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Self-fields in thin superconducting tapes: implications to the thickness effect in coated conductors

Most applications of superconductors, such as power transmission lines, motors, generators, and transformers, require long cables through which large currents circulate. Impressive progress has recently been achieved in the current-carrying capability in conductors based on high-temperature superconductors. Coated conductors are likely the best examples, consisting of very good quality thin layers of YBCO superconductor grown on top of a metallic tape with some intermediate layers. However, there is an important problem for achieving large currents: a large decrease in transport critical-current density Jc when increasing film thickness has been observed in coated conductors made by all available techniques. Here, we theoretically explain the nature and the ubiquitous presence of this so-called thickness effect by analyzing the self-field created by the transport currents in the superconductor, assuming a realistic field-dependent Jc. This knowledge can help finding new ways to improve transport current in thick superconducting films.

The most typical manifestation of the thickness effect occurs when measuring the transport J c of superconducting tapes at self-field (no externally applied magnetic field) as function of the tape thickness. Several explanations have been proposed to partially account for some of the effects, such as enhancement of flux pinning near the heteroepitaxial interface between YBCO and the substrate or interlayer [12], difference in intergrain coupling for films with different thicknesses [13,14], or non-uniform distribution of pinning sites across the thickness [15] . However, the fact that this effect is common for all combination of substrates, preparation technologies, and HTS deposition techniques [16,17] suggests that there is an underlaying phenomenon that is present in all the studied cases. In this work, we argue that the effect of the self-field of transport currents in thin films is an underlying basis for all the thickness dependence experiments in coated conductors. The existence of this 'universal' geometry-related thickness effect does not mean that the other mentioned effects (inhomogeneities, interface effects, etc) do not play an important role in particular cases. However, only after understanding the geometrical effects can the rest of phenomena be properly studied.
We consider an infinitely long type-II superconducting strip of rectangular cross-section located at −w/2 ≤ y ≤ w/2 and −t/2 ≤ z ≤ t/2, which obeys the critical-state model, 2 with a field-dependent critical-current density J c (H i ), where H i is the modulus of the total magnetic field. We calculate the self-field and the local current density at every point of the superconductor cross-section by an iterative process described in the Methods section (similar procedures were used in [18,19]). What can the model explain about the thickness effect? To answer this question we plot in Fig. 1 as dots the experimental points corresponding to collected data for different YBCO coated conductors, obtained from the review paper Ref. [1]. These data can be considered a typical systematic study of thickness dependence in state-of-the-art YBCO conductors. The line in Fig. 1 corresponds to the J c calculated by our model, assuming an intrinsic Kim-type with values J c0 = 12 MA/cm 2 and µ 0 H 0 = 0.006 T, and using the actual film dimensions of the experiments (Kim model is chosen becauese it has been shown to fit well the J c (H a ) measured in thin YBCO films [20]). The calculated critical-current density follows well the drop observed in the thickness-dependence experiments. The standard thickness dependence is therefore very well reproduced just by assuming a (realistic, see below) J c (H i ) function and calculating the transport current at self-field for the exact dimensions used in the experiments, without any extra parameter. The explanation has no need resorting to a spatially dependent J c (r) function including an extra distance parameter, as in some previous models [9,12]; although this spatial dependence may indeed by present in some cases, it is not logical that it appears for so many sample preparation conditions in which the effect has been observed [17].
The next question to answer is how the Kim-model J c (H i ) function we have used represents the actual field dependence in the real samples. In Fig. 2  up to an applied field µ 0 H a ∼ 0.01 T and then decreasing with applied field [1]. In contrast, the intrinsic J c (H i ) dependence is larger at low fields than the plateau value and merges with the calculated J c (H a ) for fields larger than around 0.03 T, because in this region the self-field created by currents is much less than the applied field value, so the internal field is basically H a [18,19,21,22].
Another conclusion can be obtained from the results in Fig. 2: although the measured value of J c at applied field H a =0 is the plateau value J c,plat ∼3.3 MA/cm 2 for a thickness t = 1µm, our model shows that intrinsic J c (H i ) can have larger values (a maximum of 12 MA/cm 2 in this case). This means that, following our model, when measuring transport current at self-field in the coated conductor, there are regions that actually have a current density larger than the value of the plateau. This is shown in Fig. 3 where we plot the local values of J c (Fig. 3a) and µ 0 H z (Fig. 3b)  interesting to note that the maximum magnetic field in the sample for H a =0, which occurs at film edges, corresponds rather well to the product J c,plat t [18], and not to J c0 t (in the studied case J c0 = 12 MA/cm 2 , around four times larger than J c,plat ∼ 3.3MA/cm 2 ) . When the applied field is increased, the maximum in local J c first moves to one side and then eventually smears out and local J c and field become both basically constant.
The model for the thickness effect presented in this work also explains the results deduced from the incremental model presented in [9,12]. This model artificially assumed a spatially dependent J c (r) function, which decreases linearly with the vertical distance to the substrate up to a given distance, and then it is constant. From our results, the drop of incremental J c naturally arises from the field dependence of J c , with the property that J c would not decrease in general with the distance from the substrate but with the distance from the center of the sample.
Because the described effects are geometrical and then always present in films, is it therefore impossible to have a thickness-independent J c ? The answer is not; a possible solution is to try to achieve superconductors such that its intrinsic J c (H i ) is roughly constant from 0 to at least the self-field, of the order J c,plat t. Probably this is the situation in some measurements of films that show almost no thickness dependence of J c [12].
In view of the above results, we can conclude that an explanation for the ubiquitous thickness effect has been given. It is because of its geometrical nature that it is present in striking similar form for a variety of deposition techniques (PLD, MOCVD, ex situ MOD films, etc. [17]). However, we would like to remark that our model, However, these strategies can only succeed if the geometrical effects, for which we have provided analyzing tools in this work, are properly taken into account.
In conclusion, we have explained the existence of a thickness effect in many measurements in superconducting tapes and why this effect appears in samples prepared and measured under many different conditions. Only after this geometrical effect is correctly analyzed in a given experiment, the intrinsic properties of the studied superconductor can be extracted, and in general paths towards large values of transport current in thick superconductors can be properly explored.