Granular superconductivity occurs when microscopic superconducting grains are separated by non-superconducting regions through which they communicate by Josephson tunneling to establish the macroscopic superconducting state [1]. Although crystals of the cuprate high-Tc superconductors are not granular in a structural sense, theory indicates that at low hole densities the holes can become concentrated at some locations resulting in hole-rich superconducting domains [2-5]. Granular superconductivity due to Josephson tunneling through 'undoped' regions between such domains would represent a new paradigm for the underdoped cuprates. Here we report studies of the spatial interrelationships between STM tunneling spectra in underdoped Bi2Sr2CaCu2O8+x. They reveal an apparent spatial segregation of the electronic structure into ~3nm diameter domains (with superconducting characteristics and local energy gap delta 50(2.5) meV. These observations suggest that underdoped Bi2Sr2CaCu2O8+x is a mixture of two different short-range electronic orders with the long-range characteristics of a granular superconductor.

Imaging the granular structure of high-Tc superconductivity in underdoped Bi2Sr2CaCu2O8+x

Quantum phase transitions and vortex dynamics in superconducting networks

Quantum phenomena include the fractional quantum Hall effect, whose quasiparticle is the composite fermion. Theorists show that composite fermions possess different quantum numbers than the electrons or holes they were derived from.

https://link.aps.org/pdf/10.1103/PhysRevX.5.031027

Is the Composite Fermion a Dirac Particle

As a superconducting thin film becomes disordered and subject to an increasing magnetic field, a point is reached when it undergoes a transition from a superconducting to an insulating state. Dubi et al. use numerical simulations to study this transition — or, as it turns out, two types of transition. The key effect of disorder is to create 'islands' of strong superconductivity, coupled by regions that are only weakly superconducting. In the case of weak disorder, an increasing magnetic field eventually destroys the superconducting state throughout the material, leading to an insulator. When disorder is strong, superconductivity persists in the islands, and the effect of a magnetic field is to suppress the coupling between them — again leading to an insulating state. These findings may be relevant to the high-temperature superconductors, where intrinsic disorder may play a role. As a superconducting thin film is made more disordered, or subject to an increasing magnetic field, a point is reached when the material undergoes a transition from a superconducting to an insulating state. Numerical simulations have been used to shed light on the nature of this transition, or, as it turns out, two types of transition. The interplay of superconductivity and disorder has intrigued scientists for several decades. Disorder is expected to enhance the electrical resistance of a system, whereas superconductivity is associated with a zero-resistance state. Although superconductivity has been predicted to persist even in the presence of disorder1, experiments performed on thin films have demonstrated a transition from a superconducting to an insulating state with increasing disorder or magnetic field2. The nature of this transition is still under debate, and the subject has become even more relevant with the realization that high-transition-temperature (high-Tc) superconductors are intrinsically disordered3,4,5. Here we present numerical simulations of the superconductor–insulator transition in two-dimensional disordered superconductors, starting from a microscopic description that includes thermal phase fluctuations. We demonstrate explicitly that disorder leads to the formation of islands where the superconducting order is high. For weak disorder, or high electron density, increasing the magnetic field results in the eventual vanishing of the amplitude of the superconducting order parameter, thereby forming an insulating state. On the other hand, at lower electron densities or higher disorder, increasing the magnetic field suppresses the correlations between the phases of the superconducting order parameter in different islands, giving rise to a different type of superconductor–insulator transition. One of the important predictions of this work is that in the regime of high disorder, there are still superconducting islands in the sample, even on the insulating side of the transition. This result, which is consistent with experiments6,7, explains the recently observed huge magneto-resistance peak in disordered thin films8,9,10 and may be relevant to the observation of ‘the pseudogap phenomenon’ in underdoped high-Tc superconductors11,12.

Nature of the superconductor-insulator transition in disordered superconductors.

The investigation of superconductivity in the presence of disorder began 60 years ago with the work of Alexander Shal'nikov at the Institute for Physical Problems in Moscow. The subject has played an ongoing role in condensed matter physics over the years. Interest has recently been heightened by the possibility that the disorder‐driven or magnetic‐field‐driven quenching of superconductivity in systems at the limit of zero temperature and two dimensions might be quantum phase transitions. That would link the physics of the superconductor‐insulator transition in thin films to other systems believed to exhibit quantum phase transitions—for example, helium‐4 in porous media, high temperature superconductors, Josephson‐junction arrays, two‐dimensional electron gases and various spin systems.

Superconductor‐Insulator Transitions in the Two‐Dimensional Limit

We report the first direct observation of the cyclotron motion of new quasiparticles in a 2D electron system near 1/2 filling factor. Just as electrons in a metal geometrically resonate with a sound wave when a magnetic field is applied perpendicular to the sound propagation direction, near 1/2 filling factor we observe resonance of a surface sound wave with cyclotron orbits of charge carriers. The presence of a Fermi surface and geometric resonance of a sound wave with Chern-Simons gauge transformed fermions were explicitly predicted in a recent theory of the half filled Landau level

Experimental demonstration of a Fermi surface at one-half filling of the lowest Landau level.

Surface-acoustic-wave propagation on high-quality AlGaAs/GaAs heterostructures containing a two-dimensional electron system has been examined in the fractional quantum Hall effect (FQHE) regime. The response of the electron system to the sound wave is found to be similar in the FQHE states to that previously studied in the integral quantum Hall states. However, a striking disparity is observed at Landau-level filling \ensuremath{\nu}=1/2, where sound propagation is distinctly different from that in the neighboring filling-factor range. We propose that phase separation of the 2DEG may be responsible for the feature at \ensuremath{\nu}=1/2.

Anomalous sound propagation at ν=1/2 in a 2D electron gas: Observation of a spontaneously broken translational symmetry\?

Temperature and magnetic-field-dependent measurements of the longitudinal ${\mathit{R}}_{\mathit{x}\mathit{x}}$ and transverse ${\mathit{R}}_{\mathit{x}\mathit{y}}$ resistance of amorphous indium oxide thin-film superconductors reveal the presence of distinct insulating phases for sufficiently high disorder and/or magnetic field. For field-swept transitions at fixed disorder and low temperatures there is a critical field where ${\mathit{R}}_{\mathit{x}\mathit{x}}$ diverges and the superconductor is transformed into a Bose-glass insulator with localized Cooper pairs. At higher fields there is a second critical field where ${\mathit{R}}_{\mathit{x}\mathit{y}}$ appears to diverge, the pairs unbind, and localized single electrons characterizing a Fermi-glass insulator dominate.

Low-temperature insulating phases of uniformly disordered two-dimensional superconductors

High-frequency surface acoustic waves (g2 GHz, \ensuremath{\lambda}2 \ensuremath{\mu}m) have been propagated on high mobility ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As/GaAs heterostructures, allowing measurement of the finite-wave-vector conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{x}}$(q,\ensuremath{\omega}). We find enhanced ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{x}}$(q,\ensuremath{\omega}) at multiple even-denominator filling factors \ensuremath{\nu}=1/2, 3/2, 1/4, and 3/4 with ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{x}}$(q,\ensuremath{\omega}) increasing linearly in q. The enhanced ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{x}}$(q,\ensuremath{\omega}) at \ensuremath{\nu}=1/2 persists to high temperatures (g4 K), above the temperatures at which the fractional quantum Hall effect is observed. Empirically, the effects represent a series of robust states in two-dimensional electron systems that support gapless excitations. These data are strikingly consistent with a recent theory describing Fermi-surface formation at even denominator \ensuremath{\nu}.

R. R. Ruel

Papers

Experimental demonstration of a Fermi surface at one-half filling of the lowest Landau level.

Anomalous sound propagation at ν=1/2 in a 2D electron gas: Observation of a spontaneously broken translational symmetry\?

Low-temperature insulating phases of uniformly disordered two-dimensional superconductors

Enhanced finite-wave-vector conductivity at multiple even-denominator filling factors in two-dimensional electron systems.

Experimental observation of a two-dimensional Heisenberg nuclear ferromagnet