Coulomb gap triptychs,2effective charge, and hopping transport in periodic arrays of superconductor grains

Abstract

In granular superconductors, individual grains can contain bound Cooper pairs while the system as a whole is strongly insulating. In such cases the conductivity is determined by electron hopping between localized states in individual grains. Here we examine a model of hopping conductivity in such an insulating granular superconductor, where disorder is assumed to be provided by random charges embedded in the insulating gaps between grains. We use computer simulations to calculate the single-electron and electron pair density of states at different values of the superconducting gap ∆, and we identify "triptych" symmetries and scaling relations between them. At a particular critical value of ∆, one can define an effective charge √ 2e that characterizes the density of states and the hopping transport. We discuss the implications of our results for magnetoresistance and tunneling experiments.

Vortex variable range hopping in a conventional superconducting film

“…At the highest fields the magnetoresistance changes sign, possibly because of the destruction of Cooper pairs and a crossover to VRH of single electrons. 34,35 In the range of fields 1 B 6 T we verified the duality symmetry of our resistance data on either side of critical field B c following the "geometric mean" method suggested by Shahar. 36 For that we defined several pairs of "geometrically dual" magnetic fields B sc < B c and B ins > B c such as for each pair (B sc B ins ) 1/2 = B c .…”

Electrostatic tuning of the properties of disordered indium-oxide films near the superconductor-insulator transition

“…3, can be explained qualitatively within the context of the theory of Ref. [31][32][33]. Increasing the carrier density presumably increases the density of states at the Fermi level within the superconducting grains, thereby driving up the zero-field superconducting gap, ∆ 0 .…”

“…On the other hand, reduction of the superconducting pairing energy ∆ within islands can itself lead to a tradeoff between conduction by Cooper pairs and conduction by unpaired electrons, and thus (potentially) to a MR peak, even when the concentration and the size of the islands are fixed. Recent theoretical works [31,32] studied a model with fixed size and concentration of superconducting grains (islands), and show how a MR peak deep in the insulating state can arise as a result of the reduction of the superconducting gap ∆ with increased H. This predicts that near the MR peak and at low temperature the conduction should be described by ES VRH, as shown in Figs. 1(a2) and 1(b2).…”

“…4 shows values of the resistance of a simulated 2D array of regularly-spaced, monodispersed superconducting grains as a function of H, calculated using the method described in Ref. [32]. At small ∆ 0 /E c , the conductivity is primarily due to hopping of unpaired electrons, and there is a monotonic negative MR [as seen, for example, in Fig.…”

Magnetoresistance of an Anderson Insulator of Bosons

“…These and other similar systems feature a giant peak in magnetoresistance (MR) [8][9][10][11][12]. This is often interpreted as a crossover from bosonic to fermionic transport [13,14], even though the details remain controversial. Bosonic localization problems arise also in disordered granular superconductors in the insulating regime, in cold bosonic atoms in speckle potentials (where artificial gauge fields can mimic a magnetic field) as well as in disordered quantum magnets.…”