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Instability of flux flow and production of vortex-antivortex pairs by current-driven Josephson vortices in layered superconductors

Ahmad Sheikhzada and Alex Gurevich
Phys. Rev. B 99, 214512 – Published 26 June 2019
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  • INTRODUCTION
  • COUPLED SINE-GORDON EQUATIONS
  • CHERENKOV RADIATION AND INSTABILITY
  • SINGLE VORTEX
  • VORTEX CHAIN IN AN ANNULAR STACK
  • VORTEX LATTICE
  • FINITE-SIZE EFFECTS AND VORTEX BOUNCING
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • Supplemental Material
    References

    Abstract

    We report numerical simulations of the nonlinear dynamics of Josephson vortices driven by strong dc currents in layered superconductors. Dynamic equations for interlayer phase differences in a stack of coupled superconducting layers were solved to calculate a drag coefficient η(J) of the vortex as a function of the perpendicular dc current density J. It is shown that Cherenkov radiation produced by a moving vortex causes significant radiation drag increasing η(v) at high vortex velocities v and striking instabilities of driven Josephson vortices moving faster than a terminal velocity vc. The steady-state flux flow breaks down at v>vc as the vortex starts producing a cascade of expanding vortex-antivortex pairs evolving into either planar macrovortex structures or branching flux patterns propagating both along and across the layers. This vortex-antivortex pair production triggered by a rapidly moving vortex is most pronounced in a stack of underdamped planar junctions where it can occur at J>Js well below the interlayer Josephson critical current density. Both vc and Js were calculated as functions of the quasiparticle damping parameter, and the dc magnetic field was applied parallel to the layers. The effects of vortex interaction on the Cherenkov instability of moving vortex chains and lattices in annular stacks of Josephson junctions were considered. It is shown that a vortex driven by a current density J>Js in a multilayer of finite length excites self-sustained large-amplitude standing waves of magnetic flux, resulting in temporal oscillations of the total magnetic moment. We evaluated a contribution of this effect to the power W radiated by the sample and showed that W increases strongly as the number of layers increases. These mechanisms can result in nonlinearity of the c-axis electromagnetic response and contribute to THz radiation from the layered cuprates at high dc current densities flowing perpendicular to the ab planes.

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    • Received 18 February 2019
    • Revised 15 May 2019

    DOI:https://doi.org/10.1103/PhysRevB.99.214512

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Josephson effectProximity effectSuperconductivityVortices in superconductors
    1. Physical Systems
    Collective dynamicsHigh-temperature superconductorsJosephson junctionsSolitonsSuperconducting devicesSuperconductors
    1. Techniques
    Finite-difference time-domain methodMethods in superconductivity
    Condensed Matter, Materials & Applied PhysicsNonlinear Dynamics

    Authors & Affiliations

    Ahmad Sheikhzada* and Alex Gurevich†

    • Department of Physics and Center for Accelerator Science, Old Dominion University, Norfolk, Virginia 23529, USA

    • *asheikhz@odu.edu
    • †gurevich@odu.edu

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    Issue

    Vol. 99, Iss. 21 — 1 June 2019

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    Images

    • Figure 1
      Figure 1

      Stack of intrinsic Josephson junctions (yellow) between superconducting layers (blue).

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    • Figure 2
      Figure 2

      Color map of Cherenkov radiation cone in the magnetic field Bl(x) produced by a vortex moving uniformly in the middle layer in a stack of N=101 junctions. Here Bl(x) is obtained by simulations of Eqs. (1) with β=0.25, ζ=71111, α=1, η=0.05, and B0=ϕ0/2πsλc. Only solutions for 15 neighboring junctions above and below the vortex are shown. Note that Lz=Ns∼10−3λc so the vortex is strongly elongated along the x direction.

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    • Figure 3
      Figure 3

      Phase profile of a static vortex in the middle junction (l=11) and θl(x) induced by the vortex on the layers with l=10 and 1. Here θl(x) are symmetric with respect to the central layer.

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    • Figure 4
      Figure 4

      Phase profiles of a single vortex propagating along the middle junction (l=11) and the trailing tail of Cherenkov radiation produced on the neighboring junctions (l=1 and 10) calculated from Eq. (5) at β=0.615 and η=0.05.

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    • Figure 5
      Figure 5

      A color map of the magnetic field in the vortex moving along the central junction calculated from Eq. (2) at β=0.615 and η=0.05. Here Cherenkov radiation behind the vortex manifests itself as color ripples. Since Lz∼10−4λc, the vortex is strongly elongated along the x direction.

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    • Figure 6
      Figure 6

      Stationary velocities of a vortex moving along the central JJ as a function of the bias current at different η. The instability occurs at the end points of the curves. The sharp change in the slope of v(β) at η≪1 indicates the transition from the Ohmic to radiation vortex drag.

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    • Figure 7
      Figure 7

      The threshold instability current (a) and the terminal velocity (b) as functions of η calculated for ζ=71111.

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    • Figure 8
      Figure 8

      Initial stages of generation of V-AV pairs by a vortex moving along the central junction (top panel), and snapshots of field distribution solutions showing the two-dimensional growth of instability for junctions with l=9,10, and 11 at three different times (bottom panel). The results are calculated at η=0.05, ζ=71111, and β=0.62.

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    • Figure 9
      Figure 9

      Cross-sectional view of the field distribution profiles in the stack after the instability (top panel, t=125) along with a closeup view of giant vortices moving to the left (bottom panel, t=225). Similar macrovortices with opposite polarity form at the other side of the stack (as shown in the top panel).

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    • Figure 10
      Figure 10

      Snapshots of representative solutions for θl(x,t) (top) and Bl(x,t) (bottom) along the middle JJ at the critical current β=βs=0.62.

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    • Figure 11
      Figure 11

      Snapshots of the magnetic field (top) and electric field (bottom) in junctions 1–11 calculated at β=βs=0.62, where E0=ϕ0ωJ/2πcs. Here the largest oscillation amplitude corresponds to the middle junction and the lowest amplitude corresponds to the top/bottom junction.

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    • Figure 12
      Figure 12

      Comparison between θl(x) in a single vortex and a chain of nine vortices (only three are shown) moving along the central junction at β=0.6.

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    • Figure 13
      Figure 13

      Snapshots of θl(x,t) in a moving chain of 50 vortices at β=0.44 near the instability threshold. The two profiles are superimposed for ease of comparison. Interaction of vortices with Cherenkov wakes causes temporal variations in the shape and velocity of moving vortices.

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    • Figure 14
      Figure 14

      Snapshots of the final form of the solution in electric-field (top) and magnetic-field (bottom) representations in junctions 1–11 for instability current β=0.445. The oscillations are both in phase and periodic for all layers with amplitudes decaying from the middle junction across the stack.

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    • Figure 15
      Figure 15

      Color map of the magnetic field across the stack for a stationary vortex lattice with one fluxon per layer (top) and Bl(x) for the middle and surface JJs (bottom).

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    • Figure 16
      Figure 16

      Color map of the magnetic field across the stack for a uniformly moving vortex lattice with one fluxon per layer (top) and Bl(x) for the middle and surface JJs (bottom) calculated at β=0.54.

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    • Figure 17
      Figure 17

      Color map of the magnetic field across the stack for a stationary vortex lattice with ten fluxons per layer (top) and Bl(x) for the middle and surface JJs (bottom).

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    • Figure 18
      Figure 18

      Calculated dependence of Js on the linear density of vortices per length λc along the layer in a vortex lattice.

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    • Figure 19
      Figure 19

      Color map of the magnetic field in a stationary vortex lattice composed of 50 fluxons per layer. The closeup in the top left corner shows a triangle formed by three vortices in two adjacent layers.

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    • Figure 20
      Figure 20

      Magnetic-field color map in moving vortices in junctions 10, 11, and 12 at β=0.1 (top). Bottom panel illustrates how a transient triplet is formed out of the conversion of the image of vortices from the 10th and 12th junctions in the central JJ to a pair of V-AV at β=0.175.

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    • Figure 21
      Figure 21

      Temporal oscillations of a magnetic moment M(t) due to periodic reflections of driven vortices and antivortices from the sample edges at η=0.1. (a) M(t) caused by a vortex shuttle in which a single vortex gets reflected from the edges as an antivortex at β=0.585<βs. The features marked by the arrows result from Cherenkov and bremsstrahlung radiation after reflection of a V or AV. (b) M(t) caused by a bouncing flux structure with one vortex per layer at β=0.53<βs.

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    • Figure 22
      Figure 22

      Magnetic-field color map in moving macrovortices colliding with the edge of the stack at x/λc=−0.5. Top: A chain of macrovortices reaching the edge just before the collision. Bottom: The same chain after the leading macrovortex collided with the edge and got transformed into a counterpropagating anti-macrovortex.

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    • Figure 23
      Figure 23

      Temporal magnetic moment M(t) due to bouncing macrovortices excited by a single V-AV shuttle. Inset shows M(t) caused by self-sustained MV standing waves superimposed onto M(t) due to stable oscillations of the flux structure with one vortex per layer taken from Fig. 21.

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    • Figure 24
      Figure 24

      A snapshot of beats standing waves of Bl(x,t) on different layers in a finite stack with N=41 calculated for self-sustained oscillations of M(t) shown in Fig. 23.

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    • Figure 25
      Figure 25

      Self-sustained oscillations of M(t) calculated for N=21, 41, and 81 at β=0.6 and η=0.1 after complete decay of initial transient spikes in M(t).

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    • Figure 26
      Figure 26

      Parts of M(t) at N=41 and 81 shown in Fig. 25 but calculated with the finer time steps Δt=0.01ωJ−1 to reveal high-frequency harmonics in M(t).

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