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Irreversible magnetisation of a superconducting ring containing a series array of Josephson junctions
Physica C 235-240 0994)3325-3326 North-Holland
P~YSIgA
Irreversible Magnetisation of a superconducting ring containing a series array of Josephson junctions. J. Martinek, J. Stankowski Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznafi, Poland. Metastable states of an n-junction loop are investigated as a function of external magnetic field, loop inductance and critical current. The magnetic response of such a system is analytically delivered. The conditions of the hysteretic behaviour of the loop are investigated. The obtained curve of magnetic hysteresis is very similar to the magnetisation data for high temperature superconductors. 1. INTRODUCTION Soon after the discovery of high temperature superconductors it was found that these material show irreversible magnetic effect. This phenomenon was investigated on the basis of the critical-state model of conventional hard superconductors [1,2] or because of the granular structure of ceramic samples as a positionally disordered system of grains coupled via Josephson weak links [3-7]. Recently, there have been published numerical simulations of the Josephson junctions network magnetization accounting for the magnetic energy of the screening currents [3,4]. In all these paper huge networks of Josephson junctions have been considered. In this work we investigate the magnetization curves for superconducting ring containing a series array of Josephson junctions. No external current is applied to this system. We found that the hysteresis loops of this system have similar shape to that typical for high temperature superconductors [8]. 2. MODEL Let us consider the n-junction loop of the area S, where q~x is the applied magnetic flux in units of ~o/2rt threading the ring inductance L. We will consider the case when all junctions critical currents are equal (I 1 = ,..- = In = Ic). In stationary state, the Gibbs energy of the interferometer is given by [9,10]: U=gc
Vn
E ( 1 - C O S ~ i ) + ( q : ) - q ~ x ) 2 /213 Li=l
]
(1)
where the first term is the junctions coupling energy and the second is the magnetic energy stored in the inductance, qoi is the phase difference at the i-th junction, q~ is the total magnetic flux in units of • o/27t, the constant ec = Ic~o/27t is the energy uni! and 13=27tLIc/~o is the dimensionless inductance. The junction phases and the flux fulfil the fluxoid equation: n
(p = Eq~i i=l
(2)
Only these values of q0i for which the free energy has a local minimum correspond t~ stationar) states. The q~(q~x) dependence in most cases is multivalued, and hysteretic. We have studied the conditions for the hysteretic behaviour and two different kinds of hysteretic behaviour arc: recognized. In the first one the qo(q~x) dependence is not multivalued in the whole range of q~x (A), cp - ( P x .
0.Sn
-O,Sn -6n -4n -2n 2n 4n 6n Fig. 1. Plot of the magnetic hysteresis for n = 2 and [~ =0.5 7t.
0921-4534D4/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0921-4534(94)02228-3
J Martinek, J Stankowski/Physica C 235 240 (1994) 3325 3326
3326
(Fig. 1.), in the second case (B) it is multivalued for all applied values of the magnetic field (Fig.Z). (P - .q)X
2n 7/
,
\\\" \
0
2,
-27/
¢P- (Px 27/
\
-67/ -47/ -2n 2n 47/ 6n Fig.2. Plot of the magnetic hysteresis for n = 2 and
13=2x. The kind of this dependence in a given situation is determined by the values of n and 13. For the single junction loop the case A occurs for I3 > 1 [10] while case 13 for (132-1)l/2+arccos(-13-1)>2n (13 > 4.60334). For n = 2 the system is in the mode A (Figl.) or for i3 > n in the mode B (Fig.2.). For n = 3 the interferometer is in the mode B for [3 > n/2. A superconducting ring containing more then 3 junctions with the same value of the critical current of the junction will work only in the mode 13 (Fig.3.). q ) - qOx
The maximal amplitude of magnetization (qD - q~x) for any value of n is equal to I3. For large n, partial magnetization jumps disappear (Fig.4.) an( magnetization becomes smooth.
.
0.Snl
-0.57/i -6n -4n -27/ 0 27/ 4n 6n Fig.3. Plot of the magnetic hysteresis for n = 4 and 13 =0.5 n.
7/
0
\
\
\
\
\
-7/
-27/ -12~ -Sn -47/ 0 47/ 8n 12~r Fig.4. Plot of the magnetic hysteresis for n = 20 and 13= 2 n. This work was supported in part by KBN Grant No 2 P 302 006 05. REFERENCES 1. K H. Muller, C. Andrikidis, Phys. Rex,. B 49. 1294 (1994). 2. C. J. Lin, C. Schlenker, J. Schubert, and t3 Stritzker, Phys. Rev. B 48, 13911 (1993). 3. C. Auletta, R. De Luca, S. Pace, and G. Raiconi Phys. Rex,. B 47, 14326 (1993). 4. T. Wolf, A. Majhofer, Phys. Rev. 13 47, 5383 (1993). 5. S. J. Lewandowski, Phys. Rev. 13 45, 231',' (1992). 6. D. Dominiguez, J. V. Jose, Phys. Rev. Lett. 69. 514 (1992). 7. D. X. Chen, A. Hernando, Phys. Rev. B 49, 465 (1994). 8. J. Martinek, J. Stankowski (to be published). 9. A. Barone, G. Paterno, Physics and Applications of the Josephson Effect (Wiley, New York, 1982). i 0. K. K. Likharev, Dynamics of Josephson Junctior, and Circuits (Gordon and Breach, New York. 1988).
P~YSIgA
Irreversible Magnetisation of a superconducting ring containing a series array of Josephson junctions. J. Martinek, J. Stankowski Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznafi, Poland. Metastable states of an n-junction loop are investigated as a function of external magnetic field, loop inductance and critical current. The magnetic response of such a system is analytically delivered. The conditions of the hysteretic behaviour of the loop are investigated. The obtained curve of magnetic hysteresis is very similar to the magnetisation data for high temperature superconductors. 1. INTRODUCTION Soon after the discovery of high temperature superconductors it was found that these material show irreversible magnetic effect. This phenomenon was investigated on the basis of the critical-state model of conventional hard superconductors [1,2] or because of the granular structure of ceramic samples as a positionally disordered system of grains coupled via Josephson weak links [3-7]. Recently, there have been published numerical simulations of the Josephson junctions network magnetization accounting for the magnetic energy of the screening currents [3,4]. In all these paper huge networks of Josephson junctions have been considered. In this work we investigate the magnetization curves for superconducting ring containing a series array of Josephson junctions. No external current is applied to this system. We found that the hysteresis loops of this system have similar shape to that typical for high temperature superconductors [8]. 2. MODEL Let us consider the n-junction loop of the area S, where q~x is the applied magnetic flux in units of ~o/2rt threading the ring inductance L. We will consider the case when all junctions critical currents are equal (I 1 = ,..- = In = Ic). In stationary state, the Gibbs energy of the interferometer is given by [9,10]: U=gc
Vn
E ( 1 - C O S ~ i ) + ( q : ) - q ~ x ) 2 /213 Li=l
]
(1)
where the first term is the junctions coupling energy and the second is the magnetic energy stored in the inductance, qoi is the phase difference at the i-th junction, q~ is the total magnetic flux in units of • o/27t, the constant ec = Ic~o/27t is the energy uni! and 13=27tLIc/~o is the dimensionless inductance. The junction phases and the flux fulfil the fluxoid equation: n
(p = Eq~i i=l
(2)
Only these values of q0i for which the free energy has a local minimum correspond t~ stationar) states. The q~(q~x) dependence in most cases is multivalued, and hysteretic. We have studied the conditions for the hysteretic behaviour and two different kinds of hysteretic behaviour arc: recognized. In the first one the qo(q~x) dependence is not multivalued in the whole range of q~x (A), cp - ( P x .
0.Sn
-O,Sn -6n -4n -2n 2n 4n 6n Fig. 1. Plot of the magnetic hysteresis for n = 2 and [~ =0.5 7t.
0921-4534D4/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0921-4534(94)02228-3
J Martinek, J Stankowski/Physica C 235 240 (1994) 3325 3326
3326
(Fig. 1.), in the second case (B) it is multivalued for all applied values of the magnetic field (Fig.Z). (P - .q)X
2n 7/
,
\\\" \
0
2,
-27/
¢P- (Px 27/
\
-67/ -47/ -2n 2n 47/ 6n Fig.2. Plot of the magnetic hysteresis for n = 2 and
13=2x. The kind of this dependence in a given situation is determined by the values of n and 13. For the single junction loop the case A occurs for I3 > 1 [10] while case 13 for (132-1)l/2+arccos(-13-1)>2n (13 > 4.60334). For n = 2 the system is in the mode A (Figl.) or for i3 > n in the mode B (Fig.2.). For n = 3 the interferometer is in the mode B for [3 > n/2. A superconducting ring containing more then 3 junctions with the same value of the critical current of the junction will work only in the mode 13 (Fig.3.). q ) - qOx
The maximal amplitude of magnetization (qD - q~x) for any value of n is equal to I3. For large n, partial magnetization jumps disappear (Fig.4.) an( magnetization becomes smooth.
.
0.Snl
-0.57/i -6n -4n -27/ 0 27/ 4n 6n Fig.3. Plot of the magnetic hysteresis for n = 4 and 13 =0.5 n.
7/
0
\
\
\
\
\
-7/
-27/ -12~ -Sn -47/ 0 47/ 8n 12~r Fig.4. Plot of the magnetic hysteresis for n = 20 and 13= 2 n. This work was supported in part by KBN Grant No 2 P 302 006 05. REFERENCES 1. K H. Muller, C. Andrikidis, Phys. Rex,. B 49. 1294 (1994). 2. C. J. Lin, C. Schlenker, J. Schubert, and t3 Stritzker, Phys. Rev. B 48, 13911 (1993). 3. C. Auletta, R. De Luca, S. Pace, and G. Raiconi Phys. Rex,. B 47, 14326 (1993). 4. T. Wolf, A. Majhofer, Phys. Rev. 13 47, 5383 (1993). 5. S. J. Lewandowski, Phys. Rev. 13 45, 231',' (1992). 6. D. Dominiguez, J. V. Jose, Phys. Rev. Lett. 69. 514 (1992). 7. D. X. Chen, A. Hernando, Phys. Rev. B 49, 465 (1994). 8. J. Martinek, J. Stankowski (to be published). 9. A. Barone, G. Paterno, Physics and Applications of the Josephson Effect (Wiley, New York, 1982). i 0. K. K. Likharev, Dynamics of Josephson Junctior, and Circuits (Gordon and Breach, New York. 1988).