Nonlinear Dynamics of Two-Dimensional Josephson Junction Arrays

197 Pages·2007·10.27 MB·English

Nonlinear Dynamics of Two-Dimensional
Josephson Junction Arrays
VIjASSACHUSETTS INSTITUTE
OF TECHNOLOGY
by
JUN 0 5 1996
Mauricio Barahona Garcia
LIBRARIES
Licenciado, Universidad Complutense de Madrid
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1996
@ Massachusetts Institute of Technology 1996. All rights reserved.
Author ..
............
Department of Physics
May 24, 1996
Certified by .
- -------
" , ... ... ............ .... ....... ....
v.U ...... - -Steven H. Strogatz
Associate Professor of Theoretical and
Applied Mechanics, Cornell University
Thesis Supervisor
Certified by...
Mehran Kardar
Professor of Physics
-Thesis Supervisor
Accepted by.........
George F. Koster
Chairman, Physics Graduate Committee

Nonlinear Dynamics of Two-Dimensional Josephson
Junction Arrays
by
Mauricio Barahona Garcia
Submitted to the Department of Physics
on May 24, 1996, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Arrays of superconducting Josephson junctions can be modelled as systems of cou-
pled nonlinear oscillators. We present analytical and numerical studies of the spatio-
temporal behavior of two-dimensional, open-ended, frustrated, dc-driven arrays of
Josephson junctions at zero temperature, no self-fields. We explore the crossover be-
tween arrays in one and two dimensions and clarify the role of the horizontal junctions,
which are perpendicular to the direction of current injection. A ladder array with
perpendicular current injection and two-dimensional square arrays are considered.
For the ladder, we obtain analytical approximate solutions which include correc-
tions from the edges and/or vortices present in the array. The perturbations decay
exponentially in space with a calculated characteristic length. The depinning of the
array, and its field dependence, is explained by the edge-dominated instability of the
superconducting solutions. This critical current does not change under the inclusion
of vortices since they are expelled before depinning at calculated currents. The in-
stability of the whirling solution is analytically explained by a cascade of parametric
resonances of the driving frequency with the eigenfrequencies of the lattice which,
in this case, produces no steps. At zero field, a new step is observed at this insta-
bility. This state is reduced to a system of two coupled nonlinear oscillators and
characterized analytically as a subwhirling mode where horizontal junctions oscillate
non-negligibly. In conclusion, the horizontal junctions modify the dynamics by in-
troducing an intrinsic inductance through the fluxoid quantization; by modifying the
eigenfrequencies; and by effectively enabling fully two-dimensional modes.
Simulations of underdamped and overdamped square two-dimensional arrays for
varying frustration are presented. The numerical analysis of period, type of motion
of the pendula, phase velocity and spatial distribution characterizes the dynamics.
In underdamped arrays, an analytical approximation for the spatially inhomogeneous
partially row-switched states predicts the critical currents for some row-switching
~eu~*--·--aPlr~--cl-a~-r~·r~-·~-·--·l~-- -r·----··--··I·--- --- - ---- ··- I-
events. The flux-flow region in the overdamped case exhibits a similar phenomenon
of row-activation, alternation of periodic and aperiodic solutions and a final transition
to a rigid whirling phase.
Thesis Supervisor: Steven H. Strogatz
Title: Associate Professor, Cornell University
Thesis Supervisor: Mehran Kardar
Title: Professor of Physics
Acknowledgments
In these years, I have learnt about collaboration and exchange.
I thank my advisor, Steve Strogatz, for his trust and the special effort in keeping
a long distance advisee. I value most his clarity, dissective power, and purposeful
approach to research.
I thank very warmly my adoptive advisor, Terry Orlando. The collaboration with
his group, the weekly meetings, his knowledge of the field have made this thesis
possible.
Mehran Kardar, my co-supervisor, was always available close to any deadline and
made very relevant comments to the thesis. After enjoying his faultless teaching and
the reading of his articles, this could only be taken as a compliment.
One miniparagraph goes to the research group: Herre, Shinya, Amy, Enrique. It is
not easy to find people with whom vortices and Yellowstone can be equally enjoyable.
I gratefully acknowledge full support during four years through a Spanish MEC-
Fulbright Fellowship.
As for the rest, no long list applies: family and friends. They who should be here,
already know. The curious browser need not.

A mis padres

Contents
1 Introduction 11
1.1 General background and motivation . .................. 11
1.2 Overview of the thesis .......................... 20
2 Ladder array of Josephson junctions 23
2.1 Background ................................ 23
2.2 Introduction: model and methods . .................. . 27
2.2.1 Physical description and model equations . ........... 27
2.2.2 Experimental I-V characteristics and numerical simulations . 36
2.3 Superconducting solutions ........................ 43
2.3.1 Observed and approximate superconducting solutions .... . 43
2.3.2 Instability of the superconducting solutions and depinning . 62
2.3.3 Other superconducting solutions with vortices . ........ 70
2.3.4 Summary and discussion ................... .. 87
2.4 W hirling solution ............................. 91
2.4.1 Observed and approximate whirling solutions . ........ 91
2.4.2 Instabilities in the whirling branch and repinning ....... 99
2.4.3 Novel subharmonic whirling state at zero field . ........ 110
2.4.4 Summary and discussion ................... .. 128
3 Two-dimensional Josephson junction arrays 131
3.1 Background .................... ............ 131
3.2 Introduction: model and methods . .................. . 137
3.2.1 Physical description and model equations . ........... 137
3.2.2 Numerical analysis of simulated I- V characteristics ...... 144
3.3 Array of underdamped junctions ................... .. 152
3.3.1 Superconducting state and depinning . ............. 158
3.3.2 Totally row-switched state ................... . 159
3.3.3 Partially row-switched states . .................. 161
3.3.4 Flux-flow regime ................... ...... 168
3.4 Array of overdamped junctions ................... .. 175
3.5 Summary and discussion ................... ...... 184