Dynamics of small superconductors

DYNAMICS

OF SMALL

H. A. NOTARYS

SUPERCONDUCTORSt

and J. E. MERCEREAU

California Institute of Technology,

Pasadena,

California

Synopsis

A phenomenological model is discussed for the phase-slip process in small superconducting structures. In this model, Josephson type phenomena are developed without the concept of transport of flux quanta in the film, but arise from the localized periodic quenching of the superconducting wavefunction by an electric field.

1. Introduction. Josephson-like effects have been reported in a number of purely superconducting circuits including the point contact I), Dayem bridge2) and the solder blob.3) Possibly some of these structures contain a tunneling barrier, but others are clearly entirely superconducting circuits. Common and crucial, to all of these superconducting circuits is a small superconducting volume connecting two superconductors. This report examines some of the properties of such small superconductors and presents a model characterizing the Josephson-like behavior. 2. Dimensional considerations. In order to be able to concentrate on the behavior of a particular small section of superconductor, this section was deliberately weakened relative to the surrounding area. This weakening was usually achieved utilizing the proximity effect of a normal or ferromagnetic metal on the superconductor. The geometry usually examined is shown in fig. 1. A superconducting film is overlaid in a small area (of length, I, much less than its width, w,) by a normal metal. As reported previously4), this procedure gives rise to a small section of film (total thickness 6) with well-defined geometry, characteristics and depressed transition temperature. Here we wish to consider the characteristics of this structure specifically for an overlay section whose area is sufficiently small that a flux quantum cannot be maintained by its circulating current. For this geometry, flux in the overlay region will be maintained by a circulating current around the region. Since its length is small relative to the width, the inductance is approximately pow and a circulating current will

t Work supported

by the Office of Naval Research

and the Ford Motor Company.

DYNAMICS

OF

SMALL

SAMPLE

Fig.

1. Sample

over a narrow

geometry.

A superconducting

strip of evaporated current

direction

normal

425

SUPERCONDUCTORS

GEOMETRY

film in a four terminal

metal. The resulting

is then scratched

bridge

down to desired

arrangement of length,

is evaporated

I, in the measuring

width, W.

generate flux -.~ powjAG. Maximum flux is generated at the critical current, which leads to an expression for the minimum width to contain a flux quantum cpO:w, ( T) = cpO/B,( T) 6 where B, is the critical field. Normally, for a film carrying transport current, when critical current is reached, flux quanta are developed in the film and voltage appears associated with flux flow5). However, in this geometry, if w < w, flux quanta are not allowed (even though quantized circulation may still be achieved) and voltage by flux transport gives way to a stationary flux configuration with time varying amplitude. Thus the dynamics of the process become independent of any material inhomogeneities. This condition on w means a width smaller than p,,/B,G. This minimum width is of order 10 p for our 1000 A thick overlay film with a typical critical field of order ten gauss. In these structures, these dimensional conditions are easily obtainable and we will report characteristic properties of Au-Sn bridges 1-l 0 ~1 long and 1O-20 p wide. Other metallic combinations for the overlay section such as Pb-Cu and Sn-Permalloy give similar effects. Similar behavior has also been found for pure tin structures which satisfy this size condition: 1 ,u. As this where 6 z 300 A, B, E 300 gauss and width approximately size criterion is exceeded, Josephson-like phenomena rapidly disappear. The dynamic behavior to be discussed later has also been observed6) in

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H. A. NOTARYS

AND J. E. MERCEREAU

sufficiently small contacts between a superconductor and a normal metal. In this situation the concept of relative phase across the junction becomes undefined and the oscillation must involve motion of the phase boundary between normal and super regions. 3. “Interference” phenomena. When a magnetic field is applied normal to the film, the requirement of phase coherence contributes a term to the free energy 7, proportional to (vi - n(p,J2. Usually the internal flux (pr can adjust to ncp, (and minimize the free energy) by generating the appropriate circulating currents. But in these small volumes, by definition, this shielding flux is less than ‘p,, and thus the free energy, and hence critical current, depend on (cp,ncp,)” as in the Parks and Little experiments). This predicts a periodic dependence of critical current with magnetic field, since for any (pi, n assumes a value to minimize the energy as best it can. Figure 2 shows experimental I

I

MAGNETIC _CRITICAL BRIDGE

I -600

I

I

I

I

FIELD DEPENDENCE OF CURRENT OF 4 MICRON

I

I

-400

I

I

I

I

I

I

I

I

I

I

I

’

I

1

-200

0

I

200

400

I 600

6 (mG)

Fig. 2. Critical-current dependence on magnetic field normal to bridge 4 microns long and 25 microns wide. The measured periodicity corrected for demagnetizing effects is the flux quantum, 9,.

results of critical current VS. magnetic field, measured for a 4~ long and 25~ wide bridge near T,. There is a periodic dependence of I, on B, with the current decreasing at higher fields. This behavior is in qualitative agreement both as to amplitude and periodicity, with an analysis based on the integrated free energy of the total overlay region. For these flat film structures, demagnetizing -effects are quite large and must be taken into account in determining the flux period. The corrected flux period of this structure is actually ‘p,,. It should be specifically noted this is not a diffraction pattern, since all peaks are the

DYNAMICS

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same width, as suggested by this model. At lower temperatures, where w > s h ie Id ing is permitted, the periodicity disappears. &B,(T)Sand These devices have been used as replacements for Josephson tunnel junctions in several applications. Two small bridges connected in parallel by superconducting links form the usual dc interferometer arrangementg) and yield interference phenomena in the expected fashion. Also one of these bridges inserted into a thin film superconducting ring forms an ac interferometer’O) with the usual interference modulation in both ac and dc magnetic fields. The principal advantage of these bridges is the reproducibility and reliability of the structure. In the following sections, some of the parameters affecting the noise and frequency of operation will be developed. 4. Dynamic behavior (‘nite voltages). If the superconductor can maintain thermal equilibrium, supercurrent may also flow at finite voltages as indicated in the following model. In the condition of velocity limited superconductivity, the presence of an electric field accelerates the supercurrent until it reaches the critical current level. This supercurrent must then spontaneously decay lo) due to the unique current- velocity relation in this type of superconductor by driving the order parameter to zero in a second-order transition. As the supercurrent decays an induction field must appear to transfer the kinetic energy of the current to an electric field. Once the decay has taken place the superconductor may recondense at zero velocity and the process will repeat. This process is just a relaxation oscillation, charging and discharging the bridge as a capacitor, giving rise to a voltage supporting superconductivity which contains an oscillating supercurrent. We propose the following kinetic two-fluid model to describe these oscillations. We presume that the current can be divided into superfluid and normal components. Further, for simplicity since dimensions are small relative to wave length, we assume that lumped circuit parameters can be used to describe the circuit as long as we stay well below the gap frequency. For these junctions the normal resistance is = 10m2ohms and normal inductance - bw - lo-“H so that the normal response time = 1O-s s. Thus for frequencies above about 1O8cps (or voltages > 1O+ volts) the normal current can be assumed constant. However since the supercurrent oscillates as described previously displacement currents must also appear. Since the dimensions are so small, we also assume that the time for decay of superconductivity when the supercurrent reaches critical (and the subsequent’ recondensation) is essentially (gap frequency)-’ or h/A = 1O-12s. and is presumed “instantaneous” on this time scale. In this model the supercurrent accelerates at a constant rate in the electric field necessary to drive the normal current until it reaches its critical value; at which instant it decays and the process repeats. The time to accelerate from zero velocity to critical velocity is thus the period of the oscillation.

428

H. A. NOTARYS

AND

J. E. MERCEREAU

In a superconductor, electric fields can exist only across dimensions of the coherence distance otherwise charge would redistribute to cancel the field. However, within this distance, on a free-electron model: eE = rnfi. Thus the time to reach critical velocity, r~,, is T - (mv,/eE) (h/mu,) the frequency of the oscillation is:

and since e -

These arguments as to the physical mechanism of oscillation are of course inexact, however the final equality must be true as shown by Anderson”) and Josephson. Equation (1) can also be written in terms of a kinetic inductance Lk(l) = j_~“f (h*/w6) as:

and where the penetration depth A = (m/pone2)112 and I, = neu,w& Thus estimates of kinetic inductance for the process can be obtained from the measured critical current as:

L(E) = CpJc-‘. Thus if the bridge length is approximately the coherence length, the oscillating supercurrent will have a frequency 2eV = hv. For longer bridges the identification of E> with V is no longer assured and the behavior becomes more erratic and noisy. Besides giving rise to induction or radiation this oscillation implies a time average dc supercurrent, I, = I,/2 at large voltages. Figure 3 illustrates an experimental Z-V curve from a 1 micron bridge. As is shown, the excess current above the normal current is approximately Z,/2 as expected: Further evidence of this supercurrent at finite voltage is given by being able to synchronize these supercurrents with an rf field in order to give “steps” in the Z-V characteristics similar to those found by Dayem. Experimentally it is found that these steps can be stimulated by radiation in a frequency band where the upper frequency is somewhat less than the gap frequency and the lower frequency is related to noise currents in the structure and depends on the normal-state resistance. 5. Equivalent circuit and noise. On the basis of this previous information, at low frequencies an equivalent circuit for the bridge can be drawn as in fig. 4, where L,(c) is the kinetic inductance, R the normal-state resistance,

DYNAMICS I-V

OF SMALL

INTEGRATED

FROM

SUPERCONDUCTORS dV/dI

vs I OF I MICRON

429

BRIDGE

Fig. 3. I-V characteristics of 1 micron bridge driven by constant current source. Crosses give I-V curve integrated from measured dV/dl vs. I. Solid line gives I-V characteristic for the normal resistance of the bridge. The dashed line extrapolation shows the bridge carried an average dc supercurrent of approximately I,/2 at large voltages. EQUIVALENT

CIRCUIT

Fig. 4. Equivalent circuit at low frequencies for bridges driven by constant current source. The parameters are switch S, which interrupts the circuit instantaneously at a frequency 2eV = hv; Lkr the kinetic inductance; R, the normal-state resistance; VN, the noise voltage which leads in this circuit to the given noise current I&J). I/ the noise voltage in R, and S a switch (symbolizing the decay of supercurrent when I, reaches critical) interrupting the circuit instantaneously at a frequency 2eV = hv. To the extent that other noise sources can be neglected, when the switch is closed, noise current IN*(w) in bandwidth dw is given by z,2(w)

-

2 kT WR)d6J TT L ~+(wL/R)~’

430

H. A. NOTARYS

AND

J. E. MERCEREAU

while the total noise current is ZN2= (kT/L) and the noise bandwidth For circuits of our dimensions

1’2-

is R/L.

lo+ amps and R/L = &I, = 108s-1. ‘PO

These values correspond well to experimental values of the lower synchronization frequency and the noise rounding of the I-V curves for several bridges of varying dimensions and resistance. If the switch were operated at a frequency P (R/L)the noise would not have time to build to its maximum amplitude between cycles; the switch having the effect of a high pass filter. And in the limit of very high frequency this would remove the effect of low-frequency noise currents from the circuit. If properly phased, an ac voltage at the switch will pass a dc current and we speculate that the use of rf to open and close the switch at high frequencies may be an explanation of the Wyatt effect 12). In this description, the microwaves drive high-frequency currents to produce high-frequency switching and eliminate the lower-frequency noise currents in the bridge. These noise currents normally degrade the critical current and cause it to be less than ideal. In fact, in these structures the temperature variation of the critical current near the transition is (AT)2. However, under irradiation at sufficiently high frequency, the critical current increases and the temperature dependence becomes the expected (AT)3/2. Figure 5 shows the ratio of maximum critical current with radiation Z,(rf) to critical current without radiation I,(O), as a function of temperature for bridges of equivalent cross section but varying length.

0.7

0.8

0.9

I.6

T/T, Fig. 5. Ratio tion.

of maximum

I,(O), as a function

section.

Data

long bridge,

is for triangles.

critical

IO GHz The

conducting

current

of reduced applied

I k bridge above

with radiation,

temperature

to 7 p long bridge, ratio goes to infinity

its T,. because

I,(rf),

for various

to critical

length

circles;

bridges,

current

without

radia-

of equivalent

cross

3.2 p long bridge,

at T, since the bridge

of the application

of the radiation.

crosses;

remained

1p super-

DYNAMICS

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SUPERCONDUCTORS

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Essentially our model represents a one-dimensional limit to flux flow in which. we have attempted to characterize the dynamic superconducting behavior in terms of measurable parameters. It proposes a physical model for Josephson-like effects which is fundamentally a relaxation process periodically quenching superconductivity: where the time necessary to gain sufficient kinetic energy from an electric field to quench superconductivity is related to the potential creating the field. Although we have concentrated our discussion on the controllable geometry of fig. 1, these ideas should apply to other superconducting structures such as Dayem bridges and point contacts. This dynamic model may also be useful to characterize Josephson-like effects in other superfluid situations where tunneling is not a likely process.

REFERENCES 1) ZimmermanJ. E. and Silver, A. H., Phys. Letters 10 (1964) 47. 2) Anderson P. W. and Dayem, A. H., Phys. Rev. Letters 13 (1964) 195. 3) Clarke, J., Phil. Mag. 13(1966) 115. 4) Friebertshauser, P. E., Notarys, H. A. and Mercereau, J. E., Bull. Am. Phys. SOC. Ser. 11 13(1968) 1670. 5) Giaver, I., Phys. Rev. Letters 15 (1965) 825. 6) Eck, R. E. et al., Proceedings of the Conference on Fluctuations in Superconductors, Ed. W. S. Govee and F. Chilton (Stanford Research Institute, Menlo Park, California, 1968). 7) Byers, N. and Yang, C. N., Phys. Rev. Letters 7 (196 1) 46. 8) Parks, R. D. and Little, W. A., Phys. Rev. 133A (1964) 97. 9) Jaklevic, R. C., Lambe, J., Silver, A. H. and Mercereau, J. E., Phys. Rev. Letters 12 (1964) 159. 10) Nisenoff, M., private communication. Mercereau, J. E., J. appl. Phys. 40( 1969) 1994. 11) Anderson, P. W., Rev. mod. Phys. 38 (1966) 298. 12) Wyatt, A. F. G., Dmitriev, V. M., Moore, W. S. and Sheard, F. W., Phys. Rev. Letters 16 (1966) 1166.