Confocal Annular Josephson Tunnel Junctions

Abstract

The physics of Josephson tunnel junctions drastically depends on their geometrical configurations and here we show that also tiny geometrical details play a determinant role. More specifically, we develop the theory of short and long confocal annular Josephson tunnel junctions in the presence of an in-plane magnetic field of arbitrary orientations. The behavior of a circular annular Josephson tunnel junction is then seen to be simply a special case of the above result. For junctions having a normalized perimeter less than one the threshold curves are derived and computed even in the case with trapped Josephson vortices. For longer junctions a numerical analysis is carried out after the derivation of the appropriate motion equation for the Josephson phase. We found that the system is modeled by a modified and perturbed sine-Gordon equation with a space dependent effective Josephson penetration length inversely proportional to the local junction width. Both the fluxon statics and dynamics are deeply affected by the non-uniform annulus width. Static zero-field multiple-fluxon solutions exist even in presence of a large bias current. The tangential velocity of a traveling fluxon is not determined by the balance between the driving and drag forces due to the dissipative losses. Furthermore, the fluxon motion is characterized by a strong radial inward acceleration which causes electromagnetic radiation concentrated at the ellipse equatorial points.

Flux-flow effects in annular Josephson tunnel junctions

“…In the limit c → 0, the elliptic coordinates (ν, τ ) reduce to polar coordinates (r, θ), where θ is the angle relative to the Y -axis; the correspondence is given by τ → θ and c cosh ν → r (note that ν itself will becomes infinite as c → 0). For closely spaced inner and outer ellipses, ∆ν ≡ ν o − ν i << 1, the expression of the local annulus width is 25 :…”

“…In the small-width approximation, ∆w max << λ J , the system becomes one-dimensional and the ν-independent Josephson phase, φ(τ,t), of a CAJTJ in the presence of a spatially homogeneous in-plane magnetic field H of arbitrary orientation,θ, relative to the Y -axis, obeys a modified and perturbed 1+1 sine-Gordon equation with a space dependent effective Josephson penetration, λ J /Qν(τ ), length inversely proportional to the local junction width 25 :…”

“…Resonances may occur between the fluxon and the plasma waves 49,50 whose strength drastically depends on the waves amplitudes which, in turn, are strictly related to fluxon velocity and to the system's dissipation as well as to the steepness of the potential that is determined by the annulus eccentricity. The interaction between the fluxon and the small amplitude waves destabilizes its forward advancement and prevents it from reaching relativistic speeds 25 . The dispersion relation of plasma waves in confocal AJTJs has been recently investigated in the absence of trapped fluxons 51 .…”

“…Recently, the circular geometry in which the internal and external boundaries of the annulus are closely spaced concentric circumferences has been generalized to the so-called confocal geometry in which the annulus boundaries are confocal ellipses 25,26 , rather than concentric circles. The circular AJTJs can be seen as a special case of the confocal AJTJ (CAJTJ)…”

The Fraction of Broken Waves in Natural Surf Zones

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