Physica A 157 (1989) 135-139

North-Holland, Amsterdam

ANOMALOUS AHARONOV-BOHM EFFECT IN PERCOLATING SUPERCONDUCTING FILMS

A. GERBER and G. DEUTSCHER

School of Physics and Astronomy, Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

We have studied the superconducting transition in thin films of Pb and Al as a function of

temperature and magnetic field. Examination of the films in a transmission electron mi-

croscope shows a percolative structure with typical small loop sizes of a few thousand

angstroms in Pb and about a thousand in Al. An ac current passing through the sample is

found to cause an anomalous dc voltage across the sample in the vicinity of the superconduct-

ing transition. The measured voltage consists of a symmetric and an antisymmetric part as a

function of an external magnetic field. The antisymmetric signal shows a sharp oscillation

pattern identified with an Aharonov-Bohm-like flux quantization effect in submicron loops of

the sample.

The study of the physical properties of disordered media has recently

focused on the Aharonov-Bohm effect in the cylinder or ring geometry. The

basic phenomenon [l] states that the physical quantities measured on a cylinder

must be periodic as a function of the flux along the cylinder axis. The

application of the Aharonov-Bohm effect to the study of electron-electron

interactions in two-dimensional systems caused the theoretical prediction of

periodic oscillations in the magnetoresistance of small normal metal cylinders

[2] and rings which have diameters smaller than the phase coherence length of

the electrons in the metal. The period of the oscillations can be h/e or h/2e,

with the amplitude of the h/e oscillations decreasing with increasing number of

conductance channels in a sample. Clear experimental evidences for these

effects were observed in normal metal cylinders [3], superconducting metal

cylinders above T, [4], single normal rings [5] and in periodic networks of

normal metal rings [6,7]. Magnetic flux quantization effects were found to be

responsible for the oscillations of resistive critical temperature T, of a regular

[8], fractal [9] d an random [lo] network of superconducting metal.

We want to report here on the observation of a dc voltage across macros-

copic superconducting percolating films in the vicinity of their superconducting

transition, when an alternating current of frequencies from 10 Hz up to 1 MHz

is passing through the sample. We can distinguish two contributions to the dc

voltage measured as a function of an applied magnetic field: symmetric

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136 A. Gerber and G. Deutscher I Aharonov-Bohm effect in superconducting films

V(H) = V(-H) and antisymmetric V(H) = - V(-H). The antisymmetric sig- nal shows a highly reproducible and sharp characteristic pattern, quasi-periodic as a function of the applied magnetic field. Usually it consists of a superposi- tion of signals with various periods, but in some cases only one period dominates. The period of the oscillations is individual for every sample and is of the order of 200-250 gauss for Pb and 600-1500 gauss for Al. The measured samples are macroscopic strips, 12 mm long and 4 mm wide, with a typical average thickness of respectively 160 A and 30 A for Pb and Al. Lead samples were prepared by evaporation on pre-evaporated germanium films of 400 A thick, and Al ones by direct evaporation onto the glass substrate. Examination of the samples in a transmission electron microscope shows a percolative structure with typical small loop sizes of a few thousand angstroms ion Pb and about a thousand in Al. The average dimensions of the smallest loops, defined as the half sum of the internal and external diameters, is of the order of 3000 A and 1500 A for Pb and Al films respectively. These loop sizes and the field periodicities are in good agreement with the flux quantization condition for the respective films HS = c#+,, c#+, = 2.07 X lo-’ gauss cm2.

The behavior of the induced dc voltage is studied by measuring the dc current passing through an external resistance connected in parallel with the sample (fig. 1). The sample is a lead film close to its percolation threshold with

I,

0 I I I I 1 I I I I

-4 -3 -2 -I 0 I2 34 B ( KgOUSS)

Fig. 1. Variation of the dc current passing through an external resistance connected in parallel with

the sample, as function of a magnetic field applied perpendicular to the film. The sample is

connected to an ac current source delivering about 1 mA rms.

A. Gerber and G. Deutscher I Aharonov-Bohm effect in superconducting films 137

B (gauss )

V(,V)

14

ti ‘2 , i”;; OnA 8 6-

4-

2-

lb I I I I I I I I I I I

-4000 -3000 -2000 -1oco 0 ImO 2cm 3000 4cm

B (QOUSS 1

Fig. 2. (a) Voltage oscillations in an Al sample at different applied dc currents through the sample. The magnetic field is applied perpendicular to the sample’s surface. T, - T= 0.1 K. All the measurements are carried out with the same wires areangement. (b) Same measurement as that in (a), but with an angle of 60” between the applied field and the normal to the sample’s surface. The period of the oscillations is doubled as compared to that shown in (a). (See the width of the central feature in this figure as compared to that in (a).)

138 A. Gerber and G. Deutscher I Aharonov-Bohm effect in superconducting films

a wide superconducting transition and non-zero resistance even far below the

transition temperature of Pb. At applied fields smaller than the critical field of

the sample, a quasi-periodic dc current is observed in the external circuit. This

current drops to zero when the critical field is reached. The induced dc current

oscillates as a function of H around a non-zero I,,, and relative to this value is

antisymmetric as a function of the applied field, Z(H) = -1(-H). Note that no

dc current has been applied.

Measurements for different applied dc currents and two values of the angle 8

between the normal to the sample and the magnetic field are shown figs. 2a, b.

Changing the amplitude and the polarity of the dc current passing through the

sample does not affect the periodicity nor the magnitude of the measured

oscillations for small enough applied currents. The behavior remains the same

as when the applied dc current is zero.

Identification of the observed oscillations with the Aharonov-Bohm effect is

demonstrated by varying the angle 8. The period of the oscillations is found to

be inversely proportional to cos 8. For example, it is doubled when 8 is

changed from 0” (perpendicular field) to 60“ (fig. 2a, b), and the oscillation

vanishes when the field is applied parallel to the film.

The sample’s response has been studied as a function of the amplitude and

frequency of the applied ac current. The average oscillation amplitude of the

induced dc voltage is found to be proportional to the rms magnitude of the

applied ac current. The oscillation’s dependence on the frequency of the ac

current is relatively weak and more complicated.

We believe that we have reported here the direct observation of the

Aharonov-Bohm effect in a macroscopic percolating sample, through loops of

similar size which are part of the infinite cluster. It is remarkable that, in spite

of the spread in the size of the loops, a sharp superposition pattern from the

loops up to a micron size, or in the extreme cases the pattern corresponding to

the average size of the smallest loops is observed.

Many aspects of the reported phenomena remain not understood. First, we

are not familiar with any theory of a low frequency rectification mechanism in

superconducting percolating film in the vicinity of the superconducting transi-

tion. An explanation based on the inverse Josephson effect seems to be

misleading because of very low frequencies of the ac current. Second, the

measured voltage oscillations are antisymmetric as function of an applied

magnetic field. This is in contrast to any reported phenomenon connected to

flux quantization, like oscillating magnetoconductance or the Little-Parks

effect.

However, we have observed a quantum effect of flux quantization in a

macroscopic disorder superconducting system, which raises an interesting

question of physical order in geometrically disordered material.

A. Gerber and G. Deutscher / Aharonov-Bohm effect in superconducting films 139

Acknowledgements

We acknowledge fruitful conversations with Y. Aharonov. This research was partly supported by the Oren Family Chair for experimental Solid State Physics and by the US-Israel Binational Science Foundation.

References

[l] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. (21 B.L. Altshuler, A.G. Aronov and B.Z. Spivak, Zh. Eksp. Teor. Fiz. Piz’ma. Red. 33 (1981)

101 [JETP Lett. 33 (1981) 941. [3] D. Yu. Sharvin and Yu. V Sharvin, Zh. Eksp. Teor. Fiz. Piz’ma. Red. 34 (1981) 285 [JETP

Lett. 34 (1981) 2721. [4] M. Gijs, C. Van Haesendock and Y. Bruynseraede, in: PTB-PG-1, L. Schweitzer and B.

Kramer, eds. (Braunschweig, FRG, 1984), p. 111 (unpublished); J.M. Gordon, Phys. Rev. B 30 (1984) 6770.

[5] R.A. Webb, S. Washburn, C.P. Umbach and R.B. Laibowitz, Phys. Rev. Lett. 54 (1985) 2696; V Chandrasekhar, M.J. Rooks, S. Wind and D.E. Prober, Phys. Rev. Lett. 55 (1985) 1610.

[6] B. Pannetier, J. Chaussy, R. Rammal and P. Gandit, Phys. Rev. Lett. 53 (1984) 718. [7] D.J. Bishop, J.C. Lucini and G.J. Dolan, Appl. Phys. Lett. 46 (1985) 1000. [8] B. Pannetier, J. Chaussy, R. Rammal and J.C. Villegier, Phys. Rev. Lett. 53 (1984) 1845. [9] J.M. Gordon, A.M. Goldman, J. Maps, D. Costello, R. Tiberio and B. Whitehead, Phys.

Rev. Lett. 56 (1986) 2280. [lo] R.G. Steinmann and B. Pannetier, Europhys. Lett. 5 (1988) 559.