Cooper-pair Insulator Research Articles

Origin of topological order in a Cooper-pair insulator

We unveil the microscopic origin of the topologically ordered counterpart of the s-wave superconductor in this work. For this, we employ the recently developed unitary renormalisation group (URG) method on a generalised model of 2D electrons attractive interactions. The effective Hamiltonian obtained at the stable low-energy fixed point of the RG flow corresponds to a gapped, insulating state of quantum matter we call the Cooper pair insulator (CPI). We show that the CPI ground state manifold displays several signatures of topological order, including a four-fold degeneracy when placed on the torus. Spectral flow arguments reveal the emergent gauge-theoretic structure of the effective Hamiltonian, as it can be written entirely in terms of non-local Wilson loops. It also contains a topological $\theta$-term whose coefficient is quantised, in keeping with the requirement of invariance of the ground state under large gauge transformations. We find that the long-ranged many-particle entanglement content of the CPI ground state is driven by inter-helicity two-particle scattering processes. Analysis of the passage from CPI to BCS superconducting ground state shows the RG flow promotes fluctuations in the number of condensed Cooper pairs and lowers those in the conjugate global phase. Consequently, the distinct signatures of long-ranged entanglement in the CPI are replaced by the well-known short-ranged entanglement of the BCS state. Finally, we study the renormalisation of the entanglement in $k$-space for both the CPI and BCS ground states. The topologically ordered CPI state is shown to possess an emergent hierarchy of scales of entanglement, and that this hierarchy collapses in the BCS state. Our work offers clear evidence for the microscopic origins of topological order in this prototypical system, and lays the foundation for similar investigations in other systems of correlated electrons.

Corrigendum to “Effects of Ion Beam Irradiation on Nanoscale InOx Cooper-Pair Insulators”

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Nonequilibrium Second-Order Phase Transition in a Cooper-Pair Insulator.

In certain disordered superconductors, upon increasing the magnetic field, superconductivity terminates with a direct transition into an insulating phase. This phase is comprised of localized Cooper pairs and is termed a Cooper-pair insulator. The current-voltage characteristics measured in this insulating phase are highly nonlinear and, at low temperatures, exhibit abrupt current jumps. Increasing the temperature diminishes the jumps until the current-voltage characteristics become continuous. We show that a direct correspondence exists between our system and systems that undergo an equilibrium, second-order, phase transition. We illustrate this correspondence by comparing our results to the van der Waals equation of state for the liquid-gas mixture. We use the similarities to identify a critical point where an out of equilibrium second-order-like phase transition occurs in our system. Approaching the critical point, we find a power-law behavior with critical exponents that characterizes the transition.

Parity effect and single-electron injection for Josephson junction chains deep in the insulating state

We have made a systematic investigation of charge transport in one-dimensional chains of Josephson junctions where the characteristic Josephson energy is much less than the single-junction Cooper-pair charging energy, ${E}_{J}\ensuremath{\ll}{E}_{CP}$. Such chains are deep in the insulating state, where superconducting phase coherence across the chain is absent, and a voltage threshold for conduction is observed at the lowest temperatures. We find that Cooper-pair tunneling in such chains is completely suppressed. Instead, charge transport is dominated by tunneling of single electrons, which is very sensitive to the presence of BCS quasiparticles on the superconducting islands of the chain. Consequently, we observe a strong parity effect, where the threshold voltage vanishes sharply at a characteristic parity temperature ${T}^{*}$, which is significantly lower than the critical temperature ${T}_{c}$. A measurable and thermally activated zero-bias conductance appears above ${T}^{*}$, with an activation energy equal to the superconducting gap, confirming the role of thermally excited quasiparticles. Conduction below ${T}^{*}$ and above the voltage threshold occurs via injection of single electrons/holes into the Cooper-pair insulator, forming a nonequilibrium steady state with a significantly enhanced effective temperature. Our results explicitly show that single-electron transport dominates deep in the insulating state of Josephson junction arrays. This conduction process has mostly been ignored in previous studies of both superconducting junction arrays and granular superconducting films below the superconductor-insulator quantum phase transition.

Effects of Ion Beam Irradiation on NanoscaleInOxCooper-Pair Insulators

This paper examines the effects of irradiating indium oxide films of nanoscale thickness by ion beams, when these films are in the Cooper-pair insulator state. Radiation effects are predicted on the basis of Monte Carlo simulations of ion transport. Results of numerical experiments are interpreted within the theoretical model of a Cooper-pair insulator. The study suggests that radiation-induced changes inInOxfilms exposed to ion beams could significantly alter their current-voltage characteristics and that a transition to a metallic state is possible, due to radiation-induced perturbation of the fine-tuned granular structure. Furthermore, incident and displaced ions can break up enough Cooper pairs inInOxfilms to cause dissolution of this specific insulating state.

Cooper pair insulator phase induced in amorphous Pb0.9Bi0.1 thin films

A Cooper pair insulator (CPI) phase emerges near the superconductor–insulator transitions of a number of strongly-disordered thin film systems. Much recent study has focused on a mechanism driving the underlying Cooper pair localization. We present data showing that a CPI phase develops in amorphous Pb0.9Bi0.1 films deposited onto nano-porous anodized aluminum oxide surfaces just as it has been shown to develop for a-Bi films. This result confirms the assertion that this CPI phase emerges due to the structure of the substrate. It supports the picture that nanoscale film thickness variations induced by the substrate drive the localization. Moreover, it implies that the CPI phase can be induced in any superconducting material that can be deposited onto this surface.

Cooper pair islanding model of insulating nanohoneycomb films

We first review evidence for the Cooper pair insulator (CPI) phase in amorphous nanohoneycomb (NHC) films. We then extend our analysis of superconducting islands induced by film thickness variations in NHC films to examine the evolution of island sizes through the magnetic field-driven SIT. Finally, using the islanding picture, we present a plausible model for the appearance and behavior of the CPI phase in amorphous NHC films.

Radiation hardness of indium oxide films in the Cooper-pair insulator state

This paper investigates possible radiation effects in the Cooper-pair insulator state of indium oxide films. Radiation effects are predicted on the basis of Monte Carlo simulations. Results of a combined theoretical and numerical analysis suggest that radiation-induced changes in the investigated films could significantly affect their current-voltage characteristics, and that a transition to a metallic state is possible, due to radiation-induced disruption of the fine-tuned granular structure. Dissociation of Cooper pairs, caused by both the incident radiation and the ions displaced within InOx films, can also destroy the conditions for this specific insulating state to subsist.

Cooper-pair insulator phase in superconducting amorphous Bi films induced by nanometer-scale thickness variations

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.

Cooper pair insulators and theory of correlated superconductors

The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the $s$-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or $\mathit{XY}$ universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple $s$-wave picture.

Pair density wave instability and Cooper pair insulators in gapped fermion systems

By analyzing simple models of fermions in lattice potentials we argue that the zero-temperature pairing instability of any ideal band-insulator occurs at a finite momentum. The resulting supersolid state is known as "pair density wave". The pairing momentum at the onset of instability is generally incommensurate as a result of phase-space restrictions and relative strengths of interband and intraband pairing. However, commensurate pairing occurs in the strong-coupling limit and becomes a Cooper-channel analogue of the Halperin-Rice exciton condensation instability in indirect bandgap semiconductors. The exceptional sensitivity of incommensurate pairing to quantum fluctuations can lead to a strongly-correlated insulating regime and a non-BCS transition, even in the case of weak coupling as shown by an exact renormalization group analysis.

Observation of Giant Positive Magnetoresistance in a Cooper Pair Insulator

Ultrathin amorphous Bi films, patterned with a nanohoneycomb array of holes, can exhibit an insulating phase with transport dominated by the incoherent motion of Cooper pairs (CP) of electrons between localized states. Here, we show that the magnetoresistance (MR) of this Cooper pair insulator (CPI) phase is positive and grows exponentially with decreasing temperature T, for T well below the pair formation temperature. It peaks at a field estimated to be sufficient to break the pairs and then decreases monotonically into a regime in which the film resistance assumes the T dependence appropriate for weakly localized single electron transport. We discuss how these results support proposals that the large MR peaks in other unpatterned, ultrathin film systems disclose a CPI phase and provide new insight into the CP localization.

Instanton Glass Generated by Noise in a Josephson-Junction Array

We compute the correlation function of a superconducting order parameter in a continuous model of a two-dimensional Josephson-junction array in the presence of a weak Gaussian noise. When the Josephson coupling is large compared to the charging energy, the correlations in the Euclidian space decay exponentially at low temperatures regardless of the strength of the noise. We interpret such a state as a collective Cooper-pair insulator and argue that it resembles properties of disordered superconducting films.

Evidence of local superconductivity in granular Bi nanowires fabricated by electrodeposition

An unusual enhancement of resistance (i.e., a ``superresistivity'') below a certain characteristic temperature ${T}_{\text{sr}}$ was observed in granular Bi nanowires. This ``superresistive'' state was found to be dependent on the applied magnetic field as well as the excitation current. The suppression of ${T}_{\text{sr}}$ by magnetic field resembles that of a superconductor. The observed superresistivity appears to be related to the nucleation of local superconductivity inside the granular nanowire without long-range phase coherence. The phenomenon is reminiscent of the ``Cooper-pair insulator'' observed previously in ultrathin two-dimensional superconducting films and three-dimensional percolative superconducting films.

Superinsulator and quantum synchronization

Synchronized oscillators are ubiquitous in nature, and synchronization plays a key part in various classical and quantum phenomena. Several experiments have shown that in thin superconducting films, disorder enforces the droplet-like electronic texture--superconducting islands immersed into a normal matrix--and that tuning disorder drives the system from superconducting to insulating behaviour. In the vicinity of the transition, a distinct state forms: a Cooper-pair insulator, with thermally activated conductivity. It results from synchronization of the phase of the superconducting order parameter at the islands across the whole system. Here we show that at a certain finite temperature, a Cooper--air insulator undergoes a transition to a superinsulating state with infinite resistance. We present experimental evidence of this transition in titanium nitride films and show that the superinsulating state is dual to the superconducting state: it is destroyed by a sufficiently strong critical magnetic field, and breaks down at some critical voltage that is analogous to the critical current in superconductors.