Upper Critical Field Based on a Width of ΔH = ΔB region in a Superconductor

We studied a method of measuring upper critical field (Hc2) of a superconductor based on a width of ΔH = ΔB region, which appears in a superconductor that volume defects are many and dominant. Here we show basic concepts and details of the method. Although Hc2 of a superconductor is fixed according to a kind of superconductor, it is difficult to measure Hc2 experimentally. Thus, results are different depending on experimental conditions. Hc2 was otained by a theory on a width of ΔH = ΔB region, which is that pinned fluxes at volume defects are picked out and move into an inside of the superconductor when the distance between pinned fluxes is the same as that at Hc2 of the superconductor. Hc2 of MgB2 obtained by the method was 65.4 Tesla at 0 K, which is quite same as that of Ginzburg-Landau theory. The reason that Hc2 obtained by the method is closer to ultimate Hc2 is based on that Fpinning/Fpickout is more than 4 when pinned fluxes at volume defects of 163 nm radius are depinned, which means that the Hc2 is less sensitive to fluctuation. The method will help to find the ultimate Hc2 of volume defect-dominating superconductors.

Generally, it was reported that H c2 of MgB 2 is 20-30 T [7][8][9] . However, this is equivalent to the statement that H c2 of MgB 2 is more than 20 T and the upper limit is unknown.
No matter how high H c2 was measured, it was the H c2 that was appropriate for the condition of measurement and the state of the specimen, thus it has its own meaning. However, it is also important to measure ultimate H c2 experimentally. In previous study, we have asserted that a ΔH = ΔB region is formulated if volume defects are many enough in volume defect-dominating superconductor, which is the region that increased applied magnetic field is the same as increasing magnetic induction 10 . Two conditions are suggested for that pinned fluxes have to be picked out from the volume defect, which are F pinning < F pickout or the distance between pinned fluxes at a volume defect is equal to that of H c2 . We have calculated H c2 using the theory with experimental results, and have obtained a fairly reasonable result. Thus, we would introduce the method of obtaining H c2 of superconductor based on a ΔH = ΔB region.

Results
A model for flux quanta distribution at H c2 and problems of measurement. Figure 1 shows field dependences of magnetization (M-H curves) for various temperature, and applied magnetic field is the maximum at 6.5 T, and they are cut to focus H c2 . M-H curves below 15 K are not considered because the diamagnetic properties at the temperatures are quite meaningful at 6.5 T. The field that diamagnetic property changes from − to + is H c2 , and results are shown in Fig. 2. The equation of extrapolating is as follows.
where t = T m /T c , ξ is coherence length 1,11 . T m is the measuring temperature and T c is critical temperature. As shown in Fig. 2, different results were obtained for three specimens, and are needed to check that they were closer to the ultimate H c2 of the specimen. Since the diamagnetic properties of the superconductor approach zero if the external magnetic field is high enough, the arrangements of the flux quanta in the superconductor would be that of Fig. 3. When many magnetic fluxes quanta have penetrate into the inside of the superconductor, the   www.nature.com/scientificreports www.nature.com/scientificreports/ supercurrent circulating the surface of the superconductor is the only one related to diamagnetic property. Thus, the field that superconducting current disappears would be H c2 as the external field increases.
As shown in Fig. 1, applied magnetic field increased by 0.5 T when the external field is above 1 T in doped specimens whereas it did by 0.25 T in pure specimen. It is clear that all of increased magnetic field penetrates the inside of the superconductor when the diamagnetic property is close to zero. On the other hand, the fluxes in the superconductor exist as flux quanta, and the repulsive forces are acting between them. The repulsive force per unit length (cm) between quantum fluxes is, which is caused by vortexes where J s is the total supercurrent density due to vortices 12 . When a large field of 0.5 T increases at once, the flux quanta that try to penetrate into the superconductor will interact with the existing flux quanta in the superconductor, and the repulsive force between them will cause continuous vibration. The vibration will cause mutual interference, which are amplification and attenuation. The attenuation is expected to have little effect on the diamagnetic supercurrent, but the amplification may cause some flux quanta to rebound out from the superconductor because there is little difference in fluxes density between the inside and the outside of the superconductor and circulating supercurrent is tiny.
When rebounds of quantum fluxes occur, the supercurrents circulating the surface of the superconductor are interfered. When the number of the rebounding flux quantum exceeds a certain value, the supercurrents would disappear, which induces that the diamagnetic property does not appear. It is considered that the phenomenon increases as the distance between fluxes become closer to that of ultimate H c2 , and increases as the magnitude of magnetic field applied at once increases. Therefore, it is clear that H c2 obtained from M-H curves must be much lower than the ultimate H c2 .
On the other hand, when the magnetic field is applied first to the superconductor and next the superconducting current is supplied to determine H c2 of a superconductor, fluxes inside the superconductor would be much more stabilized. In this case, it is determined that the magnetic field in the superconductor is arranged in the form of Fig. 3(a) or ultimately stabilized form of Fig. 3(b). From the point of view, it is considered that the arrangement of the flux quanta in the H c2 state of a superconductor is not depending on what kind of superconductor is, but how long it takes after applying the magnetic field 13,14 . Since a stabilization of quantum fluxes would be achieved over time, it is natural that the arrangement of the flux quanta would be the state as shown in Fig. 3(b).
If a small amount of current flows around the surface of specimen in the state that the magnetic fluxes in the superconductor are stabilized, the increasing magnetic flux quanta in the superconductor is only a magnetic field generated by the flowing current. If the magnitude of the current is small, the interference caused by the increased magnetic flux quanta would be small, thus the magnetic flux quanta rebounding out of the superconductor will also be small. Therefore, it is considered that H c2 measured by currents method is much closer to the ultimate H c2 of the superconductor than that of M-H curves. However, it is certain that H c2 measured by this method is not the ultimate because a certain amount of current must flow, which generates a magnetic field.
Upper critical field by ΔH = ΔB region. If volume defects are spherical, their size is constant, and they are arranged regularly in a superconductor, a superconductor of 1 cm 3 has m′ 3 volume defects. Assuming that the pinned fluxes at volume defects are picked out and move into an inside of the superconductor when the distance between pinned fluxes is the same as that of H c2 as shown in Fig. 3(a), the maximum number of flux quanta that can be pinned at a spherical defect of radius r in a static state is where r, d and P is the radius of defects, the distance between quantum fluxes pinned at the volume defect of which radius is r and filling rate which is π/4 when they have square structure, respectively, as shown in Fig. 3(a) 15 . If volume defects in a superconductor are many enough, the superconductor has a ΔH = ΔB region, and the width of the region is where H final is the final field of the ΔH = ΔB region, H c1 ′ is the first field of the ΔH = ΔB region 10 . m cps is the number of defects which are in the vertically closed packed state, n 2 is the number of flux quanta pinned at a defect of radius r, m is the number of the volume defects from surface to center along an axis (m′ = 2m), M is magnetization, and Φ 0 is flux quantum. m cps is the minimum number of defects when the penetrated fluxes into the superconductor are completely pinned. Thus, 2r × m cps is unit.
The number of flux quanta pinned at a defect is Because 2r × m cps = 1, the equation is The thickness of the specimen used for the measurement is 0.25 cm. Thus, the width of the region as unit length have to be 4 × t (t is the thickness of the specimen). In addition, since applied magnetic field penetrates into both sides of the specimen, volume defects inside the superconductor have pinned the fluxes for both side until applied magnetic field reach ′ H c1 . Therefore, the width of the region as unit length is where w is experimentally obtained width of the region, n d is the number of specimen when a specimen of unit length was divided (n d t = 1). Although the width of the region is 1.3 T as shown in the Fig. 4(b), the width of ΔH = ΔB region as unit length is 5.8 T because the width of the specimen was 0.25 cm. www.nature.com/scientificreports www.nature.com/scientificreports/ If the average radius of defects, the width of ΔH = ΔB region, M, ′ H c1 , and m are 163 nm, 5.8 T, −150 emu/cm 3 , 2000 Oe, and 4000, respectively, which are experimental results of 5 wt.% (Fe, Ti) particle-doped MgB 2 as shown in Fig. 4(a,b), H c2 of the specimen is 56.7 T at 5 K. Concerning m, it is 4000 because magnetic field penetrates into the superconductor from both sides although the specimen have 8000 3 volume defects of average 163 nm radius 10 . The coherence length (ξ) is 2.41 nm when H c2 is 56.7 T at 5 K. Extrapolated by Eq. (1), ξ is 2.24 nm and H c2 is 65.4 T at 0 K. Accidentally, the value is much closer to that of Ginzburg-Landau theory, which is 68.6 T at 0 K 6 .
Discussion. As mentioned earlier, the methods of measuring H c2 of a superconductor have their own drawbacks. Supercurrents method may be close to the ultimate H c2 of the superconductor, but it is clear that there is a difference between the result and the ultimate H c2 because of the magnetic field induced by applied currents. However, we believe that H c2 measured by this method can further reduce the difference.
We could understand how stabilized the pinned fluxes are in H c2 state if inspecting the force balances of the pinned fluxes when they are picked out. Generally, pinned fluxes at volume defect move when F pickout is more than ΔF pininng . However, it was our assertion that the pinned fluxes are picked out and moved even in F pinning > F pickout state when the distance between them is equal to that of H c2 . The justification of the assumption is that there is no pinning effect if the neighborhoods of the volume defect are changed to normal state.
F pinning is where n 2 is the number of quantum fluxes pinned at a spherical volume defect of radius r, H c2 is upper critical field of the superconductor, Φ o is flux quantum which is 2.07 × 10 −7 G⋅cm 2 , c is the velocity of light, aL is an average length of quantum fluxes which are pinned and bent between defects (a is an average bent constant which is 1 < a < 1.2 and L is the distance between defects in vertically packed state) and P is the filling rate which is π∕4 when flux quanta are pinned at a volume defect in the form of square 15 . Numerically, if ′ H c1 is 2000 Oe, r is 0.163 nm, n is 45, and aL is 1.1 × 3.9 × 10 −4 cm, which are results of idealized 5 wt.% (Fe, Ti) doped MgB 2 specimen, F pinning is 5.3 × 10 −4 dyne and F pickout is 1.4 × 10 −4 dyne. Comparing F pinning with F pickout , F pinning /F pickout is more than 4. Generally, when fluxes are approaching a volume defect, they have a velocity. If F pinning are similar with F pickout , the pick-out of pinned fluxes from the volume defect is easier than that of calculation because fluxes have a velocity when they move in the superconductor. However, if F pinning is more than 4 times of F pickout , it is considered that the depinning occurs after the distance between pinned fluxes is same as that of H c2 even if fluxes had some velocity.

conclusion
We have investigated characteristics of several methods for obtaining H c2 of type II superconductors and explained that any experimental method to obtain H c2 would be different from the ultimate H c2 . In addition, no matter how high H c2 was obtained, it has its meaning because it was affected by the state of the specimen and measurement conditions. We suggested a method to obtain H c2 , which is that H c2 of volume defect-dominating superconductor could be obtained from a width of ΔH = ΔB region. We used the property that ΔH = ΔB region is formed in the M-H curve when volume defects in the superconductor are many enough. It is based on the theory that pinned fluxes at the volume defects would be picked out from the volume defects and move when the distance between them is equal to that at H c2 . From the results of 5 wt.% (Fe, Ti) doped MgB 2 , H c2 was 56.7 T at 5 K, which is quite same as that of Ginzburg-Landau theory. We obtained that F pinning /F pickout is more than 4 in ΔH = ΔB region, which means that fluxes had been pinned at the volume defect were depinned even though F pinning is much larger than F pickout . The behavior means that the H c2 is less sensitive to fluctuation. Therefore, it is determined that the obtained H c2 by the method is much closer to the ultimate H c2 of the superconductor.

Method
Pure MgB 2 and (Fe, Ti) particle-doped MgB 2 specimens were synthesized using the nonspecial atmosphere synthesis (NAS) method 16 . Briefly, NAS method needs Mg (99.9% powder), B (96.6% amorphous powder), (Fe, Ti) particles and stainless steel tube. Mixed Mg and B stoichiometry, and (Fe, Ti) particles were added by weight. They were finely ground and pressed into 10 mm diameter pellets. (Fe, Ti) particles were ball-milled for several days, and average radius of (Fe, Ti) particles was approximately 0.163 μm 10 . On the other hand, an 8 m-long stainless-steel (304) tube was cut into 10 cm pieces. Insert holed Fe plate into stainless-steel (304) tube. One side of the 10 cm-long tube was forged and welded. The pellets and pelletized excess Mg were placed at uplayer and downlayer in the stainless-steel tube, respectively. The pellets were annealed at 300 °C for 1 hour to make them hard before inserting them into the stainless-steel tube. The other side of the stainless-steel tube was also forged. High-purity Ar gas was put into the stainless-steel tube, and which was then welded. Specimens had been synthesized at 920 °C for 1 hour. They are cooled in air and quenched in water respectively. The field and temperature dependence of magnetization were measured using a MPMS-7 (Quantum Design).