The effect of size and aspect ratio on the trapped field properties of single grain, Y–Ba–Cu–O bulk superconductors

Bulk, single grain (RE)Ba2Cu3O7−δ [(RE)BCO, where RE is a rare earth element or yttrium] high temperature superconductors exhibit significant potential for use in a variety of engineering applications due to their ability to trap large magnetic fields, which can be up to ten times greater than those generated by conventional, iron-based magnets. Limitations on the maximum size to which single grains can be grown, however, are a major obstacle to the further development of these materials. Indeed, multiple samples are often required to achieve the required superconducting properties in particular applications. The geometry of bulk (RE)BCO single grain samples plays an important role in determining the superconducting properties of a given technical arrangement. In order to gain a better understanding of the full application potential of bulk single grain superconductors, three relatively long, cylindrical YBCO single grains of different diameters were fabricated and their trapped field and total trapped flux measured at 77 K as a function of sample height. The effects of size and aspect ratio of YBCO single grains on these key applied properties have been investigated experimentally and the results compared qualitatively with the predictions of an established theoretical model. Conclusions based on the trapped field measurements on a variety of single grain samples are presented in this study and the possibilities of using assemblies of smaller samples for engineering devices, in particular, are discussed.


Introduction
Bulk, single grain (RE)Ba 2 Cu 3 O 7−δ [(RE)BCO, where RE is a rare earth element or yttrium] high temperature superconductors exhibit significant potential for exploitation in a variety of engineering applications due to their ability to trap large magnetic fields of up to ∼18 T [1]. Unresolved issues, such as poor mechanical properties and difficulty of magnetisation, however, continue to present challenges to the wide-scale realisation of practical applications of these materials. Limitations to the size of samples that can be fabricated practically are another important set of issues that need to be resolved, with the maximum diameter of single grain samples currently in production being 140 mm for GdBa 2 Cu 3 O 7−δ (GdBCO) [2]. Importantly, an increase in single grain size does not, typically, increase the magnitude of trapped field, as might be expected from simple magnetic moment considerations and the assumption of uniform materials properties. As a result, a multiple sample arrangement is often used to achieve the required superconducting properties, such as the stack of disks used to generate world record fields [1,3].
It is known from previous modelling that the trapped field in a cylindrical bulk superconductor saturates at a given height for a sample of a fixed diameter [4]. This is consistent with the observation by Eisterer that the best aspect ratio, height/radius (H/R), of a YBCO single grain is between 0.67 and 1, which 'seems to be a good compromise between a high trapped field and a reasonable amount of superconductor' [4]. This conclusion is based on the results of a model developed to reproduce a measured trapped field by systematic variation of the critical current density J c in samples of toroidal geometry, assuming that J c is constant in each toroid. The value of J c in the samples was varied iteratively in the model until it yielded the magnetic field at the centre of a toroid of rectangular cross section, based on an assumed J c (B) behaviour. However, in reality, J c (B) varies significantly within one single toroidal grain and from toroid to toroid. In addition, the optimum single grain aspect ratio may vary from sample to sample and between samples within different (RE)BCO systems (where RE=Nd, Sm, Eu, Gd and Y, or combination of RE elements). Bulk superconductors generally require bespoke geometries for different applications, which obviously requires a better understanding of the effect of aspect ratio on the superconducting properties, so that the bulk material can be used efficiently and economically for a specific application. Furthermore, reports of trapped field in bulk (RE)BCO have generally remained limited to the maximum field B max measured at the surface of the sample, since B max varies generally in proportion to the average J c within the sample, at least according to the Bean model [5], which has, therefore, been assumed to provide information on this key superconducting parameter. The maximum value of trapped field at the centre of the top surface of a bulk superconductor is certainly a useful indicator of the magnetic properties of the sample, given that this value is unique for a fully magnetised, single grain and can, therefore, be used effectively to compare the applied properties of different samples. However, a real sample cannot be infinitely long, as assumed by the Bean model, so the spatial distribution of trapped field becomes more relevant to engineering the applied properties of the superconductor, and knowledge of the magnitude of the field generated away from the exposed, and therefore potentially accessible, surfaces of the sample.
Here, we report an investigation of the effect of size and aspect ratio on the superconducting properties of YBa 2 Cu 3 O 7−δ (YBCO) single grains. This is directly relevant to applications that require high field, high total flux and a known field distribution. Three, tall cylindrical YBCO single grain superconductors of diameter 16, 20 and 25 mm were fabricated for this purpose. Each sample was sliced successively from its bottom surface between six and seven times in order to obtain specimens of the same single grain but of successively decreasing height. The trapped field distributions at both the top and bottom surfaces of these samples and subsamples were measured at distances of 1.5, 3.0 and 4.5 mm above the exposed surface of the sample. The total flux was calculated in arbitrary units for each measurement and used to compare the distribution of flux for the different samples. The effect of size and aspect ratio on the superconducting properties, trapped field and total flux are discussed based on the relatively large quantity of data generated as part of this investigation. The results of this study have been used successfully to assemble smaller sample arrangements of single grain bulk (RE)BCO superconductors that generate either higher maximum or more uniform trapped fields.

Experimental
Three YBCO single grains of diameter 16, 20 and 25 mm were fabricated via the well-established top-seeded meltgrowth (TSMG) [6,7] process using 20, 32 and 55 g of mixed precursor powder. The heights of the as-processed samples (1-3) were 16 mm in diameter and 15.3 mm in height, 20 mm in diameter and 19.3 mm in height and 25 mm in diameter and 18.9 mm in height, as shown in figures 1(a)-(c). The precursor powder, of composition of 75 wt% YBa 2 Cu 3 O 7−δ (Y-123)+25 wt% Y 2 BaCuO 5 (Y-211)+0.5 wt% CeO 2 , was mixed using a Turbula ® mixer for 2.5 h prior to melt processing. Each precursor powder batch was poured into a die to obtain the target geometry and pressed isostatically at 2000 bar. A small amount of Yb 2 O 3 powder was added to the bottom of each pellet prior to compaction to act as a substrate to prevent the undesirable nucleation and growth of subgrains during TSMG from the bottom of the sample. A NdBa 2 Cu 3 O 7-δ seed was placed at the centre of the top surface of each pellet and the whole arrangement loaded into a box furnace to allow the samples to grow into single grains in air. Heating profiles used for the TSMG process have been reported elsewhere [8,9]. Briefly, each sample was heated at about 75°C h −1 to 1060°C, held for one hour to allow the superconducting Y-123 phase in the precursor to decompose completely to Y-211 and Ba 3 Cu 5 O 8 (liquid phase), cooled rapidly to 1015°C at 50°C h −1 and then slow-cooled to 975°C at between 0.6 and 0.3°C h −1 to, allowing single grains to form. The larger samples took a longer time to grow, and hence required a slower cooling rate than the smaller samples. Finally, the single grain samples were cooled to room temperature at 100°C h −1 . The single grains were oxygenated in a subsequent, separate process at 450°C for 8 d in order to transform their crystallographic structure from tetragonal to the superconducting orthorhombic phase.
The trapped fields at both the top and bottom surfaces of the three single grain samples were measured following a field-cooling (FC) magnetisation process, which involved applying a field of 1.4 T to each sample and cooling to 77 K in a liquid nitrogen bath. The trapped field profiles of the magnetised samples were measured using a rotating array of 18 Hall probes with the array positioned in turn at 1.5, 3.0 and 4.5 mm above the top and bottom surfaces of each single grain. The total flux in arbitrary units was determined by summing the positive values of trapped field in the data matrix generated by the Origin TM software used to construct graphically each trapped field distribution. Samples 1-3 were sliced successively following this initial trapped field measurement at about 2.5 mm from their bottom surface, as indicated in figure 1(d). The measurements of trapped field and total flux were repeated for the upper part of the three samples before the slicing process was repeated successively until the remaining height of each sample was not sufficient to sustain further cutting. The height and thickness of every subsample is shown in table 1. In total, 21 sub-samples were measured in this research specifically to investigate the effect of aspect ratio on the superconducting properties of the individual single gains.
In a complementary study, the trapped field B t at 77 K at the top and bottom surfaces of a number of stacks of different, pre-processed single grain (RE)BCO samples was also measured in order to investigate whether a higher maximum trapped field or more uniform field distribution could be generated by a single grain stacking process. The trapped fields and the dimensions of these individual, single grain samples before stacking are listed in table 2.

The effect of aspect ratio on trapped field
The maximum trapped fields at the top and bottom surfaces of the three successively-sliced YBCO samples of different diameters were measured initially using a hand-held Hall probe after they had been fully magnetised by the FC process, as shown in figures 2(a)-(c). The black and red solid dots represent the trapped fields at the top and bottom surfaces of the samples, respectively.
It can be seen that the highest trapped fields at the top surfaces of these three samples are 0.67, 0.79 and 0.813 T, and that these values do not change significantly when the samples are sliced successively from the bottom (i.e. from thick to thin) until sample 1 reaches a thickness 5.79 mm (H/R: 0.72), sample 2 a thickness of 8.16 mm (H/R: 0.82) and sample 3 a thickness of 11.34 mm (H/R: 0.91), as indicated by the dashed blue lines in figure 2. It can be seen that the trapped field deceases abruptly for an aspect ratio (H/R) lower than the values indicated above, which represent optimum values for these three samples. In general, the results of  these measurements are consistent with Eisterer's observation that a ration of H/R of between 0.67 and 1 'seems to be a good compromise between a high trapped field and a reasonable amount of superconductor' [4] and with the predictions of his model (although this assumed a uniform J c -B distribution in each sample). The data suggest further that trapped field saturates when the single grain sample reaches a certain height to radius ratio. Clearly, it is not necessary to grow tall samples for applications where the target property is simply to achieve a given B max , in which case it is important only to achieve the sample height required to generate the required level of trapped field. However, it is apparent from figure 2 that the trends in the trapped field data fluctuate, and that this is inconsistent with the predictions of Eisterer's model [4]. The trapped field measured at the bottom of each of the three samples tends to be low at the beginning of the slicing process, and particularly so in samples of full height and after only one cut. This is due to an increased concentration of Y-211 particles and solidified liquid phase residue at the bottom of the samples [8] (and an associated lower concentration of the Y-123 superconducting phase) due to the nature of the single grain growth process, which can be explained by particle 'pushing/trapping' theory [10]. As a result, both T c and J c tend to be very low at the bottom of each sample, resulting, inevitably, in low trapped field for that region of the single grain.
The trend in observed trapped field for the three samples at the top and bottom surfaces is similar for the remaining portions of each single grain when its height is lower than that indicated by the red dotted line (i.e. to the right of each red line in figure 2). The trapped field at the bottom surface of some of the specimens (indicated by the red circle in figures 2(b) and (c)) can be either higher or lower than the saturated value of trapped field measured at the top surface. This suggests that there could be layers of material with higher or lower J c than the average value at positions close to the bottom of the samples, given that T c in these regions is constant. This fluctuation in J c reflects the non-uniformity of the distribution of flux pinning centres in the single grain microstructure, which is related mainly to the distribution of Y-211 inclusions in the superconducting Y-123 phase matrix [8,11,12]. The size, shape and concentration of Y-211 inclusions will change from sample to sample and from location to location within each sample. Samples of height (or aspect ratio) equal to or less than the optimum value exhibit the same value of trapped field at both the top and bottom surfaces of the sample. This suggests that the single grain samples exhibit generally predictable, uniform trapped fields and superconducting properties along the direction perpendicular to their top surface, or, at least, properties that change only gradually over the range within which the dimensions of the sample are close to the optimum aspect ratio. Both the top and bottom surfaces of samples of these aspect ratios can be used effectively in engineering applications with only minimal characterisation and sample selection, which is significant in the development of a commercial bulk superconductor.
It is worth noting that the present study suggests that the saturated field is obtained at a higher aspect ratio (H/R) for the larger, as-processed single grain sample. This is not predicted by the model, which assumes a uniform J c -B behaviour throughout the sample. This observation may suggest that there are significant differences in the distribution of J c within a given sample and between samples of different heights, which, in turn, relates directly to issues that affect the growth of single grains, such as heating profile, precursor powder composition, the presence of impurities, particle size and the size and distribution of the cracks and porosity in samples of different size and composition.
It appears from the highest values of trapped fields measured at the top surfaces of the three samples (0.67, 0.79 and 0.813 T) that the smaller sample exhibits the highest J c . This can be verified, at least to a first approximation, using Chen's formula [13]:  It can be seen that the smaller sample exhibits the higher calculated average value of J c , in good agreement with previous measurements on YBCO single grains. This can be understood easily by considering the growth process of single grains, which are effectively quasi-single crystals, in that there is an increased probability of the formation of larger and a greater number of defects, such as, large cracks, pores and high angle sub-grains, in larger samples. These defects are so large that, not only do they not act as effective pinning centres due to their size, they form significant obstacles to the flow of superconducting current, which, in turn, decreases the average J c . On one hand, the techniques that are used for growing single grain samples specifically to yield improved superconducting properties require continual improvement, which can only be achieved by understanding the process limitations of single grains of a particular diameter, height and superconducting properties. Until the processing of bulk YBCO samples with large geometries can be achieved reliably and reproducibly, the use of multiple single grains with appropriate dimensions for practical applications, therefore, is the only feasible way to realise such applications with reasonable material consumption. Hence, it can be concluded that assemblies of smaller samples with better superconducting properties per unit volume will be used for applications in the short-term. By comparing the total flux at the same distance from the surface of the three samples for different diameters, it is also observed that the larger the surface area of the sample, the higher the maximum total flux at a similar height. An exception to the above observation is apparent in the data that are circled in figures 3(a)-(c). The trapped field at the bottom of each sample increases after effectively removing the layer with poor superconducting properties, as can be seen in figure 2. This sudden increase in total flux results from an increment in the trapped field in that layer. However, it is interesting to note that the total flux at the top surface decreases at the same time (which is true for all three samples), suggesting that the flux at the top surfaces diverges more at the edge of each sample, with the observed vertical component of trapped field, B z , deceasing due to the increase in flux at the bottom of the sample. This can be confirmed by the trapped field profiles measured 1.5 mm above at the surfaces of these samples shown in figure 4, which shows the trapped field distribution of sample 2 measured at 1.5 mm above both the top and bottom surfaces of the sample when the height of the sample was 14.72 mm, 17.07 mm and 19.2 mm (samples 1, 2 and 3, respectively). It can be seen clearly from figures 4(b) and (e) that the trapped field at the bottom of the sample increases significantly as the superconducting properties of the bottom layer increase. Although the same top surface remains constant throughout the trapped field measurements and the magnetisation of the sample has saturated, the trapped field at the top surface is decreased, which is why the total flux of B z changes abruptly. The increments in trapped field at the bottom of sample 2 reduce either the height or width of the shape of the cone of the trapped field profile measured at the top surface of the sample, which contributes jointly to the observed total flux and confirms the results in figure 3 (note that the results shown in figure 4 are consistent for all three samples). This indicates that the total flux is sensitive to the presence of layer of material within the bulk material where the superconducting properties change. This should be taken into account in designing the arrangement of samples for practical applications. Significantly, it is not necessary to remove the bottom of the sample for applications where only a large trapped field at the top surface of the sample is required.

Arbitrary total flux and maximum trapped fields at different heights above the top and bottom surfaces
It can also be seen from figure 3 that the rate of increase in total flux becomes lower when the height of the sample increases, presumably reaching a maximum in the limit of infinite length (height). This indicates that the trapped field above the top and bottom surface of the single grain will decrease relatively slowly for tall samples, which is both important for the use of bulk YBCO in most engineering devices and addresses a common concern that the trapped field diverges too quickly at positions distant from the sample surface for most practical applications. Figure 5 shows the trapped field distribution measured at distances of 1.5 mm, 3.0 mm and 4.5 mm above the top surface of sample 2 at heights of 19.2 mm (H/R=1.92) and 8.16 mm (H/R= 0.82), respectively. It can be seen that the cone-shaped trapped field distributions in the top row of figure 5 are all higher than those in the bottom row, which indicates that trapped field in the tall samples generally decreases less rapidly in the vertical space above the single grain (this trend is observed in all three samples). Figure 6 shows the maximum trapped fields measured at 0, 1.5, 3.0 and 4.5 mm above the top surfaces of all three samples at two sets of heights: one is the full height of the sample, indicated by black squares, and the other is the height where the best aspect ratios are observed, indicated by red dots. It can be seen from figures 6(a)-(c) that the trapped fields of the thinner samples (red dots) decrease more quickly than those of the tall samples (black squares). The trapped field of the thin samples of diameter 20 and 25 mm decrease to 60% of their peak value at a distance of 1.5 mm above the top surface of the sample (figures 6(e) and (f)), although the trapped field of the thick samples at the same distance above the top surface still exhibit 80% of their peak trapped field on the surface of the sample (figures 6(e) and (f)). This suggests that increasing the height of bulk single grains may not help increase the trapped field at the surface, but, instead, will help improve the trapped field at a position several millimetres away from the top surface, which is consistent with the notion that the trapped field at the positions above the top surface of the sample should not change if the bulk single grain is infinitely long in the perpendicular direction. In this case, taller samples would have greater potential for practical applications due to the requirement that the trapped field is generated in an accessible space, for example several millimetres away from any surface.
It can be concluded that the total flux of single grain YBCO samples increases when either the height or diameter of the sample or proximity to the sample surface increase. However, the total flux can change abruptly when the quality of material at the top or bottom surface of the single grain changes. The total flux of the tall samples increases more slowly, however, suggesting that the trapped field also decreases more slowly away from the top (bottom) surface.
As a result, the addition of a thin layer of superconducting material to engineer the trapped field at the surface of the sample may not decrease significantly the maximum trapped field but may shape the field distribution to meet the requirements of a specific application.

Attempts to stack three pairs of thin single grains of diameter 25 mm
Two attempts to stack two bulk samples were performed and analysed based on the observation that smaller samples tend to exhibit a higher average J c , as shown in figure 7. Two single YBCO grains, Y1 and Y2 of height 12.0 and 9.0 mm with maximum trapped fields of 0.86 and 0.81 T at their top surfaces were used to form stack 1. Trapped fields of 0.91 and 0.86 T at the top and bottom surfaces of this stack were measured, which are both higher than the values of either of those of the constituent bulk single grains. This result is anticipated, although the observed increment in trapped field of the stack is not significant at 77 K. That said, the improvement in trapped field of bulk samples is not easy to achieve given that, as observed in the present study, they saturate at a critical aspect ratio, although this approach does help to realise the full field trapping potential of bulk YBCO, and particularly at temperatures below 77 K when the increment of the trapped field could be an order of magnitude higher.
Two sliver-containing, GdBCO bulk single grains, G1 and G2, fabricated separately, were stacked together (stack 2) in the same geometrical arrangement as stack 1. Maximum trapped fields at the top surface and bottom surfaces of this stack of 1.06 T and 1.04 T were observed, respectively, which are higher than the average trapped field 0.9 T of the individual batch-processed GdBCO-Ag samples [14]. Interestingly, based on measurement of the trapped field of all four possible combination of the Y1, Y2, G1 and G2 bulk samples, the highest values of trapped field at both the top and bottom surfaces of the stack were obtained when the individual grains in the stack were of the same composition (i.e. Y1+Y2 and G1+G2). This, clearly, is a very preliminary observation, and more research is required to understand how to stack individual samples optimally to obtain higher fields.
In a final experiment, two recycled GdBCO-Ag single grains, #600 and #601, were stacked together and the magnitude and distribution of their trapped fields measured. Two perpendicular slots were cut into the top surface of sample #600 using a diamond blade of diameter 150 mm, as shown in figure 8, following an initial measurement of trapped field. The depth H of the cut in figure 8 is estimated to be 1.0 mm. It can be seen from figure 8(b) that sample #600 has a maximum trapped field of 0.58 T at its top surface and that the distribution of trapped field at a distance of 1.5 mm above the sample surface is in the form of a cone, which is typical for a single grains, such as #601 in figure 8(a). The values of the trapped fields of these two samples are not very high because the properties of recycled grains are usually inferior to those grown using a primary melt process [15,16]. However, it can be seen from figure 8(c) that the shape and magnitude of the trapped field of sample #600 changes significantly from 0.58 to 0.49 T, with the cone becoming flatter at the top and wider. In other words, a rather uniform trapped field has been achieved 1.5 mm above the top surface by cutting the sample, with only a relatively small decrease in maximum trapped field, by changing slightly the shape of sample. The alignment of the crystallographic c-axis with the height (or thickness) of (RE)BCO single grains means, effectively, that they are composed of a stack of ab planes, which are the main supercurrent charge carriers [17]. Sample #600, with the two perpendicular cuts in its surface, therefore, can be modelled as a composite of two separate, thin single grains A and B, as illustrated in figures 8(e) and (f). The shape of the trapped field of single grain A will not be cone-shaped with a single peak since the cuts prevent the supercurrent from flowing in concentric loops. However, shape of the trapped field of grain A compensates for that of grain B, resulting in the observed, remarkably flat trapped field topography produced by the combination of trapped fields from sub-grains A and B, which may be desirable for some specific application, such as MRI where a very flat field profile is desired [18]. A stack of sample #600, with the perpendicular cuts in its surface, and sample #601 was then assembled, as shown in figure 8(f). It can be seen from figure 8(d) that the maximum trapped field increases to 0.56 T for this arrangement, and that a uniform distribution of field is retained (the cone-shape remains wide with a flat top). This experiment confirms the result of figure 7, that the trapped field of tall samples generally remains high. This experiment also suggests that the distribution of the trapped field can be tuned, especially to uniform values, without significant loss of magnitude (B z ) compared to the original, uncut sample. This is potentially extremely useful for applications requiring such a profile [18].

Conclusions
The size and aspect ratios of bulk, single grain (RE)BCO superconductors are important parameters in determining the field generating properties of these materials. The use of multiple high temperature superconducting bulk samples allows much better control and optimisation of flux density and distribution than the use of individual single grains. Measurements of trapped fields and total flux of three YBCO single grains of diameter 16, 20 and 25 mm and sub-specimens of these grains are consistent with the predictions of Eisterer's model that the trapped field saturates for aspect ratios where 0.67<H/R<1, although the microstructures and properties of the samples used in this study are significantly less uniform than that assumed in this model.
Measurements on actual samples indicates that the best aspect ratio (H/R) is lower for smaller samples, with optimum H/R values of 0.72, 0.82 to 0.91 observed for the samples of diameter 16, 20 and 25 mm. We have demonstrated clearly that smaller samples have higher average J c estimated using Chen's formula [13], suggesting that higher trapped field may be obtained if thinner, multiple samples are stacked together.
Measurements of the total flux of three single grain YBCO samples have revealed some interesting trends in the relation between sample geometry and trapped field. In general, the taller and larger the sample and the closer to the top or bottom surfaces, the higher the value of the total flux in the direction perpendicular to sample surface (i.e. the ab plane). This study has revealed that the total flux may change abruptly if a layer of material with varying superconducting properties is introduced to the sample microstructure. Removing the bottom layer of a single grain sample is observed to cause a change in trapped field associated with that layer, with an observed decrease in total flux at the top surface of the sample, suggesting that the bottom layer, which does not generally exhibit good superconducting properties, actually aids the generation of magnetic field at the top surface of the sample. The observed increment of the total flux becomes lower when the height of the sample increases, and, therefore, the observed trapped field decreases less rapidly at positions more distant from the top (or bottom) surfaces of the sample. This observation can be very important for the design of practical applications.
This study shows that stacking thin bulk samples can improve the trapped field of the resultant composite bulk as compared to a monolithic bulk superconductor of the same diameter. Furthermore, we have demonstrated that this approach can be combined with the use of a tailored arrangement of smaller samples to tune the resultant flux profile, to generate a flatter, more uniform field profile. This is of particular significance for a variety of applications such as MRI based on bulk superconductors.