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Pinned fluxons in a Josephson junction with a finite-length inhomogeneity

Published online by Cambridge University Press:  26 August 2011

GIANNE DERKS ,
ARJEN DOELMAN ,
CHRISTOPHER J. K. KNIGHT  and
HADI SUSANTO
GIANNE DERKS
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK emails: g.derks@surrey.ac.uk, christopher.knight@surrey.ac.uk
ARJEN DOELMAN
Affiliation:
Mathematisch Instituut, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands email: doelman@math.leidenuniv.nl
CHRISTOPHER J. K. KNIGHT
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK emails: g.derks@surrey.ac.uk, christopher.knight@surrey.ac.uk
HADI SUSANTO
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK email: hadi.susanto@math.nottingham.ac.uk
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Abstract

We consider a Josephson junction system installed with a finite length inhomogeneity, either of micro-resistor or micro-resonator type. The system can be modelled by a sine-Gordon equation with a piecewise-constant function to represent the varying Josephson tunneling critical current. The existence of pinned fluxons depends on the length of the inhomogeneity, the variation in the Josephson tunneling critical current and the applied bias current. We establish that a system may either not be able to sustain a pinned fluxon, or – for instance by varying the length of the inhomogeneity – may exhibit various different types of pinned fluxons. Our stability analysis shows that changes of stability can only occur at critical points of the length of the inhomogeneity as a function of the (Hamiltonian) energy density inside the inhomogeneity – a relation we determine explicitly. In combination with continuation arguments and Sturm–Liouville theory, we determine the stability of all constructed pinned fluxons. It follows that if a given system is able to sustain at least one pinned fluxon, a microresistor has exactly one pinned fluxon, i.e. the system selects one unique pinned stable pinned configuration, and a microresonator has at least one stable pinned configuration. Moreover, it is shown that both for micro-resistors and micro-resonators this stable pinned configuration may be non-monotonic – something which is not possible in the homogeneous case. Finally, it is shown that results in the literature on localised inhomogeneities can be recovered as limits of our results on micro-resonators.

Keywords

Josephson junctionInhomogeneous sine-Gordon equationPinned fluxonStability
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 2 , April 2012 , pp. 201 - 244
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Ablowitz, M. J., Kaup, D. J., Newell, A. C. & Segur, H. (1973) Method for solving the sine-Gordon equation. Phys. Rev. Lett. 30, 1262–1264.CrossRefGoogle Scholar
[2]Akoh, H., Sakai, S., Yagi, A. & Hayakawa, H. (1985) Real time fluxon dynamics in Josephson transmission line. IEEE Trans. Magn. 21, 737–740.CrossRefGoogle Scholar
[3]Andreeva, O. Yu., Boyadjiev, T. L. & Shukrinov, Yu. M. (2007) Vortex structure in long Josephson junction with two inhomogeneities. Physica C 460–462, 1315–1316.CrossRefGoogle Scholar
[4]Benabdallah, A., Caputo, J. G. & Flytzanis, N. (2002) The window Josephson junction: A coupled linear nonlinear system. Physica D 161, 79–101.CrossRefGoogle Scholar
[5]Benabdallah, A. & Caputo, J. G. (2002), Influence of the passive region on zero field steps for window Josephson junctions. J. Appl. Phys. 92, 3853–3862.CrossRefGoogle Scholar
[6]Boyadjiev, T. L., Semerdjieva, E. G. & Shukrinov, Yu. M. (2007) Common features of vortex structure in long exponentially shaped Josephson junctions and Josephson junctions with inhomogeneities. Physica C 460–462, 1317–1318.CrossRefGoogle Scholar
[7]Boyadjiev, T. L., Andreeva, O. Yu., Semerdjieva, E. G. & Shukrinov, Yu. M. (2008) Created by current states in long Josephson junctions. Europhys. Lett. 83, 47008.CrossRefGoogle Scholar
[8]Caputo, J. G., Efraimidis, N., Flytzanis, N., Lazaridis, N., Gaididei, Y., Moulitsa, I. & Vavalis, E. (2000) Static properties and waveguide modes of a wide lateral window Josephson junction. Int. J. Mod. Phys. C 11, 493–518.CrossRefGoogle Scholar
[9]Carr, L. D., Mahmud, K. W. & Reinhardt, W. P. (2001) Tunable tunneling: An application of stationary states of Bose–Einstein condensates in traps of finite depth. Phys. Rev. A 64, 033603.CrossRefGoogle Scholar
[10]Derks, G., Doelman, A., van Gils, S. A. & Susanto, H. (2007) Stability analysis of π-kinks in a 0-π Josephson junction. SIAM J. Appl. Dyn. Syst. 6, 99–141.CrossRefGoogle Scholar
[11]Derks, G., Doelman, A., van Gils, S. A. & Visser, T. (2003) Travelling waves in a singularly perturbed sine-Gordon equation. Physica D 180, 40–70.CrossRefGoogle Scholar
[12]Goldobin, E., Vogel, K., Crasser, O., Walser, R., Schleich, W. P., Koelle, D. & Kleiner, R. (2005) Quantum tunneling of semifluxons in a 0-π-0 long Josephson junction. Phys. Rev. B 72, 054527.CrossRefGoogle Scholar
[13]Goodman, R. H. & Haberman, R. (2007) Chaotic Scattering and the n-Bounce Resonance in Solitary-Wave Interactions Phys. Rev. Lett. 98, 104103.CrossRefGoogle ScholarPubMed
[14]Goodman, R. H. & Weinstein, M. I. (2008), Stability and instability of nonlinear defect states in the coupled mode equations – Analytical and numerical study. Physica D 237, 2731–2760.CrossRefGoogle Scholar
[15]van Heijster, P. J. A., Doelman, A., Kaper, T. J., Nishiura, Y., Ueda, K.-I. (2011) Pinned fronts in heterogeneous media of jump type. Nonlinearity 24, 127–157.CrossRefGoogle Scholar
[16]Hilgenkamp, H. (2008) π-phase shift Josephson structures. Supercond. Sci. Technol. 21, 034011.CrossRefGoogle Scholar
[17]Kivshar, Yu. S., Kosevich, A. M. & Chubykalo, O. A. (1988) Finite-size effects in fluxon scattering by an inhomogeneity. Phys. Lett. A 129, 449–452.CrossRefGoogle Scholar
[18]Kivshar, Y. S. & Malomed, B. A. (1989) Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61, 763–915; (1991) 63, 211 (Addendum).CrossRefGoogle Scholar
[19]Kivshar, Y. S., Fei, Z. & Vázquez, L. (1991) Resonant soliton-impurity interactions Phys. Rev. Lett. 67, 1177–1180.CrossRefGoogle ScholarPubMed
[20]Knight, C. J. K. (2008) Microresistor Pinning of 2kπ-Fluxons in Long Josephson Junctions, MMath Thesis, University of Surrey, Guildford, UK.Google Scholar
[21]Knight, C. J. K., Derks, G., Doelman, A. & Susanto, H.Stability of stationary fronts in inhomogeneous wave equations, in preparation. Preprint http://personal.maths.surrey.ac.uk/st/G.Derks/Publications/preprint_knight.pdf).Google Scholar
[22]Kontos, T., Aprili, M., Lesueur, J., Genet, F., Stephanidis, B. & Boursier, R. (2002) Josephson junction through a thin ferromagnetic layer: Negative coupling. Phys. Rev. Lett. 89, 137007.CrossRefGoogle ScholarPubMed
[23]Mann, E. (1997) Systematic perturbation theory for sine-Gordon solitons without use of inverse scattering methods. J. Phys. A: Math. Gen. 30, 1227–1241.CrossRefGoogle Scholar
[24]Marangell, R., Jones, C. K. R. T. & Susanto, H. (2010) Localized standing waves in inhomogeneous Schrödinger equations. Nonlinearity 23, 2059.CrossRefGoogle Scholar
[25]McLaughlin, D. W. & Scott, A. C. (1978) Perturbation analysis of fluxon dynamics. Phys. Rev. A 18, 1652–1679.CrossRefGoogle Scholar
[26]Ortlepp, T., AriandoMielke, O. Mielke, O., Verwijs, C. J. M., Foo, K. F. K., Rogalla, H., Uhlmann, F. H. & Hilgenkamp, H. (2006) Flip-flopping fractional flux quanta. Science 312, 1495–1497.CrossRefGoogle ScholarPubMed
[27]Parker, N. G. (2004) Numerical Studies of Vortices and Dark Solitons in Atomic Bose–Einstein Condensates, PhD Thesis, Durham University.Google Scholar
[28]Pegrum, C. M. (2006) Can a fraction of a quantum be better than a whole one? Science 312, 1483–1484.CrossRefGoogle ScholarPubMed
[29]Piette, B., Zakrzewski, W. J. & Brand, J. (2005) Scattering of topological solitons on holes and barriers. J. Phys. A 38, 10403–10412.CrossRefGoogle Scholar
[30]Piette, B. & Zakrzewski, W. J. (2007) Scattering of sine-Gordon kinks on potential wells. J. Phys. A 40, 5995–6010.CrossRefGoogle Scholar
[31]Sakai, S., Akoh, H. & Hayakawa, H. (1985) Fluxon transfer devices. Japan. J. Appl. Phys. 24, L771–L773.CrossRefGoogle Scholar
[32]Scharinger, S., Gürlich, C., Mints, R. G., Weides, M., Kohlstedt, H., Goldobin, E., Koelle, D. & Kleiner, R. (2010) Interference patterns of multifacet 20 × (0-π) Josephson junctions with ferromagnetic barrier. Phys. Rev. B 81, 174535.CrossRefGoogle Scholar
[33]Serpuchenko, I. L. & Ustinov, A. V. (1987) Experimental observation of the fine structure on the current-voltage characteristics of long Josephson junctions with a lattice of inhomogeneities. Sov. Phys. JETP Lett. 46, 549–551; (1987) Pisma Zh. Eksp. Teor. Fiz. 46, 436–437.Google Scholar
[34]Susanto, H., van Gils, S. A., Visser, T. P. P., Smilde, H. J. H. & Hilgenkamp, H. (2003) Static semifluxons in a long Josephson junction with π-discontinuity points. Phys. Rev. B 68, 104501–104508.CrossRefGoogle Scholar
[35]Susanto, H., Goldobin, E., Koelle, D., Kleiner, R. & van Gils, S. A. (2005) Controllable plasma energy bands in a one-dimensional crystal of fractional Josephson vortices. Phys. Rev. B 71, 174510.CrossRefGoogle Scholar
[36]Titchmarsh, E. C. (1962) Eigenfunction Expansions Associated with Second-Order Differential Equations, 2nd ed., Oxford University Press, Oxford.CrossRefGoogle Scholar
[37]Vystavkin, A. N., Drachevskii, Yu. F., Koshelets, V. P. & Serpuchenko, I. L. (1988) First observation of static bound states of fluxons in long Josephson junctions with inhomogeneities. Sov. J. Low Temp. Phys. 14, 357–358; (1988) Fiz. Nizk. Temp. 14, 646–649.Google Scholar
[38]Weides, M., Kemmler, M., Goldobin, E., Koelle, D., Kleiner, R., Kohlstedt, H. & Buzdin, A. (2006) High quality ferromagnetic 0 and π Josephson tunnel junctions. Appl. Phys. Lett. 89, 122511.CrossRefGoogle Scholar
[39]Weides, M., Kemmler, M., Goldobin, E., Kohlstedt, H., Waser, R., Koelle, D. & Kleiner, R. (2006) 0-π Josephson tunnel junctions with ferromagnetic barrier. Phys. Rev. Lett. 97, 247001.CrossRefGoogle ScholarPubMed
[40]Weides, M., Kohlstedt, H., Waser, R., Kemmler, M., Pfeiffer, J., Koelle, D., Kleiner, R. & Goldobin, E. (2007) Ferromagnetic 0-π Josephson junctions. App. Phys. A 89, 613–617.CrossRefGoogle Scholar