Skip to main content Accessibility help

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


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.

Hide All
[1]Ablowitz, M. J., Kaup, D. J., Newell, A. C. & Segur, H. (1973) Method for solving the sine-Gordon equation. Phys. Rev. Lett. 30, 12621264.
[2]Akoh, H., Sakai, S., Yagi, A. & Hayakawa, H. (1985) Real time fluxon dynamics in Josephson transmission line. IEEE Trans. Magn. 21, 737740.
[3]Andreeva, O. Yu., Boyadjiev, T. L. & Shukrinov, Yu. M. (2007) Vortex structure in long Josephson junction with two inhomogeneities. Physica C 460–462, 13151316.
[4]Benabdallah, A., Caputo, J. G. & Flytzanis, N. (2002) The window Josephson junction: A coupled linear nonlinear system. Physica D 161, 79101.
[5]Benabdallah, A. & Caputo, J. G. (2002), Influence of the passive region on zero field steps for window Josephson junctions. J. Appl. Phys. 92, 38533862.
[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, 13171318.
[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.
[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, 493518.
[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.
[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, 99141.
[11]Derks, G., Doelman, A., van Gils, S. A. & Visser, T. (2003) Travelling waves in a singularly perturbed sine-Gordon equation. Physica D 180, 4070.
[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.
[13]Goodman, R. H. & Haberman, R. (2007) Chaotic Scattering and the n-Bounce Resonance in Solitary-Wave Interactions Phys. Rev. Lett. 98, 104103.
[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, 27312760.
[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, 127157.
[16]Hilgenkamp, H. (2008) π-phase shift Josephson structures. Supercond. Sci. Technol. 21, 034011.
[17]Kivshar, Yu. S., Kosevich, A. M. & Chubykalo, O. A. (1988) Finite-size effects in fluxon scattering by an inhomogeneity. Phys. Lett. A 129, 449452.
[18]Kivshar, Y. S. & Malomed, B. A. (1989) Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61, 763915; (1991) 63, 211 (Addendum).
[19]Kivshar, Y. S., Fei, Z. & Vázquez, L. (1991) Resonant soliton-impurity interactions Phys. Rev. Lett. 67, 11771180.
[20]Knight, C. J. K. (2008) Microresistor Pinning of 2kπ-Fluxons in Long Josephson Junctions, MMath Thesis, University of Surrey, Guildford, UK.
[21]Knight, C. J. K., Derks, G., Doelman, A. & Susanto, H.Stability of stationary fronts in inhomogeneous wave equations, in preparation. Preprint
[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.
[23]Mann, E. (1997) Systematic perturbation theory for sine-Gordon solitons without use of inverse scattering methods. J. Phys. A: Math. Gen. 30, 12271241.
[24]Marangell, R., Jones, C. K. R. T. & Susanto, H. (2010) Localized standing waves in inhomogeneous Schrödinger equations. Nonlinearity 23, 2059.
[25]McLaughlin, D. W. & Scott, A. C. (1978) Perturbation analysis of fluxon dynamics. Phys. Rev. A 18, 16521679.
[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, 14951497.
[27]Parker, N. G. (2004) Numerical Studies of Vortices and Dark Solitons in Atomic Bose–Einstein Condensates, PhD Thesis, Durham University.
[28]Pegrum, C. M. (2006) Can a fraction of a quantum be better than a whole one? Science 312, 14831484.
[29]Piette, B., Zakrzewski, W. J. & Brand, J. (2005) Scattering of topological solitons on holes and barriers. J. Phys. A 38, 1040310412.
[30]Piette, B. & Zakrzewski, W. J. (2007) Scattering of sine-Gordon kinks on potential wells. J. Phys. A 40, 59956010.
[31]Sakai, S., Akoh, H. & Hayakawa, H. (1985) Fluxon transfer devices. Japan. J. Appl. Phys. 24, L771L773.
[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.
[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, 549551; (1987) Pisma Zh. Eksp. Teor. Fiz. 46, 436–437.
[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, 104501104508.
[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.
[36]Titchmarsh, E. C. (1962) Eigenfunction Expansions Associated with Second-Order Differential Equations, 2nd ed., Oxford University Press, Oxford.
[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, 357358; (1988) Fiz. Nizk. Temp. 14, 646–649.
[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.
[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.
[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, 613617.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed