# Quantum phase transition in overdoped La2-xSrxCuO4 evinced by the superfluid
density

T. Schneider  Physik-Institut der Universität Zürich,Winterthurerstrasse 190,
CH-8057 Zürich, Switzerland

###### Abstract

The superfluid density
$\rho_{\|sf}\left(T\right)=\lambda_{\|}^{-2}\left(T\right)$ of overdoped
La2-xSrxCuO4 thin films of high quality have been measured with $T_{c}$
(defined by the onset of the Meissner effect ) from $5.1$ to $41.6$ K by
Božović et al. [1]. Given this $T_{c}$ the superfluid density
$\rho_{\|sf}\left(T\right)$ shows no clear evidence of critical fluctuations
and no indication of vortex unbinding as $T\rightarrow T_{c}$. Nevertheless,
$\rho_{\|sf}\left(T\right)$ displays the behavior expected for a quantum phase
transition (QPT) in the $\left(3+1\right)$D-xy dimensional universality class,
with $\rho_{\|sf}\left(T\right)\propto T_{c}^{-2}$ as $T\rightarrow 0$.
However, this relation is also a hallmark of dirty superconductors, treated in
the mean-field approximation. Here we attempt to clear out the nature of the
suppression of $\rho_{\|sf}\left(T\right)$ as $T_{c}\rightarrow 0$. Noting
that for any finite system the continuous transition will be rounded we
perform a finite size scaling analysis. It uncovers that the
$\rho_{\|sf}\left(T\right)$ data are consistent with a finite length limited
$3$D-xy transition. In some films it is their thickness and in others their
inhomogeneity that determines the limiting length. Having established the
precondition for the occurrence of a QPT mapping on the $\left(3+1\right)$D-xy
model, we explore the consistency with the hallmarks of this transition. In
particular with the relations
$\rho_{\|sf}\left(T\right)/\rho_{\|s}\left(0\right)=1-y_{c}T/T_{c}$,
$T_{c}\propto 1/\lambda_{\|}\left(0\right)$ as $T_{c}\rightarrow 0$ and
$y_{c}=\alpha\lambda_{\parallel}^{2}\left(0\right)T_{c}$, where $\alpha$ is
the coefficient in $\rho_{\|sf}\left(T\right)=\rho_{\|s}\left(0\right)-\alpha
T$. The emerging agreement with these characteristics points clearly to a
quantum fluctuations induced suppression, revealing the crossover from the
thermal to the quantum critical regime as $T_{c}\rightarrow 0$. In the
classical-quantum mapping it corresponds to a $3$D to $(3+1)$D crossover.

###### pacs:

74.40.Kb,74.25.Bt,74.20.De,74.62.-c,74.78.-w

##  I Introduction

An essential unsolved puzzle in cuprate superconductivity is that, despite the
evidence of more conventional normal state properties [2, 3, 4], the
superfluid density $\rho_{\|s}\left(T\right)=\lambda_{\|}^{-2}\left(T\right)$
of overdoped La2-xSrxCuO4 seems to be controlled by phase fluctuations [1, 5,
6, 7]. Furthermore, the vanishing superfluid density in overdoped La2-xSrxCuO4
films is consistent with $\rho_{\|s}\left(T\right)\propto$ $T_{c}^{2\text{ }}$
as $T_{c}\rightarrow 0$ [1], expected for a quantum phase transition in the
$(3+1)$-dimensional (D)- xy universality class [8, 9]. This implies $3$D-xy
critical behavior at any $T>0$, characteristic for uncharged superfluids [10],
as observed near optimum doping in YBa2Cu3O6+y single crystals [11, 12] and
other cuprates [9, 13, 14, 15]. However, the measurements of
$\rho_{\|s}\left(T\right)$ on the overdoped La2-xSrxCuO4 films show no clear
evidence of 3D-xy critical behavior [1]. Rather the approach to $T_{c}$ is
mean-field-like. Taking this mean-field-like behavior of
$\rho_{\|s}\left(T\right)$ for granted the suppression of $T_{c}$ is also
attributable to pair breaking in a d-wave superconductor. In the dirty limit
it leads to the same scaling relation, namely
$\rho_{\|s}\left(T\right)\propto$ $T_{c}^{2}$ [16, 17]. The puzzle, namely
what suppresses the superfluid density in overdoped cuprates, is what we
attempt to solve.

First we perform a finite size scaling analysis of the superfluid density data
of Božović et al.[1]. It turns out that the observed mean-field-like behavior
is a finite size effect. It stems from the fact that in an infinite and
homogeneous 3D-xy system the transverse correlation length diverges as
$\xi_{\perp}^{t}\left(T\right)=\xi_{\perp 0}^{t}t^{-\nu}$, where
$t=\left(1-T/T_{c}\right)$ and $\nu\simeq 2/3$ [18], while in the films
$\xi_{\perp}^{t}\left(T_{c}\right)$ is set by the limiting length $L_{c}$. In
homogeneous films it is determined by the effective film thickness $L_{cf}$
and in inhomogeneous films by $L_{ci}<L_{cf}$, the size of the homogenous
domains. Nevertheless, in the films analyzed here, the $3$D-xy critical regime
is reached. This analysis provides estimates for the transition temperatures
and the critical amplitude $\lambda_{\parallel 0}\left(T_{c}\right)$ of the
infinite system appearing in
$\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel 0}^{-2}t^{\nu}$ as
$T\rightarrow T_{c}$. Because the 3D-xy transition temperatures are lower than
those estimated from the onset of the Meissner effect [1], the mean-field-like
behavior of $\rho_{\|sf}\left(T\right)=\lambda_{\|f}^{-2}\left(T\right)$ is
traced back to a finite size rounded transition. In addition the estimates for
$T_{c}$ and $\lambda_{\parallel 0}$ uncover the relationship $T_{c}\propto
1/\lambda_{\parallel 0}$ as $T_{c}\rightarrow 0$. Accordingly, the system
approaches the extreme type II limit ($\lambda_{\parallel}\rightarrow\infty$)
as $T_{c}\rightarrow 0$. In this regime it is expected that the static
critical properties at zero external field are dominated by phase fluctuations
[19, 20].

Second, having established that 3D-xy fluctuations determine the finite
temperature critical behavior, limited by the finite size effect, we turn to
the analysis of the low temperature data extending down to $0.3$ K. Therefore,
an extrapolation is necessary to estimate
$\lambda_{\parallel}^{-2}\left(0\right)$. It turns out that the linear
extrapolation with $\lambda_{\parallel
f}^{-2}\left(T\right)=\lambda_{\parallel}^{-2}\left(0\right)-\alpha T$ fits
the data up to $1$ K very well. Given then the estimates of $T_{c}$,
$\lambda_{\parallel}^{-2}\left(0\right),$ $\alpha$ , and the $T_{c}$
dependence of the latter, their consistency with a QPT, mapped on the
$(3+1)$D-xy universality class, can now be tested. This mapping requires that
the scaling relations $T_{c}=f_{\|}(0)/\lambda_{\parallel}\left(0\right)$,
$\alpha=y_{c}T_{c}/f_{\|}^{2}\left(0\right)$ are satisfied and the data
plotted as $\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ vs . $T/T_{c}$ collapse on a single curve. In the limit
$T/T_{c}\rightarrow 0$ and sufficiently small $T_{c}$ it should tend to the
line $\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)=1-y_{c}T/T_{c}$ and in the $T/T_{c}\rightarrow 1$ limit
to $t^{2/3}$. However, the rounded transition leads around $T/T_{c}\approx 1$
to characteristic deviations.
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ exceeds $T/T_{c}=1$ up to the temperature set by the
onset of the Meissner effect. As $T_{c}\rightarrow 0$ the rounding becomes
stronger because the ratio
$L_{c}/\xi_{\perp}^{t}\left(T\right)=\left(L_{c}/\xi_{\perp
0}^{t}\right)t^{2/3}$, controlling the finite size effect, decreases because
$\xi_{\perp 0}^{t}$ grows like $\xi_{\perp 0}^{t}\propto 1/T_{c}$. On the
other hand, as $T/T_{c}$ drops the collapse is no longer affected by the
finite size effect. The crossover from the thermal to the quantum fluctuation
dominated regime implies that
$\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel 0}^{-2}t^{2/3}$
approaches the low $T_{c}$ behavior and thus expands the critical thermal
regime
$\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel}^{-2}\left(0\right)t^{2/3}$
towards lower temperatures. Noting that the QPT has been mapped on the
(3+1)D-xy model it corresponds here to a dimensional crossover from $3$D to
$(3+1)$D. Our analysis reveals that the
$\rho_{\|sf}\left(T\right)=\lambda_{\|f}^{-2}\left(T\right)$ data of the thin
and overdoped La2-xSrxCuO4 films are remarkably consistent with the outlined
hallmarks of this QPT. In summary, our analysis reveals that the suppression
of the superfluid density is driven by quantum phase fluctuations and uncovers
the crossover from the thermal to the quantum critical regime. The thermal
regime shows finite size limited 3D-xy criticality and the quantum regime is
compatible with $(3+1)$D-xy criticality.

In Sec. II we sketch the theoretical background including the finite size
scaling approach, the scaling relations and scaling forms of the QPT mapped on
the classical $(3+1)$D-xy model. Sec. III is devoted to the analysis of the
superfluid density data of Božović et al.[1]. We close with a brief summary
and conclusions.

##  II Theoretical background

A precondition for the occurrence of 3D-xy critical behavior in homogeneous
superconducting films with effective thickness $L_{c}$ originates from the
vortex interaction. Indeed, a Berezinskii [21] -Kosterlitz-Thouless [22] (BKT)
transition might occur, provided that the Pearl length [23]
$\lambda_{pearl}=2\lambda_{\parallel}^{2}\left(T_{BKT}\right)/L_{c}$ exceeds
$L_{L}$, the lateral extent of the film. In this case, a precondition for a
BKT transition, the logarithmic intervortex interaction, is satisfied.
Invoking the Nelsom-Kosterlitz relation [24], $T_{BKT}=\pi\Lambda
L_{c}/\left(2\lambda_{\parallel}^{2}\left(T_{BKT}\right)\right)$ we obtain the
precondition

$\lambda_{pearl}=\frac{\pi\Lambda}{T_{c}}>L_{L},$ (1)

where $\Lambda=\Phi_{0}^{2}/\left(16\pi^{3}\text{k}_{B}\right)\simeq
6259~{}\mu m$ K. For the films considered here $L_{L}=1$cm and with that
$T_{BKT}<\pi\Lambda/L_{L}\simeq 1.97$ K, which is considerably below
$T_{c}\simeq 4.4$ K, the lowest value attained by Božović et al.[1]. In a
homogeneous and sufficiently large system 3D-xy critical behavior implies an
infinite correlation length below $T_{c}$ [25]. One can extract an alternative
diverging length the so called transverse correlation length defined in terms
of the helicity modulus $\Upsilon\left(T\right)$ as $\xi^{t}\left(T\right)=$
$\Upsilon\left(T\right)/k_{B}T$, where
$\Upsilon\left(T\right)\lambda^{2}\left(T\right)=\Phi_{0}^{2}/\left(16\pi^{3}\right)$
and $\lambda$ denotes the penetration depth [8, 9]. It probes the global
superconducting phase coherence across the system. In superconductors with
uniaxial symmetry ($\lambda_{x}=\lambda_{y}=\lambda_{\|}$,
$\lambda_{z}=\lambda_{\perp}$) it leads with

$\xi_{\perp}^{t}\left(t\right)=\xi_{\perp
0}t^{-\nu},\lambda_{\|}^{2}\left(t\right)=\lambda_{\|0}^{2}t^{-\nu},t=1-T/T_{c}\text{,
}\nu\simeq 2/3\text{,}$ (2)

to the universal relation [9]

$T_{c}=\Lambda\xi_{\perp 0}^{t}/\lambda_{\parallel 0}^{2}.$ (3)

It controls the $T_{c}$ dependence of the critical amplitudes $\xi_{\perp
0}^{t}$ and $\lambda_{\parallel 0}$.

Other hallmarks of 3D-xy criticality follow from the derivative
$d\lambda_{\|}^{-2}/dT$ and the crossing of the curves
$\lambda_{\|}\left(T\right)^{-2}$ and $\lambda_{\|}^{-3}\left(T\right)$. From
Eq. (2) we obtain

$d\lambda_{\|}^{-2}/dT=-2/\left(3\lambda_{\|0}^{2}T_{c}\right)t^{-1/3}.$ (4)

Contrariwise there is no singularity in the mean-field treatment where
$\lambda_{\|}^{2}\left(t\right)=\lambda_{\|0}^{2}t^{-1}$ yields
$d\lambda_{\|}^{-2}/dT=-1/\lambda_{\|0}^{2}T_{c}$. Given 3D-xy criticality
(Eq. (2)) the curves $\lambda_{\|}\left(T\right)^{-2}$ and
$\lambda_{\|}^{-3}\left(T\right)$ cross at

$T_{cr}=T_{c}\left(1-\left(\lambda_{\|0}/a\right)^{3}\right),$ (5)

as long as $\lambda_{\|0}/a<1$ with $a=1$ in units of $\lambda_{\|0}$. In this
case $\lambda_{\|}^{-3}\left(T\right)<\lambda_{\|}\left(T\right)^{-2}$ for
$T<T_{cr}$ and
$\lambda_{\|}^{-3}\left(T\right)>\lambda_{\|}^{-2}\left(T\right)$ for
$T>T_{cr}$. Contrariwise, if $\lambda_{\|0}/a>1$ there is no crossing point
but

$\lambda_{\|}\left(T\right)^{-2}>\lambda_{\|}^{-3}\left(T\right).$ (6)

Given then magnetic penetration depth data
$\lambda_{\parallel}^{-2}\left(T\right)$ of a homogeneous and infinite system
the transition temperature $T_{c}$ and the critical amplitude
$\lambda_{\parallel 0}$ can be estimated from fits to Eq. (2), its derivative
(Eq. (4)) and crosschecked with the crossingpoint relations (5) and (6).

In homogeneous films the outlined critical behavior will be modified because
the transverse correlation length $\xi_{\perp f}^{t}\left(T\right)$ does not
diverge at $T_{c}$ and is analytic for all $t$. However, for a sufficiently
large limiting length $L_{c}$ the finite size scaling theory predicts a
scaling behavior of the form [26, 27]

$\xi_{\perp
f}^{t}\left(t\right)/\xi_{\perp}^{t}\left(t\right)=g\left(L_{c}/\xi_{\perp}^{t}\left(t\right)\right).$
(7)

In homogeneous films the limiting length is set by the effective film
thickness $L_{cf}$ $=L_{c}$ and in inhomogeneous films by $L_{c}=L_{ci}$
$<L_{cf}$, the size of the homogeneous domains. $g\left(x\right)$ is the
finite size scaling function. The critical $3$D-xy critical regime is reached
when $L_{c}/\xi_{\perp}^{t}\left(t\right)>1$ and
$g\left(L_{c}/\xi_{\perp}^{t}\left(t\right)\right)\rightarrow 1=\xi_{\perp
f}\left(t\right)/\xi_{\perp}\left(t\right)$. In contrast if
$L_{c}/\xi_{\perp}^{t}\left(t\right)<1$ the transition is rounded because
$g\left(L_{c}/\xi_{\perp}^{t}\left(t\right)\right)\rightarrow
L_{c}/\xi_{\perp}^{t}\left(t\right)$ and with that $\xi_{\perp
f}\left(t\right)\rightarrow L_{c}$. In principle, thick films are required for
observing the signatures of criticality such as Eq. (4), and for accurate
determination of $T_{c}$, and the critical amplitudes $\xi_{\perp 0}$ and
$\lambda_{\|0}$. Noting that the divergence of $\xi_{\perp}^{t}\left(t\right)$
is cut off at $\xi_{\perp}^{t}\left(t_{L_{c}}\right)=\xi_{\perp
0}\left(1-T_{L_{c}}/T_{c}\right)^{-2/3}=L_{c}$ we obtain from
$T_{L_{c}}/T_{c}=1-\left(\xi_{\perp 0}/L_{c}\right)^{3/2}$ the precondition

$\xi_{\perp 0}<<L_{c},$ (8)

for a reliable finite size scaling analysis. Given the penetration depth data
and the universal relation (3) it is convenient to transform the finite size
scaling form (7) to

$\lambda_{\parallel
f}^{2}\left(t\right)/\lambda_{\parallel}^{2}\left(t\right)=g\left(L_{\lambda\|}^{2}/\lambda_{\parallel}^{2}\left(t\right)\right),$
(9)

with

$L_{c}=L_{\lambda\|}^{2}T_{c}/\Lambda.$ (10)

The critical $3$D-xy critical regime is reached when
$L_{\lambda\|}^{2}/\lambda_{\parallel}^{2}\left(t\right)<1$ and
$g\left(L_{\lambda\|}^{2}/\lambda_{\parallel}^{2}\left(t\right)\right)\rightarrow
1=\lambda_{\parallel
f}^{2}\left(t\right)/\lambda_{\parallel}^{2}\left(t\right)$. In contrast if
$L_{\lambda\|}^{2}/\lambda_{\parallel}^{2}\left(t\right)>1$ the transition is
rounded because
$g\left(L_{\lambda\|}^{2}/\lambda_{\parallel}^{2}\left(t\right)\right)\rightarrow
L_{\lambda\|}^{2}/\lambda_{\parallel}^{2}\left(t\right)$ and
$\lambda_{\parallel f}\left(t\right)\rightarrow L_{\lambda\|}$ by which
$L_{c}$ can be determined with Eq. (10). Using the estimates for $T_{c}$,
$1/\lambda_{\parallel 0}^{2}$ and $L_{\lambda\parallel}^{2}$ the plot
$\left(\lambda_{\|f}\left(\left|t\right|\right)/\lambda_{\|}\left(\left|t\right|\right)\right)^{2}$
vs. $L_{\lambda\|}^{2}/\lambda_{\|}^{2}\left(\left|t\right|\right)$ should
then let the experimental data ($\lambda_{\|f}\left(\left|t\right|\right)$)
collapse onto a single line. For this to happen, the estimates for $T_{c}$,
$1/\lambda_{\parallel 0}^{2}$ and
$L_{\lambda\parallel}^{2}=L_{c}\Lambda/T_{c}$ need to be correct. On the one
hand, this plot enables the reliability of the estimated values $T_{c}$,
$\lambda_{\|0}$ and $L_{c}$ to be checked, and on the other hand it uncovers
the nature of the limiting length. If $L_{c}<L_{cf}$ it is traced back to the
size of the homogenous domains $L_{ci}$ and if $L_{c}\simeq L_{cf}$ to the
effective film thickness $L_{cf}$.

In summary, the finite scaling analysis provides crosschecked estimates of
$T_{c}$, the critical amplitudes $\xi_{\bot 0}^{t}$ and $\lambda_{\|0}$ of the
infinite and homogeneous system and the $T_{c}$ dependence of the latter. Of
particular interest is the resulting phase transition line
$T_{c}(1/\lambda_{\|0})$ which shows how the finite temperature superfluid
density depends on $T_{c}$. Supposing that $T_{c}$ vanishes as

$T_{c}=f_{\|0}/\lambda_{\parallel 0},$ (11)

the universal relation (3) points to a QPT because the transverse correlation
length

$\xi_{\perp 0}^{t}=f_{\|0}^{2}/\Lambda T_{c}.$ (12)

diverges as $T_{c}\rightarrow 0$, while the temperature dependence of
$\lambda_{\parallel}^{-2}\left(t\right)=\lambda_{\parallel 0}^{-2}t^{-2/3}$
and $\xi_{\perp}^{t}\left(t\right)$ is not affected. In this case the system
approaches the extreme Type II limit where
$\lambda_{\parallel}\rightarrow\infty$ as $T_{c}\rightarrow 0.$

Approaching the corresponding quantum critical point at $T=0$ we can invoke
the scaling theory of quantum critical phenomena [8, 9]. A hallmark of a QPT
mapped on the classical $(3+1)$ D-xy model is

$T_{c}=f_{\|}\left(0\right)/\lambda_{\parallel}\left(0\right),$ (13)

the quantum counterpart of Eq. (11). Since D$=4$ is the upper critical
dimension of the XY model, mean-field critical behavior follows, supplemented
by logarithmic corrections [28, 29]. The scaling relations of this transition
include the quantum analog of Eq. (3)

$T_{c}=R\Lambda\xi_{\perp}^{t}\left(0\right)/\lambda_{\parallel}^{2}\left(0\right),$
(14)

where $R$ is a nonuniversal coefficient. It yields with Eq. (13) the important
relations

$T_{c}=f_{\|}\left(0\right)/\lambda_{\parallel}\left(0\right)=f_{\|}^{2}\left(0\right)/\left(R\Lambda\xi_{\perp}^{t}\left(0\right)\right),$
(15)

In addition there is the scaling form [8, 9]

$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=F\left(y_{c}T/T_{c}\right),\text{
}F\left(0\right)=1$ (16)

valid in the limit $T\rightarrow 0$ and low $T_{c}$. $y_{c}$ $=ab$ denotes the
universal critical value of the scaling variable $y=aT\delta^{-1/2}$ at which
the scaling function $F\left(T=0,\delta\right)$ exhibits a singularity at

$T_{c}=b\delta^{1/2}\propto 1/\lambda_{\parallel}\left(0\right),$ (17)

where $\delta$ denotes the tuning parameter. A crossover to 3D-xy criticality
(Eq. (2)) then takes place as the temperature increases. In $3$D it is the
crossover from the quantum to the thermal critical regime and in the quantum
3D- xy model mapped onto the ($3+1$)D-xy model a dimensional crossover from
($3+1$) to $3$D. To check the compatibility with scaling relations (13), (14
and the scaling form (16) estimates for $\lambda_{\parallel}\left(0\right)$
are required. Given the data extending down to $0.3$ K we have to relay on an
extrapolation. A linear temperature dependence
$\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel}^{-2}\left(0\right)-\alpha
T$ leads with Eq. (16) to the scaling form

$\displaystyle\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)$
$\displaystyle=$ $\displaystyle F\left(y_{c}T/T_{c}\right)=1-y_{c}T/T_{c}$
(18) $\displaystyle=$ $\displaystyle
1-y_{c}T_{c}\lambda_{\parallel}\left(0\right)/f\left(0\right),$

and the scaling relation

$y_{c}=\alpha\lambda_{\parallel}^{2}\left(0\right)T_{c}=\alpha
f_{\|}\left(0\right)\lambda_{\parallel}\left(0\right)=\alpha
f^{2}\left(0\right)/T_{c},$ (19)

setting a stringent constraint on the coefficient $\alpha$. Given then
estimates for $\alpha$ and $1/\lambda_{\parallel}^{2}\left(0\right)$,
consistency with the $(3+1)$D- xy QPT requires that the data, plotted as
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ vs. $T\lambda_{\parallel}\left(0\right)$ or vs.
$T/T_{c}$, collapse for sufficiently low $T_{c}$ as $T\rightarrow 0$ on the
line given by Eq. (18), provided that the estimates for
$\lambda_{\parallel}\left(0\right)$, $\alpha$ and $T_{c}$ are correct and
$\alpha$ scales according to Eq. (19) as

$\alpha=y_{c}T_{c}/f_{\|}^{2}\left(0\right).$ (20)

This linear temperature dependence
$\left.d\lambda_{\|}/dT\right|_{T\rightarrow
0}=\alpha\lambda_{\|}^{3}\left(0\right)/2$ was proposed by several authors in
terms of phase fluctuations [30, 31, 32].

Finally, the relationship between the zero temperature penetration depth
$\lambda_{\parallel}\left(0\right)$, the correlation length
$\xi_{\perp}\left(0\right)$ and the respective critical amplitudes,
$\lambda_{\parallel 0}$ and $\xi_{\perp 0}^{t}$, need to be clarified. The
crossover from quantum to classical critical behavior implies
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=F\left(y_{c}T/T_{c}\right)\rightarrow$
$\lambda_{\parallel 0}^{2}/\lambda_{\parallel}^{2}\left(T\right)t^{2/3}$ as
$T/T_{c}$ increases and with that the crossover from
$\lambda_{\parallel}\left(0\right)$ to the finite temperature amplitude
$\lambda_{\parallel 0}$. This crossover does not modify the characteristic
temperature dependence of 3D-xy criticality ($t^{2/3}$) but reflects the
quantum fluctuation induced change of the critical amplitude
$\lambda_{\parallel 0}$. With $\lambda_{\parallel
0}\rightarrow\lambda_{\parallel}\left(0\right)$ as $T_{c}\rightarrow 0$ the
scaling relations (11) and (13) reduce then to

$T_{c}=f_{\|}\left(0\right)/\lambda_{\parallel}\left(0\right){}=f_{\|}\left(0\right)/\lambda_{\parallel
0}.$ (21)

Combining this result with the scaling relations (3), (15) we obtain the
desired relationship

$\displaystyle T_{c}$ $\displaystyle=$ $\displaystyle
f_{\|}\left(0\right)/\lambda_{\parallel}\left(0\right)=f_{\|}^{2}\left(0\right)/\left(R\Lambda\xi_{\perp}^{t}\left(0\right)\right)$
(22) $\displaystyle=$ $\displaystyle f_{\|}\left(0\right)/\lambda_{\parallel
0}=f_{\|}^{2}\left(0\right)/\Lambda\xi_{\perp 0}^{t}.$

An essential implication of this crossover is the growth of the amplitude
$\xi_{\perp 0}^{t}$ as $T_{c}\rightarrow 0$. It crosses over to
$\xi_{\perp}^{t}\left(t\right)=\xi_{\perp
0}^{t}t^{-2/3}=\left(f_{\|}^{2}\left(0\right)/\Lambda T_{c}\right)t^{-2/3}$
which leads to an inevitable enhancement of the finite size effect controlled
by the ratio $L_{c}/\xi_{\perp}^{t}\left(T\right)=\left(L_{c}/\xi_{\perp
0}^{t}\right)t^{2/3}$ (Eq. (7)) because
$L_{c}/\xi_{\perp}^{t}\left(T\right)=\left(L_{c}/\xi_{\perp
0}^{t}\right)t^{2/3}\propto L_{c}T_{c}t^{2/3}$ as $T_{c}\rightarrow 0$. In
contrast, the zero temperature counterpart
$L_{c}/\xi_{\perp}^{t}\left(0\right)$ remains large in spite of
$\xi_{\perp}^{t}\left(0\right)\propto 1/T_{c}$ for any reasonable $L_{c}$
because $t^{2/3}$ does not enter.

Another manifestation of the quantum fluctuations comes from their effect on
the condensate and Cooper-pair density. In the Ginzburg-landau version of the
D-xy-model, where amplitude $\left|\Psi\left(R\right)\right|$and phase
$\varphi\left(R\right)$ are taken into account, these densities can be
expressed in terms of the complex order parameter
$\Psi\left(R\right)\exp(i\varphi\left(R\right)$. The superfluid density
$\rho_{\|s}\left(0\right)=\lambda_{\|}^{-2}\left(0\right)$, the helicity
modulus $\Upsilon_{\|}$, the condensate density
$\left\langle\left|\Psi\right|\right\rangle^{2}$ and the Cooper-pair
counterpart $\left\langle\left|\Psi\right|^{2}\right\rangle$ are found to
scale as [20]

$\displaystyle\rho_{\|s}\left(t\right)$ $\displaystyle=$
$\displaystyle\lambda_{\|0}^{-2}t^{\nu}=(16\pi^{3}/\Phi_{0}^{2})\Upsilon_{\|0}t^{\nu}\neq\left(\hbar^{2}/2m_{\|}\right)\left|\left\langle\Psi\right\rangle\right|^{2},$
$\displaystyle\text{ }\left\langle\left|\Psi\right|\right\rangle^{2}$
$\displaystyle\propto$ $\displaystyle t^{2\beta},\text{
}\left\langle\left|\Psi\right|^{2}\right\rangle>\left\langle\left|\Psi\right|\right\rangle^{2},\text{
}\left\langle\left|\delta\Psi\right|^{2}\right\rangle>0.$ (23)

Accordingly, the superfluid and condensate densities vanish as $T\rightarrow
T_{c}$ while the Cooper pair density remains finite at and above $T_{c}$.
Indeed, at $T_{c}$ it scales as
$\left\langle\left|\delta\Psi\right|^{2}\right\rangle_{T=T_{c}}\propto T_{c}$.
The difference between the superfluid and condensate density stems from the
fact that in the 3D-xy model $\nu=2\beta-\eta$ [33, 34] with $\eta\simeq
0.0381$ [35]. The appearance of 3D-xy critical behavior stems from the fact
that in extreme type II superconductors the amplitude fluctuations are
dominated by phase fluctuations over a sizeable temperature regime [20]. Note
that in the QPT considered here, where $T_{c}\propto
1/\lambda_{\parallel}\left(0\right){}\propto 1/\lambda_{\parallel 0}$ (Eq.
(21)), the extreme type II limit ($\lambda_{\parallel}\rightarrow\infty$) is
attained as $T_{c}\rightarrow 0$.

The mapping of the QPT on the $\left(3+1\right)$ D-xy model implies mean-field
critical behavior. Therefore, taking the quantum fluctuations in the 3D-xy
model into account, the superfluid density
$\rho_{\|s}\left(0\right)=\lambda_{\|}^{-2}\left(0\right)$, the helicity
modulus $\Upsilon_{\|}\left(0\right)$, the condensate density
$\left\langle\left|\Psi\right|\right\rangle_{T=0}^{2}$ and the Cooper-pair
counterpart $\left\langle\left|\Psi\right|^{2}\right\rangle_{T=0}$ scale as

$\displaystyle\rho_{\|s}\left(0\right)$ $\displaystyle=$
$\displaystyle\lambda_{\|}^{-2}\left(0\right)=(16\pi^{3}/\Phi_{0}^{2})\Upsilon_{\|}\left(0\right)$
$\displaystyle=$
$\displaystyle\left(\hbar^{2}/2m_{\|}\right)\left|\left\langle\Psi\right\rangle\right|_{T=0}^{2},$
$\displaystyle\left\langle\left|\Psi\right|\right\rangle_{T=0}^{2}$
$\displaystyle=$
$\displaystyle\left\langle\left|\Psi\right|^{2}\right\rangle_{T=0}\propto\delta\propto
T_{c}^{2\text{ }},$ (24)

where we used Eq. (17). $\delta$ denotes the tuning parameter of the QPT.
Remarkably, not only the superfluid density is suppressed by quantum
fluctuations as $T_{c}\rightarrow 0$ but also the condensation and Cooper pair
density.

##  III Data Analysis

We are now prepared to analyze the penetration depth data. In the nearly
optimally doped film S1 evidence for the relevance of 3D-xy fluctuations
emerges from Fig. 1, revealing a crossing point in the plots
$\lambda_{\parallel f}^{-2}\left(T\right)$, $\lambda_{\parallel
f}^{-3}\left(T\right)$ vs. $T$ (Eq. (6)). Indeed,
$\lambda_{\|}^{-2}\left(t\right)=\lambda_{\|0}^{-2}\left(1-T/T_{c}\right)^{2/3}$
is a characteristic of 3D-xy criticality. The experimental data is seen to
cross at $T_{crF}\simeq 40.2$ K, a little higher than $T_{cr}\simeq 40$ K, the
value that corresponds to the estimated $3$D-xy critical behavior $T_{c}=40.4$
K (Eq. (5) Table 1). This and the fact that $\lambda_{\parallel
f}^{-2}\left(T\right)$ does not vanish at and extends beyond $T_{c}$ points to
a finite size effect that rounds the sharp 3D-xy-transition, marked by the
black line. It implies that the 3D-xy transition temperature $T_{c}=40.4$ K is
considerably lower than the $T_{c}\simeq$ $41.6$ K, defined by the onset of
the Meissner effect[1]. Additional evidence for a finite size induced rounding
of the transition comes from $d\lambda_{\parallel f}^{-2}\left(T\right)/dT$
vs. $T$ , shown in the inset of Fig. 1. As the estimated $T_{c}=40.4$ K is
approached from below the data tends to the characteristic 3D-xy singularity,
$d\lambda_{\|}^{-2}/dT=-2/\left(3\lambda_{\|0}^{2}T_{c}\right)t^{-1/3}$ (Eq.
(4)), marked by the solid line. However, close to $T_{c}$ the singularity is
cut off and $\lambda_{\parallel f}$ approaches $\lambda_{\parallel
f}\left(T_{c}\right)=$ $L_{\lambda\|}\propto L_{c}$(Eqs. (9), (10)).

Figure 1: $\lambda_{\parallel f}^{-2}\left(T\right)$, $\lambda_{\parallel
f}^{-3}\left(T\right)$ vs. $T$ of film S1 marked by the red and blue lines.
The solid black and the dash-dot lines mark the 3D-xy critical behavior in
terms of $\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel
0}^{-2}t^{-2/3}$, $\lambda_{\parallel}^{-3}\left(T\right)=\lambda_{\parallel
0}^{-3}t$ with $\lambda_{\parallel 0}^{-2}$ and $T_{c}$ values listed in Table
I. The crossing point of the experimental data occurs at $T_{crf}\simeq 40.2$
K and the 3D-xy counterpart at $T_{cr}\simeq 40$ K . The inset shows
$d\lambda_{\parallel f}^{-2}\left(T\right)/dT$ vs. $T$ and the solid line
marks the 3D-xy critical behavior (Eq. (4)) with the parameters listed in
Table 1.

To unravel the nature of the weakly rounded transition we performed the finite
size scaling analysis of the data. Fig. 2 shows $\lambda_{\parallel
f}^{2}\left(t\right)/\lambda_{\parallel}^{2}\left(t\right)$ vs .
$1/\lambda_{\parallel}^{2}\left(t\right)$ for the films S1, S62, and S97. From
the limiting behavior $\lambda_{\parallel
f}^{2}\left(t\right)/\lambda_{\parallel}^{2}\left(t\right)\rightarrow
L_{\lambda\|}^{2}/\lambda_{\parallel}^{2}\left(t\right)$ (Eq. (9)) we derive
$L_{\lambda\|}^{2}\simeq 1.2$ $\mu m^{2}$ for the film S1, yielding with
Eq.(14) $L_{c}\simeq 8$ nm for the limiting length, which is slightly larger
than the quoted film thickness $L_{cf}=6.6$ nm [1]. Nevertheless, the $3$D-xy
transition is only slightly rounded in this thin film because the amplitude of
the transverse correlation length $\xi_{\parallel 0}=0.3$ nm (Table 1) is
small compared to the film thickness (Eq. (8)). Therefore, the analysis
compiled in Figs. 1 and 2 shows clearly that this film is homogeneous and
exhibits due to its thickness a weakly rounded 3D-xy-transition. Moreover, the
approximately linear temperature dependence of $1/\lambda_{\parallel f}^{2}$,
over a rather wide temperature range, is then traced back to the film
thickness induced finite size effect and is not the fingerprint of a d-wave
superconductor in the dirty limit [4, 16, 17]. Nevertheless, associating the
drop of $T_{c}$ with increased dopant concentration, it seems plausible that
in the limit $T_{c}\rightarrow 0$ the size of the homogenous domains instead
of the film thickness set the limiting length.

Figure 2: Finite size scaling plots $\lambda_{\parallel f}^{2}\left(t\right)/\lambda_{\parallel}^{2}\left(t\right)$ vs. $1/\lambda_{\parallel}^{2}\left(t\right)$ for the films S1, S62, and S97. The limiting behavior $\left(\lambda_{\parallel f}\left(t\right)/\lambda_{\parallel}\left(t\right)\right)^{2}\rightarrow\left(L_{\lambda\|}/\lambda_{\parallel}\left(t\right)\right)^{2}$ yields $L_{\lambda\|}^{2}$ and with Eq. (14) the estimates for the limiting length $L_{c}$ listed in Table 1. The black solid, dashed, and dashed-dot lines are $\left(\lambda_{\parallel f}\left(t\right)/\lambda_{\parallel}\left(t\right)\right)^{2}=\tanh(L_{\lambda\|}^{2}/\lambda_{\parallel}^{2}\left(t\right))$ which describe the crossover from the finite size $\left(\lambda_{\parallel f}\left(t\right)/\lambda_{\parallel}\left(t\right)\right)^{2}<1$ to the 3D-xy critical regime $\left(\lambda_{\parallel f}\left(t\right)/\lambda_{\parallel}\left(t\right)\right)^{2}\rightarrow 1$ remarkably well. Table 1: Estimates for $T_{c}$, the 3D-xy critical amplitudes $\lambda_{\parallel 0}$, $\xi_{\bot 0}^{t}$ (Eq. (2)), the limiting length $L_{\lambda\|}$ of the penetration depth $\lambda_{\parallel}\left(T\right)$, the resulting limiting length $L_{c}$ (Eq. (10)), the zero temperature penetration depth $\lambda_{\parallel 0}\left(0\right)$, and the coefficient $\alpha$ entering the linear temperature dependence of $\lambda_{\parallel}^{-2}\left(T\right)$ in the limit $T\rightarrow 0$ (Eq. (18)). The quoted film thickness is $L_{cf}=6.6$ nm [1]. | $T_{c}$ | $\lambda_{\parallel 0}^{-2}$ | $\xi_{\bot 0}^{t}$ | $L_{\lambda\|}^{2}$ | $L_{c}$ | $\lambda_{\parallel}^{2}\left(0\right)$ | $\alpha$
---|---|---|---|---|---|---|---
| K |  $\mu$m-2 | nm |  $\mu$m2 | nm |  $\mu$m-2 |  $\mu$m-2 K-1
S1 | 40.4 | 21.70 | 0.30 | 1.2 | 7.7 |  $\simeq$23.20 |  $\simeq$0.35
S62 | 9.7 | 2.62 | 0.59 | 2.0 | 3.1 | 2.54 | 0.124
S73 | 6.73 | 1.10 | 0.98 | 4.0 | 4.3 | 1.07 | 0.080
S87 | 5.64 | 0.76 | 1.19 | 8.5 | 7.7 | 0.75 | 0.082
S97 | 4.45 | 0.49 | 1.45 | 9.5 | 6.8 | 0.47 | 0.061
S98 | 4.27 | 0.47 | 1.45 | 10.0 | 6.8 | 0.46 | 0.070

To clarify this issue we turn to the analysis of the films S62 and S97 with
$T_{c}\simeq 9.7$ K and $T_{c}\simeq 4.45$ K, respectively. Fig. 3 shows
$\lambda_{\parallel f}^{-2}\left(T\right)$, $\lambda_{\parallel
f}^{-3}\left(T\right)$ vs. $T$ of the film S62. The crossing point at
$T_{crf}\simeq 7.5$ K uncovers again the presence of 3D-xy fluctuations.
However, the deviations from the estimated 3D-xy critical behavior near and
above $T_{c}$ reveal a strongly rounded transition. This behavior is also
visible in $d\lambda_{\|}^{-2}\left(T\right)/dT$ shown in the inset of Fig. 3.
The critical behavior
$d\lambda_{\|}^{-2}\left(T\right)/dT=-2/\left(3\lambda_{\|0}^{2}T_{c}\right)t^{-1/3}$
(Eq. (4)) is attained at rather low temperatures. Both phenomena are
consistent with the crossover from the thermal to the quantum critical regime
where the transverse correlation length scales as
$\xi_{\perp}^{t}\left(t\right)\propto t^{-2/3}/T_{c}$ (Eq.(22)). Indeed, the
crossover $\xi_{\perp}^{t}\left(t\right)=\xi_{\perp
0}^{t}t^{-2/3}\rightarrow\xi_{\perp}^{t}\left(0\right)t^{-2/3}$ extends the
3Dxy critical regime to lower temperatures and enhances the finite size effect
for fixed $L_{c}$, because
$L_{c}/\xi_{\perp}^{t}\left(t\right)=L_{c}t^{2/3}/\xi_{\perp 0}^{t}\rightarrow
L_{c}t^{2/3}/\xi_{\perp}^{t}\left(0\right)\propto L_{c}T_{c}t^{2/3}$ and weak
rounding requires that $L_{c}/\xi_{\perp}^{t}\left(t\right)>1$(Eq. (7)). On
the other hand, it seems plausible that stronger rounding with reduced $T_{c}$
is attributable to a shrinkage of the homogenous domains with increasing
doping. In fact the limiting length $L_{c}\simeq 3.1$ nm (Table 1), derived
from the finite size scaling plot shown in Fig. 2, is considerably smaller
than the quoted film thickness $L_{cf}=6.6$ nm [1] and appears to confirm this
scenario. Nevertheless, Fig. 3 shows that the film attains the 3D-xy critical
regime. Supposing that inhomogeneities stem from superconducting and metallic
regions, with the fraction of metallic regions increasing on overdoping [3],
the limiting length $L_{c}$ is expected to decrease as $T_{c}\rightarrow 0$.
To clarify this issue we turn to the analysis of the film S97 with
$T_{c}\simeq 4.45$ K.

Figure 3: $\lambda_{\parallel f}^{-2}\left(T\right)$, $\lambda_{\parallel
f}^{-3}\left(T\right)$ vs. $T$ of film S62 marked by the red and blue lines.
The solid black and the dash-dot lines mark the 3D-xy critical behavior in
terms of $\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel
0}^{-2}t^{-2/3}$, $\lambda_{\parallel}^{-3}\left(T\right)=\lambda_{\parallel
0}^{-3}t$ with $\lambda_{\parallel 0}^{-2}$ and $T_{c}$ values listed in Table
1. The crossing point of the experimental data occurs at $T_{crf}\simeq 7.5$ K
and the 3D-xy counterpart at $T_{cr}\simeq 7.4$ K .

Fig. 4 shows $\lambda_{\parallel f}^{-2}\left(T\right)$, $\lambda_{\parallel
f}^{-3}\left(T\right)$ vs. $T$ for the film S97. The solid black line and the
dash-dot line indicate the estimated 3D-xy critical behavior of ,
$\lambda_{\parallel}^{-2}\left(T\right)$ and
$\lambda_{\parallel}^{-3}\left(T\right)$. Essential features include the
absence of the crossing point, the strongly rounded transition and the
associated shift of the attainable 3D-xy critical regime to lower
temperatures. The absence of the crossing point is according to Eq. (5) a
consequence of 3D-xy criticality because the critical amplitude
$\lambda_{\parallel 0}^{-}$ of this film satisfies the condition
$\lambda_{\parallel 0}^{-2}\simeq 0.49~{}\mu$m${}^{-2}<1~{}\mu$m-2 whereupon
$\lambda_{\|}\left(T\right)^{-2}$ $>\lambda_{\|}^{-3}\left(T\right)$ follows
(Eq. (6)). To clarify the origin of the strongly rounded transition, also
visible in $d\lambda_{\parallel f}^{-2}\left(T\right)/dT$ vs. $T$ depicted in
the inset of Fig. 4, we turn to the results of the finite size scaling
analysis shown in Fig. 2. The essential result is that the finite size limited
3D-xy criticality mimics the heavily smeared transition well. But in this case
with a limiting length $L_{c}=7.1$ nm (Table 1), compatible with the quoted
film thickness. Therefore, in spite of the pronounced overdoping, the finite
size scaling analysis uncovers the homogeneity of this film. We have carried
out the same analysis for the films S73, S87 and S98. The result is that all
show agreement with a film thickness limited 3D-xy criticality. Comparing the
limiting lengths $L_{c}$ listed in Table 1 with the film thickness,
$L_{cf}=6.6$ nm it follows that the limiting length responsible for the
rounded 3D-xy transitions in S87 and S98 stems from the film thickness,
because $L_{c}\simeq L_{cF}$, while in S73, where $L_{c}$ is considerably
smaller than $L_{cf}$, inhomogeneities set the limiting length. However, the
missing systematic correlation between $T_{c}$ and $L_{c}$ suggests that
$L_{c}<L_{cf}$ simply points to worse film quality and not to an unavoidable
doping-induced phase separation, according to which the size of the
homogeneous domains should shrink with reduced $T_{c}$.

Figure 4: $\lambda_{\parallel f}^{-2}\left(T\right)$, $\lambda_{\parallel
f}^{-3}\left(T\right)$ vs. $T$ of film S97 marked by the red and blue lines,.
The solid black line and the dash-dot line indicate the estimated 3D-xy
critical behavior, $\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel
0}^{-2}t^{-2/3}$, $\lambda_{\parallel}^{-3}\left(T\right)=\lambda_{\parallel
0}^{-3}t$ with $\lambda_{\parallel 0}^{-2}$ and $T_{c}$ listed in Table 1.
There is no crossing point because $\lambda_{\parallel 0}^{-2}\simeq
0.49~{}\mu$m${}^{-2}<1\mu$m-2. Nevertheless there is still evidence for 3D-xy
critical behavior emerging from $d\lambda_{\parallel F}^{-2}\left(T\right)/dT$
vs. $T$ shown in the inset. The solid line represents the corresponding 3D-xy
critical behavior (Eq. (4)) with the parameters listed in Table 1.

Finally we crosscheck the finite size scaling results. In Fig. 5 we depicted
the finite size scaling plot $\lambda_{\parallel
F}\left(T\right)/\lambda_{\parallel}\left(T\right))^{2}$ vs
$\left(L_{\lambda}/\lambda_{\parallel}\left(T\right)\right)^{2}$ of the films
listed in Table I. Given the estimates for $T_{c}$, $1/\lambda_{\parallel
0}^{2}$ and $L_{\lambda\parallel}^{2}$ this plot should then let
$\lambda_{\parallel F}\left(T\right)/\lambda_{\parallel}\left(T\right))^{2}$
collapse onto a single line. As Fig. 5 shows we have a good collapse of the
data onto the scaling function $g\left(x\right)$ (Eq. (9)). For this to
happen, the estimates for $T_{c}$, $1/\lambda_{\parallel 0}^{2}$ and
$L_{\lambda\parallel}^{2}=L_{c}\Lambda/T_{c}$ need to be correct. In addition
it uncovers the nature of the limiting length. If $L_{c}=L_{ci}<L_{cf}$ it is
traced back to inhomogeneities and if $L_{c}\simeq L_{cf}$ to the effective
film thickness $L_{cf}$. Notably, this collapse is almost perfect regardless
of the nature of the limiting length. In the films S1, S87, S97 and S98 it is
the film thickness $L_{cf}$, while in the films S62 and S73 inhomogeneities
are what causes the limiting length $L_{ci}<L_{cf}$ (see Table 1).
Nevertheless, the films attain the critical regime where $g(x)\rightarrow 1$.
The reason for this is that the amplitude of the transverse correlation length
$\xi_{\perp 0}^{t}$ remains small compared to the film thickness $L_{cf}=6.6$
nm (Eq. (8), Table 1).

Figure 5: Finite size scaling plot $\left(\lambda_{\parallel
f}\left(T\right)/\lambda_{\parallel}\left(T\right)\right)^{2}$ vs
$\left(L_{\lambda}/\lambda_{\parallel}\left(T\right)\right)^{2}$ of the films
listed in Table 1. The line is $\left(\lambda_{\parallel
F}\left(T\right)/\lambda_{\parallel}\left(T\right)\right)^{2}=$tanh$\left(L_{\lambda}/\lambda_{\parallel}\left(T\right)\right)^{2}$.
Figure 6: Scaling plots $\left(\lambda_{\parallel 0}/\lambda_{\parallel
f}\left(T\right)\right)^{2}$ vs $T/T_{c}$ of the films listed in Table 1 using
the estimates for $T_{c}$ and $\lambda_{\parallel 0}$. The data collapsing on
$\left(\lambda_{\parallel
0}/\lambda_{\parallel}\left(T\right)\right)^{2}=(1-T/T_{c})^{2/3}$ (black
curve) uncover the consistency with $3$D-xy criticality. The inset shows the
$T_{c}$ dependence of the amplitude of the transverse correlation length
$\xi_{\perp 0}^{t}$ according to $\xi_{\perp}^{t}\left(t\right)=\xi_{t\perp
0}t^{-2/3}\propto t^{-2/3}/T_{c\text{ }}$(Eq. (22)). The line is $\xi_{\perp
0}^{t}=6.5/T_{c}$ with $\xi_{\perp 0}^{t}$ in nm and $T_{c}$ in K.

Next we turn to the analysis of the low-temperature penetration depth data
extending down to $0.3$ K. Therefore, an extrapolation is necessary to
estimate $\lambda_{\parallel}^{-2}\left(0\right)$. Fig. 7 shows that the
linear temperature dependence
$\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel}^{-2}\left(0\right)-\alpha
T$, using the data up to $1$ K, fits the data remarkably well. In
$\lambda_{\parallel}^{-}\left(T\right)$ it corresponds to
$\left.d\lambda_{\parallel}^{-}\left(T\right)/dT\right|_{T\rightarrow
0}=\alpha\lambda_{\parallel}^{3}\left(0\right)/2$ attributed by several
authors to phase fluctuations[30, 31]. In addition, the mapping on the (3 +
1)D-xy model implies a linear temperature dependence because
$\Upsilon_{\parallel}\left(T\right)\propto\left\langle\left|\Psi\right|\right\rangle^{2}\propto\left(1-T/T_{c}\right)$.
The resulting $\lambda_{\parallel}^{-2}\left(0\right)$ and $\alpha$ are listed
in Table 1. With the estimates for
$T_{c},\lambda_{\parallel}^{-2}\left(0\right)$, $\lambda_{\parallel
0}^{-2}\left(0\right)$ and $\alpha$ it is now possible to clarify their
compatibility with a quantum transition belonging to the (3+1)D-xy
universality class. Indeed, consistency with this transition requires that the
relationships
$T_{c}=f_{\|}\left(0\right)/\lambda_{\parallel}\left(0\right)=f_{\|}\left(0\right)/\lambda_{\parallel
0}$ (Eq. (21)), $\alpha=y_{c}T_{c}/f_{\|}^{2}\left(0\right)$ (Eq. (20)), and
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=1-y_{c}T/T_{c}=1-\alpha\lambda_{\parallel}^{2}\left(0\right)T$
(Eq. (18)) are met. In Fig. 8, depicting $T_{c}$ vs.
$1/\lambda_{\parallel}\left(0\right)$ and $1/\lambda_{\parallel 0}$, we
observe that the first relation is confirmed with

$f_{{}_{\|}}\left(0\right)\simeq 6.45~{}\text{K}\mu m\text{.}$ (25)

Consequently, as $T_{c}$ drops quantum fluctuations reduce
$1/\lambda_{\parallel}\left(0\right)$ and $1/\lambda_{\parallel 0}$ in the
same manner. Further consistency emerges from the $T_{c}$ dependence of the
coefficient $\alpha$ shown in the inset of Fig. 8, yielding with
$\alpha=\left(y_{c}/f_{\|}^{-2}\left(0\right)\right)T_{c}$ the estimates

$y_{c}/f_{\|}^{2}\left(0\right)\simeq 0.014~{}(\mu
m^{-2}\text{K}^{-1})\text{,}$ (26)

and with $f\left(0\right)\simeq 6.45$ K$\mu m$

$y_{c}\simeq 0.58,$ (27)

for the universal coefficient. Noting that
$\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel}^{-2}\left(0\right)-\alpha
T$ can be rewritten in the form
$\lambda_{\parallel}^{-2}\left(T\right)=\left(T_{c}/f_{\|}\left(0\right)\right)^{2}-\alpha
T_{c}\left(T/T_{c}\right)$ it is evident that $\alpha$ scales in the limit
$T/T_{c}\rightarrow 0$ as $\alpha\propto T_{c}$.

Figure 7: $\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ vs. $T$ for the quoted films. The lines mark the fits to
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=1-\alpha\lambda_{\parallel}^{2-}\left(0\right)T$
yielding the estimates for $\lambda_{\parallel}^{2-}\left(0\right)$ and
$\alpha$ listed in Table 1 using the data from $0.3$ to $1$ K. Figure 8:
$T_{c}$ vs. $1/\lambda_{\parallel}\left(0\right)$ and $1/\lambda_{\parallel
0}$ using the estimates listed in Table 1. The solid black line is
$T_{c}=f_{\|}\left(0\right)/\lambda_{\parallel}\left(0\right)$ with
$f_{\|}\left(0\right)=6.45$ K$\mu$m. The inset shows $\alpha$ vs. $T_{c}$. The
straight line is $\alpha=\left(y_{c}/f_{\|}^{2}\left(0\right)\right)T_{c}$(Eq.
(20)) with $y_{c}/f_{\|}^{2}\left(0\right)=0.014\mu$m-2 K-1.

To complete the consistency checks we show in Fig. 9 the plot
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ vs. $T/T_{c}$. The outlined linear temperature
dependence of $\lambda_{\parallel f}^{-2}\left(T\right)$ implies that the data
should fall on the line
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=1-y_{c}T/T_{c}$
with $y_{c}\simeq 0.58$ (Eq. (27)). The approach to this line is clearly seen
as $T_{c}\rightarrow 0$. A crosscheck comes from the relation
$T_{c}=f\left(0\right)/\lambda_{\parallel}\left(0\right)$ (Eq. (15)) after
which $T/T_{c}$ can be replaced by
$T\lambda_{\parallel}\left(0\right)/f_{\|}\left(0\right)$. This plot is shown
in Fig. 9. Here the data should collapse on the line
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=1-uT\lambda_{\parallel}\left(0\right)$
with

$u=y_{c}/f_{\|}(0)=0.09,$ (28)

using the estimates given in Eqs. (25) and (27). Again, as $T_{c}$ drops the
data are seen to approach this line.

Figure 9: Scaling plot
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ vs. $T/T_{c}$ in the limit $T/T_{c}\rightarrow 0$. The
line is
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=1-y_{c}T/T_{c}$
with the universal coefficient $y_{c}=0.58$ (Eq. ( 27)). Figure 10: Scaling
plot $\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ vs. $T\lambda_{\parallel}\left(0\right)$ in the limit
$T\lambda_{\parallel}\left(0\right)\rightarrow 0$. The line is
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=1-uT/T_{c}$
with $u=y_{c}/f\left(0\right)\simeq 0.09$ (Eq. (28)).

To complete the analysis of the selected overdoped La2-xSrxCuO4 thin films we
show in Fig. 11 the plot
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ vs. $T/T_{c}$. It contains the data from $0.3$ K up to
the temperature defined by the onset of the Meissner effect. Because the limit
$T/T_{c}\simeq 1$ has already been dealt with (see Fig. 6), we are
concentrating on the limit $T/T_{c}\rightarrow 0$. It can be reached along the
phase transition line $T_{c}\left(1/\lambda_{\parallel}\left(0\right)\right)$
where
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=t^{2/3}$
and $\lambda_{\parallel 0}\rightarrow\lambda_{\parallel}\left(0\right),$ or
along the line $1/\lambda_{\parallel}\left(0\right)\propto\delta^{1/2}\propto
T_{c}$ at $T=0$ where $\delta$ is the tuning parameter (Eq. (17)). The dashed
line in Fig. 11 marks this limit in terms of
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=1-y_{c}T/T_{c}$
(Eq. (18)). Taking the Sr concentration $x$ in La2-xSrxCuO4 as tuning
parameter it implies that the empirical relation $T_{c}\propto\delta=x_{o}-x$
of Presland at al. [36] is violated as $x\rightarrow x_{o}$. Approaching the
QPT along the phase transition line
$T_{c}\left(1/\lambda_{\parallel}\left(0\right)\right)$ the data is seen to
collapse up to rather large $T/T_{c}$ values on the curve
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=t^{2/3}$
because $\lambda_{\parallel 0}$
$\rightarrow\lambda_{\parallel}\left(0\right)$. However, as
$T/T_{c}\rightarrow 1$ the finite size induced rounding of the transition sets
in.

Figure 11: Complete scaling plot
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ vs. $T/T_{c}$. The the dash- dot line marks the limiting
quantum behavior
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=1-y_{c}T/T_{c}$
with $y_{c}=0.58$ (Eq. (27)) and the dashed line
$\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel}^{2}\left(T\right)=(1-T/T_{c})^{2/3}$
the thermal 3D-xy criticality with
$\lambda_{\parallel}\left(0\right)=\lambda_{\parallel 0}$.

What remains is the question of which material parameters control the 3D-xy
transition. Fundamental parameters are the anisotropy
$\gamma_{0}=\xi_{\|0}^{t}\left(0\right)/\xi_{\perp 0}^{t}$ and the ratio
$\kappa_{\|0}=\lambda_{\parallel 0}/\xi_{\|0}^{t}$. Noting that the universal
relation (3) transforms with these parameters to

$\gamma_{0}\kappa_{\|0}=\Lambda/T_{c}\lambda_{\parallel 0}$ (29)

it is evident that $T_{c}\lambda_{\parallel 0}$ is fixed by
$\gamma_{0}\kappa_{\|0}$. In the films considered here we obtain with
$f_{{}_{\|}}\left(0\right)\simeq f_{{}_{\|0}}\simeq 6.45$ K$\mu m$ (Eq. (25))

$\gamma_{0}\kappa_{\|0}=\Lambda/T_{c}\lambda_{\parallel
0}=\Lambda/f_{{}_{\|0}}\simeq 91.5,$ (30)

and with $\gamma_{0}\simeq 10.6$ [37], $\kappa_{\|0}\simeq 91.5$ as
$T_{c}\rightarrow 0$. This differs drastically from the behavior of highly
underdoped YBa2Cu3O6+y, where $f_{{}_{\|0}}\approx 6.5$ K$\mu m$ [38] but
$\gamma_{0}\approx 100$ as $T_{c}\rightarrow 0$ [39] and with that
$\kappa_{\|0}\approx 10$ as $T_{c}\rightarrow 0$.

##  IV Summary and Conclusions

In summary, our analysis of the penetration depth data from Božović et al.[1]
for thin overdoped La2-xSrxCuO4 films revealed: First, the observed mean-
field-like behavior of $\lambda_{\parallel f}^{-2}\left(T\right)$ is
attributable to a finite size effect. It stems from the fact that in an
infinite and homogeneous 3D-xy system the transverse correlation length
diverges as $\xi_{\perp}^{t}\left(T\right)=\xi_{\perp 0}^{t}t^{-\nu}$, where
$t=\left(1-T/T_{c}\right)$ and $\nu\simeq 2/3$, while in the films $\xi_{\perp
f}^{t}\left(T_{c}\right)$ the divergence is cut off. In homogenous films the
limiting length is set by the effective film thickness $L_{cf}$ and in the
inhomogeneous ones by $L_{ci}<L_{cf}$, the size of the homogenous domains.
Nevertheless, in the films analyzed here, the $3$D-xy critical regime is
reached. The finite size scaling analysis of the rounded transition provided
estimates for the $T_{c}$’s and the critical amplitudes $\lambda_{\parallel
0}\left(T_{c}\right)$ of the infinite and homogenous counterpart. Because the
$T_{c}$’s are lower than those estimated from the the onset of the Meissner
effect [1], the mean-field-like behavior of
$\rho_{\|sf}\left(T\right)=\lambda_{\|f}^{-2}\left(T\right)$ was traced back
to a finite size effect. In addition the estimates for $T_{c}$ and
$\lambda_{\parallel 0}$ uncovered the relationship
$T_{c}=f_{\|0}/\lambda_{\parallel 0}$. Second, having established that the
films exhibit 3D-xy critical behavior, rounded by the finite size effect, we
performed the analysis of the low temperature data extending down to $0.3$ K.
Therefore, an extrapolation was necessary to estimate
$\lambda_{\parallel}^{-2}\left(0\right)$. It was shown that the linear
extrapolation with $\lambda_{\parallel
f}^{-2}\left(T\right)=\lambda_{\parallel}^{-2}\left(0\right)-\alpha T$ fits
the data up to $1$K very well. Given the estimates of $T_{c}$,
$\lambda_{\parallel}^{-2}\left(0\right),$ $\alpha$ , and the $T_{c}$
dependence of the latter, their consistency with a QPT mapping on the
$(3+1)$D-xy universality class was tested. This mapping requires that the
scaling relations $T_{c}=f_{\|}(0)/\lambda_{\parallel}\left(0\right)$,
$\alpha=y_{c}T_{c}/f_{\|}^{2}\left(0\right)$ are satisfied and the data
plotted as $\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ vs. $T/T_{c}$ collapses on a single curve. In the limit
$T/T_{c}\rightarrow 0$ and sufficiently small $T_{c}$ it should tend to the
line $\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)=1-y_{c}T/T_{c}$ and in the $T/T_{c}\rightarrow 1$ limit
to $t^{2/3}$. However, due to the rounded transition is this limit not fully
attainable and $\lambda_{\parallel}^{2}\left(0\right)/\lambda_{\parallel
f}^{2}\left(T\right)$ exceeds $T/T_{c}=1$. Nevertheless, as $T_{c}$ drops the
collapse improved because
$\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel 0}^{-2}t^{2/3}$
approached
$\lambda_{\parallel}^{-2}\left(T\right)=\lambda_{\parallel}^{-2}\left(0\right)t^{2/3}$,
due to the crossover from the thermal to the quantum fluctuation dominated
regime. The consistency shown with these characteristics of 3D-xy- and
(3+1)D-xy criticality clearly revealed that the suppression of the superfluid
density in the thin and overdoped La2-xSrxCuO4 films of Božović et al. [1] is
a quantum effect. It uncovers the crossover from the thermal to the quantum
fluctuation dominated regime as $T_{c}$ drops. In the classical-quantum
mapping it is the crossover from $3$D to $(3+1)$D. The thermal regime shows
finite size limited 3D-xy criticality and the quantum regime is compatible
with ($3+1$)D-xy criticality. Another manifestation of the relevance of
quantum fluctuations comes according to Eq. (24) from the observation that in
the overdoped La2-xSrxCuO4 films conductivity measurements show that below
$T_{c}$ a large fraction of the Drude weight remains uncondensed [7].
Moreover, consistent with the importance of thermal fluctuations (Eq. (23)) in
the overdoped regime are Nernst, torque, magnetization [40], recent specific
heat and ARPES measurements [41], revealing in overdoped Bi-2212 single
crystals Cooper pairs which are already formed above $T_{c}$. However, it
should be kept in mind that the observed strong diamagnetism and Nernst signal
in a wide temperature window above $T_{c}$ exceeds the fluctuation dominated
regime drastically. Indeed, taking the inhomogeneities of single crystals into
account, the effective dimensionality is reduced to zero and reduced
dimensionality enhances the thermal fluctuations [42]. Nonetheless, the proven
existence of copper pairs above $T_{c}$ implies that they contribute to the
transport properties and with that modify the standard Fermi liquid picture.

###### Acknowledgements.

I thank Stefano Gariglio and Jean-Marc Triscone for their support.

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