Josephson equations for the simplest superconducting multilayer system

PHYSICA ELSEVIER

Physica C 259 (1996) 349-355

Josephson equations for the simplest superconducting multilayer system G. C a r a p e l l a , G. C o s t a b i l e , R. D e L u c a *, S. P a c e , A. P o l c a r i , C. S o r i a n o 1NFM Research Unit and Department of Physics, University of Salerno, 1-84081 Baronissi (SA), Italy

Received4 January 1996

Abstract We extend Feynman's model of a single Josephson junction to the case of three stacked superconducting layers. In order to derive the correct voltage-phase relation, a self consistent approach to this model is used. By this analysis we notice that, if the intermediate electrode is thin enough to allow the overlap of the superconducting order parameters of the outer layers, an additional term appears in the Josephson current-phase relation. Two representative applications are presented for these types of junctions. PACS: 74.50.+r Keywords: Josephsonjunctions; Multilayeredstructures;Josephsoneffect

1. Introduction Ever since the discovery of high-To superconductors [ 1 ] the interest in Josephson-coupled superconducting systems has been continuously increasing. Indeed, the high degree of anisotropy and short coherence length in the structure of these superconducting compounds suggests the use of Josephson-coupled multilayer models [2]. The physical properties predicted by these types of models can be experimentally investigated by means stacks of low-Te metallic junctions [ 3 ]. The simplest multilayer structure is represented by two strongly coupled Josephson junctions; therefore, in the present paper a phenomenological analysis of the constitutive Josephson equations for two strongly coupled junctions is given.

* Correspondingauthor. Fax: +39 89 95 3805; e-mail: [email protected].

After the predictions made by Josephson [4] in his original work on superconducting tunnel junctions, Feynman [5] showed that a single Josephson junction (JJ) can be seen as a coupled two-level quantum system. This well known model of the JJ, though, falls to give consistent answers if the density of Cooper pairs in the two electrodes is not the same. Ohta [6] therefore developed a version of the same model which explicitly accounts for the interaction between the junction and the external bias circuit. This approach predicts the validity of the strict Josephson voltage-frequency relation, and provides a self-consistent procedure to solve the problem from a semi-classical point of view. We summarize Ohta's reasoning, in a slightly modified form, as follows. The semiclassical Hamiltonian associated to the two level quantum system is the following:

no=(¢,;,¢,¢)

0921-4534/96/$15.00 @ 1996 ElsevierScienceB.V. All fights reserved PII S0921-4534(96)001 15-3

-ic/2 e2 ]

¢2

G. Carapella et al./Physica c 259 (1996) 349-355

350

=ElI~blI2+E21~212 - ½K(¢~¢2 + ¢ ~ 0 1 ) ,

(1)

where K is a positive constant and ~k is the order parameter of the kth electrode (k = 1,2), Ek being the associated energy. By taking the order parameter of the kth electrode to be (2)

Ok = V ~ k e-iok ,

where Ok is the superconducting phase associated to ~k and Nk is the density of Cooper pairs, and substituting (2) into (1), one has

Ho = E1NI + E2N2 - K v ~ I N 2 cos(O1 - 02).

(3)

The full Hamiltonian H of the system is obtained by subtracting from H0 the energy of interaction Wex of the JJ with the external circuit, where

If we suppose that both electrodes are immersed in a constant temperature thermal bath, we can write /V1m/V2-- 0,

(11)

so that Eq. (10) now reads: Ec - Wex = constant.

We now notice that the Eq. (6) reduces to a mere equality, and Eq. (7) gives Ol - 02 = ~o = El - E2/h,

(4)

V being the voltage across the junction and I the current flowing through the barrier. The Hamiltonian H can thus be written as follows:

(14)

The above equation is the strict voltage-phase relation. The Josephson current-phase relation is obtained by noticing that

IV =Wex=/~c=S-" ~ ~aEc Ok,

2

H =E

(13)

where q~ = O1 - 02 is the phase difference across the barrier. The energy difference across the junction is just 2 eV for the tunneling electrons, so that ~b = 2eV/h.

Wex = f IV dt,

(12)

(15)

k

EkNk + Ee(NI, N2, Ol - 02)

so that by Eq. (14)

k=l

--Wex (N1, N2, O1,02),

(5)

where Ec = - K ~ cos(O1 - 02). Treating now the phases Ok and the quantities hNk as conjugate variables, one can determine the dynamics of the system by Hamilton's equations h/Vk = -- OH = _ O._.O_(Ec _ Wex),

aOk aOk 1 OH Ek+ 1 0 Ok= hON~ - 1~ -h"~k (Ee -- Wex).

(6) (7)

By imposing energy conservation we write

n0= E1 Uj +E2 N2 + Ec=We~.

(8)

On the other hand, since the total number of the particles must be conserved, we must set N1 + /72= 0. Therefore, Eq. (8) may be rewritten as follows: Wex= (El -- E2)

/VI "t- Ec.

(9)

Integrating with respect to time, we have Wex = Ee + [ ( E l -- E2) J

hi

dt.

(10)

I -- ~ - ~ - ~

sin~o.

(16)

In what follows we shall generalize the above analysis to the case of three electrodes, assuming that the presence of the intermediate electrode is thin enough to allow an overlap of the order parameters of the two outer electrodes. In Section 3 we shall analyze some characteristic electromagnetic responses of the system. In particular, on the basis of the modified Josephson equations obtained in Section 2, we predict (1) an enhancement of the maximum Josephson current of the series connection of the two junctions with respect to that expected from the usual Josephson relations of the junctions, (2) the appearance of half-integer current steps in the I - V characteristic, if the system is irradiated by an rf source. Conclusions will be drawn in the last section.

2. The modified Josephson equations We now turn to the study of the constitutive equations for the system of three stacked superconduct-

G. Carapella et al./Physica C 259 (1996) 349-355

Is1 Is21 $3

where the coupling energy Ee and the energy of the interaction with the external circuit Wex are taken to depend on all the variables Nk and Ok. The expression for the coupling energy Ee is thus the following:

I I

Ee = -

I2

+Q) v1

ing layers, represented in Fig. 1. Usually, the Josephson current-phase relation is described only in terms of tunneling Cooper pairs from one electrode to its next neighbor. However, if the intermediate electrode is thin enough to allow superposition of the superconducting order parameters of the outer electrodes, one can infer that a direct coupling between the latter two electrodes is present. Following this physical idea, we extend the previous analysis to the case of two strongly coupled JJs by adopting the following form of the semi-classical Hamiltonian H0:

Ho = (¢,~, ¢,~', ¢,;) -K~2/2 ~2 -K23/2] -K~3/Z-K23/2 E3 ]

= ~ Ekl~,kl2 - ½~ Kij~l; ~]j, k

(17)

where, again, Ek is the energy of the kth electrode and the K~i are positive coupling constants relative to the ith and the jth electrodes (Kij = Kji). The particular form of the external circuit, represented in Fig. 1, leads us to write the energy of interaction Wex as follows: (18)

where Vl and ½ are the voltages across the first and second junction, respectively, and 11 and 12 are the currents flowing in these same junctions. Therefore, by setting ~k = x/-~e -i°~, (k = 1,2,3), we may write the full Hamiltonian H as follows: 3

H =~ k=l

(j = 1,2,3),

h Nk=

OH O'Ok =

I OH

0 -- OOk ( Ee --

Ek

Wex),

1 0 "E

Ok= h0-'-~k = ~ + h~'~k ( e -- Wex),

(19)

(21 ) (22)

with the index k running from 1 to 3. By conservation of energy, particle conservation and the constant temperature thermal bath hypothesis, we find that the quantity Ec - Wex is a constant that does not depend on the dynamic variables. Therefore, Eq. (21 ) reduces to an identity and Eq. (22) gives

Ol - 02 = (El - Ez)/h,

(23)

E3)/h.

(24)

By now defining ~ol = Ol - Oz and ~o2 = 02 - 03 and setting E1 - E2 = 2e~] and E2 - E3 = 2 e ½ , we may write:

qik = 2eVk/h,

k = 1,2.

(25)

The modified Josephson current phase relations are obtained by setting Ilr~ + 12½ =Wex

=~

~

/c=-I

ok = + ~

~

ok.

(26)

k=-I

In fact, from Eqs. (20) and (25) we get 2e 11 = - - ~ - ~

KI2 sin ~Ol

2e +'--~x/NIN3 K13 sin(q~l + Pz),

EkNk + E c ( N j , Oj) -- Wex(Nj, O j ) ,

(20)

In analogy with the previous analysis, the dynamics of the conjugate variables hNk and Ok is given by Hamilton's equations:

9 2 -- 0 3 = ( E 2 --

i~ j

Wex = f (I~V~ + I2Vz) dr,

K12 c o s ( O l - 0 2 )

-- Nv/-N-~IN~/(13cos(a1 - 03).

V2

Fig. 1. External bias circuit for three superconducting layers.

×

~

-v/N'zN3 K23 cos(a2 -- 03)

+Q)-

-

351

(27)

352

G. Carapella et al./Physica C 259 (1996) 349-355

2e 12 = -h-"~

and 9 o constant. In this way, Eq. (29) can be written as follows:

K23 sin 92

+~-~~K13

sin(91 + 92).

(28)

Eqs.(25), (27), and (28) represent the modified Josephson equations, predicted by the extension of Feynman's model to the case of two strongly coupled JJs. These equations can be recast in a more compact form if we define: Icl = ( 2 e / h ) ~ K 1 2 , Ic2 = ( 2 e / h ) Nv/-~2-~aK23, and I13 = (2e/h) Nv~1N3 K13. In fact, making use of the above definitions, we can rewrite the current-phase relation as follows: lk=IckSingi+Ii3sin(91+92),

k = 1,2.

(29)

Moreover, the relation Wex = Ec +const. and Eq. (20) we get the free energy of the Josephson stack structure in terms of the phase variables 91 and 92: Vex = ~/~Id e ( 1 - cos 91 ) + -hI~2 ~ e ( 1 -- cos 92 ) Ml3 [ 1 +-T$-e

c o s ( 9 1 + 92) ]

(30)

Alternatively, one can evaluate this expression for Wex by startingfrom the Eq. (18) and carrying out the integrations with the aid of Eqs. (29) and (25).

1 ( 9 ° , 9 °) =/el sing° + 1,3 sin(9° + 9°),

(32)

sin ~o° = (Icl/le2) sin 9 °.

(33)

The maximum of 1(9°,90 ) in the domain of the phase variables fixed by Eq. (33) gives the desired maximum supercurrent. In the 91-92 plane Eq. (33) has the following solutions: 9° =arcsin ( ~ 2 sin9 °) +27rk,

(34)

9° = ~--arcsin (~e-e~ sin90) + 2~-k,

(35)

where k is an arbitrary integer. The above solutions, when substituted in Eq. (32), give two different restrictions for the supercurrent I (9 °, 9 °). We indicate with I+ (90) the restriction of I ( 9 °, 9 °) to the subdomain specified by the first solution and with I_ (90) the restriction of 1 (9 °, 9 °) to the subdomain specified by the second solution. Therefore, we obtain I+ =Iel sin9 ° +1,3 sin ( 9 ° + arcsin ( / ~ sin 9 ° ) ) ,

(36)

I_ =/el sin9 ° 3. Some characteristic responses

-1,3 sin ( 9 ° - a r c s i n (/~2 sin91°)) .

3.1. Supercurrent in the series connection stack

Eqs. (25) and (29) describe, respectively, the voltage-phase and current-phase relations in stacked JJs. In what follows the maximum Josephson current of the series connection will be found, assuming that the critical currents of the single junctions in Fig. 1 are such that Icl <_ Ic2. If both junctions are in the zero voltage state, one can write the additional relation Il = 12 = I from charge conservation, so that Icl sin 91 = Ic2 sin 92.

(31)

The above expression represents a constraint: only the points in the 91-92 plane which satisfy Eq. (31) are allowed. To calculate the supercurrent with the junctions in the zero voltage state, we set ~ = V2 -- 0, so that Eq. (25) gives 91 = 90 and 92 = 9 °, with 90

(37)

The functions I+ and I_ are periodic with period 27r. Therefore, we may use numerical calculations to determine their maxima in the periodicity interval. The maxima of I+ (I_) versus I13 are shown in Fig. 2 (Fig. 3) for different values of the ratio le2/Icl. By comparison of Fig. 2 and Fig. 3 we notice that I+ max > I_ max SO that the maximum Josephson current of the series connection is always determined by I+. From this result it is also evident that the modified Josephson current-phase relations as given in the Eq. (29) predict, at zero voltage, a maximum Josephson current value Imax which is always greater than lcl. Moreover, if the critical current values Icl and le2 are known, by the knowledge of/max from the experimental I - V characteristic, one can estimate the coupling current 1,3. In particular, referring to recent experiments on stacked JJs by Monaco et al. [7], it has been found

G. Carapella et al.IPhysica C 259 (1996) 349-355

2.0

•

i

,

•

,

Ic2 / I c 1.8 ~-~

1

. . . . .

1.6

I~

lO

~ ................

~

following relations between the phase variables:

,

i .............

........ 1"2i .... 1.4 .......... 2

1.4 ..... ~

)

.......:

...............!- ::.':::.-~-~-

i

353

....... .;:'J'"

~o = ~0 + 2~rk,

(38)

~o = ~r - ~0 + 2~rk.

(39)

"

.......................iY:": .>.;:.'5~ ............. .......i/'f "~'" : ........... .:::"i./'2"fi~i ........... .............

Similary to what has been previously reported, these two solutions give

+

1.2 1.0

,

0.0

0.2

,

0.4 0.6 113 / Icl

0.8

1.0

Fig. 2. M a x i m u m value of supercurrent I + as a function of the coupling current I13 for different values of the critical current Ie2 shown in the inset. All current values are normalized to the critical current v a l u e / e l .

that to a critical current lel ~ 1.60 mA, there correspond the ratios Imax/Iel ~ 1.11, and Ie2/Iel ~ 1.41, so that, by Fig. 2, one can estimate the ratio I13/Iel to be approximately 0.15. In an analogous experiment by Nevirkovets et al. [8] it has been found that to a critical current lel ~ 1.22 mA, there correspond the ratios Imax/Icl "~ 1.8, and le2/lel "~ 1.2, so that, by Fig. 2, one can estimate the ratio I13/lel to be approximately 0.97. In the case of identical junctions (Ie2 = / e l = I0) the value of/max can be derived analytically. In this case the constraint represented by Eq. (33) gives the

1.0 ......

~

.......

~ ....

o.81 fiOrD

0.6" ~0.4" - ~ O.

l+(~P0) = I0 sin~p + I~3 sin2~,

(40)

I - ( ~ 0 ) -- I0 sin~.

(41)

As before the maximum of I+(~o0) determines the value of/max which can be found to appear at the following phase value ~max: cos max =

I0

............ ~--I02

............i...............i............. ~ i .............

0.(3 0,0

0'.2

0.4

0.6

018

1.0

I13 / Icl Fig. 3. M a x i m u m value o f supercurrent 1_ as a function o f the coupling current 113 for different values o f the critical current le2 shown in the inset. All current values are normalized to the critical current value 1¢1.

1+32~

) .

(42)

This solution gives the following expression for/max: /max = (I0 + 21~3 COStPmax)v t l -- COS2~max.

(43)

This curve is seen to be identical to the one calculated numerically for Ie2 = let and reported as the bottom curve in Fig. 2.

3.2. Half-integer Shapiro steps In the present section we report a simple analysis of the response of the stack of two strongly coupled Josephson junctions to an applied rf signal. In the same hypotheses of the voltage bias model [9] we assume that the effect of the external radiation is to produce an alternating electric field of the same frequency across both junctions. We focus our attention on the case when the junctions are in the same dynamical state; i.e., when ~1 (t) = ~02(t)

"'- ....... 1.2 ........ !................. ! .............. ::!......::.........

-1+

=

~o(t).

(44)

Let us consider the series-connected stack and denote by I the current flowing in both junctions, assumed to be equal, so that, by Eq. (29) and Eq. (44), we have I = I0 sinq~ + I13 sin2~o,

(45)

where I0 = Ici = Ie2 is the critical current of both junctions. We may then write the voltage V(t), as measured on the first junction as

V ( t ) = VDc + v cos tort,

(46)

G. Carapella et aL /Physica C 259 (1996) 349-355

354

where VDC is the DC voltage across the barrier, and v is the amplitude and mr is the angular frequency of the applied rf signal. By integrating the voltage-phase relation for the first junction, we get

~( t ) = ~o + tojt + a sincort,

(47)

where ~o0 is a constant, oaj = (2e/h)VDc and a = 2ev/hoar. In this way, Eq. (45) can be expressed in terms of the time variable as follows: I (t) = Im{I0 e i(~°°+wjt+a sin tort) +I13 e 2i(q~°+~°jt+a sin tort) }.

(48)

Let us now make use of the following series expansion of the quantity e ia sin o,rt in terms of Bessel functions of integer order k:

voltage-frequency relation differs from the classical one by a factor !2" It is worth mentioning that, if these half-integer Shapiro steps will be experimentally observed, they could provide a direct measurement of the coupling current I~3. The value of 113 SO obtained could then be compared with that derived by a different method. For example, one can estimate the value of 113 obtained by the dependence of the amplitude 11 of the first halfinteger step versus the square root of the rf power 11 = I13lJl (2a)1,

(53)

and compare it with the one derived by the analysis of the maximum Josephson current in the zero-voltage state described in Section 3.1.

oo

eia sin o~,.t = E

J k ( a ) e ik°~rt,

(49)

4. Conclusion

k=---o~

and rewrite Eq. (48) in the following form: a ) e i(~°l+l°ar)tei~

I ( t ) = Im { I o ~ J l ( /----oo

"t-]13 E

J~(2a)ei(2°~J+k~°r)te 2i~ ,

(50)

k----~

where we have purposely distinguished the indices of the two infinite sums. As in the case of a single junction, we expect to see current steps of amplitude I, (n integer) in the I - V characteristic of the first junction of the stack when the argument of the oscillatory term in Eq. (50) vanishes for all t. Imposing the above condition to Eq. (50), we find that current steps appear at the following voltage values: n(h) Vn = ~

/'r,

(51)

where n is an integer and /~r = Oar/27r, and find that they have the following amplitudes:

ln=1131Jn(2a) l,

n odd

=max[IoJ,/2(a) sin ~00 + I13Jn(2a) sin2~00], n even.

(52)

From Eq. (51 ) we notice the appearance of half integer Shapiro steps, i.e., of current steps for which the

A set of modified Josephson equations for a system of two strongly coupled junctions has been derived by extending Feynman's model to the case of three superconducting layers. We found that, if the intermediate electrode is thin enough to allow the superconducting order parameters of the outer layers to propagate through, an additional term in the Josephson currentphase relations appears. This term is proportional to the coupling energy constant between the outer layers and is modulated by the sine of the sum of the superconducting phase differences across the junctions. We propose two independent experimental procedures to detect the effect of this additional term. The first consists in measuring the enhancement of the maximum Josephson current of the series connection of the two strongly coupled junctions in the zerovoltage state, with respect to the corresponding value given by the usual Josephson current-phase relation. The second consists in detecting half-integer Shapiro steps in the single junction I - V characteristics as the system response to an rf electromagnetic radiation. We may finally notice that the strong coupling hypothesis is particularly relevant in the case of Josephson-coupled multilayer models of high-T¢ superconductors. In fact, in this latter case the electrode widths are of the order of a single atomic layer, given the particular structure of this class of superconductors. Generalization of our results to more complex structures than three superconducting thin layers is

G. Carapella et al./Physica C 259 (1996) 349-355

straightforward. Therefore, in extending Josephson coupled multilayer models [ 10] to second next neighbor interaction, the modified Josephson current-phase relation should be taken into account.

Acknowledgements We highly appreciate the numerous discussions and the constant input of suggestions by Professor R.D. Parmentier and by R. Monaco. One of the authors (S.C.) gratefully acknowledges CNR fellowship support.

355

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