films used in the present study.

(D) was 140 nm thick film of Y0.8Ca0.2BCO produced by

PLD on (001) STO.

The main systematic uncertainty in our measurement

is the uncertainty in the film thickness, d, which affects

the absolute value of λ−2at T = 0, but not its T de-

pendence. The 36 and 20 K films were nominally 40 unit

cells thick (d = 0.048µm), while the 10K film was 20 unit

cells thick (d = 0.025µm). We know the film thickness to

10%, therefore we know λ−2(0) to within 10% , and λ(0)

to 5%. This uncertainty does not affect the conclusions

of the present work.

Figure 1 shows the superfluid fraction, ρs∝ λ−2, along

with peaks in σ1, as functions of T. The narrowest tran-

sition is that of the 36 K film.

”severely” underdoped because the pseudogap tempera-

ture is greater than 300 K11.

In figure 2 we compare T2power law vs. T3/2law for

our films. The black dotted curves are experimental data,

while red lines are the power law fits to the first 20% drop

of the superfluid density. Left panels are the data plotted

vs. T2, while right panels are the data plotted against

T3/2. The two power laws are close to each other and

are hard to distinguish. Yet, as far as we can tell, the

quadratic fit is better over larger temperature interval.

This is most evident in 36K film (top panel, fig.2), where

the quadratic fit is almost perfect over as much as 50% of

the superfluid density drop and 70% of the temperature

interval.

This film qualifies as

Sample Thickness, µm TC,K λ(0), µm

A0.048

B0.048

C0.025

D0.14

X2

T3/2

X2

T2

36

20

10

28

0.45

0.76

2.4

0.94

3.0 · 10−54.6 · 10−6

2.0 · 10−51.2 · 10−6

8.3 · 10−53.6 · 10−5

5.2 · 10−55.8 · 10−6

TABLE I: Parameters for the films presented in this report.

X2comes from fits to the first 20% drop in λ−2(T).

FIG. 2: (Color online) Low-T part of the superfluid density

for films A, B, C and D (top to bottom). In each pair of

graphs, left one shows data against T2, while the right one

is against T3/2. The red lines are linear fits to the first 20%

decrease of the superfluid density. In each case, quadratic T

dependence provides better fit over larger temperature range.

To be more specific, we can describe the fit quality by

calculating a fractional fit error:

X2=

1

N

?(ρs,fit− ρs,data)2

ρ2

s,fit

(1)

over first 20% of the drop in superfluid density ρs, where

N is the number of data points in that interval. Values of

X2are given in the table for both power laws. One can

see that X2is about an order of magnitude smaller for T2

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fit (good to about 0.1%), than for T3/2fit, which is good

only to about 1% over same temperature interval. The

quadratic fit quality at lowest temperatures for sample

A is actually slightly worse than that for sample B, even

though at higher temperatures sample A fits the T2law

better.

However, T3/2gives a reasonable agreement with data

over some temperature range (10-15K). In principle, the

fit can be improved somewhat by including T2term.

However, as mentioned above, the quadratic term should

be 100 times smaller than the T3/2term because T∗is

about 10 times larger than TC.

One can be concerned about possible modification of

the temperature dependence by spread in TC across a

given sample. It is easy to show that such spread would

cause only a change in a prefactor, but no new power

law behavior, as long as the temperature is below about

a half of the lowest TC, which is the case here.

While the 36K film provides the cleanest comparison

with ”pair fluctuation” theory because of its very sharp

transition, the same qualitative conclusion is supported

by data on 20 and 10 K films, as well as Ca doped film

(three lower panel, fig. 2). The transitions in these films

are not narrow enough to draw more quantitative con-

clusions from those data.

It should be noted that if our films were cleaner, i.e.

if they displayed T-linear dependence of ρs, it would be

more difficult to reject the pair-fluctuations picture. In

very clean crystals of D. Broun et al.13where T-linear

dependence clearly yields to T2at lower temperatures,

these two power laws together produce a good fit to T3/2

behaviour over as much as 40% drop in superfluid density.

We believe that this is accidental and is not caused by

preformed pairs condensation.

In conclusion, we made high precision measurements of

superfluid density in heavily underdoped YBCO to test a

novel prediction of a particular model of the pseudogap.

We did not find persuasive evidence for the predicted

1− a(T/TC)3/2behavior in superfluid fraction at low T.

In every case, a quadratic T dependence fitted the data

better. It is possible that the predicted nonanalytic be-

havior is confined to just the first few percent drop in

superfluid density, but we do not see why this should be

so.

This work has been supported by NSF DMR grant

0203739. We acknowledge help from A. Weber, G. Ham-

merl, C.W. Schneider and J. Mannhart of the University

of Augsburg, who made the sample D for this study.

∗Present address: Materials Science and Technology Divi-

sion, Oak Ridge National Laboratory, Oak Ridge, TN

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