Creation and annihilation of fluxons in ac-driven semi-annular Josephson junction
aa r X i v : . [ n li n . PS ] J un Creation and annihilation of fluxons in ac-drivensemi-annular Josephson junction
Chitra R N † and V C Kuriakose ‡ Department of Physics, Cochin University of Science and Technology, Kochi, 682022E-mail: † [email protected], ‡ [email protected] Abstract.
The dynamical behavior of a fluxon in a semi-annular long Josephsonjunction in the presence of an ac-drive is studied. The non-uniformity due to thenon uniform distribution of bias current is investigated. The oscillating potential isfound to increase the depinning current. Finite difference method is used for numericalanalysis and the response of the system to the ac-bias is studied. The creation andannihilation of fluxon is also demonstrated numerically for the first, second and thirdZero-Field step cases.PACS numbers: 05.45.Yv, 82.20.Wt reation and annihilation of fluxons in ac-driven semi-annular Josephson junction
1. Introduction
A fluxon in long Josephson junctions (LJJ) is a well-known physical example of a sine-Gordon fluxon. Fluxons, endemic to LJJs, have been employed in the fabrication ofdevices like constant voltage standards [1, 2], flux flow oscillators [3, 4], logic gates[5, 6]and also in qubits [7, 8]. LJJs of various geometries have been thoroughly studiedboth experimentally and theoretically in the past. Fluxon dynamical properties likefluxon pinning [9], fluxon trapping [10], and phase locked states have been studied forrectangular [11, 12] and annular [13] LJJs.The non rectangular Josephson junction has been in the focus of fluxon dynamicsstudy in recent years because of the non uniformity caused by the shape. Semicirculargeometry for Josephson junction has been proposed and fluxon dynamics has beenstudied both analytically and numerically and its various applications has been discussed[14]. It has been shown that in the presence of an external magnetic field appliedparallel to the dielectric barrier of such a geometry, the ends of the junction has oppositepolarities and because of that opposite polarity fluxons can enter the junction from theends under a properly biased dc current. If the direction of the current is reversed, fluxpenetration and progression is not possible and flux free state exists in the junction.This unique phenomenon cannot be achieved in any other geometry and thus thisjunction behaves as a perfect diode. The effect of in-plane static and rf-magnetic fieldon fluxon dynamics in a semiannular Josephson junction has also been studied [15].The response of a fluxon to an ac-drive has investigated by several authors. It wasshown that in a system with periodic boundary condition average progrssive motion offluxon commenses after the amplitude of the ac drive exceeds a certain threshold value[16]. Complex switching distributions has been obtained for ac-driven annular JJs andtheoretical explanation has been provided for the multipeaked experimental observations[17]. The behavior of fluxon under two ac forces has been studied and it was shown thatthe direction of motion of fluxon is dependent on ratio of frequencies, amplitudes andphases of the harmonic forces [18]. In this work, we study the effect of an ac-bias appliedin the plane of a semi-annular Josephson junction. in section II we discuss the equationrepresenting the junction and arrive at an expression for the potential of the junction.The numerical results are also presented. In section III we demonstrate creation andannihilation of fluxons in semiannular JJs in the presence of an ac bias and an externalmagnetic field. Section IV deals with the results and discussion.
2. Perturbation analysis of a fluxon in a semiannular junction
The dynamical equation for a semiannular LJJ in a harmonically oscillating field appliedin its plane is ϕ tt − ϕ xx + sin ϕ = − αϕ t + βϕ xxt − b cos( kx ) − γ + i sin( ωt ) (1)where ϕ ( x, t ) is the superconducting phase difference between the electrodes of thejunction with the spatial coordinate x normalized to λ J , the Josephson penetration reation and annihilation of fluxons in ac-driven semi-annular Josephson junction t normalized to the inverse plasma frequency ω − and ω = e cλ J , e c beingthe maximum velocity of the electromagnetic waves in the junction. R is resistance perunit length, L p is the inductance per unit length, C is the capacitance per unit length,and γ = jj is the normalized amplitude of a dc bias normalized to maximum Josephsoncurrent j and i sin( ωt ) is the applied ac biasing. α is the quasiparticle tunneling lossand β is the surface loss term in the electrodes and their values vary from 0 .
001 to0 . bsin ( kx ) is due to the semicircular geometry ofthe junction and k = πl and b = 2 πλ J ∆ Bk/ Φ = 2 k ( B/B c ), where B c = Φ π ∆ λ J is thefirst critical field of the Josephson junction. Φ = h e is the flux quantum and its valueis 2 . × − . The extra term bsin ( kx ) corresponds to a force that drives fluxonstowards the left and anti fluxon towards the right. Thus in the absence of an externalfield a flux free state will exist in the junction as any static trapped fluxon present inthe junction will be removed [15]. In the absence of any perturbation (1) reduces tosimple sine-Gordon equation with fluxon solution given by ϕ ( x, t ) = 4 tan − [exp σ ( x − X ) √ − u ] (2)where u is the velocity of the fluxon and X = ut + x is the instantaneous locationof the fluxon. σ = ± The Lagrangian density of Eq. 1 with γ = α = i = β = 0 is L = 12 ϕ t − ϕ x − bk sin( kx ) ! − ϕ (3)where the first term is the kinetic energy associated with the energy density of theelectric field, the second term accounts for the potential energy density associated withthe magnetic field and the third term represents the Josephson coupling energy density.From the potential energy density term, the change in potential energy due to thecombined effect of fluxon motion and the applied field can be determined by integratingthe term − bk sin( kx ) ϕ x over the length of the junction [15]. The fluxon induced potentialas a function of the fluxon coordinate X may be calculated as U ( X ) = − bk Z ∞−∞ sin( kx ) ϕ x dx (4)The integration over −∞ to ∞ may be justified as the length of the junction is verylarge as compared to the size of the fluxon. Substituting Eq. 2 in Eq. 4 and integratingwe get the expression for potential as U ( X ) = − bl sec h π l √ − u ! sin( kX ) (5) reation and annihilation of fluxons in ac-driven semi-annular Josephson junction u ≃ U ( X ) = − bl sec h π l ! sin( kX ) (6)which has a potential well form with the depth of the well depending on b and l. Thevortex will be pinned to the potential minima as long as the bias current is smaller thanthe depinning current. The pinned state of a vortex corresponds to a zero voltage state.Now we arrive at an expression for the potential function of the perturbed system.The Hamiltonian of the system can be written as a combination of the Hamiltonian ofthe unperturbed sine Gordon part plus the hamiltonian of the perturbation part [11].Energy of the unperturbed sine-Gordon system is H SG = Z ∞−∞ [ 12 ( ϕ t + ϕ x + 1 − cosϕ )] dx (7)Substituting(2) in (7) we get ddt H SG = 8 u (1 − u ) − / dudt (8)Due to the perturbational part, energy is dissipated and rate of dissipation is given as ddt ( H p ) = [ ϕ x ϕ t ] ∞−∞ − (9) Z ∞−∞ ( αϕ t + βϕ xt + [ b cos( kx ) + γ + i sin( ωt )] ϕ t ) dx Here the first term on the right hand side accounts for the boundary conditionsand vanishes. Substituting (2) in above equation we obtain the equation for rate ofdissipation as ddt ( H p ) = 2 πu ( γ + i sin( ωt )) − αu √ − u (10) − βu − u ) / − πbu sec h ( π √ − u l ) cos( kX )Following the perturbation analysis we get dudt = π γ + i sin( ωt )) (cid:16) − u (cid:17) / − αu (cid:16) − u (cid:17) (11) − βu − π b (cid:16) − u (cid:17) / sec h ( π √ − u l ) cos( kX )Eq. 11 describes the effect of perturbations on the vortex velocity. The first termrepresents the effect of applied biasing, the second and the third term representsdissipation and fourth term is the effect of the external magnetic field on the semicirculargeometry.In perturbational analysis, a vortex is considered as a non-relativistic particle ofrest mass m = 8 moving in one dimension. Therefore the effective potential can beobtained using the force relation ∂U eff ∂X = − m dudt (12) reation and annihilation of fluxons in ac-driven semi-annular Josephson junction U eff ( X ) = − bl sec h ( π l ) sin( kX ) − π ( γ + i sin( ωt )) X (13)The potential energy function U eff ( X ) has a well form in the absence of externalbiasing and the fluxon will remain pinned to the centre of the junction under such apotential. As the biasing is increased the potential gets tilted finally favoring the motionof the vortex. The dc bias at which the zero voltage switches to a finite voltage is calledthe depinning current. Fig. 1 shows the form of the potential for b = 0 . l = 15 for different external biasing. It can be seen that in the presence ofan external bias the potential gets tilted favoring motion of the fluxon. While movingthrough such a potential, the fluxon(antifluxon) after bouncing from the edge turns intoan antifluxon (fluxon) and hence will move in the opposite direction. In the presence X U ( X ) γ =0 γ =0.01 γ =0.1 γ =0.2 Figure 1.
Potential well form for a JJ of length l=15. Other parameter values are b = 0 . i = 0 . The γ value is increasing from top to bottom line. of an ac, the potential gets oscillating with a frequency equal to the frequency of theapplied field and the shape of the potential depends on the amplitude of the applied acand dc biasing. In the presence of external ac biasing along with the dc, the potentialgets time varying as shown in fig2. In Fig. 2(a), the applied ac has an amplitude of Figure 2.
Form of the oscillating potential for a JJ of length l=15. Parameter valuesare b = 0 . i = 0 . , ω = 0 . , γ = 0 .
1. a) applied dc bias is γ = 0 . γ = 0 . reation and annihilation of fluxons in ac-driven semi-annular Josephson junction . . To solve Eq. 1 numerically, we use an explicit method treating φ xx with a five point, φ tt with a three point and φ t with a two point finite difference method. The boundaryconditions are treated by the introduction of imaginary points and the correspondingfinite difference equation is solved using standard tridiagonal algorithm[19]. Numericalsimulations are carried out on the JJ of normalized length (l=15). The time step wastaken as 0 . . . The numerical results were checked bysystematically halving and doubling the time steps and space steps. Details of thesimulation can be obtained from [12, 14]. After the simulation of the phase dynamicsfor a transient time, we calculate the average voltage V for a time interval T to be V = 1 T Z T ϕ t dt = ϕ ( T ) − ϕ (0) T Also for the faster convergence of the averaging procedure, the phases ϕ ( x ) in theequation were averaged over the length of the junction. The spatial averaging increasesthe accuracy in the calculation of the voltages in cases where the the time period overwhich integration is made is not an exact multiple of the time period of oscillation.Once the voltage averaging for a current γ is complete, it is increased in small steps of0 .
01 to calculate the next point of the characteristic graph. The average velocity of thefluxons can be calculated from the average voltage using the relation u = V ( l/ π )Taking β to be zero, the velocity change with increase in dc biasing is observed.Fig.3 shows the velocity change with dc biasing for different values of amplitude of theac biasing. In the presence of ac biasing the averaging interval T was taken as a multipleof the ac drive’s period 2 π/ω [16]. If an ac-biasing is present, the depinning current isfound to increase which can be seen from Fig.3. Constant voltage steps are observedfor an ac-bias of amplitude 0 . . . In the presence of external magnetic fields, the velocity versus dc bias is shown inFig. 4. The value of dc bias to cause a finite velocity for the fluxon in a JJ with amagnetic field of b = 0 . . b = 0 the depinning currentis 0 .
04. Thus the external magnetic filed also increases the depinning current value.The depinning current increases for higher values of damping parameter β = 0 . . The depinning current to be applied to the semi-annular JJ in the absence of ac, and reation and annihilation of fluxons in ac-driven semi-annular Josephson junction u γ i =0.0i =0.1i =0.2i =0.3 Figure 3.
The velocity- bias characteristics of a LJJ of length l=15 with no externalmagnetic field applied. Other parameter values are ω = 0 . , β = 0 . , α = 0 . −0.2 0 0.2 0.4 0.6 0.8 1 1.200.10.20.30.40.5 u γ i =0.0i =0.1i =0.2i =0.3 Figure 4.
The velocity- bias characteristics of a LJJ of length l=15 in the presenceof an external magnetic field b = 0 . a magnetic field of b = 0 . . . . .
46 and 0 .
36. For dcbias values of more than these values, quasiperiodic or chaotic motion may exist in thesystem.
3. Creation and Annihilation of Fluxons
An annular LJJ preserves the number of trapped fluxons in it. However in an openended geometry the number of fluxons is not a conserved quantity. In this sectionwe investigate the creation and annihilation of fluxons in semiannular JJ with openboundary conditions in the presence of an external field and an ac and dc biasing. reation and annihilation of fluxons in ac-driven semi-annular Josephson junction u γ i =0.0i =0.1i =0.2i =0.3 Figure 5.
The velocity- bias characteristics of a LJJ of length l=15 in the presenceof an external magnetic field b = 0 .
1. The damping parameter β = 0 . The collision of fluxons with localized obstacles leads to creation and annihilation offluxons. The fluxon creation and annihilation process for a single kink solution as inputis described here. A kink solution is launched from the centre of the junction with aninitial velocity of v = 0 .
6. For each value of biasing the fluxon is allowed to propagatefor some time in order to stabilize its motion in the junction. The kink fluxon getsreflected from the boundaries and moves on till γ = 0 .
57. Above this biasing, nosolitonic propagation is observed for an external magnetic field of strength 0 . γ value of 0 . Figure 6. (a)The pattern shows annhilation of fluxon propagating in a JJ with l=15for a dc bias of γ = 0 . i ac = 0 .
2. (b) Creation of fluxon with γ = 0 . i ac = 0 . ω = 0 . , β = 0 . , α = 0 . , b = 0 . propagates through the semiannular junction, while an ac bias of 0 . γ = 0 . i ac = 0 . reation and annihilation of fluxons in ac-driven semi-annular Josephson junction Two kink solutions were launched with at differ initial points in the junction with intialvelocity v = 0 .
6. An dc bias of more than γ = 0 . γ = 0 . . − .
45 supportstwo fluxon propagation in the junction in the absence of an ac biasing. However if anac bias of 0.1 is applied along with γ = 0 .
41 creation of a fluxon occurs as shown in Fig.7 (b).
Figure 7.
The pattern shows single fluxon propagating in a JJ with l=15 for idc = 0 . iac = 0 . idc = 0 . In this case, three kink fluxons are launched at different initial points. A dc bias of γ = 0 . γ valueof 0 .
3. Also for γ = 0 .
4, if an ac bias is applied annihilation of one fluxon occurs againgiving the two solitonic propagation as shown in 8(a). The ac biasing causes annihilationand if the i value is increased to 0 .
19 or more the solitonic profile is lost.
Figure 8.
The pattern shows two fluxons propagating in a JJ with l=15. Otherparameter values are idc = 0 . idc = 0 . Creation of fluxon is observed for γ values of 0.55 as shown in Fig. 8(b) with ac reation and annihilation of fluxons in ac-driven semi-annular Josephson junction i = 0 . . it is to be noted that all these effectstakes place only in the presence of an external magnetic field in semiannular JJs. Inthe absence of magnetic fields, we were not able to observe creationa nd annihilation offluxons.
4. Conclusions
We have studied the dynamics of a fluxon trapped in a semi annular JJ in the presence ofan external magnetic field along with an ac biasing. This method of applying ac biasingoffers a much easier and controllable way to induce a harmonic periodic modulation tothe junction. In the presence of an external magnetic field the vortex remains pinnedin the potential well. The ac biasing modulates the form of the potential and we obtainan oscillating potential with frequency of oscillation equal to the driving field. In thepresence of an ac-drive and magnetic field , fluxon creation and annihilation phenomenais observed. This has been demonstrated for one, two and three fluxons and can beextended to higher number solutions. The fluxon creation and annihilation processbeing crucial for the understanding of the internal dynamics of the junctions, it will haveimportant applications in design and fabrication of superconducting digital devices.
Acknowledgments