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Institute of Physics Publishing doi:10.1088/1742-6596/43/1/269

Journal of Physics: Conference Series 43 (2006) 1105­1109 7th European Conference on Applied Superconductivity

Dynamics of underdamped Josephson junctions with nonsinusoidal current-phase relation
V. K. Kornev (1), T. Y. Karminskaya (1) Y.V. Kislinskii (2), P.V. Komissinki (3), K.Y. Constantinian (2) G.A. Ovsyannikov (2,3) (1) Physics Department, Moscow State University, Moscow 119992, Russia (2) Institute of Radio Engineering and Electronics, RAS, Moscow 125009, Russia (3) Chalmers University of Technology, S 41296, Gothenburg, Sweden kornev@phys.msu.ru
Abstract. Results on analytical and computational investigations of high frequency dynamics of Josephson junctions, characterized by nonzero capacitance and the second harmonic in the current-phase relation are presented. These attributes each influence on behaviour of integer Shapiro steps and lead to formation of non-integer Shapiro steps. Analytic theory of the integer and non-integer Shapiro steps has been developed for so-called high frequency limit. The analytical and numerical results are compared with experimental data for hybrid heterostructures YBCO/Au/Nb. Detector response for the case of high fluctuation level has been considered as well.

1. Introduction When rf signal is applied to Josephson junction its IV-curve shows a set of Shapiro steps resulting from phase-locking of Josephson oscillations. Analytical description of the Shapiro step dependence on the signal amplitude were obtained only for a high-frequency limit in the frame of Resistively Shunted Junction (RSJ) model [1] describing an overdamped junction with McCumber parameter 2 2 I C RN C / 0 1 . At the same time, many types of Josephson junctions do not meet the model. Most of all, this concerns to the junctions on the base of high-Tc d-wave superconductors. Such a junctions are usually characterized by some digression from the sinusoidal current-phase relation, which is put in RSJ model, and also by parameter value > 0. Both the factors can cause origin of the sub-harmonic steps unavailable in the frame of RSJ mode. Among the junctions one should mention SND (s-wave superconductor / normal metal / d-wave superconductor) Josepson junctions [2-3]. In this work we deliver results of analytical theory for dependence of the harmonic and subharmonic Shapiro step amplitude on amplitude of the applied rf signal taking into account the impact of both factors: and second harmonic in the current-phase relation. The theory is developed for so-called high-frequency limit, when at least one of the three following conditions is fulfilled: >> 1, or
2

>> 1,

or a >> 1

(1)

© 2006 IOP Publishing Ltd

1105


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(frequency and the rf signal amplitude a are normalized by characteristic Josephson frequency c and voltage Vc correspondingly). The analytical results are compared with data of numerical simulation and experimental data for S/N/D junctions. 2. Analytical Theory Approach The analytical consideration of Josephson junction dynamics is performed using the following master equation: (2) sin q sin 2 i a sin( t ) if , where the bias current i and fluctuation current if are normalized by critical current Ic , and factor q describes the second harmonic contribution The term (sin sin 2 ) is a small parameter in the extreme case (1), therefore Josephson-junction phase and constant component of the current i can be presented as expansions in order of vanishing:
0 1 2

... ,

i

i

0

i1

i

2

... ,

(3) (4)

and equation (2) can be reduced to the set the of equations as follows:
0
1
2

0
1
2

i
i

0

a sin( t )
sin(
1

if ,
0
1

i1
2

0

) q sin( 2
0

),
0

(5)

cos(

) 2q

cos( 2

).

(6)

The 0-order approximation (solution of eq. (4)) describes autonomous negligible fluctuations (if = 0) the first- and second-order approximations and (6) describe accordingly harmonic and sub-harmonic Shapiro steps. corresponds to large-scale fluctuations inasmuch as the term if is put approximation. In such a case the first- and second-order approximations and (6) describe detector response at high fluctuation level. 3. Negligible Fluctuations

I-V curve. In the case of that can be found from (5) The opposite case of if 0 in equation (4) for 0-order that can be found from (5)

3.1. The case q=0 At q = 0 the amplitudes of harmonic Shapiro steps results from equation (5). The step amplitudes are described by the following expressions: in 2 J n ( x) , . (7)

x
If

a/

(

)

2

1

(8)

= 0 formulas (7) and (8) coincide with the well known ones for RSJ model [1]. Amplitudes of the sub-harmonic Shapiro steps result from equation (6). The sub-harmonic step amplitudes are described by the following sum:

i(

2n 1) / 2

2
mn

J

(2n 1) m

( x) J m ( x) /

2

(2n 1) / 2 m

2

1.

(9)

Keeping only major term, one can reduce the sum as follows:

i

( 2 n 1) / 2

2J

n1

xJ

n

x /[

2

/ 4 1] .

(10)

3.2. The case q 0 Equation (5) gives the following formula for the harmonic Shapiro step amplitudes:

i

n

2 max J n ( x) sin( ) qJ

2n

(2 x) sin(2 ) ,

(11)

where x is defined by (8). This formula can be extended for the case of several harmonics in the


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3/2-step amplitude, i

1/2-step amplitude, i

0-step amplitude, i/2

0,20 0,16 0,12 0,08 0,04 0,00 0 20

1,6

1-step amplitude, i

0,12

0,09

Theory Simulation Experiment

1,6 1,2 0,8 0,4 0,0 0 20 40

Experiment Theory q=0 Theory q=1

0,06

1,2

0,03

0,00 0 30 60 90 120

0,8

60

80

Signal amplitude, a

Signal amplitude, a

0,4

0,0
40 60 80 100

0

20

40

60

80

100

Signal amplitude, a

Signal amplitude, a

Fig. 1. Left side - Dependences of the 1/2- and 3/2-step amplitudes on the applied signal amplitude a at frequency = 0.611, = 35 and q = 0. Solid line corresponds to formula (10); filled dots - numerical simulation, and empty dots - experimental results for the c-oriented Nb/Au/YBCO junctions. Right side - Dependences of the critical current amplitude i/2 (0-step) and the 1-step amplitude i (in nset) on the applied signal amplitude a at frequency = 1.62 and = 4. Dashed and solid lines correspond to formula (11) at q = 0 and q = 1correspondingly, the filled dots correspond to experimental results for the c-tilted Nb/Au/YBCO junctions.

junction current-phase relation as follows: i n 2 max{ q k J
k

kn

(kx) sin(k )} .

(12)

And finally, the sub-harmonic Shapiro step amplitudes resulting from equation (6), are given by the following expression: J 1 ( x) J 0 ( x) J (2 x) J 0 ( 2 x) (13) i1 / 2 2 max[sin( ){qJ 1 (2 x) 4q 2 2 cos( )] , ( )2 / 4 1 ( )2 1 where x is defined by (8) as well. Fig. 1 and Fig. 2 present the analytical results, as well as experimental data for both the c-oriented and c-tilted Nb/Au/YBCO junctions formed on NdGaO substrates (junction areas were ranged from 10x10 m2 to 30x30 m2) and measured at 4.2 K under electromagnetic irradiation at frequency

1/2-step amplitude, i

0,15

1/2-step amplitude, i

0,20

Experiment Theory q=0 Theory q=0.14 Theory q=0.3

0,20

Experiment Theory q=0 Theory q=0.14 Theory q=0.3

0,15

0,10

0,10

0,05

0,05

0,00 0 20 40 60 80 100

0,00 0 25 50 75 100

Signal amplitude, a

Signal amplitude, a

Fig. 2. Dependence of the 1/2-step amplitude i on the applied signal amplitude a at = 4 for frequencies = 1.62 (left side) and = 2.2 (right side). Dashed, solid and dotted lines correspond to the step behaviour given by formula (13) accordingly at q=0, q=0.14 and q=0.3. The filled dots are experimental data for the c-tilted Nb/Au/YBCO junction.


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36В120 GHz. [2-3]. In the latter case the S/N/D (1 1 20) YBCO have been prepared on specially inclined growth of epitaxial YBCO. The c-oriented = 35, while the parameters for the c-tilted junctions

heterojunctions based on single-domain films of oriented (7 10 2) NdGaO substrates, yielding in junction parameters were estimated as q = 0 and are q = 0.14 and = 4.

4. Detector response Detector response resp = i(v) - ia(v) is difference between the I-V curve under rf signal impact and the autonomous one. As a rule, it is more convenient to use the frequency difference n = n - v instead of normalized voltage v. 4.1. The case of negligible fluctuations In the case of negligible fluctuations the set of equations (4)-(6) yields the harmonic detector response for arbitrary as follows:
resp J n ( x) , if J n ( x) 2 /
n n

0
22 n

,
1 , if
n

(14)

0

4.2. Large-scale fluctuations We have considered impact of the large-scale fluctuations on detector response in the high-frequency limit. In this case, when noise-factor is much more than 1 and therefore the term if is put in equation (4), the set (4)-(6) allows to analyse detector response at arbitrary values of and q. When q = 0 and = 0, the harmonic detector response is described by the simple expression:

resp

1 J 2

n

2

( x)
n

n 2 2 1

.

(15)

At arbitrary value of and q = 0, more complicated expression takes place:

resp

1 J 2

n

2

( x)
n

n 2 2 1 n 2

n

(

1

1/ )

2

.

(16)

In the general case of arbitrary values of and q the harmonic detector response is as follows:

resp

1 J 2

n

2

( x)
n

n 2 2 1 n 2

n

(

1

1/ ) 12 qJ 2

2

(17)
2n 2

(2 x)

n 2 2 1 2

n 2

.

( 1 2/ ) n n The second harmonic in current-phase relation yields also sub-harmonic detector response:

resp
where
n

q2J

n

2

(2 x)
n

n 2 2 1 n 2

n

(

1

2/ )

2

,

(18)

2v n . In all the expressions (14)-(18) argument x is given by (8).

5. Conclusion Generalizing formulas both for harmonic and sub-harmonic Shapiro steps in the presence of nonzero junction capacitance and second harmonic in current-phase relation are obtained. The analytical theory generalizes the well-known high-frequency-limit consideration developed earlier for RSJ model [1] to the stated departures from RSJ model. The formulas are verified by numerical simulation and mainly by experimental results for YBCO/Au/Nb heterostructures. Some quantitative disagreement of the


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experimental data, which takes place mostly for sub-harmonic steps shown in Fig. 2, follows from distributed character of the junctions with the size of order of characteristic Josephson length J. At relatively small signal amplitude a, harmonic detector response is proportional to a2n i.e. linear in respect to the signal power P at n = 1, and proportional to Pn at n > 1. One should emphasize that the consideration of second harmonic in the junction current-phase relation gives the second-order contribution to the harmonic responses, and the main contribution proportional to power P to the subharmonic responses at v n / 2 . It means that observation of the sub-harmonic response enables the mostly sensitive way to detect second harmonic in current-phase relation. Acknowledgement This work was supported in part by ISTC Grant 2369, and Russian Grant for Scientific School (contract 02.445.11.7169). References [1] Likharev K K 1986 Dynamics of Josephson junctions and circuits., (New York: Gordon and Breach.) ch. 8. [2] Komissinski F V, Kislinskii Yu V, Ovsyannikov G A, et al. 2004 Low Temp. Phys. 30 599-610. [3] Kislinskii Y V, Komissinski F V, Constantinian K I, Ovsyannikov G A, Karminskaya T Y, Soloviev I I, Kornev V K 2005 Journal of Exp. and Theoretical Physics 101 3 494-503. Translated from Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 128 3575-585.