1D-transport properties of single superconducting lead nanowires

Article abstract of DOI:10.1016/S0921-4534(03)01086-4

We report on the transport properties of single superconducting lead nanowires grown by an electrodeposition technique, embedded in a nanoporous track-etched polymer membrane. The nanowires are granular, have uniform diameter of ?40 nm and a very large aspect ratio (?500). The diameter of the nanowire is small enough to ensure a 1D superconducting regime in a wide temperature range below T_c. The non-zero resistance in the superconducting state and its variation caused by fluctuations of the superconducting order parameter were measured versus temperature, magnetic field, and applied DC current (or voltage). The current induced breakdowns in the V-I characteristics may be explained by the formation of phase slip centers. Moreover, DC voltage driven measurements reveal the existence of a new S-shape behavior near the formation of these phase slip centers.

Full text of DOI:10.1016/S0921-4534(03)01086-4

Physica C 391 (2003) 369–375
www.elsevier.com/locate/physc
1D-transport properties of single
superconducting lead nanowires
a,b,
a,b
, S. Matefi-Tempfli a,b, L. Piraux
ꢀ ꢀ
*
S. Michotte
a
Uniteꢀ de Physico-Chimie et de Physique des Mateꢀriaux (PCPM), Universiteꢀ Catholique de Louvain, Place Croix du Sud 1,
B-1348 Louvain-la-Neuve, Belgium
b
Research Center on Microscopic and Nanoscopic Electronic Devices and Materials (CERMIN), Universiteꢀ Catholique de Louvain,
B-1348 Louvain-la-Neuve, Belgium
Received 5 October 2002; received in revised form 25 March 2003; accepted 24 April 2003
Abstract
We report on the transport properties of single superconducting lead nanowires grown by an electrodeposition
technique, embedded in a nanoporous track-etched polymer membrane. The nanowires are granular, have uniform
diameter of ꢀ40 nm and a very large aspect ratio (ꢀ500). The diameter of the nanowire is small enough to ensure a 1D
superconducting regime in a wide temperature range below Tc. The non-zero resistance in the superconducting state and
its variation caused by fluctuations of the superconducting order parameter were measured versus temperature,
magnetic field, and applied DC current (or voltage). The current induced breakdowns in the V –I characteristics may be
explained by the formation of phase slip centers. Moreover, DC voltage driven measurements reveal the existence of a
new S-shape behavior near the formation of these phase slip centers.
ꢀ 2003 Elsevier B.V. All rights reserved.
PACS: 74.40.+k; 74.25.Fy; 85.40.U
Keywords: Lead nanowire; 1D superconductivity; IV characteristic; Phase-slip-centers
1. Introduction
ity, fluctuations of the superconducting order pa-
rameter can lead to a non-zero resistance below
the critical temperature. One-dimensional (1D)
superconductors, which are well suited to observe
those fluctuations, have therefore been extensively
studied theoretically [1–6] but to a lesser extent
experimentally [7–11].
Infinitely sharp change in conductivity during a
superconducting phase transition occurs only in a
sample, which is infinite in the three dimensions
[1]. In a superconductor of reduced dimensional-
In 1D superconductor, the Ginzburg–Landau
(GL) order parameter w ¼ AðzÞ expði/ðzÞÞ has an
amplitude and a phase that can vary only along
the wire axis z. Since the current I is proportional
to r/, this current induces a spatial variation of
the phase /. This leads to the known picture of the
* Corresponding author. Address: Unitꢀe de Physico-Chimie
ꢀ
ꢀ
et de Physique des Materiaux (PCPM), Universite Catholique
de Louvain, Place Croix du Sud 1, B-1348 Louvain-la-Neuve,
Belgium.
E-mail address: michotte@pcpm.ucl.ac.be (S. Michotte).
0921-4534/$ - see front matter ꢀ 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0921-4534(03)01086-4

370
S. Michotte et al. / Physica C 391 (2003) 369–375
‘‘helical’’ order parameter in the complex space.
Different minima of the multiwelled GL free
energy are separated by 2p in phase, correspond-
ing to different helicities of w. Jumps between two
neighboring minima can occur either by thermal
activation [5,6] (thermally activated phase slips,
TAPS) or by quantum tunneling through the
energy barrier [4,10,11] (tunneling quantum phase
slips, TQPS). The resistance in the superconduct-
ing state arises from the addition of those two
effects.
On the other hand, the superconducting prop-
erties of 1D superconductors are expected to de-
pend strongly on the level of disorder. By analogy
with experiments performed on superconducting
thin films [12], two cases can be distinguished.
When the disorder increases, homogeneous films
exhibit a direct transition from superconductivity
to the localized insulating phase (S–I), where
fluctuations leads to the suppression of the am-
plitude of the order parameter and to a sharp de-
crease of Tc. On the contrary, in granular films,
disorder favors quantum fluctuations of the phase
and a superconductor–metal–insulator (S–M–I)
transition with Tc almost unchanged.
[15–17]. Using nuclear track-etched polymer mem-
branes [18] as template, arrays of parallel thin
cylinders can be obtained electrochemically. These
nanowires possess uniform dimensions with con-
trolled diameters as small as a few tens of nano-
meters.
The nanoporous polycarbonate membrane used
in this study was 22 lm thick. The pore density
was around 4 · 109 cmꢁ2 and those pores were al-
most parallel to each other, perpendicular to the
membrane surface and have a uniform diameter
around 40 nm. A thick gold layer, used as a
cathode, is deposited on one side of the membrane
to cover the pores. The other face was exposed
to an electrolyte containing Pb2þ ions (40.4 g/l
Pb(BF4)2, 33.6 g/l HBF4 and 15 g/l H3BO3). Con-
stant potential of )0.5 V versus an Ag/AgCl ref-
erence electrode was used in a three-electrode
configuration in order to reduce the Pb2þ ions into
the pores. The electrodeposition was carried out at
ambient temperature without stirring.
After dissolution of the membrane with dichlo-
rometane, lead nanowires were observed using a
SEM-FEG microscope. As can be seen in Fig. 1,
these nanowires present a nice cylindrical geome-
try and the diameter is uniform.
In order to contact a single nanowire of the
array (while still inside the membrane), a thin gold
layer was deposited on the side exposed to the
In this work, we report on a simple way to
fabricate granular lead nanowires protected from
oxidation and to measure the transport properties
of a single nanowire. We observed different stages
of the destruction of superconductivity indicating
the presence of fluctuations leading to the forma-
tion of phase slip centers. Using a voltage driven
setup, we reveal a new S-shape behavior that exists
for a range of current values near the formation of
these phase slip centers.
2. Experimental
Different ways such as step-edge technique and
lithography were used to fabricate single super-
conducting nanowires [8,11,13]. However, the
shape of the nanowires obtained using such tech-
niques was often far from being perfectly cylin-
drical, as usually assumed from theoretical studies.
In the template synthesis method [14], which is
used here, the material is grown by electrodeposi-
tion into the voids of a nanoporous host material
Fig. 1. SEM micrograph of lead nanowires after dissolution of
the hosting polycarbonate membrane.

S. Michotte et al. / Physica C 391 (2003) 369–375
371
electrolyte prior to electrodeposition. This con-
tacting layer covers the surface of the membrane
but is not thick enough to plug the pores. Its area
was somewhat larger than the electrodeposition
area (3 mm diameter). The emergence of the first
Pb nanowire from the surface, detected by a sharp
jump in the plating current, naturally interrupts
the growth in the other pores. Electrodeposition is
still performed for a while so that a thin lead film is
deposited on the top face of the porous membrane
sample. In this way, a single 22 lm long lead
nanowire with extremely large aspect ratio (length
over diameter ꢀ 500), completely embedded in the
polymer matrix and protected from oxidation, can
be easily contacted electrically with silver paint.
The final sample is depicted in the inset of Fig. 2.
In the following, we report on the transport
measurements performed on such 40 nm Pb
nanowires. The reproductibility of the results has
been verified on several samples. The resistance
versus temperature (R–T ) and versus magnetic
field (R–H) were measured down to 4.2 K with a
lock-in amplifier (PAR model 124A) by applying a
very small current (I ꢀ 0:1 lA). The magnetic field
was applied parallel to the wire axis to eliminate
demagnetising effects. The DC voltage–current
characteristics (V –I) were also measured at differ-
ent temperatures, applying either a DC current or
a DC voltage. When applying a DC voltage, the
current flowing into the nanowire was determined
by measuring the voltage across a 1 X resistance
added in series and the voltage developed across
the sample was measured separately. All the curves
presented here were measured in the forward and
backward way and no hysteresis was observed.
3. Results and discussion
As far as superconductivity is concerned, a
sample is considered 1D if its transverse dimen-
sions are smaller than the coherence length nðT Þ,
which is temperature dependent. This length di-
verges at the critical temperature Tc (7.2 K for bulk
lead) and is of the order of a few tens of nm below
Tc for lead, depending on the level of disorder. The
measured resistance at room temperature and at
8 K was 3410 and 288 X respectively. This corre-
spond to a single 40 nm Pb nanowire (22 lm in
length) as calculated using the phonon contribu-
tion q300 K ꢁ q8 K ffi 18:9 lX cm for lead [19]. The
residual resistivity, just above the transition, was
q8 K ¼ 1:7 lX cm. Using the free electron model
[20], we can estimate roughly the mean free path
length ‘ ꢀ 25 nm at low temperature, which is
of the order of the radius of the wire. The intrin-
sic coherence length n0 for lead being 82 nm,
our nanowire is in the dirty limit because ‘ < n0.
Therefore, the following relation nðtÞꢂ0:907n0ð1þ
ð1 ꢁ 0:25tÞn0=‘Þꢁ1=2ð1 ꢁ t2Þꢁ1=2 where t ¼ T=Tc ap-
plies for all temperature below Tc [21]. It results
from this relation that the diameter of our nano-
wire is smaller than nðTÞ in all the temperature
range investigated in the present study. Our lead
nanowires thus present the advantage to behave
one dimensionally as far as superconductivity is
concerned in a wide temperature range below Tc.
The low-temperature resistance of the 40 nm Pb
nanowire is shown in Fig. 2. The normal state
values obtained at the upper critical magnetic field
is also reported for comparison. The relatively
Fig. 2. Resistance of a single 40 nm lead nanowire at low
temperature measured by applying a current of 0.1 lA. The
critical temperature is around 7.2 K. The value of the resistance
when superconductivity is destroyed by a magnetic field applied
parallel to the nanowire axis is also shown and the inset depicts
a schema of the sample.

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S. Michotte et al. / Physica C 391 (2003) 369–375
long electronic mean free path imply that the
nanowire is far from the insulator transition as it
appears in the R–T curve. Two regimes can be
distinguished in the superconducting state [2,10,
11], with a smooth transition between them as in
[7]. Just below Tc, TAPS dominate but their fre-
quency reduces sharply when the temperature is
further reduced. At temperature below about 5 K,
the resistance is almost temperature independent
corresponding to the metallic side of the S–M–I
transition, though the exact contribution from the
contacts is difficult to estimate.
When a DC current is applied, the V –I charac-
teristics shown in Fig. 3 are similar to those pre-
viously reported for superconducting whiskers
[22,23]. The different jumps observed in the V –I
characteristics can be explained by the formation
of phase slip centers (PSC) [24]. These two steps
observed at 4.3 K in Fig. 3 with current drive are
thus most probably due to the successive appear-
ance of two PSCs, in which phase slips occur in
localized regions of length nðTÞ, but the voltage
developed across each PSC is governed by quasi-
particle diffusion [25]. The healing length Ln of the
non-equilibrium region over which normal current
flows is given by Ln ¼ ðLT=bPSCÞðRn=RTÞ where LT
and RT are the total length and normal resistance
of the nanowire, Rn is the normal resistance of the
region over which the normal currents flow during
the phase slip process (first step in resistance) and
bPSC ꢂ 0:65 is the ratio between the time-averaged
normal current and the total current carried by a
PSC at low voltage. In our case, e.g. at 4.3 K,
(Rn=RT) is slightly less than 0.5. The total length of
the nanowire being 22 lm, this gives Ln ꢂ 17 lm.
This value is reasonable because it is theoretically
twice the quasiparticle diffusion length which is
typically of the order of 10 lm. In spite that suc-
cessive PSCs tend to avoid those already in place,
the second PSC in our case is thus force to overlap
the first one, which explains why the second jump
in resistance is smaller. Note that the third little
jump corresponds simply to a return to the normal
state. Note also that unavoidable ‘‘weak spots’’
such as grain boundary, etc., probably only de-
termine the position of the PSC nucleation along
the nanowire and the value of the ‘‘switching’’
current. Indeed, PSCs are a general behavior
proper to all low dimensional superconductors and
have also been observed in virtually perfect single
crystals [23]. Moreover, it has been also shown in
[22] that these steps are not related to the nature of
the electrical contacts. We emphasis that in our
experiments, the surface/volume ratio is large and
that the heat produced in this non-equilibrium
state is well evacuated. The observed behaviors are
thus not limited to temperature close to Tc as it is
the case for micron-sized whiskers in which self-
heating hotspots hidden this phenomenon when
the temperature is lowered [25].
Measuring the V –I characteristics in the oppo-
site way (i.e., by applying the voltage and mea-
suring the current) was not yet reported to our
knowledge. In Fig. 3, we see that the sample be-
haves differently at the lowest temperatures in the
steps regions. In the framework of the LAMH [5,6]
model, the system is completely superconducting
and can be found in one of the metastable current-
carrying states. The resistance is then proportional
to the rate at which fluctuations drive the system
to a neighboring metastable state. Therefore a
single solution exists for a range of current values
and there is no difference between the current and
voltage driven measurements in the V –I charac-
teristics, as predicted by McCumber [26], and as
observed in our experiment far from the formation
of PSC. On the other hand in the non-equilibrium
Fig. 3. Voltage–current characteristics at different tempera-
tures. The continuous lines show the DC voltage measured
versus the applied DC current. The dotted lines show the DC
current versus the applied DC voltage.

S. Michotte et al. / Physica C 391 (2003) 369–375
373
region of a phase slip center formation, we observe
for the first time in the voltage driven experiment a
S-shape behavior which has been poorly addressed
theoretically up to now. In the work of Tucker and
Halperin [27] when the system is partially normal
but contains a low density of fluctuating Cooper
pairs, such a S-shape behavior may exist in the
dirty limit. However, a complete theoretical de-
scription of this behavior is still lacking. Notice
that the existence of such multiple solutions could
also explain the rapid switching between two sta-
ble states that we observed in current drive as in
[22] when the current was adjusted on such a jump
in the V –I characteristics.
Due to the lack of realistic model, we can only
point out some interesting experimental observa-
tions in our voltage driven experiment. At 4.3 K,
just before the formation of the first PSC, the value
of the first maximum in current density, assuming
a uniform current in the nanowire section due to
its small diameter, is j ¼ 8 Â 106 A/cm2. We use
j ¼ nsevd, where ns is the total number of super-
conducting electrons, e is the electronic charge and
vd is the depairing velocity. This last value corre-
sponds to the one expected from the expression
vd ¼ D=ðꢁhkFÞ ¼ 120 m/s using D(4.3 K) ¼ 1.25
meV and kF ¼ 1:58 Â 108 cmꢁ1 for lead. We thus
find ns ¼ 4:2 Â 1021 eꢁ/cm3. Comparing this value
with the total number of conduction electrons in
lead n ¼ 1:3 Â 1023 eꢁ/cm3, it appears that 3% of
the conduction electrons become superconducting
at T ¼ 4:3 K. This value is poorly referenced but it
is reasonable comparing with tin where in the zero
temperature limit, 16% of the conduction electrons
are superconducting [28]. Moreover, we also note
that at the first maximum in current at 4.3 K, the
voltage obtained when the linear I–V part due to
the residual resistance is removed from the mea-
sured one (see Fig. 3) corresponds nearly to the
value of the gap (ꢂ1.25 mV). All of this tends to
indicate that in this voltage driven case, Cooper
pairs are speed up until they reach their depairing
velocity. Then they reach progressively, through
the negative differential resistance branch of the
curve, the state corresponding to one PSC like in
the current driven case.
produce by Joule heating in the PSC has to be
evacuated mostly through the media surrounding
the measured nanowire. Although the thermal
conductivity of polycarbonate is rather low, the
surrounding media is in fact a composite material
consisting not only of the polymer medium but
also of metal nanowires, due to the high density of
nanopores. These other nanowires are not con-
tacted electrically to the contacting layer but they
also contribute to heat transfer in the composite
material due to their larger thermal conductivity.
The bending of the curves seen for the largest
currents of Fig. 3 is a signature of heating effects in
the measurements. If we extend linearly the 7.2 K
curve in Fig. 3, we can notice that the third little
jump in the 4.3 and 5.5 K curves correspond pre-
cisely to the temperature of 7.2 K and thus a this
point, the small heating lead back to a return to
the normal state of the resistance. As a conse-
quence, though the exact shape of the reported S-
shape behavior is probably slightly modified due
to the temperature increase, we think that the
heating effects do not throw the S-shape behavior
back into question.
Fig. 4 shows the magnetic field dependence of
the resistance below Tc with the magnetic field
parallel to the nanowire axis. The presence of the
lead film on the upper face of the membrane (see
Fig. 2) is not embarrassing as its critical magnetic
Fig. 4. Resistance versus magnetic field carried out at different
temperatures. The resistance was measured by applying a cur-
rent of 0.1 lA and the magnetic field was applied parallel to the
nanowire axis. The inset shows the numerical derivative dR=dH
at 4.3 K.
It should be noted that heating effects are un-
avoidable in such experiments. Indeed, the heat

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S. Michotte et al. / Physica C 391 (2003) 369–375
field Hc is small compared to that of the nanowire
[16,17]. The small resistance jump around 0.04 T at
4.3 K account for the resistance contribution of ꢀ3
X from this thin lead protecting film in series with
our nanowire. Bulk lead is known as a type I su-
perconductor. Due to our mean free path value,
resistance presenting different stages in the su-
perconducting state. These stages may be ex-
plained by the formation of phase slip centers
when superconductivity is destroyed by a current
or a magnetic field. A new S-shape behavior has
been observed in the V –I characteristics near the
formation region of these phase slip centers in
voltage driven measurements. Further work is
now in progress to have a better understanding
of these observations. Finally, from preliminary
work, we may also anticipate that the grain size
in such nanowires can be diminished by using
a pulsed potential during electrodeposition. The
mean free path is then reduced and fluctuations
proliferate more, at last leading to an insulating
state as in [8,11]. A more detailed study of the
influence of disorder on the fluctuations in su-
perconducting nanowires will be presented in a
future publication.
the BCS dirty value of the penetration length [21]
applies
1=2
kð‘; tÞ ꢂ kLðtÞð1 þ ð1 ꢁ 0:25 Á tÞðn0=‘ÞÞ
where kLðtÞ is the London penetration length. It
results that the Ginzburg parameter defined by
p
j ¼ kðTÞ=nðT Þ is bigger than 2=2 at all temper-
atures below Tc. This corresponds to a type II su-
perconductor. For example, at 4.3 K, we found a
lower (Hc1) and upper (Hc2) critical magnetic fields
being equal to about 0.4 and 1.25 T, respectively.
These are much larger than the thermodynamic
critical magnetic field value of bulk lead at this
temperature (ꢀ0.055 T).
As it appears clearly in Fig. 4, the destruction of
the superconducting state by the magnetic field at
4.3 K also presents different stages with different
slopes but the number of these irregularities is
larger here than in the I–V characteristics at the
corresponding temperature (see inset to Fig. 4).
Note that vortex entry inside the lead nanowire is
not favorable, because of its small section S and a
rough estimate shows that a vortex would require
a field higher than U0=S ¼ 1:56 T where U0 is the
quantum of magnetic flux. Moreover, inhomoge-
neities in diameter along the nanowire can also be
exclude because the behavior is not present close
to Tc (see for example the curve obtained at
T ¼ 6:5 K in Fig. 4). Increasing the magnetic field,
which is a pair breaking perturbation, is known
[29] to decrease the quasiparticle diffusion time and
consequently the healing length.
Acknowledgements
We thank the ‘‘Laboratoire des Hauts Poly-
ꢂ
meres’’ of UCL for providing the polycarbonate
membrane samples used in this study. S.M. is a
Research Fellow of the National Fund for Scien-
tific Research (FNRS) Belgium. We thank Profs.
F. Peeters and V. Moshchalkov for valuable dis-
cussions. This work has been partly supported
by the Belgian Interuniversity Attraction Pole
Program (PAI-IUAP P5/1/1) and by the ‘‘Com-
ꢀ
munaute Franßcaise de Belgique’’ through the Pro-
ꢀ
gram ‘‘Actions de Recherches Concertees’’.
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