Phase slips and vortex-antivortex pairs in superconducting films with constrictions
Sang L. Chu, A. T. Bollinger, and A. Bezryadin
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080
(Dated: December 11, 2003)
A system of two thin coplanar superconducting films seamlessly connected by a bridge is studied.
When such a system is cooled, two distinct resistive transitions are observed. The first one is
identified with the Berezinskii-Kosterlitz-Thouless (BKT) transition in the film, whereas the second
one is a crossover related to the quenching of thermally activated phase slips (TAPS) occurring on
the bridge. Using direct and indirect methods, we have measured the resistance R(T ) of such samples
over a range of eleven orders of magnitude. Over this broad range of resistance a good agreement
is found with the Langer-Ambegaokar-McCumber-Halperin (LAMH) theory of TAPS, which was
adopted for short bridges, as given by the simple expression R(T ) = Rn exp [−(c/t)(1 − t)3/2 ],
where t = T /Tc and Rn is the normal resistance of the bridge.
Thermal fluctuations are known to generate a
non-zero resistance in superconductors with reduced
dimensionalities.1 In two-dimensional (2D) thin films the
fluctuations which cause the dissipation are known to be
broken vortex-antivortex pairs (VAP)2–10 while in onedimensional (1D) wires the resistance is caused by thermally activated phase slips (TAPS).11–19 One important
difference between these two types of fluctuations is that
in 2D a topological phase transition takes place, i.e. the
Berezinskii-Kosterlitz-Thouless (BKT) transition, while
in 1D only a crossover occurs. The BKT transition takes
place at a finite temperature, below which the VAP are
bound and the sample resistance is zero. On the contrary,
in 1D wires the resistance is always greater than zero because the TAPS has a finite probability at any T > 0.11
These 1D fluctuations are described by the theory of
Langer-Ambegaokar-McCumber-Halperin (LAMH).12,13
In this Communication we present an experimental
study of the interplay between TAPS and VAP types
of fluctuations in structures with a mixed dimensionality. For this purpose we fabricate and measure a series of
thin superconducting films interrupted by constrictions
or “bridges” (Fig. 1a). The width of the bridges is in the
range 10-30 nm, a few times larger than the coherence
length (ξ(0) ≈ 7 nm in MoGe). Two resistive transitions
are observed in such samples, indicating that VAP and
TAPS are two independent processes. For T > TBKT
the contribution of VAPs is dominant. Below TBKT the
transport properties are well described by the LAMH theory. Using different techniques we have measured the resistance variation over eleven orders of magnitude, without detecting any significant deviation from the LAMH.
Before presenting experimental results we give a brief
review of the BKT and LAMH theories. The BKT theory applies to 2D films and predicts a universal jump in
the superfluid density at the characteristic temperature
TBKT , which is lower than the mean field critical temperature of the film, Tc0 . The voltage-current characteristics
are described by a power law via V ∼ I α . For T < TBKT
the VAP are bound and therefore the V (I) curves are
nonlinear with α ≥ 3, meaning that the zero-bias resistance R(T ) is zero. At T > TBKT one expects α = 1,
corresponding to the occurrence of a non-zero resistance
FIG. 1: (a) Sample schematic. MoGe film (black) of thickness d = 2.5–4.5 nm is deposited over a SiN membrane (gray)
substrate with a constriction of width w. (b) An SEM micrograph of a typical sample. The MoGe coated SiN bridge
(gray) is suspended over a deep trench (black).
R(T ) > 0. Above TBKT the resistance of a film is given
by the Halperin-Nelson (HN) formula6,8
h p
i
RHN = 10.8bRn,f exp −2 b(Tc0 − TBKT )/(T − TBKT )
(1)
where Rn,f is the normal state resistance of the film and
b is a non-universal constant. Note that RHN (TBKT ) = 0.
The LAMH theory1 applies to narrow superconducting channels in which thermal fluctuations can cause
phase slips, i.e. jumps in the phase difference of the
order parameter by 2π. Thermal activations of the
system over a free energy barrier ∆F occur at a rate
given by Ω(T )e−∆F/kT , where the attempt frequency
Ω(T ) = (L/ξ(T ))(∆F/kT )1/2 (τGL )−1 is inversely proportional to the relaxation time of the time-dependent
Ginzburg-Landau (GL) theory τGL = 8k(Tc − T )/π~ (L
is the wire length, ξ(T ) is the GL coherence length, and
Tc is the mean field critical temperature of the wire).13
17
The free energy
√ barrier for2 a single phase slip is given
by ∆F = (8 2/3)(Hc (T ) /8π)(Aξ(T )), which is essentially the condensation energy density multiplied by the
volume Aξ(T ) of a phase slip (where A is the wire’s
cross sectional area). A current I causes a nonzero
voltage (averaged over the rapid phase slip processes)
2
Sample
A1
B1
B2
C1
C2
w (nm) RN (Ω) Tc (K) Tc0 (K) d (nm)
27 ± 4
1380
3.88
3.90
2.5
13 ± 4
1650
4.80
4.91
3.5
28 ± 4
1320
4.81
4.91
3.5
13 ± 4
1440
5.16
5.50
4.5
27 ± 4
680
5.39
5.50
4.5
β
1.47
0.723
1.10
0.653
2.21
TABLE I: Sample parameters, including the width of the
constriction (w), normal resistance of the bridge (RN ), critical temperature (determined from RWL (T ) fits) of the bridge
(Tc ), critical temperature of the film (Tc0 ), film thickness (d),
and a geometrical fitting parameter (β).
V = (~Ω(T )/e)e−∆F/kT sinh(I/I0 ) where I0 = 4ekT /h.
In the limit of low currents I ¿ I0 Ohm’s law is recovered RLAMH (T ) ≡ V /I ≈ Rq (~Ω(T )/kT )e−∆F/kT where
Rq = h/(2e)2 .
We turn now to the experiment. The sample geometry is shown schematically in Fig. 1a. The fabrication
is done starting with a Si wafer covered with SiO2 and
SiN films. A suspended SiN bridge is formed using electron beam lithography, reactive ion etching, and HF wet
etching.20 The bridge and the entire substrate is then
sputter-coated with amorphous Mo79 Ge21 superconducting alloy, topped with a 2 nm overlayer of Si for protection (MoGe was dc sputtered while Si was rf sputtered; the sputtering system was equipped with a cold
trap; base pressure was 10−7 torr). The resulting bridges
are 100 nm long with a minimum width w ≈ 11–28 nm
as measured with a scanning electron microscope (SEM)
(Fig. 1(b)). All samples are listed in Table I.
Transport measurements19 have been performed in a
pumped 4 He cryostat equipped with a set of rf-filtered
leads. The linear resistance R(T ) was determined from
the low-bias slope of the voltage vs. current V (I) curves.
The high-bias differential resistance was measured using
an ac excitation on top of a dc current offset generated
by a low-distortion function generator (SRS-DS360) connected in series with a 1 MΩ resistor. One sample was
measured down to the mΩ level using a low temperature
transformer (LTT) manufactured by Cambridge Magnetic Refrigeration (CMR).
Compare first two samples with different geometries:
the first is simply a thin MoGe film (“film” sample) and
the second is the same type of film but with a hyperbolic
constriction (“bridge” sample). Both are fabricated on
the same substrate simultaneously. A resistive transition
measured on the film sample is shown in Fig. 2. The
HN fit generated by Eq. 1 is shown as a solid line and
exhibits a good agreement with the data. The fit gives
TBKT = 4.8 K, which is slightly less than Tc0 = 4.91 K.
The inset of Fig. 2 compares the R(T ) measurements
of the “film” (open circles) and the “bridge” (solid line)
samples. At T = 4.8 K the R(T ) curve for the film sample
crosses the R = 0 axis with a finite slope, confirming the
occurrence of the BKT transition at this point. Nevertheless, unlike the film sample, the bridge sample shows a
FIG. 2: Resistance vs. temperature dependence (open circles)
for a thin film (d = 3.5 nm) without constriction. The solid
line is a fit to the HN theory (Eq. 1). (Inset): Resistance of
the film without constriction (multiplied by a constant factor), shown as open circles, is compared to the sample with a
hyperbolic bridge, shown by a solid line.
nonzero resistance even below the BKT transition. Such
resistive tails, occurring at T < TBKT , have been found
in all samples with constrictions. Below we analyze these
tails in details and demonstrate that they are produced
by the phase slip events localized on the constrictions and
effectively independent from the adjacent films.
In Fig. 3 the R(T ) curves for five samples with bridges
are plotted in a log-linear format. The resistance of the
sample B1 has been measured down to mΩ range using an LTT. Two transitions are seen in each curve as
the temperature decreases. As explained above, the first
transition is the BKT transition when the films become
superconducting. The second “transition” is in fact the
resistive tail mentioned earlier. In order to understand
its origin it should be compared to the LAMH theory.
Such comparison is not straightforward since LAMH
results apply to long uniform 1D wires, not short bridges.
In particular, the attempt frequency Ω(T ) depends on
two factors that are not relevant to a bridge. It has a term
L/ξ(T ) that accounts for the number of independent sites
where a phase slip can occur. Since each of our samples
has only one narrow spot where a TAPS event can happen, this number equals unity. Similarly the coefficient
(∆F/kT )1/2 , which takes into account possible overlaps
of TAPS at different places along the wire, equals unity
for short bridges. So the attempt frequency for a bridge
is simply ΩWL = 1/τGL . Since exact expression for resistance is unknown, we approximate the resistance of a
constriction (weak link) as
RWL (T ) = Rn e−∆FWL /kT .
(2)
The exponential factor here is that of LAMH and the
prefactor is simply the normal resistance of the bridge.12
3
FIG. 3: Resistive transitions for five samples with bridges
(Table I). Open symbols are the experimental results. Solid
lines are fits to Eq. 3.
The Eq. 2 allows a simple interpretation: The duration of
a single phase slip (i.e. the time it takes for the order parameter to recover) equals τGL and the number of phase
slips occurring per second is ΩWL (T ) exp [−∆F (T )/kT ],
with ΩWL = 1/τGL . Therefore the fraction of time in
which the constriction is populated with a phase slip
(i.e. when the gap is suppressed) is the product of these
two values, i.e. f = (τGL )(1/τGL ) exp [−∆F (T )/kT ] =
exp [−∆F (T )/kT ]. Following Little,11 it can be assumed
that the bridge has the normal state resistance Rn during
the time when a phase slip is present. The resistance is
zero otherwise; thus we arrive at the averaged resistance
R = f Rn + (1 − f ) × 0 = Rn exp [−∆F/kT ] as in Eq. 2.
In order to compare Eq. 2 with the experimental results, an explicit expression for ∆FWL (T ) is re17
quired. Starting with the usual
√ form 2derived for a long
1D wire, ∆FLAM H (0) = (8 2/3)(Hc (0)/8π)Aξ(0) =
0.83kTc Rq L/Rn ξ(0) where L is the length of the wire,
and using Rn = ρn L/A, the
√ free energy barrier for a
weak link is ∆FWL (0) = (8 2/3)(Hc2 (0)/8π)βwdξ(0) =
0.83kTc Rq βwd/ρn ξ(0), where w is the width of the
bridge, d is the film thickness, and ρn is the normal
resistivity. The parameter β measures the ratio of the
phase slip length along the bridge to the length of a
phase slip in a 1D wire. Finally, assuming the same
temperature dependence of the barrier as in LAMH, i.e.
∆F (T ) = ∆F (0)(1 − T /Tc )3/2 , we arrive at the expression for a constriction resistance induced by thermal fluctuations as
"
µ
¶3/2 #
T
Tc
βwdRq
1−
(3)
RWL (T ) = Rn exp −0.83
ρn ξ(0)
Tc
T
The fits generated by Eq. 3 are shown in Fig. 3 as solid
lines. A very good agreement is found for all five sam-
ples. In particular, sample B1 measured with an LTT,
shows an agreement with the predicted resistance RWL
over about seven orders of magnitude. Only two fitting
parameters are used: β and Tc (listed in Table I). The
other parameters required in Eq. 3, including Rn , d, w,
ξ(0), and ρn ≈ 180 µΩ cm are known.18,21 Such good
agreement confirms the expectation that the dissipation
in a thin film with a constriction at T < TBKT is solely
due to thermal activation of phase slips localized on the
constriction. As expected, β ≈ 1 for all samples. An
observed trend, however, is that larger values of β are
found for wider constrictions.
We now discuss nonlinear properties of films with constrictions. Measurements of the differential resistance
versus DC bias current dV (I)/dI are shown in Fig. 4.
Using these results it is possible to distinguish between
the BKT mechanism, which leads to a power-law V (I)
dependence, and the LAMH mechanism, which gives an
exponential V (I). From Fig. 4 it is clear that an exponential rather than a power law dependence characterizes samples with constrictions (at T < TBKT and sufficiently low currents). Thus it is appropriate to compare the measurements with the LAMH theory which
predicts dV /dI = R(T ) cosh(I/I0 ), where R(T ) is the
zero-bias resistance. Using this relation, we fit the differential resistance data as shown in Fig. 4 for different
temperatures. A good agreement is observed, confirming
that the dissipation is due to LAMH type phase slippage
events occurring in the bridge. No indication of enhanced
VAP breaking near the bridge is found.
It is remarkable that by fitting high bias dV (I)/dI
FIG. 4: Differential resistance as a function of the DC
bias current for sample B2. Experimental data are denoted
by open symbols and the solid lines are fits to dV /dI =
R(T ) cosh(I/I0 ). Temperatures from left to right are 4.12,
3.92, 3.80, 3.64, 3.45, 3.36, 3.26, 3.16, 3.07, 2.80 and 2.68 K.
4
FIG. 5: Resistance vs. temperature for sample B2. Open circles represent a direct measurement while filled circles give the
resistance values determined by fitting the dV /dI curves using
formula dV /dI = R(T ) cosh(I/I0 ). The solid and the dashed
curves give the best fits generated by the RWL (T ) (Tc = 4.81
K) and RLAMH (T ) (Tc = 5.38 K) formulas respectively.
curves with a simple LAMH expression dV /dI =
R(T ) cosh(I/I0 ), it is possible to extract the zero-bias resistance R(T ) down to as low as ≈ 10−8 Ω. This method
was systematically applied to sample B2 (Fig. 5). Fig. 5
shows R(T ) measured directly (open circles) and the resistance determined as the best fitting parameter to the
dV (I)/dI curves (solid circles). These two sets of data
agree with each other. The solid curve in Fig. 5 is a RWL
fit obtained from Eq. 3. An excellent agreement is seen
in a wide range of resistances spanning eleven orders of
1
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magnitude. This re-confirms that the TAPS mechanism
is dominant in the bridge samples for T < TBKT . The
usual LAMH expression RLAMH (discussed above) can
also be used to fit the data. The fit (dashed curve in
Fig. 5) demonstrates a good agreement with experiment
as well as with the RWL fit. The only problem is that the
usual form of LAMH (dashed curve) gives an unreasonably high critical temperature for the bridge, Tc = 5.38 K
in this case, which is even higher than Tc0 = 4.91 K
for the film itself. Note that the original LAMH theory gives Ω(Tc ) = 0 and consequently RLAMH (Tc ) = 0
(Fig. 5, dashed line), which is unphysical. On the other
hand, the Tc extracted from the fits with Eq. 3 are almost equal and slightly lower than the film Tc0 (Table I)
as expected. Thus our modifications made to the LAMH
formulae appear justified for short bridges. We remark
that some of measured bridges were wider than ξ(0), yet
the TAPS model produces good fits. This is because the
criterion for a superconducting channel to be regarded as
one-dimensional is that its width must be w . 4.4ξ(T )
(Ref. 12, p. 510), which is true for all our samples.
In conclusion we have shown that the LAMH theory, with slight modifications, describes the behavior of
short and rather wide bridges. Good agreement between
the theory and the experiment is observed over eleven
decades of resistance. A contribution of VAPs is observed
only above the BKT transition temperature in thin film
electrodes. No amplification of vortex-antivortex pair
fluctuations in the vicinity of the bridge was detected.
A new method of resistance determination is suggested.
We thank P. Goldbart and M. Fisher for suggestions.
This work was supported by the NSF carrier Grant No.
DMR-01-34770 and PHY-0243675, the Alfred P. Sloan
Foundation, and the Center for Microanalysis of Materials (UIUC), which is partially supported by the U.S.
Department of Energy Grant No. DEFG02-91-ER45439.
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