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Fluxon motion in one-dimensional Josephson junction arrays
PHYSICA
Physica B 194-:196 (1994) 1765-1766 Noah-Holland
Fluxon motion in one-dimensional Josephson junction arrays A. V. Ustinov a*, M. Cirillo a , B. A. Malomed b , and Yu. S. Kivshar e ~Dipartimento di Fisica, Universit£ di R o m a " T o t Vergata", 00133 Roma, Italy. bDepartment of Applied Mathematics, School of Mathematical Sciences Tel Aviv University, R a m a t Aviv 69978, Israel ¢Institut fhr Theoretische Physik I, Universit/it Diisseldorf, D-4000 Diisseldorf 1, Germany Current-voltage ( I V - ) characteristics of linear and annular discrete arrays of underdamped small Josephson junctions with different values of single loop inductances are calculated numerically and compared with those obtained for continuum long Josephson junctions. The resonances between the moving fluxon and the linear waves caused by the array discreteness induce steps on t h e / V - c u r v e which do not exist in the continuum case. The voltage position of the steps is found to be in good agreement with the kinematic approach based on the Frenkel-Kontoi'ova model. Our results show that in Josephson fluxon devices the discreteness may lead to strong super-radiant emission of electromagnetic waves by moving fluxons.
1. I N T R O D U C T I O N A discretized Josephson transmission line was at the basis of the the idea of the so-called phasemode logic suggested by Nakajima et al.[1]. One of the advantages of the discrete array with respect to the long Josephson junction is the fact that the m a x i n m m velocity of electromagnetic wave propagation (so-called, Swihart velocity) is much higher than in the continuum case. We study numerically and analytically the fluxon / V - c h a r a c t e r i s t i c s in underdamped 1D parallel array of Josephson junctions and compare the results with the continuum case. In order to discriminate between the influence of the array boundaries and the effects induced by discreteness we consider both the periodic boundary conditions case and that with the open boundaries. 2. M O D E L
A 1D parallel array of Josephson junctions is described by the discretized version of the perturbed sine-Gordon equation 99n-1 -- 2 ~ n q- ~ n + l II I = ~ , + ~ G ~ + s i n ~,,~ + 7 , (1) a2 *On leave from: Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow district, 142432 Russia. Paa-tial financial support of the Sucrytec project of CNR (Italy) is acknowledged.
where 1 < n _< N, N is the number of junctions which are assumed to be identical, ~,~ is the superconducting phase difference on the n - t h junction, the spatial coordinate x is normalized to the effective aosephson penetration depth Aj, the time t is normalized to the inverse plasma frequency Uao1, c~ is the dissipation coefficient., 7 is the normalized bias current. The discreteness parameter a = fir 1/2 = (2rrLoi~/dpo)U2 is measured in units of the effective Aj, where L0 is the inductance of a single cell of the array, [~ is the critical current of each junction. In the limit of a --~ 0 the model (1) corresponds to the continuum case. The spatial discreteness leads to the radiation of small-amplitude waves ("phonons") by a moving soliton[2]. Here we focus on the conditions under which the resonances (which could be observed experimentally in / V - c u r v e of the array) between fluxons moving in the discrete system with dissipation and the linear waves radiated by them are generated. With ~ = 9' = 0 Eq. (1) correspond to the well-known Frenkel-Kontorova model. The dispersion law for linear waves is ~2 = 1 + 4 _ s i n 2 ( ~ ) , where 0 _< k _< 2~a Tile phase velocity of the linear waves excited by the moving fluxon coincides with the fluxon velocity. This leads to the following condition for a resonance between the emitted waves and the moving periodic chain of fluxons spaced by L [3]:
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93) 1514-M
1766 "
v.,-
L
2urn
1~ V
+
~
~rna
sin2(-~).
(2)
i
-
*
--A
-[
r1
where rn is an integer, rn,,~i,, < m. < L_ with mmi,~ cl corresponding to the highest velocity (2) below unity. ',5 :;
3. NUMERICAL RESULTS
Figure 1 shows the numerically integrated single-fluxon I V - characteristics of the array described by Eq. (1). The chosen parameters N = 10, a = 1.0, and c~ = 0.1 could be achieved in a typical experiment. The array voltage < ~y' > is proportional to the average fluxon velocity v. The difference between the discrete and the continuum cases with periodic boundary conditions (solid and dashed lines) is clearly seen from Fig. 1. The I V - c u r v e for the discrete array consists of a series of equally spaced current singularilies, corresponding to the resonances (2) between the moving fluxon (L = N a = 10) and its radiation induced by the array discreteness. The voltage profile ~ ( / , ) during 3 fluxon revolutions calculated for the bias point shown by an arrow is given in the irrset of Fig. 1. The oscillations corresponding to the 5-th harmonic of the fluxon revolution frequency are clearly seen. The amplitude b~ of the oscillations increases with 7. For small cv the increase of b~. leads to the nonlinear amplitude-dependent correction ~2 _~ ~ ' ~ - g ~~'J~ to the dispersion relation, which explains the backbending of some resonant steps at high ? (e.g., step m = 4 in Fig. 1). For periodic boundary conditions, the comparison of the numerically calculated fluxon w~locity for each resonance with that given by Eq.(2) showed a good agreement for a broad ranges of ct (0.2 < a < 1.5) and m [3]. Some discrepancy appears at. high velocities where the quasi-continuum approach to tire fluxon motion fails since the Lorentz contracted size of the fluxon becomes to be of the order of the array discreteness a. The same effect also leads to the shiq of the discrete array curve (from that of the continuum) towards the Rower velocities at, high bias current 7. Tire open boundary condition case (which is easier to realize experimentally than the periodic
)2
) 0
.
0 0
0.2
fluxon
0.4
.
.
.
',..6
velocity,
.
.
0.~
.
' 0
v
Figure 1. l V-characteristics of the 1D Josephsou junction array with periodic (solid line) and open (squares) boundary conditions. The dashed line shows the continuum case with periodic boundary conditions for the same normalized length.
one) calculated with the same array parameters is shown by open squares. This characteristics is the discrete array analog of the first zero field step in a long Josephson junction. The most pronounced discreteness-induced resonances of the periodic boundary condition case coincide with that for tile open boundaries. In addition, we also find the resonant steps between that for the periodic case. This behavior is expected due to halving of the fundamental cavity mode frequency provided by the open boundaries, in a similar way as it appears for the Fiske steps in long aosephson ,junctions.
REFERENCES 1. 2. 3.
K. Nakajima, H. Sugahara, A. Fujimaki, and Y. Sawada, J. Appl. Phys. 66 (1989) 949. M. Peyrard and M. D. Kruskal, Physica D 14 (1984) 88. A.V. Ustinov, M. Cirillo, and B. A. Malomed, Plays. Rev. B 47 (1993) to be published.
Physica B 194-:196 (1994) 1765-1766 Noah-Holland
Fluxon motion in one-dimensional Josephson junction arrays A. V. Ustinov a*, M. Cirillo a , B. A. Malomed b , and Yu. S. Kivshar e ~Dipartimento di Fisica, Universit£ di R o m a " T o t Vergata", 00133 Roma, Italy. bDepartment of Applied Mathematics, School of Mathematical Sciences Tel Aviv University, R a m a t Aviv 69978, Israel ¢Institut fhr Theoretische Physik I, Universit/it Diisseldorf, D-4000 Diisseldorf 1, Germany Current-voltage ( I V - ) characteristics of linear and annular discrete arrays of underdamped small Josephson junctions with different values of single loop inductances are calculated numerically and compared with those obtained for continuum long Josephson junctions. The resonances between the moving fluxon and the linear waves caused by the array discreteness induce steps on t h e / V - c u r v e which do not exist in the continuum case. The voltage position of the steps is found to be in good agreement with the kinematic approach based on the Frenkel-Kontoi'ova model. Our results show that in Josephson fluxon devices the discreteness may lead to strong super-radiant emission of electromagnetic waves by moving fluxons.
1. I N T R O D U C T I O N A discretized Josephson transmission line was at the basis of the the idea of the so-called phasemode logic suggested by Nakajima et al.[1]. One of the advantages of the discrete array with respect to the long Josephson junction is the fact that the m a x i n m m velocity of electromagnetic wave propagation (so-called, Swihart velocity) is much higher than in the continuum case. We study numerically and analytically the fluxon / V - c h a r a c t e r i s t i c s in underdamped 1D parallel array of Josephson junctions and compare the results with the continuum case. In order to discriminate between the influence of the array boundaries and the effects induced by discreteness we consider both the periodic boundary conditions case and that with the open boundaries. 2. M O D E L
A 1D parallel array of Josephson junctions is described by the discretized version of the perturbed sine-Gordon equation 99n-1 -- 2 ~ n q- ~ n + l II I = ~ , + ~ G ~ + s i n ~,,~ + 7 , (1) a2 *On leave from: Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow district, 142432 Russia. Paa-tial financial support of the Sucrytec project of CNR (Italy) is acknowledged.
where 1 < n _< N, N is the number of junctions which are assumed to be identical, ~,~ is the superconducting phase difference on the n - t h junction, the spatial coordinate x is normalized to the effective aosephson penetration depth Aj, the time t is normalized to the inverse plasma frequency Uao1, c~ is the dissipation coefficient., 7 is the normalized bias current. The discreteness parameter a = fir 1/2 = (2rrLoi~/dpo)U2 is measured in units of the effective Aj, where L0 is the inductance of a single cell of the array, [~ is the critical current of each junction. In the limit of a --~ 0 the model (1) corresponds to the continuum case. The spatial discreteness leads to the radiation of small-amplitude waves ("phonons") by a moving soliton[2]. Here we focus on the conditions under which the resonances (which could be observed experimentally in / V - c u r v e of the array) between fluxons moving in the discrete system with dissipation and the linear waves radiated by them are generated. With ~ = 9' = 0 Eq. (1) correspond to the well-known Frenkel-Kontorova model. The dispersion law for linear waves is ~2 = 1 + 4 _ s i n 2 ( ~ ) , where 0 _< k _< 2~a Tile phase velocity of the linear waves excited by the moving fluxon coincides with the fluxon velocity. This leads to the following condition for a resonance between the emitted waves and the moving periodic chain of fluxons spaced by L [3]:
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93) 1514-M
1766 "
v.,-
L
2urn
1~ V
+
~
~rna
sin2(-~).
(2)
i
-
*
--A
-[
r1
where rn is an integer, rn,,~i,, < m. < L_ with mmi,~ cl corresponding to the highest velocity (2) below unity. ',5 :;
3. NUMERICAL RESULTS
Figure 1 shows the numerically integrated single-fluxon I V - characteristics of the array described by Eq. (1). The chosen parameters N = 10, a = 1.0, and c~ = 0.1 could be achieved in a typical experiment. The array voltage < ~y' > is proportional to the average fluxon velocity v. The difference between the discrete and the continuum cases with periodic boundary conditions (solid and dashed lines) is clearly seen from Fig. 1. The I V - c u r v e for the discrete array consists of a series of equally spaced current singularilies, corresponding to the resonances (2) between the moving fluxon (L = N a = 10) and its radiation induced by the array discreteness. The voltage profile ~ ( / , ) during 3 fluxon revolutions calculated for the bias point shown by an arrow is given in the irrset of Fig. 1. The oscillations corresponding to the 5-th harmonic of the fluxon revolution frequency are clearly seen. The amplitude b~ of the oscillations increases with 7. For small cv the increase of b~. leads to the nonlinear amplitude-dependent correction ~2 _~ ~ ' ~ - g ~~'J~ to the dispersion relation, which explains the backbending of some resonant steps at high ? (e.g., step m = 4 in Fig. 1). For periodic boundary conditions, the comparison of the numerically calculated fluxon w~locity for each resonance with that given by Eq.(2) showed a good agreement for a broad ranges of ct (0.2 < a < 1.5) and m [3]. Some discrepancy appears at. high velocities where the quasi-continuum approach to tire fluxon motion fails since the Lorentz contracted size of the fluxon becomes to be of the order of the array discreteness a. The same effect also leads to the shiq of the discrete array curve (from that of the continuum) towards the Rower velocities at, high bias current 7. Tire open boundary condition case (which is easier to realize experimentally than the periodic
)2
) 0
.
0 0
0.2
fluxon
0.4
.
.
.
',..6
velocity,
.
.
0.~
.
' 0
v
Figure 1. l V-characteristics of the 1D Josephsou junction array with periodic (solid line) and open (squares) boundary conditions. The dashed line shows the continuum case with periodic boundary conditions for the same normalized length.
one) calculated with the same array parameters is shown by open squares. This characteristics is the discrete array analog of the first zero field step in a long Josephson junction. The most pronounced discreteness-induced resonances of the periodic boundary condition case coincide with that for tile open boundaries. In addition, we also find the resonant steps between that for the periodic case. This behavior is expected due to halving of the fundamental cavity mode frequency provided by the open boundaries, in a similar way as it appears for the Fiske steps in long aosephson ,junctions.
REFERENCES 1. 2. 3.
K. Nakajima, H. Sugahara, A. Fujimaki, and Y. Sawada, J. Appl. Phys. 66 (1989) 949. M. Peyrard and M. D. Kruskal, Physica D 14 (1984) 88. A.V. Ustinov, M. Cirillo, and B. A. Malomed, Plays. Rev. B 47 (1993) to be published.