Dissipation in ultra-thin current-carrying superconducting bridges; evidence for quantum tunneling of Pearl vortices
F. Tafuri$^1$
J.R. Kirtley$^2$
D. Born$^3$
D. Stornaiuolo$^3$
P.G. Medaglia$^4$
P. Orgiani$^4$
G. Balestrino$^4$
V.G. Kogan$^5$
We have made current-voltage (IV) measurements of artificially layered high-$T_c$ thin-film �bridges. Scanning SQUID microscopy of these films provides values for the Pearl lengths $\Lambda$ that exceed the bridge width, and shows that the current distributions are uniform across the bridges. At high temperatures and high currents the voltages follow the power law $V \propto I^n$, with $n=\Phi_0^2/8\pi^2\Lambda k_B T+1$, and at high temperatures and low-currents the �resistance is exponential in temperature, in good agreement with the predictions for thermally activated vortex motion. At low temperatures, the IV's are better fit by $\ln V$ linear in $I^{-2}$. This is expected if the low temperature dissipation is dominated by quantum tunneling of Pearl vortices.
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\title{Dissipation in ultra-thin current-carrying superconducting
bridges; evidence for quantum tunneling of Pearl vortices}

\author{
F. Tafuri$^1$, J.R. Kirtley$^2$, D. Born$^3$, D. Stornaiuolo$^3$,
P.G. Medaglia$^4$,
P. Orgiani$^4$, G. Balestrino$^4$, V.G. Kogan$^5$}
\affiliation{$^1$INFM-Coherentia Dip. Ingegneria dell'Informazione
Seconda Universit\'{a} di Napoli, Aversa �(CE), Italy\\
$^2$IBM Watson research center, Route 134 Yorktown Heights, NY 10598\\
$^3$ INFM-Coherentia, Dipartimento di Scienze Fisiche, Universit{\'a}
di Napoli ``Federico II"
Napoli, Italy �\\
$^4$ INFM-Coherentia, Dipartimento di Ingegnaria Meccanica,
Universit{\'a} di Roma Tor Vergata, Roma, Italy \\
$^5$Ames Laboratory - DOE and Department of �Physics and Astronomy,
Iowa State University, Ames IA �50011-3160
}

\date{\today}
\begin{abstract}
We have made current-voltage (IV) measurements of
artificially layered high-$T_c$ thin-film �bridges.
Scanning SQUID microscopy of these films
provides values for the Pearl lengths $\Lambda$ that exceed
the bridge width, and shows that the current distributions
are uniform across the bridges. At high temperatures and high currents the
voltages follow the power law   $V \propto I^n$,
with $n=\Phi_0^2/8\pi^2\Lambda k_B T+1$, and at high temperatures and
low-currents
the �resistance is exponential in temperature, in good
agreement with the predictions for thermally activated vortex motion.
At low temperatures, the IV's
are better fit by $\ln V$ linear in $I^{-2}$.
This is expected if the low temperature dissipation is dominated by
quantum tunneling of Pearl vortices.
\end{abstract}
\pacs{ 74.50.+r, 74.78.Fk, 74.78.-w }
\maketitle

Soon after the seminal work of Caldeira and Leggett \cite{Leggett}, the
possibility of quantum tunneling of vortices in thin films was
considered by Glazman and Fogel \cite{Glazman}. This idea was
studied extensively in theory and experiment in relation �to
{\it finite} flux-creep rates �in hard type-II superconductors for
$T\to 0$. For thermally activated
creep, the rates should go to zero in this limit, see review
\onlinecite{Blatter} and references therein. Within �creep models, the
barriers through which the vortices tunnel are due to material disorder and
are treated statistically.

Narrow current carrying �thin-film bridges offer a
unique opportunity to study both
thermal activation and quantum tunneling through �the
  well defined potential barriers that are tunable
by current and temperature.
Within the London approach, �the barrier shape is well known for
bridges narrow relative to the Pearl length
$\Lambda=2\lambda^2/d$, where $\lambda$ is the penetration depth and $d$ is
the film thickness.

To realize these conditions we use ultra-thin high-$T_c$ films with extremely
large Pearl lengths $\Lambda\sim 100\,\mu$m. The critical temperature of these
films $T_c\sim 40\,$K provides a relatively easy $T$-domain to work in, and the
data reported are quite robust.

When current $I$ flows through the bridge at high temperatures,
vortices are �thermally activated at the strip edges and pushed in,
causing dissipation and a non-zero voltage $V$. We show that these
processes should lead to a power-law dependence
$V\propto I^n$ �with the exponent $n$ determined by the film parameter
$\Lambda$ and temperature:
\begin{equation}
n=\frac{\phi_0^2}{ 8\pi^2\Lambda(T) k T}+1\,,\label{n}
\end{equation}
($\phi_0$ is the flux quantum and $k $ is the Boltzmann constant).
This �is �confirmed by the high-temperature  IV's and by the
independently measured Pearl length.

However �for low $T$'s, the
IV's show a slower current dependence than required by
thermal activation. Instead, a quantum tunneling model utilizing
the ideas of Refs.\,\onlinecite{Leggett,Glazman},
along with our knowledge of the barrier shape, provide a good
representation of the data.

In the present work, ultrathin sandwiches of
[Ba$_{0.9}$Nd$_{0.1}$CuO$_{2+x}$]$_{5}$/[CaCuO$_{2}$]$_{2}$/[Ba$_{0.9}$Nd$_{0.1}$CuO$_{2+x}$]$_{5}$
(CBCO 5/2/5) were used. They consist of one superconducting infinite
layer block (two CaCuO$_{2}$ unit cells), sandwiched between two charge
reservoirs ($5$\ Ba-based unit cells). The structure was grown on (001)\
SrTiO$_{3}$ substrates, with nominally zero miscut angle, by
  using a focussed KrF excimer pulsed
laser source \cite{balestrino0,balestrino apl}. The films were
photolithographically patterned into bridges. In this paper we present
measurements from a ``wide" bridge approximately 85$\mu$m across. After
measurement, this bridge was   narrowed by a factor
of 2 and then remeasured. The patterning of such a thin layer
(5\,nm with a superconducting cell  only 1\,nm thick) protected by a 100\,nm
amorphous cap layer required specially careful
  photolithography and �milling. The
use of a cap layer deposited in situ required pre-deposited Au
contacts (details will be given elsewhere). Four terminal
measurements of the current-voltage characteristics
were realized down to 300\,mK in an Oxford Heliox probe suitably
modified by superconducting wiring to minimize heating at elevated
currents. The bridges were
 imaged with a scanning SQUID microscope (SSM)
\cite{ssmapl,vartapl} to determine
current uniformity and the film Pearl length.

 
\begin{figure}[htb]
\includegraphics[ width=3.5in]{brdgecrs.ps}
\caption{\label{fig:brdgecrs}(Color online) 
(a) SSM image of the wide 5/2/5 CBCO bridge
after cooling in a 22\,mG field �at 5\,K with an 8$\,\mu$m square
pickup loop. (b) Cross-section through an isolated Pearl
vortex after cooling in a 1\,mG field. The line fit is obtained using
the field distribution $h_z(x,z)$ given in Ref.\,\onlinecite{cbcoprl},
with an effective pickup loop height $z=5\,\mu$m and
$\Lambda =400\,\mu$m. (c) SSM image with a 104\,Hz, 100$\,\mu$A r.m.s.
a.c. current passing through the bridge at $T=$20\,K.
(d) Cross-section through the data along
the dashed line in (c); the solid line is calculated for uniform current
through the bridge, with $z=5 \,\mu$m.}
\end{figure}

Figure \ref{fig:brdgecrs}a shows an SSM image of the wide bridge
after cooling in a 22\,mG field. The bridge outline is visible along
with �trapped Pearl vortices \cite{pearlapl,cbcoprl}. Fig.
\ref{fig:brdgecrs}b shows a cross-section through one Pearl vortex,
after recooling in a smaller field to reduce the density of vortices.
Fitting such cross-sections results in low temperature Pearl lengths of
400$\pm$50$\,\mu$m for this bridge. Fits to scanning susceptibility
measurements   of the same bridge gave $\Lambda=$
200$\pm$20$\,\mu$m (see Ref.\,\onlinecite{cbcoprl} for details of these
procedures). This difference may be due to the finite lateral dimensions
of the film, which are not accounted for in the modelling.
The length $\Lambda$ is therefore longer than the bridge
width, and we expect the current through the bridge to be uniform, as
confirmed by the SSM image and fit of Fig.\,\ref{fig:brdgecrs}c,d. The
field $h_z$ (normal to the film) is maximum near the edges $x=0,W$ where
vortices �or antivortices �nucleate, penetrate the strip by being pushed in
by the Lorentz �force $\phi_0
I/cW$, and �annihilate near the strip middle. We focus on the barrier for
vortices; the one for antivortices is obtained by replacing $x$ with $W-x$.

For narrow strips of width $W\ll \Lambda$ with no current, the
energy of a vortex at a position $0<x<W $ is \cite{Pearlvortex}:
\begin{equation}
\epsilon(x) =
\frac{\phi_0^2}{8\pi^2\Lambda}\ln\left(\frac{2W}{\pi \xi}\sin
\frac{\pi x}{W}\right)\,,
\label{energy}
\end{equation}
$\xi$ is the coherence length. 
The vortex energy in the presence of a uniform current is
\begin{equation}
U(x)=\epsilon(x) -\frac{\phi_0 I}{cW}\,x \,.
\label{potential}
\end{equation}
The current causes suppression of the barrier maximum $U_m$ and pushes
the maximum position $x_m$ from the middle (at $I=0$) toward the edge
$x=0$. We obtain �after simple algebra:
$x_m= (W/\pi)\,\tan^{-1}(I_0/I)$ and
\begin{eqnarray}
U_m=\epsilon_0\left( \ln \frac{2W}{\pi
\xi\sqrt{1+ I^2/I_0^2}} �-
\frac{I}{I_0}\tan^{-1}\frac{I_0}{I}\right),
\label{Um}\\
I_0=\frac{c\phi_0}{8\pi\Lambda}\,,\qquad
\epsilon_0=\frac{\phi_0^2}{8\pi^2\Lambda}\,. \label{Io}
\end{eqnarray}
For $\Lambda\approx 400\,\mu$m, $I_0\approx 2 \times 10^{-3}\,$mA at low
temperatures and decreases on warming. For the majority of our data
$I/I_0\gg 1$ and Eq.\,(\ref{Um}) simplifies to:
\begin{eqnarray}
U_m=\epsilon_0\ln\frac{2 WI_0}{e\pi\xi \, I
 }   \,,\label{barrier}
\end{eqnarray}
where $e=2.718$. The logarithmic dependence of the barrier height
on the current results in power-law  IV's, the dependence
encountered in models of vortices moving through
random potentials in disordered materials \cite{Blatter}. In our
case, it is caused by a well-defined barrier.

%\subsection{Thermal activation}

If vortices cross the strip via a thermally
activated process, a voltage $V\propto \exp(-U_m/kT)$ is generated,
i.e.,
\begin{eqnarray}
V &\propto& \exp\left(-\frac{\epsilon_0}{kT} \ln \frac{
2WI_0}{e\pi\xi �I
 } \right)   = \left(\frac{I}{I_d}\right)^{m}
\,,\label{V}\\
m&=&\frac{\epsilon_0(T)}{kT}=
\frac{\phi_0^2}{8\pi^2\Lambda(T)kT }\,,\\
I_d&=&\frac{c\phi_0}{8e\pi^2\lambda^2\xi}\, Wd\,.\label{Idp}
\label{n_estimate}
\end{eqnarray}
where $I_d$ is of the order of the bridge depairing current.
Thus, the model provides not only the power-law {\it per se}, but gives the
exponent in terms of the film parameters. We note also that our
model differs from that used in Ref.\,\onlinecite{Repaci} to 
interpret similar data.
 
\begin{figure}[htb]
\includegraphics[ width=70mm]{fig2a.eps}
\includegraphics[ width=65mm]{n.eps}
\caption{\label{f1}(Color online) The upper panel: $V(I)$ on a log-log
scale at temperatures indicated in the legend. �Lower panel: The exponent
$n$ extracted from $V\propto I^n$ for the wide bridge (circles) and the
narrow bridge (triangles). The solid line is $n(T)$ of Eq.\,(\ref{n})
with with the best-fit parameters $T_c=37.6\,$K and
$\Lambda(0)=317\,\mu$m. �}
\end{figure}


As is usually the case with thermal activation, it is difficult to fix
the ``attempt frequency" �in the probability $\exp(-U_m/kT)$. Still, the number
of vortices penetrating the strip must be proportional to the field value at
the edges, i.e., to the current
$I$. Adding a prefactor const$\cdot I$ in Eq.\,(\ref{V}), we
obtain the exponent $n=m+1$ of Eq.\,(\ref{n})
which gives the correct limit for $T\to T_c$.

The upper panel of Fig.\,\ref{f1} shows IV measurements for the wide bridge
on a log-log scale. All curves are
nearly straight (except noisy low-voltage parts), i.e., $V\propto I^n$
at high currents. The exponent $n$ extracted from power-law fits to the data is
shown in the lower panel, for both the wide and narrow bridges. The solid line
is $n(T)$ of Eq.\,(\ref{n}), in which the two-fluid
$\Lambda=2\lambda_0^2/d(1-t^4)$ has been used for simplicity;
$t=T/T_c$.
 Fitting the high-$T$ part of the data for the wide bridge we obtain
$T_c\approx 38\,$K and $\Lambda(0)\approx 320\,\mu$m in   agreement with
$\Lambda(0)\approx 200-400\,\mu$m independently measured by SSM. Thus,   the
thermal activation model works well   above
$\approx 15\,$K, where it gives �correct values of the exponent $n(T)$.
Although $T_c$ �of the narrow bridge was slightly lower after repatterning, the
high temperature exponents were quite close, which means that the Pearl
length was nearly the same in both bridges.

\begin{figure}[htb]
\includegraphics[ width=65mm]{fig3.eps} 
\caption{\label{f5}(Color online) $R(T)$ for $30<T<36\,$K and fit
to an exponential dependence.   }
\end{figure}



Another quantity which can be extracted from the IV's is the
low-voltage differential resistance, defined as $R=(dV/dI)_{I\to 0}$.
At low temperatures, the
voltages at low currents are exceedingly small and cannot be extracted
from the noise. However, reliable determination of
$R$ is possible above roughly 30\,K. Fig.\,\ref{f5} shows that in this
domain �$R\propto \exp(T/T_0)$.

To understand this result, we turn again �to the voltage expression
(\ref{V}) for thermally activated vortex motion. As mentioned above,
the prefactor in this expression is
proportional to the current �because the number of vortices attempting
to overcome the barrier is proportional to the edge fields,
i.e., to $I$. Therefore, calculating
$R=(dV/dI)_{I\to 0}$, we can leave the current dependence only in the
prefactor and set $I=0$ in �$\exp[-U_m(I)/kT]$. We then have:
\begin{equation}
U_m\approx \epsilon_0
\ln\frac{2W}{\pi\xi}=\frac{4\phi_0^2(1-t)}{8\pi^2\Lambda(0)}
\ln\frac{2W}{\pi\xi}\,,
\end{equation}
where $\Lambda(0)$ is the low temperature Pearl length,
  and the two-fluid approximation
$\Lambda =\Lambda(0) /(1-t^4)\approx \Lambda(0) /4(1-t)$ is again used. Then,
the voltage and the resistance near $T_c$ should be proportional to
\begin{eqnarray}
\exp\left(+\, \frac{
\phi_0^2\,T}{2\pi^2\Lambda(0)k T_c^2}\ln\frac{2W}{\pi\xi}\right).
\end{eqnarray}
Thus, we expect the behavior $R\propto \exp(T/T_0)$ with
\begin{equation}
T_0=\frac{2\pi^2\Lambda(0)k T_c^2}{\phi_0^2\ln(2W/\pi\xi) �}
\,, \label{To}
\end{equation}
where $T_0$ �depends weakly on $T$ via $\ln \xi$. �Note that our
model does not allow us to literally take the limit $T\to T_c$ because
the   energy expression (\ref{energy}) is valid only for $W\gg\xi$.
The data of Fig.\,\ref{f5} correspond �to $T_0\approx 0.78\,$K.
Expression (\ref{To}) yields a smaller value $T_0\approx 0.3\,$K.

We stress that �$R\propto \exp(T/T_0)$ is obtained here
for the thermally activated process, so that this �dependence {\it per
se} is not necessarily a �manifestation of quantum tunneling (compare
with \cite{Goldman}). 

At low temperatures, the thermal activation model fails.
The fit values for $n(T)$ in Fig. \ref{f1}
do not diverge as thermal activation requires �\cite{remark1}.
Hence, we turn to the possibility of quantum tunneling. �According to Ref.
\onlinecite{Glazman}, the tunneling probability for {\it overdamped}
processes   is proportional to
$\exp\left(- \gamma\eta\, x_b^2/\hbar \right)$ �where
$\gamma$ is a geometric factor related to the barrier shape,
$\eta$
%\phi_0^2d/2\pi\xi^2c^2\rho_n$
is the drag coefficient,
and $x_b$ is the barrier {\it
width}. Unlike thermal activation, for which only the barrier height is
relevant, for the tunneling �the barrier width dominates the probability.

Since the potential $U=0$ �\cite{remark2} at the position of vortex
entry, the width $x_b$ is a root of $U(x)=0$, or of
\begin{equation}
\ln\left(\frac{2W}{\pi \xi}\sin
\frac{\pi x}{W}\right) = \frac{I}{I_0}\,\frac{\pi x}{W}\,.
\label{e10}
\end{equation}
As argued above, $I/I_0\gg 1$; therefore �$\pi x/W$ must be small.
 Since the log is a   slow function, we have
\begin{equation}
x_b\approx C_0\,\frac{WI_0}{\pi I}\,,
\label{C/i}
\end{equation}
where $C_0$ is a constant of the order one.
% and only weakly depends on $I$ and $W$.
Hence, the barrier width
shrinks with increasing current. One can show by solving numerically
Eq.\,(\ref{e10}) that for   $I/I_0\sim 10^3$ (the range of our data)
$C_0\approx 5$.


We then expect the voltage for high currents to behave   according to
\begin{equation}
\ln V =C_1-C_2\,\frac{\eta W^2}{\hbar}\,\frac{I_0^2}{I^2}\,.
\label{quant}
\end{equation}
Here, $C_1$ is a constant related to the �attempt frequency, whereas
$C_2=C_0^2\gamma/\pi^2 \sim 2.5\gamma$.
 

 
In Fig.\,\ref{f4}, we plot log$V$ for a few   $T$'s {\it versus}
$I^{-2}$ for both wide and narrow bridges. Clearly, the straight parts
of the IV's for low �temperatures and high currents �are in agreement with
Eq.\,(\ref{quant}). With increasing $T $, the IV's curve �away
from a straight line much faster, signalling deviations from
predominantly quantum tunneling.
We fit the data for $0.8\,$K of the wide bridge and for $1\,$K of
the narrow one to
$V=V_0\exp(-\alpha/I^2)$ to obtain
$V_0$=0.43\,V,
$\alpha=1.0\times �10^{-4}$A$^{2}$ for the first and
$V_0$=0.97\,V,
$\alpha=0.14\times 10^{-4}$A$^{2}$ for the second.



  \begin{figure}[htb]
 \includegraphics[ width=70mm]{f4-5.eps}
% \includegraphics[ width=70mm]{vortqt3.ps}
 \caption{\label{f4} (Color online) The voltage $V$ {\it versus} $1/I^2$ for
 temperatures indicated in the legend. The left group of IV's is for
the wide bridge with $W=85\,\mu$m; the right group is for
$W=42.5\,\mu$m.}
 \end{figure}

Going back to Eq.\,(\ref{quant}) for the quantum tunneling, we note
that the coefficient of $I^{-2}$ is proportional to $W^2$. Therefore,
as an extra check of applicability of this equation, one can �compare
$\alpha$'s of the two bridges. Given the approximate character of
our model, the {\it reduction} of $\alpha$ by a factor of 7 in the
bridge $W/2$ as compared with the   expected   {\it reduction}
by a factor of 4, shows that quantum tunneling is consistent with our
data.

Moreover, the pre-exponential factor $V_0$ proportional to the
attempt frequency should go as $1/W$ because - as
magnetostatics shows - the edge field is $\propto 1/W$. The ratio of
$V_0$'s for the two bridges is 0.97/0.43=2.3 instead of the expected
factor of 2. This is yet another indication that the low-$T$ data
are consistent with the tunneling model.

Further quantitative tests of the quantum tunneling model would
be �possible if the drag coefficient $\eta$ and the barrier shape
factor $\gamma$ were well known. Unfortunately, this is not the case.
The factor $\gamma=\pi/2$ was found for a barrier $U(x)$ as a
  cubic parabola \cite{LO}, quite different from our case.
For $I_0=2.5\mu A$ and
W=85$\mu$m, �we estimate $\eta\gamma\sim 10^{-16}\, $CGS. If one takes
$\gamma\sim 1$, this gives the drag coefficient
 about two orders of magnitude smaller than the Bardeen-Stephen
value $\eta=\phi_0^2d/2\pi\xi^2c^2\rho_n$. Kramer-Pesh
contraction \cite{KP} of the vortex core at low $T$'s might be a
cause of the $\eta$ suppression.
 


  During this work, we have also searched for a
Berezunskii-Kosterlitz-Thouless transition in the domain of the
power-law IV's. We, however, did not detect any discontinuities in the
$T$ dependence of $n$. We believe this is because in
bridges �with $W<\Lambda$,   interaction of vortices is
short-range with a cutoff at distances $W$.
 


%\subsection{Discussion}

To summarize, at high temperatures we observe the power-law IV's   well
described by thermally activated motion of vortices through the
thin-film strips narrow on the scale of Pearl
$\Lambda$ for which the shape of the potential barrier is well established.
The predicted dependence of the exponent $n$ on
$\Lambda(T)$ is in good agreement with the data. The �low temperature
  data are consistent with the overdamped quantum tunneling
of Pearl vortices, the size of which in our films
is �indeed macroscopic since $\Lambda\sim   �0.1\,$mm.

\begin{acknowledgments}
The authors thank L. Bulaevskii for thoughtful remarks. 
This work has been   supported by   ESF projects 
``Pi-Shift"  and ``QUACS", and by MIUR program ``Quantum
effects in Nano-structures and Superconducting devices".

\end{acknowledgments}




\begin{thebibliography}{99}

\bibitem{Leggett}A.G. Caldeira and TA.J. Leggett, \prl {\bf 46}, 211
(1981).

\bibitem{Glazman}L.I. Glazman and N.Ya. Fogel, Sov. J. Low temp.
Phys. �{\bf 10}, 51 (1984); Fiz. Nizk. Temp. {\bf 10}, 94
(1984).

\bibitem{Blatter} G. Blatter �{\it et al.},
%M.V. Feigel'man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur,
Rev. Mod. Phys. {\bf 66}, 1125 (1994).

\bibitem{balestrino0} �G. Balestrino {\it et al.}, Phys. Rev. B {\bf
62}, 9835 (2000).

\bibitem{balestrino apl} G. Balestrino {\it et al.}, Appl. Phys.
Lett. {\bf 79}, 99 (2001).

\bibitem{ssmapl} J.R. Kirtley {\it et al.},
  Appl. Phys. Lett. {\bf 66}, 1138 (1995).

\bibitem{vartapl} J.R. Kirtley {\it et al.},
%, C.C. Tsuei, K.A. Moler, V.G. Kogan, J.R. Clem, and A.J. Turberfield,
Appl. Phys. Lett. �{\bf 74}, 4011(1999).

\bibitem{pearlapl} J. Pearl, Appl. Phys. Lett. {\bf 5}, 65 (1964).

\bibitem{cbcoprl} F. Tafuri {\it et al.},
%, J.R. Kirtley, P.G. Medaglia, P. Orgiani, and G. Balestrino,
Phys. Rev. Lett. {\bf 92}, 157006 (2004).

\bibitem{Pearlvortex} V.G. Kogan, Phys. Rev. B {\bf 49}, 15874
(1994). The part of this work dealing with vortex in a
half-plane �film is erroneous. The energy arguments and the treatment
of vortices in samples small on the scale $\Lambda$ are correct. The
 energy expression (\ref{energy}) has been confirmed by J.R. Clem,
unpublished 1996, and by G.M. Maximova, Sov. Phys. Solid State, {\bf 40}, 1607
(1998).

\bibitem{Repaci} J.M. Repaci {\it et al.}, Phys. Rev. B {\bf 54},
9674 (1996).

\bibitem{Goldman} Y. Liu {\it et al.},
%, D.B. Haviland, L.I. Glazman, and A.M. Goldman,
\prl {\bf 68}, 2224 (1992).

\bibitem{remark1} Taking a faster than the
two-fluid low-$T$ behavior of $\Lambda$ (for materials with gap
nodes) would have made the increase of $n(T)$ at low $T$'s even stronger.
The levelling of $n(T\to 0)$ is not �pronounced in
other reports, see e.g. Ref.\,\onlinecite{Repaci}.
 
\bibitem{remark2} One can take the vortex energy   �at $x\sim\xi$
near the edge, $\epsilon(\xi)=\epsilon_0\ln 2$, as the starting
energy. This results only in a renormalization of the large constant
$2W/\pi\xi \to W/\pi\xi$, but leaves the structure of Eq.\,(\ref{e10})
unchanged.

\bibitem{LO}A.I. Larkin, Yu.N. Ovchinnikov, JETP Lett.
{\bf 37}, 3821 (1983).

 \bibitem{KP}W. Pesh and L. Kramer, J. Low Temp. Phys. 
  {\bf 15}, 367 (1974).

\end{thebibliography}

\end{document}





 
cond-mat0504140.extracted_bib
\begin{thebibliography}{99}

\bibitem{Leggett}A.G. Caldeira and TA.J. Leggett, \prl {\bf 46}, 211
(1981).

\bibitem{Glazman}L.I. Glazman and N.Ya. Fogel, Sov. J. Low temp.
Phys. �{\bf 10}, 51 (1984); Fiz. Nizk. Temp. {\bf 10}, 94
(1984).

\bibitem{Blatter} G. Blatter �{\it et al.},
%M.V. Feigel'man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur,
Rev. Mod. Phys. {\bf 66}, 1125 (1994).

\bibitem{balestrino0} �G. Balestrino {\it et al.}, Phys. Rev. B {\bf
62}, 9835 (2000).

\bibitem{balestrino apl} G. Balestrino {\it et al.}, Appl. Phys.
Lett. {\bf 79}, 99 (2001).

\bibitem{ssmapl} J.R. Kirtley {\it et al.},
  Appl. Phys. Lett. {\bf 66}, 1138 (1995).

\bibitem{vartapl} J.R. Kirtley {\it et al.},
%, C.C. Tsuei, K.A. Moler, V.G. Kogan, J.R. Clem, and A.J. Turberfield,
Appl. Phys. Lett. �{\bf 74}, 4011(1999).

\bibitem{pearlapl} J. Pearl, Appl. Phys. Lett. {\bf 5}, 65 (1964).

\bibitem{cbcoprl} F. Tafuri {\it et al.},
%, J.R. Kirtley, P.G. Medaglia, P. Orgiani, and G. Balestrino,
Phys. Rev. Lett. {\bf 92}, 157006 (2004).

\bibitem{Pearlvortex} V.G. Kogan, Phys. Rev. B {\bf 49}, 15874
(1994). The part of this work dealing with vortex in a
half-plane �film is erroneous. The energy arguments and the treatment
of vortices in samples small on the scale $\Lambda$ are correct. The
 energy expression (\ref{energy}) has been confirmed by J.R. Clem,
unpublished 1996, and by G.M. Maximova, Sov. Phys. Solid State, {\bf 40}, 1607
(1998).

\bibitem{Repaci} J.M. Repaci {\it et al.}, Phys. Rev. B {\bf 54},
9674 (1996).

\bibitem{Goldman} Y. Liu {\it et al.},
%, D.B. Haviland, L.I. Glazman, and A.M. Goldman,
\prl {\bf 68}, 2224 (1992).

\bibitem{remark1} Taking a faster than the
two-fluid low-$T$ behavior of $\Lambda$ (for materials with gap
nodes) would have made the increase of $n(T)$ at low $T$'s even stronger.
The levelling of $n(T\to 0)$ is not �pronounced in
other reports, see e.g. Ref.\,\onlinecite{Repaci}.
 
\bibitem{remark2} One can take the vortex energy   �at $x\sim\xi$
near the edge, $\epsilon(\xi)=\epsilon_0\ln 2$, as the starting
energy. This results only in a renormalization of the large constant
$2W/\pi\xi \to W/\pi\xi$, but leaves the structure of Eq.\,(\ref{e10})
unchanged.

\bibitem{LO}A.I. Larkin, Yu.N. Ovchinnikov, JETP Lett.
{\bf 37}, 3821 (1983).

 \bibitem{KP}W. Pesh and L. Kramer, J. Low Temp. Phys. 
  {\bf 15}, 367 (1974).

\end{thebibliography}