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
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
superfluid density is determined by condensation of pre-
formed pairs. The underlying BCS superfluid fraction,
(T), which vanishes at T∗, is renormalized by pair
condensation such that the measured superfluid fraction
is: ρs(T) = [∆2
The T dependence of the
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
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
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
This film qualifies as
Sample Thickness, µm TC,K λ(0), µm
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:
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
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
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
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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