Relative self-phase in resonant Josephson junctions

Volume 46A, number I

PHYSICS LETTERS

19 November 1973

RELATIVE SELF-PHASE iN RESONANT JOSEPHSON JUNCTIONS* J.A. BLACKBURN** and M.A.H. NERENBERG Department ofApplied Mathematics, University of Western Ontario, London, Ontario, Canada Received 27 August 1973 Revised manuscript received 1 October 1973 A phase shift between the perturbed and unperturbed solutions to a self-resonant Josephson junction is suggested and its influence on the zero-frequency current-voltage characteristics is discussed.

It is well known that the cavity modes of a rectangular Josephson junction have a profound effect upon the zero-frequency current-coltage characteristics of such devices. The quantitative description of the voltage positions and magnetic field sensitivity of the current peaks (first observed by Fiske [1] in the form of “steps”) was given by Eck et al. [2] and was subsequently discussed in detail by Langenberg et al. [3]. They solved the lin,arized partial differential equation governing the self-induced perturbation voltage by expandingv(z, t) in terms of the strip line modes. The

solution thus indicating the necessity of a term such as We adopt the point of view that there will then exist an optimum i,Li~such that the amplitude of the zero-frequency current can maximize. If we proceed to substitute the expansion for u(z, t) obtained by Eck et al. [21into (1) and extract the zero-frequency component of the resulting frequency modulated supercurrent, we obtain: ~U) +.,(2) 2 ‘dc cos(i~0, sin~0

junction phase was than evaluated and the frequencymodulated component supercurrentJ1 were obtained. sin~and its zero-frequency We wish to suggest that eq. (3) of ref. [2] be modifled as follows:

with

.

—

~

.

2/hw2) =(8~e1j 1

j(l)

.

2

C

i/Q sin [(kL~--n~i)/2] [(kL—nir)/2]2 (8irelj~/hew2)

nøXn =

wt

-

kz

—

~

+~ifv(r’)dt’.

(1)

The extra term ~11~ implies a phase shift between the lowest order approximation and the first order corrected value. This slippage is not inconsistent with the complete solution of the p.d.e. for v(z, t) or its equivalentintermsof~. For example, when a lowest order approximation of the form 0 (o,t—kz) is made, specific initial conditions are implicity assumed: nathely ~(O) kz and 0’(O) = w. Any improved value ought to satisfy the same initial conditions. This requirement leads to a non-decaying component of the “transient”

j(2)

I —(nirë/wL)2

(3)

-

tIl+mr/kL]2 (4)

sin2 [(kL —nir)/2] [(kL—nir)/2] 2

n=O

___________

[l+nir/kL]2

and where: n’O

I =

n

‘-~

=

—

=

n

Fl_1n~/wL~212+ Fl/Q]2 I

J

In the event that

L

k

~i

I

0 = 0, eq. (2) reduces to the result given previously by Eck etal. [2] It is apparent, however, that non-zero ~Ligenerally corresponds to 0 the maximum ‘dc• As a matter of fact it can be shown that 0o)opt the optimal tan~ [J(2)/JU)] choice is (5) .

* **

Research supported by the Nationa Research Council of Canada. Present address: Physics Department, Wilfrid Laurier University, Waterloo, Ontario.

.

.

(

15

Volume 46A, number I

PIIYSICS LITTERS

also note that the posit ~oii of the maximum is shifted slightly above the voltage determined from geometrical

20

z

considerations alone. This is due to the competing influences 0fJ(D and j( 2) near a geometrical rest)nance. To summarize, we assert that a phase constant should way as to be account introduced for into the development the perturbation of a solution relative of the self-resonant Josephson junction in such a

15

)

0

05

_______________________________________

20

40

&O VOLTAGE

~

1~O

t’dc’1i1

plotted against normalized bias voltage,current, (8eLV/ht~,with Fig. 1. The normal~edzero-frequency Q = 10,11 = 3 A/cm2, / = iDA, = 4. ?~= SODA, and applied field such that kL = 2e. I. has been set equal to the Josephson penetration depth, Xj(Xj = hc2/8ired/ 1). The lower curve is with 0: the upper curve has 9~optimized at each bias voltage.

for which the condition JdC~ 0 will always hold. The necessity here of a positive current contrasts with the case of an externally excited junction where negative currents are allowed due to the possibility of photon absorption processes. Ot ought to be emphasized at this point that a different (~/o)optwill arise for each bias voltage selected. A sample comparison between the results of Eck et al. [21 (for which ~ = 0 was implicity assumed) and the present calculation given by eqs.(~) and (5) is indicated in fig. I. It is seen that the resonant peak amplitudes are essentially unaffected implying that (Oo)opt 0 at these particular voltages. whereas the value Of Ide is enhanced elsewhere. We

16

19 November 1973

phase shift between the perturbed and unperturbed solutions. The device will adjust this apparently indriving circuitry In the case of constant deterniinate ~L’~in[41 a manner controlled by thecurrent sources, the resonant “step” heights will be nearly unchanged from the values predicted by the previous calculations [21 however the apparent half-width of the leading edge of the step will be larger at given

Q The authors gratefully acknowledge the assistance of R. (‘aim and P. Forsyth in carrying out the prograniming necessary to generate the computer plot of fig. I.

References [II M.D. Fiske, Revs. Mod. Phys. 36 (1964) 22L 121 R.1’. Eck. I).J. Scalapino and B.t~.Taylor, in Proc. 9th Intern. (‘ont. on Low temperature physics. cd. by .1G.

13]

Daunt (Plenum Press. 1965) ~ 415 -420. D.N. Langcnberg, D.J. Scatapino and B.N. Taylor,

Proc. I.E.E.E. 54 (1966) 560 141 P.W. Anderson, Progress in low temperature physics V. ed. C.J. Gorter(North’llolland, 1967) pp. 1-43.