Vortex motion in superconducting ladders

10 November

ELSEVIER

1997

Physics Letters A 235 (1997)

PHYSICS

LETTERS

Taejon 305-600,

South Korea

A

408-412

Vortex motion in superconducting ladders Seunghwan Research Department,

Electronics

and Telecommunications

Kim ’

Research Institute,

P.0. Box 106, Yusong-gu,

Received 28 April 1997; revised manuscript received 25 July 1997; accepted Communicated by A.R. Bishop

for publication

28 July 1997

Abstract We study vortex motion in superconducting ladders. Since the single-vortex excitation energy is finite in this system, even in the absence of an external magnetic field there may exist excess vortices that are stabilized by external currents. The motion of the excess vortices induces localized voltage drops, given by a nonlinear power law, at currents less than the critical current of the junctions. The localized voltage drop moves with constant velocity and its amplitude is proportional to the velocity. Above the critical current of the junctions

pairs. @ 1997 Published PACS: 74.50.+r;

74.60.Ge;

there is continuous creation and annihilation

74.40.+k

Ke~lwords: Vortex; Superconducting

ladder; Josephson junction;

IV

In the past several years there have been extensive studies on two-dimensional (2D) arrays of Josephson junctions (JJs) because of their interesting phase transitions and dynamical behavior [ I]. In these systems, vortices, topological excitations, play an important role in equilibrium and dynamics. In equilibrium these systems are described by the well-known XY model showing the Kosterlitz-Thouless transition, a vortex unbinding transition, which results from the long-range nature of the vortex interaction [2]. In the presence of an external transverse magnetic field, there are pinned vortices at small external currents [ 3 1. Large currents drive the vortices into the transverse direction producing a voltage drop. In particular, underdamped JJ arrays show ballistic vortex motion 141.

characteristic

While most studies have been concentrated on 2D JJ arrays, simpler systems such as ladder arrays and single plaquettes of JJs are interesting because of their rich behavior [ 5-71. Recently, we have investigated the vortex interaction in JJ ladders leading to an exponentially decaying interaction potential [ 81. We have also shown that the excitation energy of a single vortex is finite and that excess vortices can be created. The purpose of this Letter is to investigate the vortex motion in overdamped JJ ladders. Via a numerical simulation we find that the external intermediate currents stabilize the excess vortices and thus their motion produces a localized voltage drop given by a nonlinear power law. The localized voltage drop moves with a constant velocity and its amplitude is proportional to the velocity. Above the critical current of the junctions there is continuous creation and annihilation of vortex-antivortex pairs. The implications of this peculiar behavior are also discussed.

’ E-mail: [email protected]. 0375-9601/97/$17.00 PII

SO375-960

of vortex-antivortex

by Elsevier Science B.V.

@ 1997 Published

I (97) 00646-4

by Elsevier Science B.V. All rights reserved

S. Kim/Physics

Letters A 235 (1997) 408-412

0.2

Fig. I. Schematic picture of a ladder array of Josephson junctions driven by uniform external currents I. ( x) Josephson junction; ( l) su~rconducting grain.

We consider one-dimensional ladders of parallelcoupled JJs where the superconducting islands form 2 x N square arrays as shown in Fig. 1. We take the periodic boundary condition along the longitudinal direction of the ladders, and an external current I is uniformly injected into each node on the right edge and extracted from each node on the left edge of the ladders. Neglecting capacitive and inductive effects the ladders are described by the resistively shunted junction model. Applying the current conservation law at each node, we get the following equations of motion for the superconducting phases &‘I and #:*I (a’= 1, . . ., N, with N being the number of parallel junctions) on the nth islands of the left and the right edges, respectively,

409

0.4

0.6

0.8

1

2

Fig. 2. Time-averaged voltage V,, versus applied currents I for the system with size N = 10’24.The dots including those in the inset represent the IV characteristic for the system with eight excess vortices. The solid line is obtained for the system without an excess vortex, the solid line in the inset is the IV characteristic fitted by V,, = 0.048 - 0.0576( 1- f)“.43x.Here. V,, and i are in units of f,R and Ic, respectively.

dant due to the overall U( 1) phase rotation symmetry. So one of the phases can be fixed arbitrarily and we can solve the remaining 2N - 1 equations. We have carried out direct integrations of Eqs. (1) using the second-order Runge-Kutta method with an integration time interval St = 0.01~. Simulations were performed starting from random initial phases. By rewriting Eqs. (1) in terms of +A-) 5 c#+!,~)- &” and @+I z (pf2) + #(I) we can decouple the whole set o;equatio(ls into t’wo sets of equations, one of which involves only d&-j fdt and the other only d4L+‘/dt. Using this, the tridiagonal matrix algorithm [9] can be employed, which makes the numerical integration very efficient. First, we calculate the time-averaged voltage

(2) gWti2’ dt

dc$“’ ddt;?,‘, n dr --- dt

d&,!!, dt

= 1 - sin( #L*’ - Cp:“) - sin{ #k*’ - 4$), ) - sin(f$i*’

- +:2_‘, ) .

(1)

Here, t and the currents are in units of r 3 fi/2eRI, and fC, respectively, and we assume uniform shunting resistance R and the critical current I, of the junctions. There are 2N equations for 2N phases #i” and r$i*’ (TV= 1, . _., N). Among them one equation is redun-

across the edges for various currents with initial conditions which have finite excess vortices. We started with applied currents I = 1.21, and continuously decreased the currents by 0.01 I,. Fig. 2 shows the time averaged current-voltage (IV) characteristic obtained by the numerical simulation with an initial condition which has eight excess vortices for system size N = 1024. This figure shows that for currents higher than the critical current 1, the IV relation is very similar to

410

S. Kim/Physics

Letters A 235 (1997) 408-412

Fig. 3. Time-averaged voltage V, versus the vortex number NV at external currents I = 0.7, 0.8, and 0.9 for the system with size N = 128.

Fig. 4. Time evolution of vortex excitations (0) and vohage (solid lines) for the system with size N = 1024 at i = 0.91,. The vortex is plotted at every ten time units,the voltage at every 100 time

units. that obtained from the system without any excess vortex. As the currents are decreased below ZC,while the time-averaged voltage vanishes for the system without any excess vortex, it remains finite showing a nonlinear IV relation. However, this nonlinear IV relation does not persist as the currents are decreased further. Remarkably, at low currents (I < O.%,) the timeaveraged voltage vanishes. The nonlinear fitting of a IV characteristic shown in Fig. 2 gives the critical current Z, = 0.341, below which the excess vortices decay, thus leading to a vanishing time-averaged voltage. Fig. 3 presents the time-averaged voltage versus the vortex number at external currents Z = 0.7, 0.8, and 0.9. This figure shows that the voltage is propo~ional to the density of vortices. From the above results we can conclude that excess vortices excited initially are stabilized by the large external currents, Z > I,, and contribute to the voltage by the nonlinear power law shown in Fig. 2. To investigate the contribution of the vortex motion to the voltage we monitored the time evolution of the vortex excitations which is given by the plaquette sum of the phase differences,

(3) The plaquette position x = n + l/2 just corresponds to the dual lattice site introduced in Ref. f 81, and each phase difference has been defined on the compact interval [ +T, v). The local vorticity can take on the

values 0 and f 1. We also caicufated the local voltage V(n, t) =d&,-$‘dt. Fig. 4 presents the time evolution of the vortex excitations and the voltage induced by the vortex motion for the system with size N = 1024 at I = O.SZ,. We started with an initial condition in which there are three vortices and one antivortex. Fig. 4 shows that the vortex and the antivortex move with positive and negative uniform velocities, respectively, inducing localized voltage drops. The localized voltage drops induced by the vortex and the antivortex have the same voltage profile but they move in opposite directions. Rem~kably, an antivortex produces the same voltage as a vortex does. This is because the antivortex moves in an opposite direction with the same speed as the vortex. When the vortex crosses a parallel junction of the ladder, the phase difference of the junction changes by 27r, thus giving a voltage proportional to the velocity with proportionality 2~. Hence, the localized voltage drop has a velocity proportional to its amplitude. When a vortex and an antivortex meet, they annihilate destroying the localized voltage drop. Fig. 5 shows the time evoIution of the vortex excitations and the voltage induced by the vortex motion for the system with size N = 1024 at Z = 1.OlZ,. We started with an initial condition in which there are five vortices and one antivortex. Similarly to the case of Z = 0.9Z, the vortex motion produces a localized voltage drop which moves with a nearly uniform velocity. When a vortex and an antivortex meet they annihilate destroying the localized voltage drop.

S. Kim/Physics

L.&rem A 235 (1997) 408-412

411

=f -

600

sin 4c-j n

- cos( 4i+’ - @zti) sin( c;bi-’ - 4:;; ) .

(4)

Furthermore, we observe that the condition of zero net current along the vertical direction can be satisfied by assuming that all (pi+) ‘s are the same. On these assumptions, we can reduce the equations of motion for the 2N - 1 phases in terms of only N phase differences &) n ’ Fig. 5. Time evolutions of vortex excitations (Of and voItage (solid lines) for the system with size N = 1024 at I = 1.011,. The vortex is plotted at every five time units, the voltage at every 50 time units.

in this case there is the creation of a vortex-antivortex pair producing two localized voltage drops that move in opposite directions. This vortex-antivortex creation makes the number of localized voltage drops large and thus the time-averaged voltage very high. This behavior is very different to that in the case with no vortex. If we start with the initial condition in which there is no vortex, then the vortex-antivortex creation does not occur, and the voltage spreads out in all parallel junctions of the ladder uniformly. Hence we conclude that above the critical current Z, of the junctions initial excess vortices induce vortex-antivortex creation and thus localized voltage drops leading to a large density of vortices and a high time-averaged voltage 2 . To understand the existence of the soliton-like voltage drop we investigate the equations of motion ( 1) analytically. By rewriting Eq. ( 1) in terms of +A+) = #(2) + #” and (b(-) = #*’ - 4”) we obtain the n n fo~iowing~~uation~, It is remarkable

d@+’ -$--5

that

1 d&? ( dnr’+

-dq$;i dt >

= - sin( $I;+” - c$::‘, ) cos( 4i-j _ sin(@(+f _ &+, (+) > cos($@ n ”

- 4::; ) - 4;;; > 3

*Since Fig. 5 is plotted at discrete times with a time interval of five unit times, the two opposite vortices seem to he created closer and closer together as time increases. Fig. 5 is plotted in the steady state in which the system shows periodic motion and the creation of vortex-antivortex pairs is continuous.

=I

- sin(&-)

- c$J:~;>-sin(&)

-sinib(-)N - 4:;:).

(5)

In the continuum limit, neglecting all combined space and time derivatives of order three and linearizing the sine factors in the last two terms, Eq. (5) reduces to the overdamped driven sine-Gordon equation with dissipation

(6) When the right-handed side of Eq. (6) is zero, i.e. in the absence of damping and driving, Eq. (6) has the stationary soliton solution cbs(x) - 4tan-‘exp(x) .

(7)

While the damping term -d@-’ /dt tends to destroy the soliton, the external current Z drives the soliton. Thus for small currents the damping term dominates, destroying the sohton. When large currents dominate, the soliton moves with a uniform velocity. In conclusion, we studied the vortex motion in overdamped JJ ladders. Since the single-vortex excitation energy is finite in this system [ 83, even in the absence of an external magnetic field, there may exist excess vortices that are stabilized by large currents, I > Z, = 0.34&. For inte~~iate currents, Z, < I < Z,, the IV relation is given by a power law, V,,/Z,R = [ 6.14 - 7.37( 1 - Z/ZC)o.438]pwith vortex density p. The motion of the excess vortices induces localized voltage drops, which move with constant velocity and amplitude proportional to the velocity. When Z > Z, there is continuous creation and annihilation of the

S. Kim/Physics

412

Letrers A 235 (1997)

vortex-antivortex pairs, creating and destroying localized voltage drops, respectively. This behavior may be explained in terms of overdamped driven sine-Gordon equation with dissipation. This work was supported by the Ministry of Information and Communications, South Korea. We are grateful to Dr. E.H. Lee for his support of this research.

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152 (1988)

1; B 165/166

(1990)

I

408-412

121 J.M. Kosterlitz, T.J. Thouless, J. Phys. C 6 ( 1973) I 181; J.V. Jose, L.P. Kadanoff, S. Kirkpatrick, D.R. Nelson, Phys. Rev. B 16 (1977) 1217. 131 S. Teitel, J. Jayaprakash. Phys. Rev. B 27 ( 1983) 598: W. Shih, D. Stroud, Phys. Rev. B 28 (1983) 6575. 141 H.S.J. van der Zant, EC. Fritschy, T.P. Oralndo, J.E. Mooij, Europhys. Len. 18 ( 1992) 343. (51 B.J. Kim, S. Kim, S.J. Lee, Phys. Rev. B 51 (1995) 8462. 161 J. Kim, H.J. Lee, Phys. Rev. B 47 (1993) 582. 171 L-J. Hwang, S. Ryu, D. Stroud, Phys. Rev. B 53 (1996) R506. 181 S. Kim, Phys. Lea. A 229 (1997) 190. [9] W.H. Press, BP Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1986).