EFFECT OF ORDER PARAMETER RELAXATION ON JOSEPHSON JUNCTION TRANSIENTS

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EFFECT OF ORDER PARAMETER RELAXATION

ON JOSEPHSON JUNCTION TRANSIENTS

H. Suhl, J. Hurell, A. Silver, Y. Song

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, supplPment au no 8, Tome 39, aotit 1978, page C6-550

E F F E C T OF ORDER PARAMETER R E L A X A T I O N ON JOSEPHSON J U N C T I O N T R A N S I E N T S

f f X

H. suhlt, J. Hurell

,

A.H. Silver and Y. Song

U n i v e r s i t y o f C a l i f o r n i a , San Diego, La J o l l a , C a l i f o r n i a 92093, U.S.A.

f The Aerospace Corporation, P. 0. Box 92957, Los Angeles, C a l i f o r n i a 90009, U.S.A.

RLsum6.- Une discussion complste de l'effet Josephson doit rendre compte de la dLcroissance du para- mstre d'ordre, selon une Lquation de Landau- Ginsburg ddpendant du temps. On ddmontre, par exemple, que cette dLcroissance peut sdrieusement modifier la description habituelle. En plus, il apparaft qu'un Ltat dissipatif Zi polarisation nulle devient possible.

Abstract.- A full discussion of the Josephson effect in non-stationary situations should include relaxation of the order parameter according to a time dependent Landau-Ginsburg equation. We show by means of an example that this relaxation can significantly modify the usual description. Further more, a "zero bias" type of dissipative state seems to become possible.

The behavior of circuits containing Josephson junctions is generally discussed on the assumption that the superconducting order parameters are at all times in a constrained quasiequilibrium prescri- bed by the instantaneous values of the circuit cur- rents and voltages. (For example the Josephson voltage-phase rate of change relation is assumed to hold at any one instant). When such assumptions are dropped, and the circuit equations are supplemented by a time-dependent Landau-Ginsburg equation /l/

(in place of the Josephson relation), significant changes occur relative to the usual theory. In addition, it seems that a new dissipative steady state solution may arise that has some of the features of the so-called zero-bias tunneling anomalies. As simplest non-trivial example we analyse the two- junction squid in figure 1. The magnitudes of the order parameters are assumed constant in the two links connecting the junctions and the phase-drop along them due to the current is neglected, leaving only the drop due to electromagnetic potentials :

if ; are the order parameters just

above and just below the two junctions (see figure 1)

. . .

(i) where A = Vector potential, = ch/2e, the flux quantum. The Landau-Ginsburg energy (with T a tunneling energy) is written

plus the same term with a', b', replacing a, b. In view of (i), this becomes, with r$ = 2n

I

A ~ X / $ ~ ,

The general from of the time dependent L.G.

M.

equation is taken to be -(a$/at)-ic*) = SF/&$". K

Here v is the local voltage (regardless of origin), multiplied by 2 ~ / $ ~ , and K is a dimensionless constant between .l and 1. In the present case this gives :

iB

Setting$ a,b = f a,b a'b, we then find

'

Supported by the National Science Foundation NSF DMR77-24957

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This evidently exists only if

L

Ofa

=

2f T cos- @ cos (-+ @

eb- ea)

+

~tf,,~- ~f:,~

K at b,a 2 2

Maxwell's second equation gives vb-va-(vb,-val) =

-

61C

SO that

Vb-va+ vbl-val = 2(vb-va) + @/C. Therefore, adding and subtracting $12 on the lefthand side of equation

@

(IV) gives, with o =

7

+ Ob-ea,

where v is one-half the sum of the voltages across the two junctions. In addition, we have the circuit

2ac equations I = i + i and $ = + L(il

- i2)

-

1 2

$0 (see figure 1) where $ is 2a/@ times the external-

X 0

ly applied flux.

Noting that i

_{I = T}

4eT f f sin(eb-ea) ; a b

4eT @

i =

91

f f sin

(T

+

eb-

'a)

.

(VI) a b

the circuit equations become, with o = +

Bb-

ea

2 c; + qcv + w 2 ~ s i n a c o s a = W'B 2 o x

. .

.

(VII) S 0 where 27rLC 8eT w2=(~c)-l, Q=(Kc)-', B=

F

fafb, ux = ~ L I / @ ~ . 0

The usual theory sets the righthand side of (V) equal to zero, so that CV =

6.

This is permissible

only if w 6, and/or 11 are large compared with ~TK/M. For circuits well below 1013Hz this will not usually be the case except for large

8.

However, it seems that the time variation of f may safely be ne-

a,b

glected ("). Note that if 0 = W , vs = 0, {IV) yields the Josephson relation : for a constantv

b-va' it gives 8

- 8

= c(yb-va)t. This is consistentwith

b a

a + 8,- 8 = nay if

2 =

-(v

-

va), i.e. if the vol-

2 2 b

tages across the two junctions are equal and oppo- site. In addition, we have the possibility of a

ti-

me-independent

,

field sensitive dissipative state with

and if this is satisfied, the equivalent resistance i S

V< Va f2+f2

R

= - - = a

103 ohms.

eg' 2a 1 47recfPf2

a b

The effect on squid behavior is easily seen in the case of heavy damping (so that

4

and may be neglected.) With O.P. relaxation neglected as u- sual, the second of (VII) reads

$J + w2 8 sina cos& = w2 a When it is not neglec-

0 2 0 X'

ted, the equation still has this form, but with

B

4TK

fk+fg

replaced by an effective B ' =

8

+

--

,

which ~ w 2 fafb

will typically be much larger than

8.

On the other hand (VIIa) is unaltered.

Equations (IV) through (VII) are correct only if, for given @ I is so large that there exists no

X'

non-dissipative time independent state (i.e. the system must be in a "voltage state"). When I is small enough, the righthand sides of IV and V are correc- ted so that the relaxation takes place towards a constrained equilibrium. (In fact, unless I exceeds the bulk critical current such a solution always exists, but may have higher free energy than the normal state). These questions can become important in a discussion of phase slippage 121, but do not appreciably affect the conclusions of this paper.

References

/l/ Langer, J.S. and Ambegaokar, Phys. Rev.

164

(1967) 498-510

In this paper a somewhat analogous situatid is considered for a uniform superconducting loof /2/ ~irkijarvi, J., Phys. Rev. B

5

(1972) 832-835

When O.P. relaxation is included in that problem, their last mentioned points become crucial