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Models for granular superconductivity: The effect of quantum fluctuations
Abstract
We develop two models for the onset of superconductivity in granular superconductors. The inclusion of a parallel shunt resistor between the grains constitutes the ohmic model, while the quasiparticle tunneling is the source of dissipation in the second. Both of them provide a phase transition in quantitative agreement with the experiments.
This chapter discusses the two-dimensional physics of Josephson junction arrays. Josephson junction arrays consist of islands of superconductor, usually arranged on an ordered lattice, coupled by Josephson junctions. They may be divided into classical and quantum arrays depending on the ratio of the Josephson coupling energy to the charging energy. Josephson junction arrays may also be divided into overdamped and underdamped arrays, referring to the fact that the equation of motion for a single Josephson junction is identical to a damped pendulum. The presence of vortices is one of the natural consequences of arranging such junctions in a two-dimensional lattice. Large arrays have proven to be very useful model systems for studying a wide variety of other physical problems, for example, phase transitions in frustrated and random systems, the dynamics of coupled non-linear systems and macroscopic quantum effects. Classical two-dimensional arrays may be shown to be isomorphic to a two-dimensional XY spin system—they are physical representations of the XY model, which is a two-dimensional lattice of spins free to rotate in the XY plane. Classical two-dimensional arrays are used in zero magnetic fields for studying phase transitions such as the Kosterlitz–Thouless–Berezinskii transition, for studying the effects of disorder on phase transitions, and for studying dimensional crossover effects in phase transitions.
2D arrays of 0.01 and 0.04 μm2 area aluminum tunnel junctions with varying junction resistance RN have been studied. For arrays with RN > 5 kΩ the resistance remains finite at low temperatures. Quasi-reentrant behaviour occurs for RN around 10 kΩ. The effect of Coulomb blockade on single electron tunneling is observed, most clearly in high resistance arrays. Preliminary results on the influence of magnetic field on the RT-curves are given.
We report on specific-heat measurements from 2 to 15 K on single crystals of Bi2CaSr2Cu2O8 and Tl2Ca2Ba2Cu3O10. We find low-temperature deviations from the Debye law that can be attributed to spin-glass behavior of a small concentration of isolated impurity copper moments. At higher temperatures, we observe contributions to the specific heat that can be attributed to a soft-phonon mode, possibly associated with the superstructure in the Bi-O and Tl-O layers. From our single-crystal data, we conclude that the thallium- and bismuth-based copper oxide superconductors show no measurable linear term in the specific heat [γ(0)≤1 mJ/mole K2].
Raman spectra taken from several well-characterized superconducting Tl2Ca2Ba2Cu3O10 (2:2:2:3) single crystals (Tc=105 K) display the characteristic phonon spectrum of the 2:2:2:3 phase on top of a broad background continuum interpreted as electronic scattering. The electronic scattering continuum decreases substantially below Tc for frequency shifts of smaller than 400 cm-1, indicating the formation of a superconducting gap. The gap does not show the expected abrupt onset but a linearly rising electronic scattering background similar to observations in YBa2Cu3O7, possibly indicating a distribution of gaps or the presence of electronic excitations within the gap.
Josephson junction arrays are ideal model systems where a variety of phenomena, phase transitions, frustration effects, vortex dynamics, chaos, to mention a few of them, can be studied in a controlled way. In this review we focus on the quantum dynamical properties of low capacitance Josephson junction arrays. The two characteristic energy scales in these systems are the Josephson energy, associated to the tunneling of Cooper pairs between neighboring islands, and the charging energy, which is the energy cost to add an extra electron charge to a neutral island. The phenomena described in this review stem from the competition between single electron effects with the Josephson effect. One example is the (quantum) Superconductor-Insulator phase transition which occurs by varying the ratio between the coupling constants and/or by means of external magnetic/electric fields. We will describe how the phase diagram depends on the various control paramters and the transport properties close to the quantum critical point. The relevant topological excitations on the superconducting side of the phase diagram are vortices. In low capacitance junction arrays vortices behave as massive underdamped particles that can exhibit quantum behaviour. We will report on the various experiments and theoretical treatments on quantum vortex dynamics. Comment: To be published in Physics Reports. Better quality figures can be obtained upon request
A functional-integral formulation is used to treat the quantum dynamics of a microscopic model of a Josephson junction, including the dissipative effects of quasiparticle tunneling. The calculation is carried to a point where it makes contact with, and therefore substantiates, recent work by Caldeira and Leggett in which the system is treated by analogy with the quantum Brownian motion of a massive particle coupled to a phenomenological heat bath.
A threshold for the onset of global phase coherence, or zero resistance, has been revealed by systematic studies of the superconductivity of ultra-thin gallium and lead films. This threshold appears to be controlled by the normal state sheet resistance of the films which is the only relevant variable. The threshold value of 6.5 kΩ□ agrees with previous observations on ultra-thin tin films and further supports the notion of a universal resistance threshold. This value also is in agreement with the predictions of recent theoretical models which indicate the importance of zero-point fluctuations coupled to dissipation in such film systems.
For granular metallic films or two-dimensional arrays of Josephson junctions, fluctuations in the number of Cooper pairs are conjugate to the superconducting phase and can lead to zero-point destruction of phase coherence at zero temperature. In two dimensions the resulting critical behavior is shown to be that of the λ point of 4He in three dimensions, with the charging energy playing the role of temperature. The nature of the quantum → classical crossover and its effect on the frequency dependence of precursor diamagnetism near the quantum critical point are discussed.
Experimental evidence from studies of the onset of superconductivity in ultrathin Sn films is presented which implies that the sheet resistance is the only relevant variable in determining the onset of global phase coherence. This result is found by a theoretical argument involving both the phase-number and energy-time uncertainty relations.
The problem of the onset of global phase coherence in granular superconductors and Josephson-junction arrays is considered, and it is shown that under conditions in which quantum fluctuations of the order parameter are important, the dissipation enters the thermodynamics as an critical parameter. This effect in superconductivity occurs because the order parameter phase is both a statistical and quantum dynamical variation, and provides a natural qualitative explanation for the results of Orr et al. (1986). Global phase coherence is found to persist for strongly coupled Josephson junctions when alpha approaches 0.
- Leggett