Magnetoresistance anisotropy in amorphous superconducting thin films: Site-bond percolation approach

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Magnetoresistance anisotropy in amorphous superconducting thin films: Site-bond percolation approach

Elkana Porat and Yigal Meir

Phys. Rev. B 92, 024509 – Published 13 July 2015

Abstract

Recent measurements of the magnetoresistance (MR) of amorphous superconducting thin films in tilted magnetic fields have displayed several surprising experimental details, in particular, a strong dependence of the MR on the field angle at low magnetic fields, which diminishes and then changes sign at large fields. Using a generalized site-bond percolation model, which takes into account both orbital and Zeeman effects of the magnetic field, we show that the resulting MR curves reproduce the main experimental features. Such measurements, accompanied by the corresponding theory, may be crucial in pinpointing the correct theory of the superconductor-insulator transition and of the MR peak in thin disordered films.

Received 22 October 2014

DOI:http://dx.doi.org/10.1103/PhysRevB.92.024509

Authors & Affiliations

Elkana Porat and Yigal Meir^,*

Department of Physics, Ben-Gurion University, Beer Sheva 84105, Israel

^*ymeir@bgu.ac.il

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Vol. 92, Iss. 2 — 1 July 2015

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Figure 1
(a) MR isotherms from Ref. [29], for a sample which is a superconductor at zero field. The traces differ in the angle θ $θ$ of the magnetic field B $B$ , with respect to the plane of the sample, from θ=0∘ $θ = 0^{\circ}$ , plotted in dark red, to 90∘ $90^{\circ}$ , plotted in dark blue. Inset: The resistance is isotropic at the crossing point at Biso=11.02T $B_{iso} = 11.02 T$ . (b) MR isotherms from Ref. [30], for a sample which is insulating at zero field. Temperatures range between T=1 $T = 1$ K and T=0.3 $T = 0.3$ K (bottom group to top group). In each group of curves, colors represent different angles of B $B$ , from θ=0∘ $θ = 0^{\circ}$ (dark red) to 90∘ $90^{\circ}$ (dark blue). (c) Site-bond percolation results for low disorder (small Δ0 $Δ_{0}$ ; see text); the sample is initially in the SC phase. Data were log-averaged [log(R)=1N∑ilog(Ri)] $[log (R) = \frac{1}{N} \sum_{i} log (R_{i})]$ over N=100 $N = 100$ disorder realizations of a sample with 25×25 $25 \times 25$ sites, at tilt angles similar to that in Fig. 1, with the parameters Δ0=0.25 $Δ_{0} = 0.25$ , x=300 $x = 300$ , n=0.5 $n = 0.5$ , W=0.4 $W = 0.4$ , Ec=4 $E_{c} = 4$ , T=1 $T = 1$ , and χ=0.05 $χ = 0.05$ . The magnetic field is normalized by the isotropic field Biso=0.6 $B_{iso} = 0.6$ . (d) Site-bond percolation results for high disorder (larger Δ0) $Δ_{0})$ ; the sample is in the insulating phase. Data were log-averaged over 200 realizations of a sample with 25×25 $25 \times 25$ sites, at tilt angles similar to that in Fig. 1, with the model parameters Δ0=0.6 $Δ_{0} = 0.6$ , x=1000 $x = 1000$ , n=0.5 $n = 0.5$ , W=0.4 $W = 0.4$ , Ec=4 $E_{c} = 4$ , and χ=0.01 $χ = 0.01$ and temperatures T=0.05 $T = 0.05$ , 0.07, 0.1, 0.15, 0.25, and 0.5 (top to bottom).
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Figure 2
Schematic of the combined effects of the Zeeman and orbital magnetic fields. (a) At a relatively low magnetic field, adjacent SCIs (dark-blue shapes) maintain phase coherence, thus forming larger SC clusters (light-blue aura). Most of the current is carried by SC paths, i.e., paths which include SCIs (thick blue arrows). (b) As a Zeeman field is added, some of the smaller islands collapse and the total resistance increases. (c) At a sufficiently high Zeeman field the SC paths become unfavorable compared to the purely normal paths (red arrows), and normal current becomes dominant. From this point on isotropic or negative MR is observed. (d) When orbital (perpendicular) field is added to the Zeeman field of frame (b), some of the intercorrelations between adjacent SCIs are destroyed due to vortex penetration. This results in an increased total resistance, compared to a parallel field of the same amplitude. (e) When normal paths are dominant, the intercoherence of adjacent SCIs, determined by the orbital field, becomes unimportant, and the resistance becomes isotropic with respect to the direction of the field.
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Figure 3
Site-bond lattice model. Left panel: The lattice sites represent either SCIs (squares) or normal regions (small circles). The concentration of SC sites is determined by disorder. Phase coherent SC sites are connected by SC bonds (wavy lines), thus forming large SC clusters [shaded (blue) areas]. A large blockade resistance (thick line) connects normal and SC sites. As orbital field is added (top-right panel), coherence between adjacent sites is suppressed, and some of the SC bonds are broken and replaced with blockade bonds. As a result, SC clusters may decompose into smaller clusters. Addition of Zeeman field (bottom-right panel) results in the collapse of some of the SC sites into normal sites, which results in an increase in normal paths (thin lines).
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Figure 4
(a) Measured SIT critical field (open squares) and peak field (filled triangles) of several films, for different orientations of B with respect to the plane of the film, taken from Ref. [29]. Solid lines correspond to solutions of Eq. (1), under the assumption of a square two-dimensional lattice and fitted Δ0 $Δ_{0}$ and x $x$ . Fitted parameters are presented in the inset in (b). Both experimental and theoretical data are scaled by the largest critical field, representing the case of a clean system and parallel field. (b) Measured perpendicular versus parallel critical magnetic fields and peak fields (open squares and filled triangles, respectively), taken according to Ref. [29], together with fitted model data (solid line). Inset: The fitted parameters, presented as log10(x) ${log}_{10} (x)$ , versus the corresponding Δ0 $Δ_{0}$ . Circles represent the data in (a), with corresponding colors.
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