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Numerical simulation of the self-pump ed Long Josephson junction using a modified Sine-Gordon model

A.S. Sobolev a,, A.L. Pankratov b, J. Mygind
a

c

Institute of Radio Engineering and Electronics (IREE), Mokhovaya st. 11/7, Moscow, 125009, Russia
b

Institute for Physics of Microstructures of RAS, Nizhny Novgorod, Russia
c

The Danish Technical University, Kgs. Lyngby, Denmark

Abstract We have numerically investigated the dynamics of a long Josephson junction (FluxFlow oscillator) biased by a DC current in the presence of magnetic field. The study is p erformed in the frame of the modified sine-Gordon model, which includes the surface losses, RC-load at b oth FFO ends and the self-pumping effect. In our model the dumping parameter dep ends b oth on the spatial coordinate and the amplitude of the AC voltage. In order to find the DC FFO voltage the damping parameter has to b e calculated self-consistently by successive approximations and time integration of the p erturb ed sine-Gordon equation. The modified model, which accounts for the presence of the sup erconducting gap, gives b etter qualitative agreement with exp erimental results compare to the conventional sine-Gordon model. Key words: Tunneling, Josephson effect, Josephson devices PACS: 74.50. + r 03.75.Lm

Preprint submitted to Elsevier Science

10 Septemb er 2005


During the last decade the flux-flow oscillator (FFO) has been considered as the most promising lo cal oscillator in superconducting integrated submillimeter receivers (SIRs) for atmosphere monitoring due to their broadband tunability and high radiation power. For making optimal FFO design, satisfying technical requirements for practical applications a proper mathematical mo del is needed, which would comprise the effects important both at low and high frequencies. For many decades the sine-Gordon mo del has been the most adequate mo del for the long Josephson junction, giving a go o d qualitative description of its basic properties, such as Fiske steps, vortices dynamics, etc. In this mo del the electro dynamics of a long Josephson junction in the presence of magnetic field is described by the perturbed sine-Gordon equation tt + t - = + (x) - sin() (1 )

xx

xxt

sub ject to the boundary conditions (0,t)x + rL cL xt - cL tt + xt = and (L, t)x +rR cR xt +cR tt +xt = . Here space and time have been normalized to the Josephson penetration length J and to the inverse plasma frequency
- p 1 , respectively, is the damping parameter, (x) is the normalized DC

bias current density and is the normalized magnetic field, is the surface losses parameter, c
L,R

and r

L,R

are the parameters of the FFO RC-load on the

left and the right ends, respectively. If the mo del parameters are determined close to the practical ones the numerical simulations of Eq.(1) give a mo derate qualitative agreement with experimental FFO IV-characteristics (see Fig.1).
Corresp onding author, phone: (7-095) 203-27-84. Email addresses: sobolev@hitech.cplire.ru (A.S. Sob olev), alp@ipm.sci-nnov.ru (A.L. Pankratov), myg@fysik.dtu.dk (J. Mygind).

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The set of IV-curves, represented in Fig. 1a for the Nb/AlN/Nb FFO has two different operational regimes. At the voltages above 1.9 (the so called "boundary voltage", Vb ) which is 1/3 of the Nb gap voltage, the damping parameter drastically increases due to the self-pumping effect [1]. It results in increase of the quasiparticle current leading to transformation of the resonant Fiske steps into a set of smo oth flux-flow curves, where continuous tuning of the FFO frequency is possible. In order to extend the sine-Gordon mo del to the "flux-flow" region one should incorporate in it the self-pumping effect, when is defined self-consistently from the amplitude of the AC-voltage and to include surface losses. If I (Vdc ) is the DC IV-curve of the unpumped Josephson junction and a high-frequency signal is applied to the junction so that the total voltage is V (t) = Vdc + V eit then according to [2] the total DC quasiparticle tunneling current Ipump of the pumped junction will be given by Ipump (Vdc , ,Vac ) =
n=+ n=-

J

2 n

eVac

I (V

dc

+ n /e), where Jn are the Bessel

functions. One can use this formula to take into account the self-pumping effect treating the Josephson radiation of the junction as an external signal. Therefore, we to ok V
dc

=

/2e and the parameter in the Sine-Gordon

mo del (which has to be dependent on the co ordinate x) is defined as the ratio of Ipump and Vdc : = Ipump (Vac (x),x)/Vdc . With these mo difications the self-pumped FFO IV-curve can be numerically computed using the iterative pro cedure combined with the implicit difference scheme for the solution of Eq. 1. The IV-curves, obtained by this approach are shown in Fig.1a (crosses, diamonds and circles). They have a step-like peculiarity on the fo ot of the curves at the "boundary voltage", similar to that observed for the experimentally measured IV-curves (Fig.1a, thin lines). The functions (x) for the three Ib values are shown in Fig.1b together with the distribution of the applied bias current (x), which was the qualitative approximation based on the topology 3


Fig. 1. (a) The FFO current-voltage characteristics. Thin lines - exp erimental measurements. Numerical simulations with the account of the self-pumping effect: for = 2.2 - crosses, for = 3.5 - diamonds, for = 3.6 - circles. (b) The function (x) for the three currents Ib , = 3.5: curve #1 - Ib =0.3, curve #2 - Ib =0.2, curve #3 - Ib =0.1. The dashed line represents the distribution (x).

of the FFO being studied. Other parameters had the following values: rL = 3, rR = 8, cL = cR = 10, = 0.04, the junction's length L = 40. In our simulations we to ok I (Vdc ) of the unpumped FFO as I = 0.028 V and as I = 0.08 V
dc dc

for V

dc


g

for Vdc < Vg .

The work has b een supp orted by the RFBR (Pro jects No. 03-02-16533 and 03-0216748), INTAS (Pro ject No. 01-0367), ISTC (Pro jects No. 2445 and 3174), Russian Science Supp ort Foundation, the President Grant for Scientific Schools 1344.2003.2, the Danish Natural Science Foundation and the Hartmann Foundation.

References

[1] V.P. Koshelets, S.V. Shitov, A.V. Shchukin, L.V. Filipp enko, J. Mygind, A.V. Ustinov. Phys. Rev. B 56, 5572 (1997). [2] J.R. Tucker, M.J. Feldman, Rev. Mod. Phys. 57, 1055 (1985).

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