Theory of the magnetoresistance of disordered superconducting films

Recent experimental studies of magnetoresistance in disordered superconducting thin films on the insulating side of the superconductor-insulator transition reveal a huge peak (about 5 orders of magnitude compared with the resistance at the transition). While it may be expected that magnetic field destroys superconductivity, leading to an enhanced resistance, attenuation of the resistance at higher magnetic fields is surprising. We propose a model which accounts for the experimental results in the entire range of magnetic fields, based on the formation of superconducting islands due to fluctuations in the superconducting order parameter amplitude in the disordered sample. At strong magnetic fields, due to Coulomb blockade in these islands, transport is mainly through the normal areas, and thus a decrease is the size and density of the superconducting islands leads to an enhanced conductance and a negative magnetoresistance. As the magnetic field is reduced and the size and density of these islands increase, the conductance is eventually dominated by transport through the superconducting islands and the magnetoresistance changes sign. Numerical calculations show a good qualitative agreement with experimental data.

Figure 1

Schematic representation of the model. Solid lines present the dominant mode of transport compared to the broken lines. (a) At strong magnetic fields, $B⪢B_{\max }$. The system is composed of small superconducting islands with large charging energy. In this regime transport through normal paths (solid lines) is always preferable than transport through the superconducting islands (SCIs) (see text). (b) As the magnetic field is decreased, but still $B>B_{\max }$, more SCIs appear, resulting in a decrease in available trajectories for transport [bottom solid line in (a) and bottom dashed line in (b)] and hence negative magnetoresistance. (c) At a certain field $B_{\max }$ the resistance of normal paths and paths that include SCIs is comparable, resulting in a peak in the resistance. (d) For even lower fields, $B_c<B<B_{\max }$, transport through SCIs is always favored, resulting in positive magnetoresistance.

Figure 2

(Color online) The lattice model is composed of regular sites and superconducting sites. Clusters of the latter form SCIs (shaded islands). The resistance between two normal sites (small circles) is given by Eq. (2). The resistance between two SC sites (squares) is very small (and becomes smaller with decreasing temperature, see text) for neighboring sites (wavy lines) and is infinite (no link) between non-neighboring sites. The resistance between neighboring normal and SC sites (thick lines) is much higher than the resistance between two normal sites, and is exponential in the charging energy of the SC island [Eq. (3)].

Figure 3

(Color online) Numerical results: resistance (on a log scale) as a function of probability $p$ for different temperatures, with the parameters $W=0.4,E_c=4,{\xi }_{\mathrm{loc}}=0.1,T=0.1,0.2,\dots ,4$. This is to be (qualitatively) compared with the experimental data (inset) of Ref. 13.

Figure 4

The activation energy $T_0$ as extracted from fitting the resistance to an activation behavior, obtained from the numerical calculation and from the experimental data of Ref. 13 (inset). Lower inset: an Arrhenius plot of the resistance as a function of temperature for $p=p_{\max }=0.5$. A deviation from an activated behavior is clearly seen.