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Diamagnetic respinse of a very simple superconducting granular system
PHISICAG
Physica C 185-189 (1991) 2739-2740 North-Holland
DIAMAGNETIC RESPONSE OF A VERY SIMPLE SUPERCONDUCTING GRANULAR SYSTEM S. PACE and R. DE LUCA Dipartimento di Fisica, Universita' deg!i Studi di Salerno, 84081 Baronissi (Salerno) Italy The diamagnetic response of superconducting granular systems is studied by means of a network of Josephson junctions (JJs) and inductances. We study the diamagnetic properties of the simplest possible granular superconductor: a three-grain system which reduces to a superconducting loop closed by three JJs. We numerically determine: the irreversibility line of the system in the H vs T plane and the temperature dependent lower threshold field Hgcl(T). In addition, we remark that the metastable shielding states realized above a thermal activation line in the H vs T diagram decay in experimentally detectable times toward thermodynamic equilibrium. Finally, we mention how this simple model can be extended to more complex systems. I . INTRODUCTION The diamagnetic properties of granular superconductors can be described by networks of Josephson junctions (JJs) and inductances 1-4. However, given the great number of grains in a sample, an analysis of the problem is possible only under rather restrictive conditions. Nevertheless, certain features of these systems are still retained by a simple planar sample made of three grains in a loop weakly coupled by three identical JJs, studied in the presence of an external magnetic field H, perpendicular to the plane and applied after zero-field coo!ing (ZFC). Fluxoid quantization 5 relates the gauge invariant phase difference (pj across the contact surfaces of the grains to the total magnetic flux • in the superconducting loop given by: ~) =LeffIa +p.oSeffH, where Leffand Serf are effective ,,alues of the loop inductance and area parameters respectively, assuming that the grains are seen as perfect diamagnets. The dynamics of penetration of flux quanta in the loop is determined through the dynamical equation for the supercond,,n÷;n~
nhaaa
rl~ffaranna
in ~h~
R.~.l
2. DIAMAGNETIC CHARACTERISTICS In order to find the value of the field Hgc; at which first penetration of flux quanta occurs, we set the first and second functional derivative of the system's Gibbs potential equal to zero, so that we obtain: 8G/5(i::)=0=~6@osin (2~;d#/3@0)+ @- @ext
(2)
62G/602=0=1 +(2~x~/3)cos (2r~@/3@o)
tar.dAIS
In this case the thermodynamic pot,~ntial can be written as follows: G(([)) = (~[)-~;)ext)2/2Lef[ + (31 j(h ,T)a#o)[1 - c o s ( 2 ~ ¢ / 3 ¢ 0 ) ]
and temperature dependent maximum Josephson current of the junctions. In what follows we shall assume that the maximum Josephson current dependence on h and T is given by the usual Fraunhofer Hke pattern 5 and the Ambegaoka: i-Jaratoff formula 5 respectively. It is well knowr~ that superconducting loops closed by JJs present two types of magnetic regimes5: a global reversible regime when the Gibbs potential has only one minimum, which corresponds to thermodynamic equilibrium, and an irreversible one when more than one minimum appears in the washboard parabolic potential G(@).
(1;
where ~ e x t = # o S t o t H iS the geometrical applied flux, and ]j(h,T) is the local field (h)
where [3=Ij(h,T)Lcfr/~o. To find an analytical expression for ~{gc~. we neglect the dependence of ~3 from h, since first penetration takes p~ace at extremely low fieids, so that: Hgc 1(T)=[~cosy+3/4 +3y/2nlqSo/FtoSefr
0921-4534D1/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights ~ser.cd.
(3)
S. Pace, R. De Luca / Diamagnetic response of a very simple superconducting granular system
2740
where x = s i n ' l ( 3 / 2 n ~ ) , with 0<3/2~_<1 and ~=~(T). A representation of the lower threshold field line Hgcl(T) is given in fig.1 (broken line) for Ijo=Ij(0,0)=10~A and ~o=I joLeff/(~o=1 0. The crossover line from global reversible to irreversible behavior in the H vs T plane can be found by considering the derivative of • with respect to ~ext, which is found by implicitly differentiating the first of eqs.2: de/d~ext= 1/[l+(2~/3)cos(2x~/3~o)]
(4)
This expression is everywhere nonsingular for values of the parameter ~ such that (2n~/3)<1. Therefore, in the H vs T plane this irreversibility line is given by : (27¢~/3)=(2~I j(h,T)Leff/3~o)=l
(5)
We approximate h with H, so that a numerical solution Hrev=Hrev(T) to eq.5 is easily found and is shown in fig.1 as a solid line for the same parameters as before and the effective area of the junction Sj=10-2Seff.
1~1
-- 7 Globally
Thermal
,
i
......
~
trapping
i
1
1
region
activation
Flux
0.0
reversible
i
0.5
i
,
. ~
T/~c
It i
- /
1.o
FIGURE 1 Diamagnetic regimes of the three grain system for [3o=10, Sj=10"2Seff, [ j 0 = 1 0 ~ A , T c = I 0 0 K , 2keTJ&o=l ; brokenline: thre~h~Ir' field Hgcl(T); sotid line: crossover field Hrev{1); dashed line: lower bound of thermal activation of flux quanta.
The above crossover fields have been calculated by neglecting thermal activation
processes, which determine time decay of the metastable shielding states. In order to have an idea of the conditions in which ",he classical probability rate of thermal activation of flux quanta becomes comparable with the inverse of characteristic measuring times, we can set the thermal energy of the flux quanta equal to the Josephson energy barrier E j = I j ( h , T ) ~ o / 2 n . The former expression is a good approximation of the parabolic washboard potential energy barrier in the c a s e ~ - l . For H and T above this activation line, shown in fig.1 as a dashed line for the same values of the parameters, a reversible experimental behavior is attained. 3. CONCLUSION We determined the regions of different low-field diamagnetic regimes of a H vs T diagram for a three grain system. The crossover lines: reported in fig. 1 for a given choice of par~,meters, can show, if any, different intersection points for different choices of the parameters. However, the main features are that one reversible quasishielding region, one flux trapping irreversible portion, and one global reversible part is present in the H vs T diagram for values of the parameter I~ greater than 3/2n. An extension of these results to a real sample is not trivial mainly because of the local character of the field h. Nevertheless, we might notice that, if [~,,1 for any elementary n-grain system, all the intergranular loops in the sample behave independently, so that some of the above results still apply. REFERENCES 1. M.Tinkham and C.J. Lobb, Solid State Phys. 42 (1989) 91. 2. J.R. Clem, Physica 153-155C (1988) 50. 3. R. De Luca et al., Phys.Lett. !54A (t9£1) 185. 4. D Stroud and C. Ebner, Physica 153-155C (1988) 63 . A. Barone e.nd G. Patern6, Physics and Application of th~ Josephson effect (Wiley, New Yo~, ~£82).
Physica C 185-189 (1991) 2739-2740 North-Holland
DIAMAGNETIC RESPONSE OF A VERY SIMPLE SUPERCONDUCTING GRANULAR SYSTEM S. PACE and R. DE LUCA Dipartimento di Fisica, Universita' deg!i Studi di Salerno, 84081 Baronissi (Salerno) Italy The diamagnetic response of superconducting granular systems is studied by means of a network of Josephson junctions (JJs) and inductances. We study the diamagnetic properties of the simplest possible granular superconductor: a three-grain system which reduces to a superconducting loop closed by three JJs. We numerically determine: the irreversibility line of the system in the H vs T plane and the temperature dependent lower threshold field Hgcl(T). In addition, we remark that the metastable shielding states realized above a thermal activation line in the H vs T diagram decay in experimentally detectable times toward thermodynamic equilibrium. Finally, we mention how this simple model can be extended to more complex systems. I . INTRODUCTION The diamagnetic properties of granular superconductors can be described by networks of Josephson junctions (JJs) and inductances 1-4. However, given the great number of grains in a sample, an analysis of the problem is possible only under rather restrictive conditions. Nevertheless, certain features of these systems are still retained by a simple planar sample made of three grains in a loop weakly coupled by three identical JJs, studied in the presence of an external magnetic field H, perpendicular to the plane and applied after zero-field coo!ing (ZFC). Fluxoid quantization 5 relates the gauge invariant phase difference (pj across the contact surfaces of the grains to the total magnetic flux • in the superconducting loop given by: ~) =LeffIa +p.oSeffH, where Leffand Serf are effective ,,alues of the loop inductance and area parameters respectively, assuming that the grains are seen as perfect diamagnets. The dynamics of penetration of flux quanta in the loop is determined through the dynamical equation for the supercond,,n÷;n~
nhaaa
rl~ffaranna
in ~h~
R.~.l
2. DIAMAGNETIC CHARACTERISTICS In order to find the value of the field Hgc; at which first penetration of flux quanta occurs, we set the first and second functional derivative of the system's Gibbs potential equal to zero, so that we obtain: 8G/5(i::)=0=~6@osin (2~;d#/3@0)+ @- @ext
(2)
62G/602=0=1 +(2~x~/3)cos (2r~@/3@o)
tar.dAIS
In this case the thermodynamic pot,~ntial can be written as follows: G(([)) = (~[)-~;)ext)2/2Lef[ + (31 j(h ,T)a#o)[1 - c o s ( 2 ~ ¢ / 3 ¢ 0 ) ]
and temperature dependent maximum Josephson current of the junctions. In what follows we shall assume that the maximum Josephson current dependence on h and T is given by the usual Fraunhofer Hke pattern 5 and the Ambegaoka: i-Jaratoff formula 5 respectively. It is well knowr~ that superconducting loops closed by JJs present two types of magnetic regimes5: a global reversible regime when the Gibbs potential has only one minimum, which corresponds to thermodynamic equilibrium, and an irreversible one when more than one minimum appears in the washboard parabolic potential G(@).
(1;
where ~ e x t = # o S t o t H iS the geometrical applied flux, and ]j(h,T) is the local field (h)
where [3=Ij(h,T)Lcfr/~o. To find an analytical expression for ~{gc~. we neglect the dependence of ~3 from h, since first penetration takes p~ace at extremely low fieids, so that: Hgc 1(T)=[~cosy+3/4 +3y/2nlqSo/FtoSefr
0921-4534D1/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights ~ser.cd.
(3)
S. Pace, R. De Luca / Diamagnetic response of a very simple superconducting granular system
2740
where x = s i n ' l ( 3 / 2 n ~ ) , with 0<3/2~_<1 and ~=~(T). A representation of the lower threshold field line Hgcl(T) is given in fig.1 (broken line) for Ijo=Ij(0,0)=10~A and ~o=I joLeff/(~o=1 0. The crossover line from global reversible to irreversible behavior in the H vs T plane can be found by considering the derivative of • with respect to ~ext, which is found by implicitly differentiating the first of eqs.2: de/d~ext= 1/[l+(2~/3)cos(2x~/3~o)]
(4)
This expression is everywhere nonsingular for values of the parameter ~ such that (2n~/3)<1. Therefore, in the H vs T plane this irreversibility line is given by : (27¢~/3)=(2~I j(h,T)Leff/3~o)=l
(5)
We approximate h with H, so that a numerical solution Hrev=Hrev(T) to eq.5 is easily found and is shown in fig.1 as a solid line for the same parameters as before and the effective area of the junction Sj=10-2Seff.
1~1
-- 7 Globally
Thermal
,
i
......
~
trapping
i
1
1
region
activation
Flux
0.0
reversible
i
0.5
i
,
. ~
T/~c
It i
- /
1.o
FIGURE 1 Diamagnetic regimes of the three grain system for [3o=10, Sj=10"2Seff, [ j 0 = 1 0 ~ A , T c = I 0 0 K , 2keTJ&o=l ; brokenline: thre~h~Ir' field Hgcl(T); sotid line: crossover field Hrev{1); dashed line: lower bound of thermal activation of flux quanta.
The above crossover fields have been calculated by neglecting thermal activation
processes, which determine time decay of the metastable shielding states. In order to have an idea of the conditions in which ",he classical probability rate of thermal activation of flux quanta becomes comparable with the inverse of characteristic measuring times, we can set the thermal energy of the flux quanta equal to the Josephson energy barrier E j = I j ( h , T ) ~ o / 2 n . The former expression is a good approximation of the parabolic washboard potential energy barrier in the c a s e ~ - l . For H and T above this activation line, shown in fig.1 as a dashed line for the same values of the parameters, a reversible experimental behavior is attained. 3. CONCLUSION We determined the regions of different low-field diamagnetic regimes of a H vs T diagram for a three grain system. The crossover lines: reported in fig. 1 for a given choice of par~,meters, can show, if any, different intersection points for different choices of the parameters. However, the main features are that one reversible quasishielding region, one flux trapping irreversible portion, and one global reversible part is present in the H vs T diagram for values of the parameter I~ greater than 3/2n. An extension of these results to a real sample is not trivial mainly because of the local character of the field h. Nevertheless, we might notice that, if [~,,1 for any elementary n-grain system, all the intergranular loops in the sample behave independently, so that some of the above results still apply. REFERENCES 1. M.Tinkham and C.J. Lobb, Solid State Phys. 42 (1989) 91. 2. J.R. Clem, Physica 153-155C (1988) 50. 3. R. De Luca et al., Phys.Lett. !54A (t9£1) 185. 4. D Stroud and C. Ebner, Physica 153-155C (1988) 63 . A. Barone e.nd G. Patern6, Physics and Application of th~ Josephson effect (Wiley, New Yo~, ~£82).