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Two-Fluid Model for Granular Superconductors

A two-fluid model is proposed to describe the transport properties of granular superconductors. Using the resistively shunted junction model and some aspects of the two-level system theory, a statistical model is developed which takes into account the ratio between the number of normal and superconducting electrons carrying the applied current. The theoretical model reveals excellent agreement when compared to transport properties of four high-Tc superconductors. The results suggest that the two-fluid model is independent of the sample composition, critical temperature and whether the superconducting compound is electron or hole-doped.


I. INTRODUCTION
During the last 40 years, much attention has been devoted to describing transport properties of superconductors. The first description was based on the motion of Abrikosov vortices. [1][2][3][4][5][6] Nowadays flux-creep and flux-flow mechanisms are well accepted to describe, respectively, non-linear and linear parts of current-voltage ( V I − ) characteristic curves of homogeneous superconductors. [5][6][7] In the flux-flow regime, the Lorentz force on an Abrikosov vortex is much higher than the pinning force resulting in vortex motion on a viscous medium under the influence of Magnus force due to supercurrents circulating around the vortex core. 5,6 In such a model, fluxflow resistance is given by 2  is the upper critical field and N R is the normal state resistance. Dissipation in this regime is attributed to the relative motion of normal electrons inside the vortex core. [1][2][3][4][5][6] Furthermore, V I − characteristic curves reveal linear dependencies with the applied current which are strongly magnetic field dependent (see, for example, reference [5]).
After the prediction of the Josephson effect, 8 several authors have taken into account aspects of thermal fluctuations or phase slippage, [9][10][11][12][13][14][15] twofluid theory, 16 and the resistively shunted junction (RSJ) model [17][18][19][20] in order to describe transport properties of superconductors. In those models, shunt resistances are always present and normal current can flow parallel to the supercurrents in the dissipative regime ( 0 ≠ V ). [9][10][11][12][13][14][15][16][17][18][19][20] With the discovery of high-C T superconductors, several researchers have integrated aspects of weak coupling or granularity in order to understand electrical properties of this new class of superconductors. 21 Granularity in those superconductors is generally manifested through a double superconducting transition, 13 Another important aspect to be considered in determining whether dissipation and weak coupling effects are related is the observation of hysteresis loops in magneto-resistance measurements. 25,26 If the loop exhibits a clockwise hysteresis, the dissipation is related to intergranular coupling. 25,26 On the other hand, if the loop has opposite direction, dissipation results from the motion of Abrikosov vortices. 26 In order to observe dissipation due to intragranular and intergranular effects in granular high-C T superconductors we measured magneto-resistance hysteresis loops under low and high magnetic fields. Results reported previously 27 curves measured below this ohmic regime show another important feature. The differential resistances in the linear regime of the V I − curves are clearly magnetic field independent. We stress that this behavior cannot be explained by using the classical flux-flow model. [27][28][29] In order to explain these experimental observations we discuss the results within the framework of the RSJ model which predicts the existence of twofluid particles, normal and superconducting electrons, carrying the total current [19][20][21][22] by taking into account a statistical theory based on similar aspects of the two-level systems. 30,31 The statistical model is applied to transport properties measurements performed on four granular high-C T superconducting samples, including holedoped and electron-doped compounds.

II. TWO-FLUID MODEL
In the two-fluid model suggested below it is supposed that both applied magnetic field and electrical current are related to the weak coupling effects and that they are not high enough to reach the limits expected to break Cooper pairs (current density is smaller than depairing current) or induce Abrikosov vortices inside superconducting samples ( H is much smaller than intragranular lower critical field).
As well accepted, there are two carriers types, normal and superconducting electrons, coexisting in equilibrium at temperatures below C T . At a fixed temperature, there is a specific density ratio between normal particles and superconducting electrons predicted by the static two-fluid model. 32 On the other hand, if electrical current is passed through a superconducting sample, a dynamical situation is created and an interplay between the numbers of normal and superconducting electrons must take place to carry the applied current. At very low applied current, the current is supposed to be carried only by superconducting electrons (normal electrons do not contribute to the conduction) and as a consequence no voltage is observed in the sample. On the other hand, according to the RSJ model, if Josephson junctions are subjected to a high applied current, a normal current can flow parallel to the supercurrent resulting in a non-ohmic dissipative regime ( [17][18][19][20] Thus, if O I is defined as a characteristic supercurrent of a granular superconducting sample (supposed to be a network of Josephson-junctions [17][18][19] ) at a fixed temperature and applied magnetic field, when the sample is in a true superconducting zero-resistance state and the total current that crosses the sample is carried by superconducting electrons. If limit, an ohmic regime should be reached. Following this idea, we suggest that there is a statistical ratio between N N and S N which is assumed to have similar physical aspects to those described by a simple two-level system. 30,31 The dynamical ratio between where and N is the total number of carriers transporting the current at a fixed temperature and under a constant applied magnetic field.
A graphical description of the statistical model is provided in Fig. 2 where are shown the fractional populations constant. In this work we have used the simplest form for such a function which is given by where O N is a constant and β is neither zero nor 1 which can be determined by using experimental data.
Remembering that in the two-level system Since the dissipation is related to the number of normal electrons crossing the sample, we suppose that the voltage in a superconducting sample must be proportional to the normal electrons ratio ( is the normal state shunt resistance which must be magnetic field independent. 18 Thus, the dissipation voltage in a superconducting sample should be given by , it is possible to show from Eq. (7) that, independent of β , dI dV / must approach the N R value, suggesting that V I − curves should display parallel linear regimes at high applied currents if the shunt resistance is magnetic field independent, which is in agreement to the RSJ model. [17][18][19][20] Fig. 3 displays dI dV / versus I curves for the Y 0.55 Pr 0.45 Ba 2 Cu 3 O 7-δ sample. These measurements were carried out at 4.2 K under low applied magnetic fields after zero-field cooling by using a copper solenoid. We can see that the slopes approach a constant value at high current limit representing magnetic independent linear regimes in the V I − curves which is in excellent agreement with the prediction of the two-fluid model reported here. Thus, we can suppose that the slope at high current limit is related to the normal state shunt resistance providing Another way to determine N R is increase applied magnetic field up to the upper limit for weak coupling effects. 27 In such a case, O I should vanish and the magneto-resistance ( I V / ) will approach an ohmic regime with resistance equal to N R . In the inset of the Fig. 3, the magnetoresistance for the Y 0.55 Pr 0.45 Ba 2 Cu 3 O 7-δ sample at 4.2 K measured under different applied currents is shown. As expected, the sample reaches ohmic behavior at high magnetic fields with the same shunt resistance value obtained from the Another important point of Fig. 4  . It is observed that O I decreases with increasing applied magnetic field, in agreement with the assumption that the coupling effects deteriorate if magnetic field is increased in the intergranular region.
By using the Eq. ). 18 On this issue, we should observe that  Finally, in order to verify that the two-fluid model can be applied to other superconducting samples. In Fig. 6  . The experimental results collapse very close to the theoretical curve over more than one order of magnitude, suggesting that the model proposed here is independent of the sample composition, critical temperature as well as whether or not the superconducting compound is electron or hole-doped.

III. CONCLUSION
This work proposes a two-fluid model to describe transport properties of granular superconductors. The model takes into account a statistical ratio between the number of normal and superconducting electrons carrying the applied current. Several implications of the model are obtained which agree very well with transport properties of different high-C T superconductors. It is observed that only two parameters ( N R and O I ), directly obtained from experimental curves, are necessary to describe V I − curves quantitatively without fitting parameters. The discussion of the results obtained in different superconducting compounds suggest that the twofluid model is independent of the sample composition, critical temperature and whether the compound is electron or hole-doped.