Evidence of Spatially Inhomogeous Pairing on the Insulating Side of a Disorder-Tuned
Superconductor-Insulator Transition
K. H. Sarwa B. Tan, Kevin A. Parendo, and A. M. Goldman
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
(Date textdate; Received textdate; Revised textdate; Accepted textdate; Published textdate)
Abstract
Measurements of transport properties of amorphous insulating InxOy thin films have been interpreted as evidence of the
presence of superconducting islands on the insulating side of a disorder-tuned superconductor-insulator transition. Although
the films are not granular, the behavior is similar to that observed in granular films. The results support theoretical models in
which the destruction of superconductivity by disorder produces spatially inhomogenous pairing with a spectral gap.
The interplay between localization and superconduc-
tivity can be investigated through studies of disordered
superconducting films [1], originally treated by Anderson
[2], and Abrikosov and Gor’kov [3], who considered the
low-disorder regime. Several approaches have been pro-
posed for strong disorder, including fermionic mean field
theories [4, 5, 6] and theories that focus on the univer-
sal critical properties near the superconductor-insulator
transition. The latter consider the transition to belong
to the dirty boson universality class [7]. When quantum
fluctuations are included in fermionic theories for high
levels of disorder a spatially inhomogeneous pairing am-
plitude is found which retains a nonvanishing spectral
gap [8]. For sufficiently disordered systems inhomoge-
neous pairing can also be brought about by thermal fluc-
tuations [9]. A similar inhomogeneous regime has also
been considered under the rubric of electronic microemul-
sions in the context of the metal-insulator transition of
two dimensional electron gases [10]. In this letter we pro-
vide evidence of a spatially inhomogeneous order param-
eter on the insulating side of a superconductor-insulator
transition driven by structural and/or chemical disorder.
Studies of disorder and magnetic field tuned
superconductor-insulator transitions have usually been
carried out on films that are either amorphous or gran-
ular. In the former the disorder is on an atomic scale,
and in the latter, on a mesoscopic scale in which case
the films consist of metallic grains or clusters connected
by tunneling, that are either embedded in an insulating
matrix, or on a bare substrate [1]. Amorphous films can
be produced when films of metal atoms such as Pb or Bi
are grown at liquid helium temperatures on substrates
precoated with a wetting layer of amorphous Ge or Sb
[11], or by careful vapor deposition of MoxGey, InxOy,
or TiN using a variety of techniques.
Granular films, are known to develop superconductiv-
ity in stages. If the grains are small and weakly con-
nected, the film is an insulator. For grains larger than
some characteristic size, and sufficiently close together,
“local superconductivity” will develop below some tem-
perature. The opening of a spectral gap in the density
of states of the grains [12] results in a relatively sharp
upturn in the resistance below this temperature, which
is usually close to the transition temperature of the bulk
material. For well enough coupled grains, there may be
a small drop in resistance at that temperature, followed
by this upturn. This is in contrast with the “global su-
perconductivity” that occurs when a sufficient fraction of
the grains or clusters are strongly enough Josephson cou-
pled to form a percolating superconducting path across
the film.
We have measured the temperature and magnetic field
dependence of the resistance, and nonlinear conductance-
voltage characteristics of amorphous InxOy films pre-
pared by electron-beam evaporation. These films are
not granular, but nevertheless exhibit local supercon-
ductivity at the lowest temperatures. At low temper-
atures, the application of a magnetic field results in a
dramatic rise in resistance exhibiting a maximum that
with decreasing temperatures is found at decreasingly
small fields. The conductance-voltage characteristics in
this high resistance regime are nonlinear in a manner
suggestive of single-particle tunneling between supercon-
ductors. We argue that these observations are consistent
with the presence of droplets, or islands of superconduc-
tivity, characterized by a nonvanishing superconducting
pair amplitude and coupled by tunneling. Many of the
droplets are Josephson coupled, but their density is not
high enough to produce a superconducting path across
the film.
The 22 nm thick films used in this study were deposited
at a rate of 0.4 nm/s by electron beam evaporation onto
(001) SrTiO3 epi-polished single crystal substrates. Plat-
inum electrodes, 10 nm in thickness, were deposited prior
to growth. The starting material was 99.999 % pure
In2O3. A shadow mask defined a Hall bar geometry in
which the effective area for four-terminal resistance mea-
surements was 500 x 500 µm2. As-grown films exhibited
sheet resistances of about 4600 Ω at room temperature
and about 23 kΩ at 10 K. By annealing at relatively low
temperatures (55-70 ◦C) in a high vacuum environment
(10−7 Torr), film resistances were lowered, and depending
upon the annealing time either insulating or supercon-
ducting behavior at low temperatures could be induced
Typeset by REVTEX 1
http://arxiv.org/abs/0704.0765v2
FIG. 1: Resistance vs. temperature for Films 1 and 2.
[13]. Low-temperature rather than high-temperature an-
nealing avoids changes in morphology that would result
in granular or microcrystalline films. As reported by
Gantmakher, et al. [14], at room temperature the re-
sistances of annealed films were found to be unstable.
However, at low temperatures (40-1400 mK) and in vac-
uum, they were stable. The films of the present study
had resistances of 2600 Ω at room temperature.
Film structure was studied using atomic force mi-
croscopy (AFM), X-ray diffraction (XRD) analysis, and
high resolution scanning electron microscopy (SEM).
From the XRD there was no indication of the presence
of crystalline In or In2O3. The SEM did not detect any
In inclusions, and could be correlated with AFM stud-
ies which revealed for a 22 nm thick film, roughness in
the form of surface features with a height of 8.5 nm, and
with bases about 18 nm in diameter. The conclusion
from these characterization efforts is that the films were
homogeneous and amorphous, and did not contain iso-
lated grains or In inclusions.
Measurements were carried out in an Oxford Kelvinox-
25 dilution refrigerator housed in a screen room, with
measuring leads filtered at room temperature using π-
section filters and RC filters. For measurements of resis-
tance, the applied current was set in the range of 10-100
pA, to avoid the possibility of heating. Figure 1 shows a
plot of R (T ) for two films which were studied in detail.
For each, dR/dT is negative at the lowest temperatures.
In the case of Film 1 there is a local minimum in R(T )
at about 350 mK. Both films exhibit a sharp upturn in
R(T ) between 200 and 300 mK, with the effects to be
discussed below, occurring for Film 1 at higher tempera-
tures than for Film 2. These behaviors are suggestive of
a regime of local superconductivity [12].
The sheet resistances of Films 1 and 2 were both ap-
proximately 78 kΩ at 40 mK. In a perpendicular magnetic
field of only 0.2 T, their sheet resistances increased by up
to a factor of 40 at 40 mK. The maximum in R(B) as
shown in Fig. 2(a) for Film 1 is followed, at the lowest
temperatures, by a relatively slow decrease in resistance
with increasing field. The resistance maximum moves to
higher fields, with increasing temperature. The behavior
of Film 2 resembled the higher temperature data for Film
1, presumably because Film 2 exhibited weaker traces of
superconductivity as evidenced by the absence of a lo-
cal minimum in R(T ) in the zero-field. This variation in
properties from film to film is expected, as small changes
in chemistry and/or morphology can have a large effect
on disordered film properties. The temperature depen-
dencies of the fields, Bpeak and resistances Rpeak are pre-
sented in Fig. 2(b). A qualitatively similar, but weaker
enhancement of resistance was previously reported for in-
sulating InxOy films by Gantmakher and coworkers [14].
A larger enhancement was reported for ultrathin insulat-
ing Be thin films [15]. However neither of these works
demonstrated the systematic effects shown in Fig. 2(b).
To probe the nature of the high resistance state, differ-
ential conductance-voltage characteristics were also stud-
ied [15, 16, 17, 18, 19, 20]. These are shown in Fig. 3
for Film 2 which was studied in detail. Film 1 exhibited
qualitatively similar features.
All of the nonlinear conductance-voltage characteris-
tics are reminiscent of the single-particle tunneling char-
acteristics of superconductor-insulator-superconductor
(SIS) tunneling junctions. The effects of electron heating
are found at voltages well above the observed conduc-
tance thresholds [21]. The fact that the low voltage non-
linearities vanish at temperatures above approximately
200 mK, suggests that they are associated with the pres-
ence of a nonvanishing pairing amplitude occurring in the
disconnected superconducting regions.
We can model these thin disordered films exhibiting
spatially inhomogeneous pairing as random networks of
tunneling junctions of various (random) levels of conduc-
tivity, connecting superconducting clusters imbedded in
an insulating matrix. Some of these junctions are su-
perconducting because they are Josephson coupled. As
a consequence there are “superclusters,” which are ag-
gregates of Josephson coupled smaller clusters that may
cover a macroscopic fraction of the film area. Charge
may flow through “superclusters” with zero electrical re-
sistance. However, as long as these do not span the film
the resistance will be determined by single-particle tun-
neling. The sheet resistance of the resultant network can
be inferred using the following simple argument [22]. Dis-
connect all of the junctions in the network whose conduc-
tance involves single particle tunneling, and then recon-
nect them one by one in ascending order of resistance.
A stage will be reached at which the next junction com-
pletes an infinite cluster connecting the ends of the net-
work. Let the normal state resistance of this last junction
FIG. 2: (a) Resistance vs. magnetic field for Film 1. The
temperatures are 40 (top), 80, 100, 120, 130, 140, 150, 170,
180, 200, 230, 250, 300, 350, 400, and 500 mK (bottom).
(b)The fields (left axis) and the resistances (right axis) of the
peaks in R(B) are plotted as a function of temperature. The
flattening of Rpeak at the lowest temperatures may be the
result of a failure to cool the electrons.
FIG. 3: Differential conductance vs. voltage of Film 2 at 100
mK for 0 (top), 0.01, 0.02, 0.03, 0.04, 0.06, 0.1, 0.175, 0.25,
0.5, and 1 T (bottom).
be Rn. The actual value will depend upon the nature of
the distribution of single-particle tunneling resistances in
the film. The measured normal-state sheet resistance of
the entire network will then be Rn, as this junction is the
bottleneck. Junctions with R > Rn are irrelevant since
they are always shunted by junctions with resistances
of order Rn. Junctions with R < Rn only form finite
clusters over macroscopic distances. They don’t affect
the conductivity because current must still pass through
junctions with resistance of order Rn to get from one
“supercluster” to the next. The action of a magnetic
field is to quench the Josephson coupling within the “su-
perclusters.” When this happens, the resistance at each
temperature will be governed by the new, higher, value
of the bottleneck resistance as there will no longer be any
Josephson-coupled “superclusters,” and the distribution
of junction resistances will shift towards higher values of
resistance.
The fact that the magnetic field that induces higher
resistance decreases with decreasing temperature is a
counter-intuitive result, implying a divergent magnetic
length scale in the zero temperature limit, possibly of
the form [φ0/B]
where φ0 is the flux quantum. This re-
sult suggests enhanced quantum fluctuations in the zero
temperature limit. A heuristic argument can be made
to demonstrate that this is plausible by treating the in-
homogeneous pairing state of the film as a collection of
superconducting grains or islands coupled by tunneling
junctions. Without the inclusion of quantum fluctuations
the argument may not capture all of the features of the
data. It is first useful to consider the magnetic field de-
pendence of the in-plane Josephson coupling between two
planar thin film square islands with an area L2. This is
a geometry resembling the grain boundary geometry of
high temperature superconductor junctions. A magnetic
field applied perpendicular to the plane will completely
penetrate both electrodes of such a junction. As a conse-
quence the minima of the diffraction pattern will be gov-
erned by the field corresponding to a single flux quantum
over the full area of the structure or B [L(2L+ d)] = φ0
where d is the width of the barrier or gap [23]. Since
L >> d, the field at the first minimum of the diffrac-
tion pattern would be found at a value proportional to
−2 . For a “supercluster” consisting of a square ar-
ray of square islands that are Josephson coupled, with
some degree of randomness in the coupling, one would
expect coherence to vanish at the first minimum. For a
random array and with clusters that are not square, one
might expect a similar dependence on L−2. If the char-
acteristic size of the islands increased with decreasing
temperature, which is a plausible assumption, the field
quenching the Josephson coupling, would be expected to
decrease as is observed. For the films studied, at the
lowest temperatures, the peak in the resistance occurs in
a field of 0.2 T, which would correspond to a length of
approximately 100 nm.
The fall off of the resistance at fields above that pro-
ducing the maximum slows with decreasing temperature,
consistent with the strengthening of the pairing ampli-
tude with decreasing temperature. The fact that this
remnant of superconductivity persists to fields up to 12
T, far above the bulk critical field of InxOy, implies that
the superconducting islands are much smaller than the
penetration depth. It should be possible to develop a
detailed percolation model of this effect, similar to that
developed for granular superconductors [24], which in-
cludes the quenching of the Josephson coupling by mag-
netic field and quantum fluctuations.
Although the resistance of the films of the present work
increases with decreasing temperatures in zero magnetic
field there is no guarantee that at some temperature lower
than the minimum value accessed in these measurements,
the resistance will not fall to zero. This could result
from the percolation of Josephson coupling across the
film as the size of the clusters increases. In that event
the inhomogeneous pairing implied by the data would be
more likely governed by a theory including both quantum
[8] and thermal [9] fluctuations.
The large peaks in the magnetoresistance found at
fields above the magnetic field-induced superconductor-
insulator transition of superconducting amorphous
InxOy thin films may result from a similar inhomogeneity
of the pairing amplitude, in that case induced by mag-
netic field rather than disorder. Such peaks were first re-
ported by Hebard and Paalanen [25] who suggested that
the state induced when superconductivity was quenched
was a Bose insulator, characterized by localized Cooper
pairs. They proposed that the peak was the signature of
a crossover to a Fermi insulating state of localized elec-
trons. This resistance peak has been the subject of more
recent studies of InxOy films [14, 16, 26], of microcrys-
talline TiN films where the high field limit appears to be
a “quantum metal” [18], and of high-Tc superconductors
[27]. The fact that inhomogeneous pairing can be in-
duced in disordered superconductors by magnetic fields
has been recently established using a Hubbard Model
[28].
The notion that disorder implies inhomogeneity of su-
perconducting order on some length scale was first dis-
cussed by Kowal and Ovadyahu [29], and as was men-
tioned earlier emerges naturally in a fermionic model of
the superconductor-insulator transition that exhibits a
disorder-tuned inhomogeneity of the pairing amplitude
[8]. The films of Kowal and Ovadyahu differ from those
of the present work in that they are presumably more
disordered, and thus further into the insulating regime.
Their magnetoresistance is always negative as there is no
Josephson coupling between islands and the main effect
of magnetic field is to weaken the inhomogeneous pairing
amplitude, leading to negative magnetoresistance.
This work was supported by the National Science
Foundation under grant no. NSF/DMR-0455121. The
authors would like to thank Zvi Ovadyahu for providing
samples and for critical comments, and Leonid Glazman
and Alex Kamenev for useful discussions.
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Phys. Rev. Lett. 98, 127003 (2007).
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http://arxiv.org/abs/cond-mat/0510422

  Measurements of transport properties of amorphous insulating indium oxide
thin films have been interpreted as evidence of the presence of superconducting
islands on the insulating side of a disorder-tuned superconductor-insulator
transition. Although the films are not granular, the behavior is similar to
that observed in granular films. The results support theoretical models in
which the destruction of superconductivity by disorder produces spatially
inhomogenous pairing with a spectral gap.

Evidence of Spatially Inhomogeous Pairing on the Insulating Side of a Disorder-Tuned
Superconductor-Insulator Transition
K. H. Sarwa B. Tan, Kevin A. Parendo, and A. M. Goldman
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
(Date textdate; Received textdate; Revised textdate; Accepted textdate; Published textdate)
Abstract
Measurements of transport properties of amorphous insulating InxOy thin films have been interpreted as evidence of the
presence of superconducting islands on the insulating side of a disorder-tuned superconductor-insulator transition. Although
the films are not granular, the behavior is similar to that observed in granular films. The results support theoretical models in
which the destruction of superconductivity by disorder produces spatially inhomogenous pairing with a spectral gap.
The interplay between localization and superconduc-
tivity can be investigated through studies of disordered
superconducting films [1], originally treated by Anderson
[2], and Abrikosov and Gor’kov [3], who considered the
low-disorder regime. Several approaches have been pro-
posed for strong disorder, including fermionic mean field
theories [4, 5, 6] and theories that focus on the univer-
sal critical properties near the superconductor-insulator
transition. The latter consider the transition to belong
to the dirty boson universality class [7]. When quantum
fluctuations are included in fermionic theories for high
levels of disorder a spatially inhomogeneous pairing am-
plitude is found which retains a nonvanishing spectral
gap [8]. For sufficiently disordered systems inhomoge-
neous pairing can also be brought about by thermal fluc-
tuations [9]. A similar inhomogeneous regime has also
been considered under the rubric of electronic microemul-
sions in the context of the metal-insulator transition of
two dimensional electron gases [10]. In this letter we pro-
vide evidence of a spatially inhomogeneous order param-
eter on the insulating side of a superconductor-insulator
transition driven by structural and/or chemical disorder.
Studies of disorder and magnetic field tuned
superconductor-insulator transitions have usually been
carried out on films that are either amorphous or gran-
ular. In the former the disorder is on an atomic scale,
and in the latter, on a mesoscopic scale in which case
the films consist of metallic grains or clusters connected
by tunneling, that are either embedded in an insulating
matrix, or on a bare substrate [1]. Amorphous films can
be produced when films of metal atoms such as Pb or Bi
are grown at liquid helium temperatures on substrates
precoated with a wetting layer of amorphous Ge or Sb
[11], or by careful vapor deposition of MoxGey, InxOy,
or TiN using a variety of techniques.
Granular films, are known to develop superconductiv-
ity in stages. If the grains are small and weakly con-
nected, the film is an insulator. For grains larger than
some characteristic size, and sufficiently close together,
“local superconductivity” will develop below some tem-
perature. The opening of a spectral gap in the density
of states of the grains [12] results in a relatively sharp
upturn in the resistance below this temperature, which
is usually close to the transition temperature of the bulk
material. For well enough coupled grains, there may be
a small drop in resistance at that temperature, followed
by this upturn. This is in contrast with the “global su-
perconductivity” that occurs when a sufficient fraction of
the grains or clusters are strongly enough Josephson cou-
pled to form a percolating superconducting path across
the film.
We have measured the temperature and magnetic field
dependence of the resistance, and nonlinear conductance-
voltage characteristics of amorphous InxOy films pre-
pared by electron-beam evaporation. These films are
not granular, but nevertheless exhibit local supercon-
ductivity at the lowest temperatures. At low temper-
atures, the application of a magnetic field results in a
dramatic rise in resistance exhibiting a maximum that
with decreasing temperatures is found at decreasingly
small fields. The conductance-voltage characteristics in
this high resistance regime are nonlinear in a manner
suggestive of single-particle tunneling between supercon-
ductors. We argue that these observations are consistent
with the presence of droplets, or islands of superconduc-
tivity, characterized by a nonvanishing superconducting
pair amplitude and coupled by tunneling. Many of the
droplets are Josephson coupled, but their density is not
high enough to produce a superconducting path across
the film.
The 22 nm thick films used in this study were deposited
at a rate of 0.4 nm/s by electron beam evaporation onto
(001) SrTiO3 epi-polished single crystal substrates. Plat-
inum electrodes, 10 nm in thickness, were deposited prior
to growth. The starting material was 99.999 % pure
In2O3. A shadow mask defined a Hall bar geometry in
which the effective area for four-terminal resistance mea-
surements was 500 x 500 µm2. As-grown films exhibited
sheet resistances of about 4600 Ω at room temperature
and about 23 kΩ at 10 K. By annealing at relatively low
temperatures (55-70 ◦C) in a high vacuum environment
(10−7 Torr), film resistances were lowered, and depending
upon the annealing time either insulating or supercon-
ducting behavior at low temperatures could be induced
Typeset by REVTEX 1
http://arxiv.org/abs/0704.0765v2
FIG. 1: Resistance vs. temperature for Films 1 and 2.
[13]. Low-temperature rather than high-temperature an-
nealing avoids changes in morphology that would result
in granular or microcrystalline films. As reported by
Gantmakher, et al. [14], at room temperature the re-
sistances of annealed films were found to be unstable.
However, at low temperatures (40-1400 mK) and in vac-
uum, they were stable. The films of the present study
had resistances of 2600 Ω at room temperature.
Film structure was studied using atomic force mi-
croscopy (AFM), X-ray diffraction (XRD) analysis, and
high resolution scanning electron microscopy (SEM).
From the XRD there was no indication of the presence
of crystalline In or In2O3. The SEM did not detect any
In inclusions, and could be correlated with AFM stud-
ies which revealed for a 22 nm thick film, roughness in
the form of surface features with a height of 8.5 nm, and
with bases about 18 nm in diameter. The conclusion
from these characterization efforts is that the films were
homogeneous and amorphous, and did not contain iso-
lated grains or In inclusions.
Measurements were carried out in an Oxford Kelvinox-
25 dilution refrigerator housed in a screen room, with
measuring leads filtered at room temperature using π-
section filters and RC filters. For measurements of resis-
tance, the applied current was set in the range of 10-100
pA, to avoid the possibility of heating. Figure 1 shows a
plot of R (T ) for two films which were studied in detail.
For each, dR/dT is negative at the lowest temperatures.
In the case of Film 1 there is a local minimum in R(T )
at about 350 mK. Both films exhibit a sharp upturn in
R(T ) between 200 and 300 mK, with the effects to be
discussed below, occurring for Film 1 at higher tempera-
tures than for Film 2. These behaviors are suggestive of
a regime of local superconductivity [12].
The sheet resistances of Films 1 and 2 were both ap-
proximately 78 kΩ at 40 mK. In a perpendicular magnetic
field of only 0.2 T, their sheet resistances increased by up
to a factor of 40 at 40 mK. The maximum in R(B) as
shown in Fig. 2(a) for Film 1 is followed, at the lowest
temperatures, by a relatively slow decrease in resistance
with increasing field. The resistance maximum moves to
higher fields, with increasing temperature. The behavior
of Film 2 resembled the higher temperature data for Film
1, presumably because Film 2 exhibited weaker traces of
superconductivity as evidenced by the absence of a lo-
cal minimum in R(T ) in the zero-field. This variation in
properties from film to film is expected, as small changes
in chemistry and/or morphology can have a large effect
on disordered film properties. The temperature depen-
dencies of the fields, Bpeak and resistances Rpeak are pre-
sented in Fig. 2(b). A qualitatively similar, but weaker
enhancement of resistance was previously reported for in-
sulating InxOy films by Gantmakher and coworkers [14].
A larger enhancement was reported for ultrathin insulat-
ing Be thin films [15]. However neither of these works
demonstrated the systematic effects shown in Fig. 2(b).
To probe the nature of the high resistance state, differ-
ential conductance-voltage characteristics were also stud-
ied [15, 16, 17, 18, 19, 20]. These are shown in Fig. 3
for Film 2 which was studied in detail. Film 1 exhibited
qualitatively similar features.
All of the nonlinear conductance-voltage characteris-
tics are reminiscent of the single-particle tunneling char-
acteristics of superconductor-insulator-superconductor
(SIS) tunneling junctions. The effects of electron heating
are found at voltages well above the observed conduc-
tance thresholds [21]. The fact that the low voltage non-
linearities vanish at temperatures above approximately
200 mK, suggests that they are associated with the pres-
ence of a nonvanishing pairing amplitude occurring in the
disconnected superconducting regions.
We can model these thin disordered films exhibiting
spatially inhomogeneous pairing as random networks of
tunneling junctions of various (random) levels of conduc-
tivity, connecting superconducting clusters imbedded in
an insulating matrix. Some of these junctions are su-
perconducting because they are Josephson coupled. As
a consequence there are “superclusters,” which are ag-
gregates of Josephson coupled smaller clusters that may
cover a macroscopic fraction of the film area. Charge
may flow through “superclusters” with zero electrical re-
sistance. However, as long as these do not span the film
the resistance will be determined by single-particle tun-
neling. The sheet resistance of the resultant network can
be inferred using the following simple argument [22]. Dis-
connect all of the junctions in the network whose conduc-
tance involves single particle tunneling, and then recon-
nect them one by one in ascending order of resistance.
A stage will be reached at which the next junction com-
pletes an infinite cluster connecting the ends of the net-
work. Let the normal state resistance of this last junction
FIG. 2: (a) Resistance vs. magnetic field for Film 1. The
temperatures are 40 (top), 80, 100, 120, 130, 140, 150, 170,
180, 200, 230, 250, 300, 350, 400, and 500 mK (bottom).
(b)The fields (left axis) and the resistances (right axis) of the
peaks in R(B) are plotted as a function of temperature. The
flattening of Rpeak at the lowest temperatures may be the
result of a failure to cool the electrons.
FIG. 3: Differential conductance vs. voltage of Film 2 at 100
mK for 0 (top), 0.01, 0.02, 0.03, 0.04, 0.06, 0.1, 0.175, 0.25,
0.5, and 1 T (bottom).
be Rn. The actual value will depend upon the nature of
the distribution of single-particle tunneling resistances in
the film. The measured normal-state sheet resistance of
the entire network will then be Rn, as this junction is the
bottleneck. Junctions with R > Rn are irrelevant since
they are always shunted by junctions with resistances
of order Rn. Junctions with R < Rn only form finite
clusters over macroscopic distances. They don’t affect
the conductivity because current must still pass through
junctions with resistance of order Rn to get from one
“supercluster” to the next. The action of a magnetic
field is to quench the Josephson coupling within the “su-
perclusters.” When this happens, the resistance at each
temperature will be governed by the new, higher, value
of the bottleneck resistance as there will no longer be any
Josephson-coupled “superclusters,” and the distribution
of junction resistances will shift towards higher values of
resistance.
The fact that the magnetic field that induces higher
resistance decreases with decreasing temperature is a
counter-intuitive result, implying a divergent magnetic
length scale in the zero temperature limit, possibly of
the form [φ0/B]
where φ0 is the flux quantum. This re-
sult suggests enhanced quantum fluctuations in the zero
temperature limit. A heuristic argument can be made
to demonstrate that this is plausible by treating the in-
homogeneous pairing state of the film as a collection of
superconducting grains or islands coupled by tunneling
junctions. Without the inclusion of quantum fluctuations
the argument may not capture all of the features of the
data. It is first useful to consider the magnetic field de-
pendence of the in-plane Josephson coupling between two
planar thin film square islands with an area L2. This is
a geometry resembling the grain boundary geometry of
high temperature superconductor junctions. A magnetic
field applied perpendicular to the plane will completely
penetrate both electrodes of such a junction. As a conse-
quence the minima of the diffraction pattern will be gov-
erned by the field corresponding to a single flux quantum
over the full area of the structure or B [L(2L+ d)] = φ0
where d is the width of the barrier or gap [23]. Since
L >> d, the field at the first minimum of the diffrac-
tion pattern would be found at a value proportional to
−2 . For a “supercluster” consisting of a square ar-
ray of square islands that are Josephson coupled, with
some degree of randomness in the coupling, one would
expect coherence to vanish at the first minimum. For a
random array and with clusters that are not square, one
might expect a similar dependence on L−2. If the char-
acteristic size of the islands increased with decreasing
temperature, which is a plausible assumption, the field
quenching the Josephson coupling, would be expected to
decrease as is observed. For the films studied, at the
lowest temperatures, the peak in the resistance occurs in
a field of 0.2 T, which would correspond to a length of
approximately 100 nm.
The fall off of the resistance at fields above that pro-
ducing the maximum slows with decreasing temperature,
consistent with the strengthening of the pairing ampli-
tude with decreasing temperature. The fact that this
remnant of superconductivity persists to fields up to 12
T, far above the bulk critical field of InxOy, implies that
the superconducting islands are much smaller than the
penetration depth. It should be possible to develop a
detailed percolation model of this effect, similar to that
developed for granular superconductors [24], which in-
cludes the quenching of the Josephson coupling by mag-
netic field and quantum fluctuations.
Although the resistance of the films of the present work
increases with decreasing temperatures in zero magnetic
field there is no guarantee that at some temperature lower
than the minimum value accessed in these measurements,
the resistance will not fall to zero. This could result
from the percolation of Josephson coupling across the
film as the size of the clusters increases. In that event
the inhomogeneous pairing implied by the data would be
more likely governed by a theory including both quantum
[8] and thermal [9] fluctuations.
The large peaks in the magnetoresistance found at
fields above the magnetic field-induced superconductor-
insulator transition of superconducting amorphous
InxOy thin films may result from a similar inhomogeneity
of the pairing amplitude, in that case induced by mag-
netic field rather than disorder. Such peaks were first re-
ported by Hebard and Paalanen [25] who suggested that
the state induced when superconductivity was quenched
was a Bose insulator, characterized by localized Cooper
pairs. They proposed that the peak was the signature of
a crossover to a Fermi insulating state of localized elec-
trons. This resistance peak has been the subject of more
recent studies of InxOy films [14, 16, 26], of microcrys-
talline TiN films where the high field limit appears to be
a “quantum metal” [18], and of high-Tc superconductors
[27]. The fact that inhomogeneous pairing can be in-
duced in disordered superconductors by magnetic fields
has been recently established using a Hubbard Model
[28].
The notion that disorder implies inhomogeneity of su-
perconducting order on some length scale was first dis-
cussed by Kowal and Ovadyahu [29], and as was men-
tioned earlier emerges naturally in a fermionic model of
the superconductor-insulator transition that exhibits a
disorder-tuned inhomogeneity of the pairing amplitude
[8]. The films of Kowal and Ovadyahu differ from those
of the present work in that they are presumably more
disordered, and thus further into the insulating regime.
Their magnetoresistance is always negative as there is no
Josephson coupling between islands and the main effect
of magnetic field is to weaken the inhomogeneous pairing
amplitude, leading to negative magnetoresistance.
This work was supported by the National Science
Foundation under grant no. NSF/DMR-0455121. The
authors would like to thank Zvi Ovadyahu for providing
samples and for critical comments, and Leonid Glazman
and Alex Kamenev for useful discussions.
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http://arxiv.org/abs/cond-mat/0510422
