Physica B 194-196 (1994) 1551-1552 North-Holland PHYSICA

Simpl i f ied Josephson junc t ion arrays as mode l s o f granular superconduc tors

Sandro Pace, Roberto De Luca, Aniello Saggese

Department of Physics, University of Salerno, 1-84081 Baronissi (Salerno), Italy

We describe the simplest Josephson junction array models that can be adopted in studying the diamagnetic properties of granular superconductors. We consider cylindrical superconducting granular samples in the presence of an axial external magnetic field H. The electrodynamic response of these systems can be analyzed by means of three-dimensional (3D) equivalent networks of Josephson junctions coupled through inductances. However, in the low field reversible regime a 2D Josephson junction array can be used. In order to study irreversible flux penetration in 3D systems, we give the equivalent network and qualitatively discuss the dynamics of flux penetration. We notice that, in analogy to 2D networks, First Irreversible Flux Penetration (FIFP) occurs when the value of the shielding current approaches the maximum Josephson current in the weakest external junction of the network.

1. THE M O D E L S

The magnetic properties of sintered granular su- perconductors can be studied by means of equivalent networks of inductively coupled Josephson junctions (JJs) [1-3]. The dynamics of the JJ associated to any two adjacent grains can be simulated by the resis- tively shunted junct ion (RSJ) model [4]. Furthermore, the junctions are coupled through in- ductances, which explicitly contain the magnetic en- ergy term and the self field generated by the transport currents. In general, three-dimensional (3D) networks are required. However, since the degrees of freedom to be considered in this problem are very numerous, it is necessary to resort to simplified Josephson junction arrays, making use of particular symmetry properties of the problem. We therefore start by con- sidering a cylindrical superconducting sintered sam- ple with longitudinal dimensions much larger than the radius. A magnetic field H is applied along the axial direction after zero-field-cooling (ZFC) as shown in fig.la, and the stationary magnetic states of the system are discussed for LIj>>~o [5,6], where L is the inductance associated to a single intergranu- lar loop, and Ij is the average value of the maximum Josephson currents of the junctions. The grains are assumed to be perfect diamagnetic entities, given the very low values of the applied field H considered.

2. FLUX PENETRATION

We can distinguish between two low-field dia- magnetic regimes after ZFC: a reversible regime for applied fields less than a lower threshold field Hgcl

nN©

H I

a b Figure 1. a) Schematic representation of the current distribution in a sintered superconducting sample for H<Hgcl . b) Equivalent network of inductively coupled JJs for the system shown in a).

and an irreversible regime for H>Hgcl. In the low- field reversible regime, given the symmetry of the problem, a 2D network can be adopted to define the stationary magnetic states of the system. Indeed, in this field range shielding currents circulate in the an- gular direction as shown in fig.la. If we view the superconducting granular sample as a collection of identical spherical grains, we may group the grains in different layers, each layer carrying the same cur- rent distribution. In this reversible shielding regime,

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then, it is possible to adopt a very simple circularly symmetric JJ network (fig. lb), which is equivalent to a 1D JJ array. With the aid of this equivalent net- work the lower threshold field value Hgcl can be an- alytically calculated to be [7]:

Hgcl={ic~+Y[1/4+(1/2rt)sin-l(1/2nfJ)]}/go S (1)

where 13=LIj/~o >>1, ~/=(l+2rcl3)/2rr~=l, and

ic= [1- (1/2n[3) 211/2.

Figure 2. a) First irreversible flux penetration in a sintered superconducting sample, b) Equivalent network of inductively coupled JJs for the system shown in a).

For H>Hgcl flux quanta penetrate the sample, so that the cylindrical symmetry is lost. In analogy with what happens in 2D systems, First Irreversible Flux Penetration (FIFP) occurs when the value of the shielding current approaches the maxi- mum Josephson current in the weakest external junction of the 3D network, as shown schematically in fig.2a. In fig.2b the minimal equivalent 3D net- work of inductively coupled JJs is given. In this circuit we notice three types of inductances. The inductance L 1 is relative to the loop where the penetrated flux is located and to the loops immediately above and below. The white box in each of these loops represents the external horizontal junction, while the shaded box represents the remaining three JJs. The inductance L2, the same for each layer, is relative to the outermost loop, which embraces the whole sample. The large black boxes on each branch containing L 2 represent the peripher-

al junctions in that layer, except for the weakest one for the central layer, and for the corresponding JJs belonging to the adjacent layers (white boxes). Finally, the inductances L3 and L4 are relative to the loops which couple the central layer to the adjacent layers and which contain four JJs (white and shaded boxes). Analogously to the 2D problem, numerical simulation show that shielding currents in L2 increase up to the value of the maximum Josephson current of the weakest J J, leading to the trapping of a flux quantum in L1, L 2 and L 3. Moreover, as the magnetic field increases, if the spreading of the maximum Josephson currents of the JJs is small, flux penetration proceeds very rapidly from this point up and down the external surface of the sample. In this way the cylindrical symmetry is restored, so that, neglecting transient phenomena, 2D networks can still be adopted. When 2D equivalent circuits are used, one sees, by numerical analysis of the magnetic states of the system[6], that fluxons penetrate the sample in a casual way, generating circularly asymmetric stationary (CAS) states. However, as the magnetic field increases, the 2D array shows circularly symmetric stationary (CSS) states also. CAS states and CSS states live in different alternating field ranges. Therefore, if one neglects all the CAS states, the concentric shell 1D model of fig.lb can still be adopted. With the use of this model magnetization curves of granular superconductors can be easily analyzed [6]. From this approach one argues that irreversible flux motion occurs whenever the value of the transport current is close to the maximum Josephson current value. In this way the currentand field distributions in the system are quite similar to those resulting from a critical state.

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7. S. Pace and R. De Luca, Physica C, 158 (1989) 69.