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## Strongly Disordered Superconductors and Electronic Segregation |

I will make a brief introduction into the
history of macroscopic quantum tunneling (MQT) in superconducting devices. Then
I will present our results on MQT in superconducting nanowires. The
superconductor-insulator transition will also be discussed.
While incoherent slips of superconducting phase in quasi-1d
wires are known for more than 50 years, the experimental evidence for coherent
ones is just emerging. To facilitate experimental work in this direction, we
consider simple "devices" where we account for coherent phase slips
at phenomenological level. Interestingly, the phase slips provide isolation of
charge in the wires, resulting in observable Coulomb blockade effects. The non-linearities found in an oscillator where superconducting
inductance is subject to coherent phase slips, oscillate as a function of
number of photons N with a period of the order of square root of N, which is
the "width" of the coherent state. We prove that such non-linearities result in multiple metastable
states encompassing few photons and study oscillatory dependence of various
responses of the resonator. The experimental realization of our proposals can
deliver an unambiguous verification of coherent quantum phase slips. We also preview some new results concerning the better
microscopic estimation of phase-slip amplitude and syncronization
of charge and phase oscillations.
The mean-field formalism of the electrodynamic response of superconductors has been in place
almost as long as the BCS theory itself. It is believed to capture the physics
of conventional superconductors quite accurately. In contrast, strongly
disordered superconductors are dominated by order parameter fluctuations and inhomogeneity effects where the mean-field theory is only
the roughest of guides. In this tutorial style talk I will review our
understanding of the electrodynamic response of
strongly disordered superconductors. I will start from the canonical mean-field
response and then extend the treatment to strongly disordered systems. In
particular I will concentrate on what we learn about fluctuation phenomena, the
role of inhomogeneities, dissipation, and the role of
coherence in these systems.
Phase-slips in superconducting nano-wires have been studied for a long time. However,
direct observation of quantum phase-slip is still a very challenging
experimental task. One possibility to realize it is to make a superconducting
loop with such a wire, which plays role of a coupler between two circulating
currents. At the degeneracy point between the circulating currents, a finite
phase-slip rate is expected to lift the degeneracy, forming a two-level system
that is a phase-slip qubit. We work on fabrication
and measurements of the phase-slip qubit and have
observed its signature. Namely, transmission spectroscopy shows a weak signal
from a two-level system dependent on magnetic field with expected energy
dependence. The energy splitting at the degeneracy point is found to be
approximately 3 GHz. Although the effect still has to be confirmed, the results
require intensive scientific discussion.
O. Crauste, F.
Couëdo,
We report on the study of the
Superconductor-to-Insulator Transition (SIT) in Nb
Three qualitatively different Insulator
to Superconductor Quantum Phase Transitions (ISTs) have been observed in
ultrathin quench condensed film systems. Amorphous Bi films deposited onto
Aluminum Oxide Nano-HoneyComb (NHC) substrates
undergo an IST with increasing thickness that can be distinguished from the
ISTs of quench condensed homogeneous and granular films on smooth
substrates. Like the granular films, the NHC film insulator consists of
localized Cooper pairs implying that superconducting order parameter phase
fluctuations dominate these ISTs. The insulator of the homogeneous films,
on the other hand, consists of unpaired electrons implying that fluctuations in
the amplitude of the superconducting order parameter dominate near their
IST. Unlike granular films, transport in NHC insulators is dominated by
Cooper pair tunneling rather than quasi-particle tunneling. I will
describe how morphological differences correlate with these 3 distinct IST
behaviors. I will also describe how the magnetic field responses of both
insulating and superconducting films vary with morphology.
Liquid/solid interfaces are attracting growing interest not only
for applications in catalytic activities and energy storage, but also for their
new electronic functions in electric double-layer transistors (EDLTs)
exemplified by high-performance organic electronic field-induced electronic phase transitions, as well as
superconductivity in SrTiO3. Broadening EDLTs to induce superconductivity
within other materials is highly demanded for enriching the materials science
of superconductors. However, it is severely hampered by inadequate choice of
materials and processing techniques. Here we introduce an easy method using
ionic liquids as gate dielectrics, mechanical micro-cleavage techniques for
surface preparation, and report the observation of field- induced
superconductivity showing a transition temperature Tc
= 15.2 K on an atomically flat film of layered nitride compound, ZrNCl. The present result reveals that the EDLT is an
extremely versatile tool to induce electronic phase transitions by
electrostatic charge accumulation and provides new routes in the search for
superconductors beyond those synthesized by traditional chemical methods. A.Grockowiak,
Short after the discovery of
superconductivity in MgB2, a superconducting transition was established in
Boron doped diamond and silicon showing that covalent semi-conductors could be
a efficient starting point to obtain superconductivity. In diamond,
superconductivity appears right at the (doping induced) metal-insulator
transition (MIT) but the critical temperature (Tc)
remains "abnormally" high down to the critical density. On the other
hand, the onset of superconductivity in silicon occurs for Boron concentration
several orders of magnitude above the critical value of the MIT but we show
that that Tc smoothly decreases with doping and does
not follow the exponential decay expected in a standard BCS model. On the
contrary we show that Tc roughly scales as \lambda^2
where \lambda is the electron-pnonon coupling
constant which can been deduced from ab-initio
calculations.
In a conventional superconductor, it is normally believed that
the transition from a superconducting to a normal state is caused by quasiparticle excitations alone and fluctuation
is the phase of the order parameter plays a negligible role on the
superconducting state. In this talk, I will review our experimental
investigations using scanning tunneling spectroscopy, penetration depth and
transport measurements, on two situations when phase fluctuations become
important in a conventional superconductor. The first situation deals with very
thin films of NbN, with thickness less than the
superconducting coherence length. The superconducting transition is these
2-dimensional films are governed by the Kosterlitz-Thouless-Berizinski
(KTB) transition, where the superconducting state is destroyed through a
proliferation of vortex-antivortex pairs. I will show
that while KTB theory provides a consistent description of the variation
of superfluid density below T
References: 1. A. Kamlapure et al., Appl. Phys. Lett. 96, 072509 (2010). 2. M. Mondal et al., Phys. Rev. Lett. 106, 047001 (2011).
Scanning tunnelling
microscopy and spectroscopy in the mK
range give simultaneously access to very high spatial and energy resolution.
This tool is therefore very well suited for studying any inhomogeneous superconducting
states. Indeed, the spatial variation of the local density of states of a
superconductor and its related superconducting gap can be measured at the mesoscopic scale and eventually down to the atomic scale. A
brief introduction to this technique will be given and illustrated with the
help of several different situations such as the superconducting proximity
effect and the Abrikosov vortex array of
superconductors in the mixed state. Highly disordered superconducting
ultra-thin films provide another situation of inhomogeneous superconductivity
which has been more recently unveiled. The case of TiN
thin films close to the superconductor-insulator transition will be described
and discussed in more details [1,2]. [1] B.
Sacépé, et al., Physical Review Letters 101, 157006 (2008). [2] B.
Sacépé, et al., Nature.
The concept of localized Cooper pairs in disordered
superconducting films close to the Superconductor-Insulator Transition (SIT) has intrigued scientists for several decades both
theoretically and experimentally. Although the interplay of localization and
superconductivity has been clearly evidenced on the macroscopic scale by
transport measurements, very little is known about the microscopic details of
the strong disorder limit. In this talk I will present tunneling
measurements of the local density-of-states on amorphous indium oxide films
close to the SIT [1]. Our results show that disorder fluctuations lead to a
mixture of superconducting and insulating regions that distinguish themselves
by the presence or absence of coherence peaks at the gap edges. Besides, using
our STM, we have continuously analyzed the local conductance between the tunneling regime and the point-contact regime. In the
latter, Andreev-Sant-James spectroscopy reveals a new
energy scale related to the superconducting coherence energy and independent
from spatial fluctuations of the pairing energy. This finding as well as other
striking anomalies will be discussed regarding recent theories of
superconductivity close to the mobility edge that interpret such a gapped state
without coherence peak as the spectral signature of localized Cooper pairs. [1] B. Sacépé,
The study of the superconductor-insulator transition has, in
recent years, experienced a shift from the initial focus on the physics of the
transition to a broader look at the properties of the insulator terminating
superconductivity as well as superconductivity itself in the presence of strong
disorder. I will attempt to review these developments focusing on results that
are in common to more than one set of materials. I will also try to point out
what I believe are the main open questions in our growing field.
I will review basic point of the approach developed recently
(Annals of Physics 325,1 (2010)) to describe Cooper pairing of electrons whose
single-particle eigenstates are nearly critical (in the
sense of the Anderson transition), whereas
Coulomb repulsion is assumed to be weak. Fractal nature of the eigenfunctions near Anderson mobility edge leads to strong
spatial fluctuations of the order parameter and to
I review the data and theoretical models of the disorder driven
superconductor-insulator transition in which Cooper pairs remain intact in the
insulating phase. I show that the remarkable feature of this transition is
apparent inhomogeneity of the state even in
homogeneously disordered materials. I discuss the properties of the insulating
state appearing at the transition, I argue that this state is characterized by
a small energy scale above which collective excitations are delocalized. As the
disorder is increased, the energy scale is increased and eventually becomes
infinite signaling the transition into the hard insulator.
In amorphous superconducting thin films of Nb0.15Si0.85 [1][2] and InOx [3][4], a finite
Nernst coefficient can be detected in a wide range of temperature and magnetic
field. Due to the negligible contribution of normal quasi-particles,
superconducting fluctuations easily dominate the Nernst response in the entire
range of study. In the vicinity of the critical temperature and in the
zero-field limit, the magnitude of the signal is in quantitative agreement with
what is theoretically expected for the Gaussian fluctuations of the
superconducting order parameter [5]. Even at higher temperatures and finite
magnetic field, the Nernst coefficient is set by the size of superconducting
fluctuations. The Nernst coefficient emerges as a direct probe of the ghost
critical field, the normal-state mirror of the upper critical field. Moreover,
upon leaving the normal state with fluctuating Cooper pairs, we show that the temperature
evolution of the Nernst coefficient is different whether the system enters a
vortex solid, a vortex liquid or a phase-fluctuating superconducting regime. [1] A. Pourret et al., Nature Physics
2, 683 - 686 (2006) [2] A. Pourret et
al., Phys. Rev. B.
76, 214504 (2007) [3] P. Spathis [4] A. Pourret et al., New Journal of
Physics 11, 055071 (2009) [5] I. Ussishkin, S. L. Sondhi and D. A. Huse, Phys. Rev.
Lett. 89, 287001 (2002)
We
analyze the recent experiments on giant I-V jumps in amorphous films of InO from the viewpoint of the model of overheated
electrons. The electron cooling rate in the localized regime and its dependence
on magnetic field and temperature is discussed. Minsoo Kim, Tailung Wu and
We present experimental results from transport
studies on amorphous indium oxide films that are driven through a
superconductor-insulator transition by applying a pair-breaking magnetic field.
The direction of the magnetic field is varied continuously from being
perpendicular to the film plane to parallel to the film plane and we identify
four distinct transport regimes when the film is rotated in a magnetic field.
We also study the evolution of these transport regimes as a function of the
disorder in the samples. Implications for our current understanding of the 2D
superconductor - insulator transition
We report a comprehensive study of the complex ac conductance of
amorphous superconducting InOx thin films. We measure
the explicit frequency dependency of the complex conductance and the phase
stiffness over a range from 0.21 to 15 GHz at temperatures down to 350 mK using a novel broadband
microwave Corbino spectrometer. The dynamic ac
measurements are sensitive to the temporal correlations of the superconducting
order parameter in the fluctuation range above Tc.
Among other aspects, we explicitly demonstrate the critical slowing down of the
characteristic fluctuation rate on the approach to the superconducting state
and show that its behavior is consistent with vortex-like phase fluctuations
and a phase-ordering scenario of the transition. If time allows I will discuss
our very recent results concerning microwave measurements across the 2D
superconductor insulator quantum phase transition. J. Paramanandam,
M. Bell, L.B. Ioffe, and
We have studied the phase transitions
induced by the magnetic field B in
arrays of Josephson junctions over a wide range of EJ/EC (EJ is the
Josephson energy, EC is the
charging energy). These unconventional arrays are characterized by a large
number of nearest-neighbor junctions connected to a single superconducting
island (typically 10). Due to the commensurability effects, the arrays
demonstrated several quantum phase transitions at different critical values of B, which is in line with earlier
observations. The critical resistance RC
for these transitions varied between 3 kΩ
and 20 kΩ. The transitions observed for RC < 5 kΩ
were consistent with the “dirty boson” scenario of the
superconductor-to-insulator transition (SIT). However, the duality was lacking
for the transitions observed at larger RC.
The activation energy E0,
extracted from the Arrhenius fitting of R(T) in the “insulating” regime, has
been compared with the quantity eVt, where Vt is the
“offset” voltage corresponding to the onset of a strong non-linearity of the
current-voltage characteristics (IVCs). Interestingly, we observed that E0 ≈ eVt for the arrays with sufficiently small values of E0. At a large current bias, the IVCs
demonstrate voltage steps 2/e indicating the onset of quasiparticle generation in the arrays. The critical power
corresponding to the threshold for quasiparticle generation
is temperature-independent below ~ 200 mK and weakly
dependent on the zero-bias resistance.
Various sample characteristics are known to exhibit mesoscopic fluctuations. In disordered superconductors, the
value of the local pairing field fluctuates across the sample which smears the
BCS peak in the density of states. We calculate the smearing of the peak and
show that it is enhanced by Coulomb repulsion, leading to strong fluctuations
in the vicinity of the quantum critical point where superconductivity is
totally suppressed by the “fermionic mechanism”.
The conductance of electron-glasses may be enhanced when exposed
to sufficiently strong electro-magnetic (EM) fields. In the microwaves (MW)
regime, this enhancement is an adiabatic effect. Data taken on several systems
will be shown to illustrate that this effect is generic and is closely related
to the non-ohmic feature commonly found in the dc
transport regime. However, in contrast with the lack of heating in the MW
regime, applying a large dc field exhibits a non-adiabatic component. A
systematic study of these effects as function of the EM frequency reveals a
crossover at a surprisingly low frequency. Possible implications of these
results to the issue of many-body-localization will be discussed.
Superinductance is a
quantum circuit element defined by its two key properties. First, it must superconduct direct current (DC). Second, it must present
to an alternating current (AC) the impedance of a frequency-independent
inductance L with sufficiently small stray capacitance C Remarkably, the experimental implementation and testing of a superinductance goes far beyond the scope of
radio-engineering. For instance, attempting to wind a regular wire, or even a
bulk superconducting wire, into a conventional coil will result in a failure,
which could be traced back to the small value of the fine structure constant.
Alternative attempts to increase the specific inductance by turning to highly
disordered superconductors would be limited by the quantum phase-slip and
eventually by some kind of superconductor-to-insulator transition. Furthermore,
in order to fully test a superinductance against
various failure modes, it is crucial to develop a setup to measure the superinductance with both DC and microwave excitations. Our implementation of a superinductance
consists of an array of Josephson tunnel junctions with properly chosen
parameters. Such array may be viewed as a special case of a highly disordered
but structurally continuous superconducting wire. We have developed two types
of measurement techniques to characterize our array-based superinductances,
which should be applicable to any other implementation. In the first technique we use the superinductance
as a semi-transparent "mirror" of a superconducting Fabry-Perot resonator, made of a conventional low kinetic
inductance superconductor. The quality factor of the resonance translates
directly into the value of the superinductance at the
resonance frequency, provided the internal dissipation inside the Fabry-Perot is negligible. This technique is nearly
insensitive to the dissipation inside the superinductance.
In the second technique we use a superinductance as a
key element of a superconducting fluxonium qubit and measure its transition frequencies and decoherence times. The former allows precise determination
of superinductance magnitude, while the latter reveals
information about the dissipation inside the superinductance
as well as the coherent quantum phase-slip frequency. Our techniques may prove useful in characterizing highly
disordered superconductors on the superconducting side of the superconductor to
insulator transition.
Influence of disorder on the temperature of
superconding transition (Tc)
is studied within the sigma-model renormalization group framework. Electron-electron
interaction in particle-hole and Cooper channels is taken into account and
assumed to be short-range. Two-dimensional systems in the weak
localization and antilocalization regime, as well as
systems near mobility edge are considered. It is shown that in all these
regimes the Anderson localization leads to strong enhancement of Tc related to the multifractal
character of wave functions.
As a prototype of
disordered superconductor we consider the attractive Hubbard model with on site
disorder. We solve the Bogoljubov-de-Gennes equations
on two-dimensional finite clusters at zero temperature and we evaluate the
electromagnetic response to a vector potential. We find that the standard
decoupling between transverse and
longitudinal response does not apply in the presence of disorder and that the superfluid density is strongly reduced by the relaxation of
the phase of the order parameter already at mean-field level when disorder is
large. We also find that the anharmonicity of the
phase fluctuations increases by increasing disorder. Going beyond mean-field,
this provides an enhancement of quantum fluctuations in producing a
zero-temperature transition to a non-superconducting phase. In the large U
limit, near half-filling, this transition is towards a phase of disordered
preformed pairs. Finally, by analyzing the distribution of the on-site order
parameter in the superconducting phase at large U we find anomalous tails like
those of a glassy (replica-symmetry breaking) phase recently suggested by Ioffe and Mezard for disorder
superconductors using a cavity method approach. References: L.Benfatto, G.Seibold, J.Lorenzana, and C.Castellani, “Superfluid density and phase relaxation in
superconductors with strong disorder
We study the electrodynamics of atomic
layer deposited and sputter deposited TiN films with
varying disorder. The films have kFℓ values between 2.6 and
11 characterizing the disorder. We measure the electromagnetic response of
these films using microwave resonators. The internal quality factor and the
shift in the resonance frequency as a function of temperature are probes for the
real and imaginary part of the complex conductivity. The response of the films
with low disorder can be described in the framework of Mattis-Bardeen,
provided that a broadening of the BCS density of states is introduced. For
increasing disorder, the needed broadening parameter is observed to increase.
This suggests that the stronger the disorder, the larger the deviation from the
conventional Mattis-Bardeen / BCS theory.
We theoretically propose
novel devices to illustrate the coherent quantum phase-slips (QPS): the QPS
oscillator, the QPS-box and QPS-transistor. The QPS oscillator can
be realized on the basis of a thin superconducting wire or a chain of Josephson
junctions1. It proves that the experimental detection of quantum phase slips is
achievable for small phase slip amplitudes, contrary to what is usually
assumed. The responses of this damped-driven oscillator exhibit a cosine
dependence on the charge induced by a gate electrode and very unusual
oscillatory dependence on the drive/frequency. The QPS-box and the
QPS-transistor are derived from the Cooper-pair box and Cooper-pair
transistor2. They exhibit sensitivity to a charge induced by a gate electrode,
this being the main signature of Coulomb blockade. Experimental realization of
such devices will prove the Coulomb blockade as an effect of coherence of QPS
processes. 1A.M. Hriscu, Y.V. Nazarov,
Phys. Rev. Lett. 106, 0
Ultrathin films near the quantum Insulator-Superconductor
Transition (IST) can exhibit Cooper pair transport in their insulating
state. This Cooper Pair Insulator (CPI) state is achieved in amorphous Bi
films evaporated onto substrates with a topography varying on lengths slightly
greater than the superconducting coherence length. We present evidence that
this topography induces film thickness and corresponding superconducting
coupling constant variations that promote Cooper pair island formation.
Analyses of many thickness-tuned ISTs show that weak links between
superconducting islands dominate the transport. In particular, the IST occurs
when the link resistance approaches the resistance quantum for pairs. These
results support conjectures that the CPI is an inhomogeneous state of matter.
Superconducting thin-_lm materials with a
high normal-state resistivity are of interest as building blocks for quantum
devices and radiation detectors. However, the very disordered nature of these
materials gives rise to a strong competition between electron localization and
superconductivity, leading to a superconductor-insulator transition with
increasing disorder1. Prior to this transition the electronic properties are
expected to become inhomogeneous even for uniform structural disorder. This
leads to the question how superconductivity manifests itself in these resistive
materials. We present measurements of the normal-to-superconducting transition
of NbTiN nanowires, with a
thick- ness of 8 nm, widths varying from 50 nm to 400
nm, and a normal-state resistivity of _ 160 _cm. Each width shows a smooth
superconductive transition at Tc = 10:5 K, consistent
with the Aslamazov-Larkin theory for two-dimensional
wires. Close to Tc however, measurements of the
critical current and the critical magnetic _eld of
the nanowires reveal that the resistive state is
reached in a series of steps, each adding a typical resistance of 5
10 k to the wire. Moreover, from the critical currents we obtain a higher
critical temperature than observed in the zero-bias resistive transition. From
this, we conclude that in a certain temperature regime the wire is resistive
while localized superconductivity is still present. 1For example Sacépé et al,
Nature Physics 7, 239 (2011)
In this
presentation I will describe a new theory of quantum phase-slips in1D
superconducting wires. This theory takes as its starting point the hypothesis
advanced by Mooij and Nazarov
[Nat. Phys. 2, 169 - 2006] that quantum phase-slips (tunneling of fluxoid quanta through narrow superconducting wires) and
Josephson tunneling of Cooper pairs are related through electromagnetic
duality. Just as Cooper pairs can tunnel through an energy barrier due to the
zero-point fluctuations associated with their finite mass (kinetic inductance),
I postulate that tunneling of fluxoid quanta through
an energy barrier (a narrow superconducting wire) arises due to a mass
associated with their electromagnetic momentum. The resulting kinetic energy is
associated with an electric field, effectively generated by moving flux, which
must charge up a capacitance. In this picture, the "kinetic
capacitance" (dual of the kinetic inductance) postulated by Mooij and Nazarov can be
connected directly with the real part of the electric permittivity (geometric
series capacitance) of the wire.
Building on this basic microscopic picture of quantum phase-slips, I
will describe a model for an extended nanowire where
polarization charge of the wire is dual to magnetic flux through a JJ barrier,
and quantum phase-slips occur coherently along the entire wire, just as Cooper pairs tunnel coherently through the entire barrier of a JJ.
This model predicts a new type of secondary quantum macroscopic effect, which I
call a type-II phase slip (intimately related to a vortex in a type-II
superconductor), that in very short 1D wires is dual
to the well-known Bloch oscillation in a Josephson junction. Excitation of
these type-II phase slips may provide a new explanation for phenomena that have
previously been identified with quantum phase slips in 1D superconducting
wires. The model also connects to the well-known LAMH model of thermal phase
slips near the critical temperature, and it provides a simple and intuitive
means to calculate the attempt rate for these phase slips. I will discuss what
this theory, if correct, would imply for the future prospects of observing
quantum phase-slips and for new types of superconducting electronic devices.
We consider fluctuation phenomena in dirty superconductors on
the basis of the superconducting sigma model [1, 2]. To begin, we reproduce
well-known results for longitudal fluctuational
conductivity. Next, we consider this effects in the
out-of-equilibrium conditions, and calculate fluctuating Hall effect, extending
existing theory [3] for arbitrary temperatures and magnetic fields. [1] A. Levchenko, A. Kamenev, Keldysh Ginzburg-Landau action of
fluctuating superconductors. Phys. Rev. B76 094518.
Localization effects on superconducting fluctuations are
investigated for 200-400 nm thick TiN films as a
function of disorder. The disorder is characterized with the product k
between 2.6 and 11.1, it is found that the resistivity increases and the free
electron density decreases with increasing disorder. _{F}ℓMeasurements on the superconducting transition at zero magnetic
field show that the resistivity is affected by superconducting fluctuations up
to 2T When a magnetic field is applied, the fluctuations are
increasingly suppressed, leading to an increased resistivity. At large fields
the resistivity saturates, sign of a characteristic field required to suppress
fluctuation effects. The width of the suppression region is constant with
sample temperature, but increases with disorder from about 0.6 T for k
= 3.7 at temperatures 0.1 K above T_{F}ℓ_{c}.
Together with the superconducting transition in zero field,_{ }this
signals a relation between fluctuation effects and disorder in TiN films.[Back] |