INFLUENCE OF BARRIER NONUNIFORMITIES ON THE RESONANCE AMPLITUDES OF HIGH-Q JOSEPHSON TUNNEL JUNCTIONS

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INFLUENCE OF BARRIER NONUNIFORMITIES ON

THE RESONANCE AMPLITUDES OF HIGH-Q

JOSEPHSON TUNNEL JUNCTIONS

E. Balsamo, G. Paternò, A. Barone, M. Russo, R. Vaglio

To cite this version:

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JOURNAL DE PHYSZQUE _Col/oqueC6, supp/&ment au no 8, Tome 39, aolit 1978, page C6-57 1

INFLUENCE OF BARRIER NONUNIFORMITIES ON THE RESONANCE AMPLITUDES OF

HIGH-Q JOSEPHSON TUNNEL JUNCTIONS

++

E.P. Balsamo, G. Paternb, A. ~arone+, M.

uss so+

and R. Vaglio

+ CNEN Centro d i Frascati, 00044 Frascati, I t a l y .

++ Laboratorio d i C i b e m e t i c a deZ CNR, 80072 Arco Felice, I t a l y . I s t i t u t o d i Fisica de Z 2 'Universitci, 84200 SaZerno, I t a l y .

Rdsum6.- On prdsente ici des mesures sur la ddpendance de l'amplitude de resonance avec le champ ma- gndtique, pour diffdrents types de jonctions tunnel Josephson avec Q Clevd. ces jonctions manifestent une distribution de densite de courant maximum qui n'est pas uniforme. Les rdsultats sont interprdtds par une forme plus gdndrale de la thdorie de Kulik des modes rdsonants.

Abstract.- Measurements of the magnetic field dependence of the resonance amplitudes for different types of high Q Josephson tunnel junctions exhibiting a nonuniform maximum current density distribution are reported. The results are interpreted by means of a more general form of Kulik's theory of resonant modes.

Self-resonant modes in Josephson junctions _{the equation} (Fiske steps/l/) offer a powerful means for inves-

a/.Jo2 (al2) = ZnF',($)

tigating several properties of the whole structure _Qn (2)

Z is a parameter defined as : Z = -(L/<x~>)*

and of the single superconductors as well. As an n n a n

example, they allow the determination of the fun- damental resonant frequency and the capacitance of the junctions, the frequency and temperature dependence of the London penetration depth of the superconductors and give important information on the quality factor, 4/21. It is worth remarking that Josephson junctions often exhibit a nonuniform tunnel barrierl31. In such a case the results obtained are strongly affected by the shape of the maximum current density distribution. To include the case of nonuniform barriers a more general form of the Kulik's theory141 of resonant modes has been re- cently developedl51. In that paper only a low Q situation was discussed. With more generally we can start from the one dimensional equation assumed in reference151 (equation 1) for the relative phase. Following closely the method by Kulik, the amplitude of the n-th resonance as a function of the ap- plied magnetic field can be written as

where Qn is the quality factor of the n-th mode and < A > = J -[2e~adTo]l'

a

.

The function F'(I$) is defined as

n

Obviously for J(x) = cost. equation (1) reproduces Kulik's results and the low Q limit (Z /4<<1) the

n

same results of reference151 are obtained. We can summarize the most important implications of the present calculations : i) the magnetic field behaviour of the resonance amplitudes depends on the shapes of J(x) ; as an example, for a current density distribution peaked at the junction edges an interferencial-like behaviour is expected. ii) Zero- field resonances occur whose amplitude is connected to the intensity of the coefficients of the Fourier expansion of J(x). iii) A higher maximum for the ra- tio I /I can occur ; in fact, as in Kulik calcula-

n o

tions we have max(In/Io) = 0.34 F1($). It can be n

easily found FA

5

2 and then I /I < 0.68. The value n o -

0.68 can be approached, for example, in a situation where

4

= @ / @ p is the magnetic flux threading the described by a J(x) highly peaked at the junction junction nopalized to the flux quantum ;

I

edges.

I

. = WL

[:I,

-

J(x)dx is the zero-field Josephson

I

The present experiments have been performed

by using both Nb-Nb 0 -Sn oxide barrier junctions

current, J(x) describes the profile of the maximum X Y

of cross type geometry and In-CdS-In semiconductor current density, and WxL is the junction area ;

barrier junctions in which the nonuniformity in J and J are the Bessel functions of order zero

1 the current density has been induced by light/6/.

and one. The coefficient a can be found by solving

In figure 1 the magnetic field dependence of both

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Fig. 1 : (up) Magnetic field dependence of the light induced maximum zero-voltage Josephson current, I

,

of an In-CdS-In junction. Experimental data (dots? are compared with the theoretical behavior (solid line) calculated by using the current density profile sketched in the inset. (down) Magnetic field dependence of the first Fiske step. The experimental data are compared with the theoretical dependence.

the Josephson current, IJ, and the amplitude,I1, of the first Fiske step are reported for and In-CdS-In structure. The junction dimensions were WxL = 0.3 x 0.4 mm2 and the zero-field Josephson current was I = 1.7 mA so that L/<XJ> = 1.1. From the IJ vs

4

pattern the shape of the function J(x) is determined by a best fitting procedure/3/, then, inserting this function in equation 3, the function F'($) is compu-

n

ted. From the data a value of z l = 0.94 is deduced (corresponding to Q c 8). Equations (1) and (2) pro-

vide the theoretical dependence of the resonance amplitude. The theoretical behavior is compared with the experiment in figure 1. In figure 2 the experimental data for a Nb-Sn junction are reported

(WxL : 0.1 x 0.3 mm2; I. = 7.2 mA ; L/<AJ> = 1.9). The theoretical dependences for the amplitudes I1 and I2 of the first two steps have been determined by inserting in equations (1),(2) and (3) a maximum current density profile, J(x), determined by the same procedure of the previous case. From the data the values z l = 12.3 and z2 = 3.0 have been deduced corresponding to Ql = Q 2 30. The agreement between

2

theory and experiment is satisfactory ; the small discrepancy in the 2nd step behavior can be proba- bly related to a not sufficiently small value of L/<XJ >j7/. A double modulation effect is present in the experimental IJ vs. $ and I1 vs. $ dependences and not in the I2 vs

4

pattern as predicted by the calculations.

In conclusion it should be noted that the quality of the fittings and the consistency of the whole theoretical scheme with the experiments gives

Fig. 2 : In the upper part it is shown the magnetic field dependence of the maximum zero-voltage Jose- phson current, IJ, of a Nb-NbxOy-Sn junction. Expe- rimental data (dots) are compared with the theoretical behavior (solid line) calculated by using the step-like current density profile shown in the inset. In the figure are also shown the magnetic field dependence of the amplitudes of the first and second Fiske steps. The experimental data are compared with the theoretical dependences.

further indications of the validity of Kulik's ana- lysis also for high Q junctions for which, only few data are available so far/7,8/.

References

/I/ Fiske,M.D., Rev. Mod. Phys.

36

(1964) 221 /2/ Broom,R.F. and Wolf,P., Phys. Rev. B16(1977)3100 /3/ Barone,A., Paterno,G., Russo,M. and Vaglio,R.,

Phys. Stat. Sol. (a)

41

(1977) 393

/4/ Kulik,I.O., Sov. Phys. Tech. Phys.

11

(1967) 1 1 1

/ 5 / Russo,M. and Vaglio,R., F%ys. Rev. B (in press) /6/ Russo,M., Phys. Lett. !3& (1977) 191

171 Nordman,J. and Paterno,G., J. Appl. Phys. (in press)