Temperature Dependence of a Width of $\Delta$H = $\Delta$B Region in 5 wt.\% (Fe, Ti) Paticle-Doped MgB$_2$ Superconductor
H. B. Lee, G. C, Kim, Y. C. Kim
TL;DR
This study investigates how the width of a specific magnetic region in Fe and Ti particle-doped MgB$_2$ superconductor varies with temperature, revealing a linear relationship and supporting a theory linking it to the upper critical field.
Contribution
It provides experimental evidence of the linear temperature dependence of the $ riangle$H = $ riangle$B region width and discusses its theoretical implications for superconductor flux behavior.
Findings
Width of the $ riangle$H = $ riangle$B region is linear with temperature.
The width's temperature dependence supports the theory relating it to the upper critical field.
Discussion includes effects of Fe doping, Bean model, and volume dependence.
Abstract
A temperature dependence of a width of H = B region has been studied for 5 wt.\% (Fe, Ti) particle-doped MgB superconductor. The result revealed that widths of the region are linear along temperature. Here we show the meaning of the result and details of the calculation. In previous report, we represented a theory that a width of H = B region is related with upper critical field of the superconductor, which is that pinned fluxes at volume defect are picked out and move in H = B region when a distance between them is the same as that of upper critical field. Thus, we inspected the relationship between a width of the region and upper critical field along temperature. The theory would gain another justification if temperature dependence of a width of the region is proportional to that of upper critical field. We discussed several topics…
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Taxonomy
TopicsSuperconductivity in MgB2 and Alloys · Physics of Superconductivity and Magnetism
††thanks: Fax: +82-51-513-7664
Temperature Dependence of a Width of H = B Region
in 5 wt.% (Fe, Ti) Paticle-Doped MgB2
Superconductor
H. B. Lee
G. C. Kim
Y. C. Kim
Department of Physics, Pusan National University, Busan 46241, Korea
Abstract
A temperature dependence of a width of H = B region has been studied for 5 wt.% (Fe, Ti) particle-doped MgB2 superconductor. The result revealed that widths of the region are linear along temperature. Here we show the meaning of the result and details of the calculation. In previous report, we represented a theory that a width of H = B region is related with upper critical field of the superconductor, which is that pinned fluxes at volume defect are picked out and move in H = B region when a distance between them is the same as that of upper critical field. Thus, we inspected the relationship between a width of the region and upper critical field along temperature. The theory would gain another justification if temperature dependence of a width of the region is proportional to that of upper critical field. We discussed several topics for H = B region of 5 wt.% (Fe, Ti) particle-doped MgB2 superconductor, which are Fe of (Fe, Ti) particle, Bean model, volume dependence of the region, etc..
MgB2, Flux pinning effect, (Fe, Ti) particles, H = B region, temperature dependence, volume defect-dominating superconductor, upper critical field
pacs:
74.60.-w; 74.70.Ad
I Introduction
H = B region in a field dependence of magnetization curve (M-H curve) is the region that appears in the volume defect-dominating superconductors. If volume defects are many enough in the superconductor, H = B region appears, which is the region that the increased applied field is the same as the increased magnetic induction after the superconductor reached the maximum diamagnetic property. The region appears because volume defects pin magnetic quantum fluxes preferentially over the entire specimen Lee4 .
A volume defect pins much more quantum fluxes than planar or line defect does because its free energy density is low, which depends on its volume. If pinned fluxes at a volume defect are many, they bend like bow because of their repulsive force between them. Thus, fluxes have been not pinned at volume defects are pushed forward, and they would be pinned at other volume defect. Therefore, there are almost no unpinned fluxes in the superconductor if volume defects are many enough. Magnetic quantum fluxes are pinned at volume defects step by step from the surface to center of the superconductor as applied magnetic field increases. In this way, magnetic quantum fluxes in a superconductor are preferentially pinned at volume defects over entire specimen.
On the other hand, when pinned fluxes are picked out from the volume defect by increase of magnetic quantum fluxes caused by applied magnetic field, the entire pinned fluxes at a volume defect are picked out all at once instead of being leaked out from the volume defects one by one Gurp ; Bonevich . We would use the term ′pick out′ instead of depinning because they are picked out at once when pinned fluxes at a volume defect are depinned. It is considered that they are picked out at once by the tension of them.
We have represented the two conditions that pinned fluxes are picked out from a volume defect Lee4 . One is when Fpickout is more than Fpinning and the other is when a distance between of pinned fluxes at a volume defect is the same as that of upper critical field (Hc2). The former is applicable before H which is defined as the applied magnetic field showing maximum diamagnetic property (the first field in H = B region), and the latter is applicable in H = B region. The justification of the latter is due to the fact that the neighborhood of the volume defect would lose superconductivity if the distance between pinned fluxes is narrower than that of Hc2, thus the concept of flux pinning will not be established in the state.
A width of H = B region decreases as temperature increases. The behavior seems to be caused by the fact that the coherence length () of a superconductor increases as temperature increases, thus the number of fluxes pinned at a volume defect decreases as temperature increases. Therefore, a width of H = B region decreases as temperature increases. Thus, it is needed to check the relationship between Hc2 and a width of H = B region as the temperature increases because Hc2 is , where is magnetic quantum flux Poole .
Many reviews do not believe the reality of H = B region and related theories. We think that the denial is mainly caused by the fact that H = B region is different from the general theory of superconductivity. If we had showed the relation between a width of H = B region and temperature, the theory and our assertion would be understood more clearly. Because a width of H = B region is related to Hc2 of the superconductor in the theory, we believe that the reality of H = B region and the theory would be proved more clearly if a width of the region depends on temperature as Hc2 does.
II results and discussion
II.1 Decisions of the width of H = B region
Widths of H = B region were obtained by taking average for the region of the first quadrant and that of the second quadrant in M-H curve. When the exact width is difficult to take due to flux jump, the value was reasonably taken. Figure 1 (a) shows M-H curve of 5 wt.% (Fe, Ti) particle-doped MgB2 at 5 K. Inspecting the first and the second quadrants, it is certain that widths of H = B region were expanded to 1.5 T. Thus the width is decided to be 1.3 T, which is 1.5 T- 0.2 T (H is 0.2 T).
M-H curve of the specimen at 10 K is shown in Fig. 1 (b). The diamagnetic property at 1.5 T of 10 K is somewhat smaller than that of 5 K in first quadrant, and diamagnetic property of second quadrant at 1.5 T is slightly lower from the horizontal line. Thus, it is hard to think the region to be expanded to 1.5 T. Taking the median value, the width of the region is 1.05 T, which is 1.25 T - 0.2 T.
Figure 2 (a) shows M-H curve of the specimen at 15 K. It is difficult to decide the exact value of the width in the first quadrant because of the flux jump. However, it is certain that the region would be up to 1 T when considered in the second quadrant. Thus, the width at 15 K is 0.8 T, which is 1.0 T - 0.2 T. M-H curve of the specimen at 20 K is shown in Fig. 2 (b). It is clear that the width of the region is up to 0.9 T in first quadrant. However, there is a little confusion that H = B region is up to 0.8 T or 0.9 T in second quadrant, thus the end of the region is conservatively decided to be 0.8 T. Therefore, the width of the region is 0.6 T, which is 0.8 T - 0.2 T.
The M-H curve at 25 K of the specimen is shown in Fig. 3 (a). The curve was made a mistake in measuring the magnetization without removing the pinned magnetic fluxes in the specimen. Estimating H at 25 K, it is certain that it is higher field than that of 20 K and lower field than that of 30 K. Thus, it will be reasonable that H at 25 K is 0.2 T because H at 30 K is not far from 0.2 T. The region has been formed before the flux jump appears if compared widths of the first quadrant with that of the second quadrant. Thus, the width of region at 25 K is determined to be 0.4 T, which is 0.6 T - 0.2 T. M-H curve of the specimen at 30 K is shown is Fig. 3 (c). H = B region completely disappears. The noting thing is the maximum diamagnetic property of the specimen also decreases a little.
The thickness of the specimen used for the measurement was 0.25 cm. Thus, the width of the region as unit thickness have to be 4t (t is the thickness of the specimen). In addition, since applied magnetic field penetrates both sides of the specimen, volume defects in the superconductor have pinned the fluxes for both side until applied magnetic field reach H. Therefore, a width of the region as unit thickness is
[TABLE]
where is a width of the region as unit thickness, is experimentally obtained width of the region, is the number of specimen when the specimen of unit thickness was divided (=1). Results for the width of H = B region along temperature are shown in Fig. 4 (a). As temperature increases, it is certain that a width of H = B region would decrease linearly.
II.2 Relation between the width of H = B region and Hc2
A width of H = B region was suggested as follows Lee4 .
[TABLE]
where is the first field of H = B region and is the final field of H = B region. , mcps, , and are the number of quantum fluxes pinned at a defect, the number of defects which are in the vertically closed packed state, the number of defects having pinned fluxes from the surface to the center of the superconductor in an axis, and flux quantum, respectively. mcps is the minimum number of defects when the penetrated fluxes into the superconductor are completely pinned. Thus, 2r$$\timesmcps is unit if is radius of spherical defect and is magnetic induction in H = B region Lee4 .
The number of flux quanta that are pinned at a spherical defect of radius r in a static state is
[TABLE]
where , and is a radius of defect, a distance between quantum fluxes pinned at the defect and filling rate which is /4 when they have square structure, respectively Lee5 .
If is 2\pi$$\xi^{2}, which is Hc2 state, a volume defect pin the maximum flux quanta. In addition, Hc2 is
[TABLE]
Inserting to Eq. (3)
[TABLE]
Thus,
[TABLE]
Thus, is proportional to Hc2 if and are fixed. Therefore, it is determined that the theory would have another justification if the behavior of the width along temperature is proportional to that of Hc2.
II.3 Discussion
It is clear that 5 wt.% (Fe, Ti) particle-doped MgB2 has H = B region in M-H curve. However, it is another problem that H = B region can be generalized in the superconductor. We insisted that the region can be generalized in the volume defect-dominating superconductor. However, many peer reviews thought that the region cannot be generalized. We suggest two papers in literature, which concerned melt-textured growth superconductor that is one of volume defect-dominating superconductor. H = B region in M-H curve was clearly shown in the papers Hsu ; Muller .
The typical equation of Hc2 along temperature is as follows.
[TABLE]
where t = /, is coherence length Tinkham . is the measuring temperature and is critical temperature. The equation shows that Hc2 is linear along temperature. Fig. 4 (b) is temperature dependence of Hc2 for 5 wt.% (Fe, Ti) particle-doped MgB2 specimen, which was air-cooled. It shows that Hc2 is also linear along temperature. Most of literatures for MgB2 were that Hc2 is linear along temperature Buzea ; Eisterer . Thus, it is clear that Hc2 is linear along temperature.
Therefore, it is certain that the pick-out of pinned fluxes at a volume defect is influenced by Hc2 because the temperature dependence of a width of H = B region is proportional to that of Hc2. On the other hand, we could understand the way that pinned fluxes are picked out from a defect in H = B region if calculating Fpinning and Fpickout. As the result of calculating Fpinning and Fpickout in H = B region for 163 nm volume defects, Fpinning/Fpickout was more than 4 Lee6 . Even though Fpinning is more than four times larger than Fpickout, pinned fluxes at the volume defect have been picked out and moved in H = B region. This is completely different from the concept that pinned fluxes are picked out and move when Fpickout is more than Fpinning.
A noting thing is that a width of H = B region is dependent on a volume of the specimen, whereas Hc2 is not. As applied magnetic field (H) increases, fluxes are quantized and penetrate into the inside of the superconductor. If H approaches Hc2, fluxes difference between inside and outside of superconductor is infinitesimal. Thus, Hc2 is not dependent of volume of the specimen. However, a width of the region decreases as much as a volume of a specimen decreases because the number of volume defects decreases as many as a volume of a specimen decreases.
Concerning Fe of (Fe, Ti) particle, H = B region was suspected to be that M-H curve is distorted because of the ferromagnetic nature of Fe. This means that it was a mistake to assert that H = B region appears in M-H curve of the specimen. We already have known the problem of Fe, which is shown in Fig. 5. It is certain that M-H curves were distorted if crystal Fe was doped on MgB2. Especially, the specimen was broken after synthesizing MgB2 because MgB2 does not match crystal Fe. The figure was measuring a broken specimen. However, there was no problem with amorphous metal. (Fe, Ti) particle is amorphous, which shows different magnetic behavior from crystal Fe. We did not find any distortion in M-H curves of specimens Lee2 . Therefore, H = B region is real in the specimen as shown in Fig. 1 - 3.
Concerning Bean model, the flux penetration method represented in the previous paper violates Bean model which is one of critical state model Lee4 ; bean . We think that Bean model does not describe the nature of superconductor properly although it is honored. Researchers having studied the difference between the critical current density (Jc) of transport and that of Bean model would notice that the former is always larger than the latter. If Bean model was to describe the nature of superconductivity properly, the transport Jc must have been less than that of Bean model because there are contact resistance, etc. Bean model seems to be a theory that suggests a minimum Jc of the superconductor. In literature, it is observed that Flkiger et al. reported transport Jc and Bean model*′s Jc* concurrently in Fe/MgB2 tapes Flukiger .
III Conclusion
We studied temperature dependence of a width of H = B region in 5 wt. % (Fe, Ti) particle-doped MgB2 superconductor. A width of the region decreases linearly as temperature increases. In previous report, we asserted the theory that a width of H = B region relates with Hc2 and pinned fluxes are picked out from the volume defects when the distance between pinned fluxes is the same as that of Hc2. Hc2 of MgB2 also decreases linearly as temperature increases. Therefore it is clear that a width of H = B region have relation with Hc2 of MgB2. It is considered that the theory has another justification.
IV Method
5 wt.% (Fe, Ti) particle-doped MgB2 and 5 wt.% nano-Fe particle-doped MgB2 were synthesized using the nonspecial atmosphere synthesis (NAS) method Lee . Briefing NAS method, Mg (99.9% powder), B (96.6% amorphous powder), (Fe, Ti) particles, and crystal nano-Fe particles were prepared for the specimen. After mixing Mg and B stoichiometry, (Fe, Ti) particles and crystal nano-Fe particles were added by weight, respectively. They were finely ground and pressed into 10 mm diameter pellets. (Fe, Ti) particles were ball-milled for several days, and average radius of the (Fe, Ti) particles was approximately 0.163 m. On the other hand, an 8 m-long stainless-steel (304) tube was cut into 10 cm pieces. One side of the 10 cm-long tube was forged and welded. The pellets and excess Mg were placed in the stainless-steel tube. The pellets were annealed at 300oC 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. The specimens was synthesized at 920oC for 1 hour and cooled in air. The field dependences of magnetization were measured using a MPMS-7 (Quantum Design).
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