contribute to unusual superconducting behaviour in MgB2 which is not observed in conventional superconductors. Since the two properties of the *σ* bands are very similar and those of the *π* bands are very similar, even through one *π* band is hole type and the other is electron type, the electronic structure is usually simplified to one *σ* band and one *π* band. Therefore, MgB2 is known and described as a two-band superconductor (Liu et al., 2001).

#### **2.2 Unique characteristic caused by two-band and two-gap structure**

MgB2 is known to be theoretically and experimentally identified as the first superconductor where two superconducting gaps clearly exist (Xi, 2008; Tajima, 2005). The unique two-band and two-gap structure has an influence on the resistivity behaviour in the normal state. In the year of discovery, it became well known that the residual resistivity ratio (RRR) was independent of *T*c in polycrystalline MgB2 samples. Specifically, many MgB2 samples showed widely different RRR values despite having similar *T*c values around ~ 39 K (Buzea & Yamashita, 2001). In the case of the high *T*c, the disorder effects on it (Eisterer, 2007) due to poor crystallinity and lattice strain can be negligible, and thus, the RRR value calculated from the residual resistivity was thought to depend on impurity contamination in a sample. This indicates that lower RRR is caused by a larger amount of impurity phase, even though it should be noted that the residual resistivity is partially attributable to extrinsic effects such as lack of connectivity associated with cracks, voids, and low mass density (Rowell, J. M. 2003). In the case of the two-gap superconductivity model (Golubov & Mazin, 1997), nonmagnetic impurities may contribute to pair breaking, which is similar to the effects of magnetic impurities in a conventional *s*-wave superconductor. Thus, it was difficult to theoretically understand the observed unclear correlation between the RRR and the *T*c. Regarding this issue, Mazin et al. clearly explained the mechanism by pointing out the very weak impurity scattering between the *σ* and the *π* bands compared with the intraband impurity scattering between the *σσ* and *ππ* sub-bands, due to the particular electronic structure in MgB2 (Mazin et al., 2002). It was also found that the *ππ* scattering rate is larger than the *σσ* scattering rate. In addition, they indicated that the *T*c is given by the maximum eigenvalue of the superconducting coupling constant *λ* for the *σπ* and the *πσ* interband coupling, and the *σσ* and *ππ* intraband coupling, while the conductivity is the sum of all the conducting channels responsible for the interband and intraband couplings. The estimated *λ* for the *σσ* intraband coupling showed the highest value, suggesting that the *T*c is affected by the *σ* band. These results provide the conclusion that because impurities have a stronger effect on the scattering rate of the *π* band in preference to the *σ* band, and thus the residual resistivity, they may not contribute to the interband scattering and *T*c.

The upper critical field behaviour of MgB2 was also found to be very different from that of a conventional phonon-mediated Bardeen, Cooper, and Schrieffer (BCS) superconductor with a single-band structure. A unique upward curvature of *B*c2(T) parallel to the *ab* plane near *T*c was observed in single crystal MgB2 samples (Zehetmayer et al., 2002; Lyard et al., 2002). This upward curvature can be well explained by using the Ginzburg-Landau theory, suggesting that the *B*c2 is determined by the Ginzburg-Landau parameter and the condensation energy (Eisterer, 2007). In the case of MgB2, which has the two-band structure, the total condensation energy depends on the sum of all the energies arising from the *σ* band, the *π* band, and the interaction between the two bands (Eisterer et al., 2005). These condensation energies in the temperature range from absolute zero to *T*c show different

contribute to unusual superconducting behaviour in MgB2 which is not observed in conventional superconductors. Since the two properties of the *σ* bands are very similar and those of the *π* bands are very similar, even through one *π* band is hole type and the other is electron type, the electronic structure is usually simplified to one *σ* band and one *π* band. Therefore, MgB2 is known and described as a two-band superconductor (Liu et al., 2001).

MgB2 is known to be theoretically and experimentally identified as the first superconductor where two superconducting gaps clearly exist (Xi, 2008; Tajima, 2005). The unique two-band and two-gap structure has an influence on the resistivity behaviour in the normal state. In the year of discovery, it became well known that the residual resistivity ratio (RRR) was independent of *T*c in polycrystalline MgB2 samples. Specifically, many MgB2 samples showed widely different RRR values despite having similar *T*c values around ~ 39 K (Buzea & Yamashita, 2001). In the case of the high *T*c, the disorder effects on it (Eisterer, 2007) due to poor crystallinity and lattice strain can be negligible, and thus, the RRR value calculated from the residual resistivity was thought to depend on impurity contamination in a sample. This indicates that lower RRR is caused by a larger amount of impurity phase, even though it should be noted that the residual resistivity is partially attributable to extrinsic effects such as lack of connectivity associated with cracks, voids, and low mass density (Rowell, J. M. 2003). In the case of the two-gap superconductivity model (Golubov & Mazin, 1997), nonmagnetic impurities may contribute to pair breaking, which is similar to the effects of magnetic impurities in a conventional *s*-wave superconductor. Thus, it was difficult to theoretically understand the observed unclear correlation between the RRR and the *T*c. Regarding this issue, Mazin et al. clearly explained the mechanism by pointing out the very weak impurity scattering between the *σ* and the *π* bands compared with the intraband impurity scattering between the *σσ* and *ππ* sub-bands, due to the particular electronic structure in MgB2 (Mazin et al., 2002). It was also found that the *ππ* scattering rate is larger than the *σσ* scattering rate. In addition, they indicated that the *T*c is given by the maximum eigenvalue of the superconducting coupling constant *λ* for the *σπ* and the *πσ* interband coupling, and the *σσ* and *ππ* intraband coupling, while the conductivity is the sum of all the conducting channels responsible for the interband and intraband couplings. The estimated *λ* for the *σσ* intraband coupling showed the highest value, suggesting that the *T*c is affected by the *σ* band. These results provide the conclusion that because impurities have a stronger effect on the scattering rate of the *π* band in preference to the *σ* band, and thus the residual

**2.2 Unique characteristic caused by two-band and two-gap structure**

resistivity, they may not contribute to the interband scattering and *T*c.

The upper critical field behaviour of MgB2 was also found to be very different from that of a conventional phonon-mediated Bardeen, Cooper, and Schrieffer (BCS) superconductor with a single-band structure. A unique upward curvature of *B*c2(T) parallel to the *ab* plane near *T*c was observed in single crystal MgB2 samples (Zehetmayer et al., 2002; Lyard et al., 2002). This upward curvature can be well explained by using the Ginzburg-Landau theory, suggesting that the *B*c2 is determined by the Ginzburg-Landau parameter and the condensation energy (Eisterer, 2007). In the case of MgB2, which has the two-band structure, the total condensation energy depends on the sum of all the energies arising from the *σ* band, the *π* band, and the interaction between the two bands (Eisterer et al., 2005). These condensation energies in the temperature range from absolute zero to *T*c show different behaviour from each other. In particular, the contribution of the *π* band to the total condensation energy at 0 T in temperatures above 30 K rapidly increases, while that below 30 K is negligible. This is the reason why the *B*c2(T) of MgB2 shows a positive curvature near *T*c. The unique slope can make the zero-temperature *B*c2(0) significantly higher than that estimated from a single-gap dirty superconductor model (Gurevich, 2003). In addition, an interesting scenario relating to *B*c2(T) for dirty two-gap superconductors was pointed out by Gurevich (Gurevich, 2007). It is assumed that MgB2 can be identified as a simple bilayer sample where two thin films corresponding to the *σ* and *π* bands are separated by a Josephson contact, which reflects the weak interband coupling in MgB2. From the two-gap dirty-limit theory for the BCS matrix constants (Golubov et al., 2002), the temperature dependence of *B*c2 is mostly determined by whether the *σ* film or the *π* film is dirtier. Provided that the *π* film is much dirtier than the *σ* film, the global *B*c2(T) of the bilayer at higher temperature is mainly controlled by the *σ* film with the higher *B*c2, while the *π* film takes over at lower temperatures. This leads to a considerable upturn in the global *B*c2 curve at low temperatures, which can not be estimated from a single-gap dirty-limit superconductor. This interesting scenario for *B*c2 at low temperatures delivered from the simple bilayer model is indirectly demonstrated by experimentally observed *B*c2 curves for alloyed MgB2 thin films (Braccini et al., 2005). Specifically, an anomalous upward curvature of *B*c2(T) at low temperatures observed in a carbon-alloyed thin film provides a record high value of *B*c2(4.2K) = 51 T parallel to the *ab* plane. The extrapolation of the *B*c2(T) curve to *T* = 0 K may reach over ~ 60 T, approaching the paramagnetic limit of 65 T which is calculated from the conventional BCS theory (Sarma, 1963). This value is significantly higher than those for NbTi and Nb3Sn superconductors, which are widely used in superconducting applications now. Moreover, Gurevich pointed out that given the case of electron-phonon coupling, the paramagnetic limit of MgB2 is estimated to be ~ 130 T (Gurevich, 2007). This means that there is still large room for a further increase in *B*c2 from optimum dirty conditions of the intraband and the interband behaviour. Therefore, we can extract a conclusion that the two-gap superconductor MgB2 is suitable for superconducting applications in higher operating temperatures and fields, which can not be reached via the use of Nb-based superconducting materials.

## **2.3 Current status in development for superconducting application**

Two major critical issues facing MgB2 polycrystalline materials such as wires and bulks for the next generation of superconducting applications are weak grain connectivity and lack of pinning centers (Rowell, J. M. 2003; Eisterer, 2007). In order to improve rapid *J*c degradation caused by its weak pinning behaviour in pure MgB2 wires and bulks, so far, many doping materials have been tested since the discovery of its superconductivity. The first successful dopant for this was silicon carbide (SiC) and its additive improved the irreversibility field (*B*irr) and *J*c (Dou et al., 2002). Numerous researchers focused on the impressive results and then started to do many studies related to carbon (C) containing dopants for real superconducting applications. Even now, SiC is still known to be one of the most effective dopants for MgB2 materials. Specifically, *B*c2 values near 0 K for the SiC-doped MgB2 wires fabricated by the *in situ* method (Sumption et al., 2005) and coated-conductors on SiC fibers prepared from hybrid physical-chemical vapor deposition (HPCVD) (Ferrando et al., 2005) were 33 T and 55 T, respectively. These values are much higher than those of NbTi and Nb3Sn. Similar doping materials, nano-carbon (Soltanian et al., 2003), carbon nanotubes

Structural Characteristic and Superconducting

high-field *J*c of MgB2.

application.

**2.4 Mg diffusion method**

tube (PIT) processed MgB2 wires (Hossain et al., 2009).

Performance of MgB2 Fabricated by Mg Diffusion Process 291

4.2 K and 6 T for *in situ* processed C4H6O5-doped MgB2 wires exceeds 1.0 × 105 A/cm2, which is comparable to that for commercial NbTi wires (Kim et al., 2010). The application of cold high pressure densification to those MgB2 wires yielded a high *J*c value of 4.0 × 104 A/cm2 at 4.2 K and 10 T, which is the record value among conventional *in situ* powder-in-

According to recent advanced microscopic analysis by using scanning transmission electron microscopy (STEM) and electron energy loss spectroscopy (EELS), un-reacted carbon agglomerates are found to be formed at grain boundaries in C-doped MgB2 (Kim et al., 2012). The residual impurity arises from inhomogeneous mixing of starting materials including carbon compound as dopant and impedes supercurrent flow. In order to address the issue, recently, a carbon containing gas diffusion method by using oxygen-free pyrene (C16H10) was proposed (Maeda et al., 2011b). Pyrene is an aromatic hydrocarbon and its boiling temperature is 404 ºC lower than the temperature of MgB2 formation. Owing to the characteristic, molecular carbon delinked from pyrene gas homogenously distributes into dense MgB2 materials and it causes intensive structural disorder without any severe deterioration of grain connectivity. In addition, more recently, an alternative carbon doping method by using carbon encapsulated boron nanopowder made from a radio frequency plasma process was also proposed and the resultant C-doped MgB2 wires showed high *J*c performance due to very thin carbon layers on boron surface for its carbon homogenous distribution (Kim et al., 2011). Thus, the weak flux pinning problem has been well addressed by these recent advanced carbon doping methods, resulting in considerable improvement of

On the other hand, the other major issue, the weak grain connectivity, has also known to be critical for the superconducting performance of MgB2 since the discovery of its superconductivity (Rowell, J. M. 2003; Eisterer, 2007). This is more serious compared with the former major issue and it occurs in MgB2 polycrystalline materials prepared from not only *in situ* reaction of Mg and B mixtures but also *ex situ* fabrication through the use of MgB2 powder. The *in situ* PIT method is the most popular methods for fabricating MgB2 conductors and the *J*c performance is currently higher than that of *ex situ* processed conductors. However, the *in situ* processed conductors show numerous voids caused by MgB2 formation (Kim et al., 2007). This is because the theoretical volume of the mixture (VMg + V2B) is larger than the volume of the reacted MgB2. The resultant mass density is limited to ~ 50 % of the theoretical density, and the porous structure significantly reduces the supercurrent-carrying area and the *J*c for MgB2 materials. In order to counter the severe problem, *in situ* Mg diffusion methods have been proposed (Canfield et al., 2001; Kang et al., 2001) and it seems to be the most effective method even though there is still room for real

Mg diffusion into a B matrix for fabricating dense MgB2 materials was suggested by Canfield et al. and Kang et al. around the same time (Canfield et al., 2001; Kang et al., 2001), soon after the discovery of superconductivity in MgB2. Canfield et al. successfully fabricated the first MgB2 wire using the Mg diffusion method (Canfield et al., 2001). The specific technique was to expose boron filaments to Mg vapour or liquid at 950 ºC for 2 hours. The obtained conductor showed a very low residual resistivity of ~ 0.38 cm and a high

(Yeoh et al., 2004), and boron carbide (Yamamoto et al., 2005a) were also found to effectively improve the in-field *J*c in MgB2 materials. Thus, it has been found that the carbon (C) element is indispensable for superconducting applications of MgB2.

From structural analysis of C-doped MgB2 single crystals (Kazakov et al., 2005), the *a*-lattice parameter was observed to decrease drastically due to the C doping effect. The lattice shrinkage is most likely due to C substitution at B sites and to the difference in atomic radius between these two atoms, which may yield substitutional defects in the crystal lattice. This is because the C replacement effect accumulates in the B layer, and dislocations and its associated stacking faults may form when the accumulated strain exceeds the critical value. The existence of lattice defects in C-doped MgB2 samples has been proven by experimental results (Kazakov et al., 2005; Avdeev et al., 2003; Zhu et al., 2007; Kim et al., 2012). Specifically, the considerable increase in the atomic displacement parameter at the Mg and B sites in C-doped single crystals reflects the enhancement of lattice disorder (Kazakov et al., 2005). The reason why reliability factors for Rietveld refinement in C-doped samples show high values compared with those for pure samples may be explained by the presence of local disorder in the lattice, which can not be described using conventional refined parameters (Kazakov et al., 2005; Avdeev et al., 2003). The C-doped thin films deposited by HPCVD, which shows a high *B*c2 at 0 K value of 70 T, are found to have intensive structural disorder caused by dislocations (Zhu et al., 2007a). In the case of SiC-doped tapes showing relatively low mass density compared with MgB2 thin films, the MgB2 grains are also found to show a large amount of interior contrast generated by intragrain defects that are likely due to lattice disorder (Zhu et al., 2007b). In addition, the relationship between lattice change and not only substitutional, but also interstitial defects in C-doped MgB2 materials can be theoretically explained by Vienna *ab* initio calculations (Bengtson et al., 2010). Recently, boron vacancies were clearly observed in C-doped MgB2 wires and it has been found that those vacancies generate intrinsic stacking faults within the MgB2 grains, together with associated lattice distortion (Kim et al., 2012). Thus, in C-doped MgB2 materials, the reduced unit cell volume caused by the lattice shrinkage leads to introduction of lattice disorder effects, which result in increasing scattering of charge carriers, reducing the mean free path, and thereby enhancing *B*c2 and *J*c (Kazakov et al., 2005; Eisterer, 2007).

However, in the case of inorganic carbon compounds as C doping materials, due to its high melting points or its high decomposition temperature, high-temperature reaction is usually required to effectively cause the lattice shrinkage of the *a*-lattice parameter. These sintering conditions generally offer the disadvantage of grain growth associated with weak grain boundary pinning, because this pinning mechanism is also known to be effective in MgB2 materials (Larbalestier et al., 2001; Eisterer, 2007). To address the problem, organic compounds as C doping materials with relatively low decomposition temperatures have been intensively studied from around 2006 (Kim et al., 2006; Hossain et al., 2009; Kim et al., 2010; Maeda et al., 2011a). One of the best known organic dopants that has been suggested to date is malic acid (C4H6O5) (Kim et al., 2006). The advantage is that the carbohydrate melts and decomposes at a low temperature below the temperature of MgB2 formation. The decomposition produces highly reactive and fresh C on the atomic scale for C substitution on B site in the lattice. Due to these excellent characteristics, the *a*-lattice parameter decreases even under low-temperature sintering conditions, and grain growth is suppressed, resulting in considerable enhancement of *J*c and *B*c2. Specifically, the *J*c value at

(Yeoh et al., 2004), and boron carbide (Yamamoto et al., 2005a) were also found to effectively improve the in-field *J*c in MgB2 materials. Thus, it has been found that the carbon (C)

From structural analysis of C-doped MgB2 single crystals (Kazakov et al., 2005), the *a*-lattice parameter was observed to decrease drastically due to the C doping effect. The lattice shrinkage is most likely due to C substitution at B sites and to the difference in atomic radius between these two atoms, which may yield substitutional defects in the crystal lattice. This is because the C replacement effect accumulates in the B layer, and dislocations and its associated stacking faults may form when the accumulated strain exceeds the critical value. The existence of lattice defects in C-doped MgB2 samples has been proven by experimental results (Kazakov et al., 2005; Avdeev et al., 2003; Zhu et al., 2007; Kim et al., 2012). Specifically, the considerable increase in the atomic displacement parameter at the Mg and B sites in C-doped single crystals reflects the enhancement of lattice disorder (Kazakov et al., 2005). The reason why reliability factors for Rietveld refinement in C-doped samples show high values compared with those for pure samples may be explained by the presence of local disorder in the lattice, which can not be described using conventional refined parameters (Kazakov et al., 2005; Avdeev et al., 2003). The C-doped thin films deposited by HPCVD, which shows a high *B*c2 at 0 K value of 70 T, are found to have intensive structural disorder caused by dislocations (Zhu et al., 2007a). In the case of SiC-doped tapes showing relatively low mass density compared with MgB2 thin films, the MgB2 grains are also found to show a large amount of interior contrast generated by intragrain defects that are likely due to lattice disorder (Zhu et al., 2007b). In addition, the relationship between lattice change and not only substitutional, but also interstitial defects in C-doped MgB2 materials can be theoretically explained by Vienna *ab* initio calculations (Bengtson et al., 2010). Recently, boron vacancies were clearly observed in C-doped MgB2 wires and it has been found that those vacancies generate intrinsic stacking faults within the MgB2 grains, together with associated lattice distortion (Kim et al., 2012). Thus, in C-doped MgB2 materials, the reduced unit cell volume caused by the lattice shrinkage leads to introduction of lattice disorder effects, which result in increasing scattering of charge carriers, reducing the mean free path, and thereby enhancing *B*c2 and *J*c (Kazakov et al., 2005; Eisterer, 2007). However, in the case of inorganic carbon compounds as C doping materials, due to its high melting points or its high decomposition temperature, high-temperature reaction is usually required to effectively cause the lattice shrinkage of the *a*-lattice parameter. These sintering conditions generally offer the disadvantage of grain growth associated with weak grain boundary pinning, because this pinning mechanism is also known to be effective in MgB2 materials (Larbalestier et al., 2001; Eisterer, 2007). To address the problem, organic compounds as C doping materials with relatively low decomposition temperatures have been intensively studied from around 2006 (Kim et al., 2006; Hossain et al., 2009; Kim et al., 2010; Maeda et al., 2011a). One of the best known organic dopants that has been suggested to date is malic acid (C4H6O5) (Kim et al., 2006). The advantage is that the carbohydrate melts and decomposes at a low temperature below the temperature of MgB2 formation. The decomposition produces highly reactive and fresh C on the atomic scale for C substitution on B site in the lattice. Due to these excellent characteristics, the *a*-lattice parameter decreases even under low-temperature sintering conditions, and grain growth is suppressed, resulting in considerable enhancement of *J*c and *B*c2. Specifically, the *J*c value at

element is indispensable for superconducting applications of MgB2.

According to recent advanced microscopic analysis by using scanning transmission electron microscopy (STEM) and electron energy loss spectroscopy (EELS), un-reacted carbon agglomerates are found to be formed at grain boundaries in C-doped MgB2 (Kim et al., 2012). The residual impurity arises from inhomogeneous mixing of starting materials including carbon compound as dopant and impedes supercurrent flow. In order to address the issue, recently, a carbon containing gas diffusion method by using oxygen-free pyrene (C16H10) was proposed (Maeda et al., 2011b). Pyrene is an aromatic hydrocarbon and its boiling temperature is 404 ºC lower than the temperature of MgB2 formation. Owing to the characteristic, molecular carbon delinked from pyrene gas homogenously distributes into dense MgB2 materials and it causes intensive structural disorder without any severe deterioration of grain connectivity. In addition, more recently, an alternative carbon doping method by using carbon encapsulated boron nanopowder made from a radio frequency plasma process was also proposed and the resultant C-doped MgB2 wires showed high *J*c performance due to very thin carbon layers on boron surface for its carbon homogenous distribution (Kim et al., 2011). Thus, the weak flux pinning problem has been well addressed by these recent advanced carbon doping methods, resulting in considerable improvement of high-field *J*c of MgB2.

On the other hand, the other major issue, the weak grain connectivity, has also known to be critical for the superconducting performance of MgB2 since the discovery of its superconductivity (Rowell, J. M. 2003; Eisterer, 2007). This is more serious compared with the former major issue and it occurs in MgB2 polycrystalline materials prepared from not only *in situ* reaction of Mg and B mixtures but also *ex situ* fabrication through the use of MgB2 powder. The *in situ* PIT method is the most popular methods for fabricating MgB2 conductors and the *J*c performance is currently higher than that of *ex situ* processed conductors. However, the *in situ* processed conductors show numerous voids caused by MgB2 formation (Kim et al., 2007). This is because the theoretical volume of the mixture (VMg + V2B) is larger than the volume of the reacted MgB2. The resultant mass density is limited to ~ 50 % of the theoretical density, and the porous structure significantly reduces the supercurrent-carrying area and the *J*c for MgB2 materials. In order to counter the severe problem, *in situ* Mg diffusion methods have been proposed (Canfield et al., 2001; Kang et al., 2001) and it seems to be the most effective method even though there is still room for real application.

## **2.4 Mg diffusion method**

Mg diffusion into a B matrix for fabricating dense MgB2 materials was suggested by Canfield et al. and Kang et al. around the same time (Canfield et al., 2001; Kang et al., 2001), soon after the discovery of superconductivity in MgB2. Canfield et al. successfully fabricated the first MgB2 wire using the Mg diffusion method (Canfield et al., 2001). The specific technique was to expose boron filaments to Mg vapour or liquid at 950 ºC for 2 hours. The obtained conductor showed a very low residual resistivity of ~ 0.38 cm and a high

Structural Characteristic and Superconducting

**3. Experimental**

evaporation processes.

Performance of MgB2 Fabricated by Mg Diffusion Process 293

and *J*c in MgB2 prepared from Mg diffusion process. In this study, therefore, we have systematically evaluated structural and superconducting properties in Mg diffusion

All samples were fabricated via Mg diffusion and evaporation processes, as shown in Fig, 1.

Fig. 1. Schematic diagram showing a fabrication method of MgB2 via Mg diffusion and

Mg ingots (purity 99.99 %, size 2-5 mm) and amorphous B powder (purity 99.995 %, size 0.25 m) were used as starting materials and the molar ratio was chosen to be 1 : 1. The reason for using the large amount of Mg ingots is to eliminate Mg loss caused by MgO formation and to more homogeneously diffuse Mg into B matrix in comparison with what occurs in a Mg diffusion process with the stoichiometric molar ratio. Amorphous B powder was pressed into slabs with dimensions of 4.5 x 21.5 x 0.5 mm under a pressure of 100 MPa. The compacted B powder and Mg ingots were placed in a Ta tube. Both ends of the Ta tube were sealed by a metal inert gas (MIG) welding method in a glove box filled with high purity Ar gas. During the MIG welding process, the Ta tube was heat-sunk with ice-cooled copper block to avoid heating the slab and Mg ingots. The sealed Ta tube was vacuumsealed in a quartz tube and then heat-treated under four major types of sintering conditions that are (1) an one-step sintering process at 1100 ºC which is higher than the boiling point of

processed MgB2 bulks sintered under various heat-treated conditions.

**3.1 Sample preparation via Mg diffusion and evaporation processes**

density above 80 % of theoretical. Even now, MgB2 conductors prepared by conventional *in situ* or *ex situ* PIT methods do not reach either value. On the other hand, Kang et al. first made high-quality MgB2 thin films via a similar method (Kang et al., 2001). The successful technique was to deposit amorphous B thin films and sinter them in Mg vapor at 950 ºC for 10-30 minutes. The self-field *J*c at 5 K for the obtained film showed a high value of ~ 6.0 × 106 A/cm2 and a low residual resistivity of ~ 2.0 cm, which was likely due to its highly dense structure. Thus, the Mg diffusion method was found to yield higher mass density of MgB2 matrix in materials compared with conventional *in situ* PIT method, and it has been modified by many researchers for further enhancement of *J*c and real applications of various materials such as wires and bulks since it was first discovered (Giunchi, 2003; Ueda et al., 2005; Togano et al., 2009).

For the synthesis of dense MgB2 bulks, the reactive liquid Mg infiltration (RLI) technique was proposed by Giunchi (Giunchi, 2003). The method has the advantage that desired forms can be fabricated on a large scale and in an inexpensive way without hot pressing methods. The following is a brief procedure of the fabrication method. First, compacted B powders in contact with an appropriate amount of Mg bulk pieces are inserted into steel containers with desirable shapes. The containers are arc-welded in Ar atmosphere using a high-temperature resistant technique and then sintered at 750-950 ºC. After sintering, the MgB2 bulks are obtained by removal of the containers. The mass density of the obtained samples is ~ 2.40 g/cm3, and the transport *J*c values at 4.2 K and 10 T reach 3.0 × 103 A/cm2. Moreover, the RLI technique in combination with *in situ* PIT methods can be applied to the fabrication of monofilament and multifilament MgB2 hollow wires (Giunchi et al., 2003). It is found that hollow cores are formed in the sintered wire due to Mg diffusion into the B matrix. The formed MgB2 core was observed to have highly dense structure from SEM observation.

In order to address the problem of the low *J*c at self field in MgB2 bulks, the powder-inclosed-tube (PICT)-diffusion method was suggested by Ueda et al. (Ueda et al., 2005). The bulks were prepared by the following procedure. The starting Mg and B powders in appropriate amounts were separately packed into a stainless steel tube. The tube was deformed into a tape-shaped sample and both ends were sealed by bending and pressing. The samples were vacuum-sealed in a silicon dioxide (SiO2) tube and then sintered at 800- 900 ºC. The obtained sample sintered at 800 ºC for 60 hours exhibits well-connected small grains with sizes of 30-100 nm. The mass density is above 95 % of the theoretical density, and the self-field *J*c value at 20 K reaches ~ 8.6 × 105 A/cm2 due to its good connectivity.

To achieve much less MgO secondary phase, as well as highly dense structure in MgB2 conductors, an interface diffusion process was proposed by Togano et al. (Togano et al., 2006). The conductors were fabricated from an interface diffusive reaction between an Fe-Mg alloy substrate and B at 950 ºC for 12 hours. The formed MgB2 layer was observed to be composed of densely packed equi-axial grains of submicron size and without oxygen contamination. This group also suggested an internal Mg diffusion (IMD) process for further enhancement of *J*c in MgB2 wires (Hur et al., 2008). Specifically, 19-filament dense MgB2 wires are fabricated by the internal Mg diffusion (IMD) process and the transport *J*c values at 4.2 K and 10 T reach ~ 1.0 × 105 A/cm2 (Togano et al., 2009). This performance is the best *J*c reported so far in MgB2 wires. However, despite these attractive performances of Mg diffusion processed materials, very little work has been done on the effects of sintering temperature on the phase composition, lattice parameters, mass density, grain size, *T*c, *B*c2, and *J*c in MgB2 prepared from Mg diffusion process. In this study, therefore, we have systematically evaluated structural and superconducting properties in Mg diffusion processed MgB2 bulks sintered under various heat-treated conditions.

## **3. Experimental**

292 Superconductors – Properties, Technology, and Applications

density above 80 % of theoretical. Even now, MgB2 conductors prepared by conventional *in situ* or *ex situ* PIT methods do not reach either value. On the other hand, Kang et al. first made high-quality MgB2 thin films via a similar method (Kang et al., 2001). The successful technique was to deposit amorphous B thin films and sinter them in Mg vapor at 950 ºC for 10-30 minutes. The self-field *J*c at 5 K for the obtained film showed a high value of ~ 6.0 × 106 A/cm2 and a low residual resistivity of ~ 2.0 cm, which was likely due to its highly dense structure. Thus, the Mg diffusion method was found to yield higher mass density of MgB2 matrix in materials compared with conventional *in situ* PIT method, and it has been modified by many researchers for further enhancement of *J*c and real applications of various materials such as wires and bulks since it was first discovered (Giunchi, 2003; Ueda et al.,

For the synthesis of dense MgB2 bulks, the reactive liquid Mg infiltration (RLI) technique was proposed by Giunchi (Giunchi, 2003). The method has the advantage that desired forms can be fabricated on a large scale and in an inexpensive way without hot pressing methods. The following is a brief procedure of the fabrication method. First, compacted B powders in contact with an appropriate amount of Mg bulk pieces are inserted into steel containers with desirable shapes. The containers are arc-welded in Ar atmosphere using a high-temperature resistant technique and then sintered at 750-950 ºC. After sintering, the MgB2 bulks are obtained by removal of the containers. The mass density of the obtained samples is ~ 2.40 g/cm3, and the transport *J*c values at 4.2 K and 10 T reach 3.0 × 103 A/cm2. Moreover, the RLI technique in combination with *in situ* PIT methods can be applied to the fabrication of monofilament and multifilament MgB2 hollow wires (Giunchi et al., 2003). It is found that hollow cores are formed in the sintered wire due to Mg diffusion into the B matrix. The formed MgB2 core was observed to have highly dense structure from SEM observation.

In order to address the problem of the low *J*c at self field in MgB2 bulks, the powder-inclosed-tube (PICT)-diffusion method was suggested by Ueda et al. (Ueda et al., 2005). The bulks were prepared by the following procedure. The starting Mg and B powders in appropriate amounts were separately packed into a stainless steel tube. The tube was deformed into a tape-shaped sample and both ends were sealed by bending and pressing. The samples were vacuum-sealed in a silicon dioxide (SiO2) tube and then sintered at 800- 900 ºC. The obtained sample sintered at 800 ºC for 60 hours exhibits well-connected small grains with sizes of 30-100 nm. The mass density is above 95 % of the theoretical density, and the self-field *J*c value at 20 K reaches ~ 8.6 × 105 A/cm2 due to its good connectivity.

To achieve much less MgO secondary phase, as well as highly dense structure in MgB2 conductors, an interface diffusion process was proposed by Togano et al. (Togano et al., 2006). The conductors were fabricated from an interface diffusive reaction between an Fe-Mg alloy substrate and B at 950 ºC for 12 hours. The formed MgB2 layer was observed to be composed of densely packed equi-axial grains of submicron size and without oxygen contamination. This group also suggested an internal Mg diffusion (IMD) process for further enhancement of *J*c in MgB2 wires (Hur et al., 2008). Specifically, 19-filament dense MgB2 wires are fabricated by the internal Mg diffusion (IMD) process and the transport *J*c values at 4.2 K and 10 T reach ~ 1.0 × 105 A/cm2 (Togano et al., 2009). This performance is the best *J*c reported so far in MgB2 wires. However, despite these attractive performances of Mg diffusion processed materials, very little work has been done on the effects of sintering temperature on the phase composition, lattice parameters, mass density, grain size, *T*c, *B*c2,

2005; Togano et al., 2009).

## **3.1 Sample preparation via Mg diffusion and evaporation processes**

All samples were fabricated via Mg diffusion and evaporation processes, as shown in Fig, 1.

Fig. 1. Schematic diagram showing a fabrication method of MgB2 via Mg diffusion and evaporation processes.

Mg ingots (purity 99.99 %, size 2-5 mm) and amorphous B powder (purity 99.995 %, size 0.25 m) were used as starting materials and the molar ratio was chosen to be 1 : 1. The reason for using the large amount of Mg ingots is to eliminate Mg loss caused by MgO formation and to more homogeneously diffuse Mg into B matrix in comparison with what occurs in a Mg diffusion process with the stoichiometric molar ratio. Amorphous B powder was pressed into slabs with dimensions of 4.5 x 21.5 x 0.5 mm under a pressure of 100 MPa. The compacted B powder and Mg ingots were placed in a Ta tube. Both ends of the Ta tube were sealed by a metal inert gas (MIG) welding method in a glove box filled with high purity Ar gas. During the MIG welding process, the Ta tube was heat-sunk with ice-cooled copper block to avoid heating the slab and Mg ingots. The sealed Ta tube was vacuumsealed in a quartz tube and then heat-treated under four major types of sintering conditions that are (1) an one-step sintering process at 1100 ºC which is higher than the boiling point of

Structural Characteristic and Superconducting

Performance of MgB2 Fabricated by Mg Diffusion Process 295

Fig. 2. Microstructure of Mg diffusion processed samples via various sintering conditions including (a) 1100 ºC 0.5 h, (b) 1100 ºC 1 h, (c) 1100 ºC 3 h, (d) 660 ºC 24 h, (e) 660 ºC 48 h, (f) 660 ºC 96 h, (g) 1100 ºC 0.1h - 660 ºC 3 h, (h) 1100 ºC 0.1 h - 660 ºC 6 h, (i) 1100 ºC 0.1h - 660 ºC 12 h, (k) 660 ºC 12 h - 1100 ºC 0.25 h, and (l) 660 ºC 12 h - 1100 ºC 0.5 h. (j) Microstructure

of compacted B powder before sintering.

Mg, (2) an one-step sintering process at 660 ºC which is higher than the melting point of Mg, (3) a two-step process sintered at 1100 ºC for a short period of 6 minutes followed by second sintering at 660 ºC, and (4) a two-step process sintered at 660 ºC for 12 hours followed by second sintering at 1100 ºC. After each sintering process for MgB2 formation, a large amount of unreacted Mg remains in the sealed Ta tubes. In order to remove the residual Mg from MgB2 sample, the Ta tube was pierced through and again vacuum-sealed in a quartz tube to undergo an Mg evaporation process at 680 °C for 0.5 hour.

## **3.2 Sample characterization**

The phase composition was investigated from X-ray powder diffraction (XRD) patterns which were collected with a Rigaku RINT 2000 diffractmeter using CuK radiation. Rietveld refinement of the crystal structure was carried out with the RIETAN-2000 program (Izumi & Ikeda, 2000) and the structural parameters were refined by fitting the observed XRD patterns. The morphology and grain size were examined using a HITACHI S–4500 Scanning Electron Microscopy (SEM). The real and the imaginary component curves of ac susceptibility and the dc magnetization loops were evaluated by using a SQUID magnetometer (MPMS-5T, Quantum Design). The *T*c and *B*c2 were determined by ac susceptibility measurements at *f* = 76.97 Hz with 0*H*ac = 10 T and 0*H*dc = 0, 0.2, 0.5, 1, 2, 3, 4, and 5 T. The *J*c values were derived from the magnetization hysteresis loops using the Bean critical state model (Chen & Goldfarb, 1989).

## **4. Results and discussion**

## **4.1 Morphology and density**

Figure 2 shows the SEM images of the surface of Mg diffusion processed samples via various sintering conditions. For reference, compacted amorphous B powder was also observed, as can be seen in Fig. 2(j). It was found that the one-step sintering process at 1100 ºC yields welldeveloped crystalline grains, as shown in Fig. 2(a), (b), and (c). The crystal grains were also found to grow rapidly with an increase in sintering time. Specifically, the average grain sizes of samples sintered at 1100 ºC for 0.5, 1, and 3 hours were observed to be approximately 0.8, 1.5, and 3.0 m, respectively. The mass densities of the corresponding samples are 2.2, 2.4, and 1.9 g/cm3, respectively, and close to the theoretical density of MgB2. However, the drastic decrease in mass density from 2.4 g/cm3 to 1.9 g/cm3 is most likely due to the observed rapid growth caused by the increase in sintering time. It is well known that grain boundaries can act as pinning sites in MgB2 materials (Larbalestier et al., 2001; Eisterer, 2007). The considerably large grains offer the disadvantage of lowering in-field *J*c, even though high mass density is easily achieved by the one-step process at 1100 ºC for the short sintering time. Contrary to the grain growth, the one-step sintering process at 660 ºC for long sintering time ranging from 24 hours to 96 hours was found to leave grain size more or less the same, as can be seen in Fig. 2(d), (e), and (f). It is important to note that the average grain size is much smaller than that of Mg diffusion processed samples at 1100 ºC and the grain shape shows poorly crystallized form. It is also well known that crystallinity associated with structural disorder in the lattice as well as grain boundaries can enhance the flux pinning effect (Yamamoto et al., 2005b; Eisterer, 2007). The low-temperature sintering process is advantageous in terms of both poor crystallinity and small grains.

Mg, (2) an one-step sintering process at 660 ºC which is higher than the melting point of Mg, (3) a two-step process sintered at 1100 ºC for a short period of 6 minutes followed by second sintering at 660 ºC, and (4) a two-step process sintered at 660 ºC for 12 hours followed by second sintering at 1100 ºC. After each sintering process for MgB2 formation, a large amount of unreacted Mg remains in the sealed Ta tubes. In order to remove the residual Mg from MgB2 sample, the Ta tube was pierced through and again vacuum-sealed in a quartz tube to

The phase composition was investigated from X-ray powder diffraction (XRD) patterns which were collected with a Rigaku RINT 2000 diffractmeter using CuK radiation. Rietveld refinement of the crystal structure was carried out with the RIETAN-2000 program (Izumi & Ikeda, 2000) and the structural parameters were refined by fitting the observed XRD patterns. The morphology and grain size were examined using a HITACHI S–4500 Scanning Electron Microscopy (SEM). The real and the imaginary component curves of ac susceptibility and the dc magnetization loops were evaluated by using a SQUID magnetometer (MPMS-5T, Quantum Design). The *T*c and *B*c2 were determined by ac susceptibility measurements at *f* = 76.97 Hz with 0*H*ac = 10 T and 0*H*dc = 0, 0.2, 0.5, 1, 2, 3, 4, and 5 T. The *J*c values were derived from the magnetization hysteresis loops using the

Figure 2 shows the SEM images of the surface of Mg diffusion processed samples via various sintering conditions. For reference, compacted amorphous B powder was also observed, as can be seen in Fig. 2(j). It was found that the one-step sintering process at 1100 ºC yields welldeveloped crystalline grains, as shown in Fig. 2(a), (b), and (c). The crystal grains were also found to grow rapidly with an increase in sintering time. Specifically, the average grain sizes of samples sintered at 1100 ºC for 0.5, 1, and 3 hours were observed to be approximately 0.8, 1.5, and 3.0 m, respectively. The mass densities of the corresponding samples are 2.2, 2.4, and 1.9 g/cm3, respectively, and close to the theoretical density of MgB2. However, the drastic decrease in mass density from 2.4 g/cm3 to 1.9 g/cm3 is most likely due to the observed rapid growth caused by the increase in sintering time. It is well known that grain boundaries can act as pinning sites in MgB2 materials (Larbalestier et al., 2001; Eisterer, 2007). The considerably large grains offer the disadvantage of lowering in-field *J*c, even though high mass density is easily achieved by the one-step process at 1100 ºC for the short sintering time. Contrary to the grain growth, the one-step sintering process at 660 ºC for long sintering time ranging from 24 hours to 96 hours was found to leave grain size more or less the same, as can be seen in Fig. 2(d), (e), and (f). It is important to note that the average grain size is much smaller than that of Mg diffusion processed samples at 1100 ºC and the grain shape shows poorly crystallized form. It is also well known that crystallinity associated with structural disorder in the lattice as well as grain boundaries can enhance the flux pinning effect (Yamamoto et al., 2005b; Eisterer, 2007). The low-temperature sintering process is advantageous in terms of both poor

undergo an Mg evaporation process at 680 °C for 0.5 hour.

Bean critical state model (Chen & Goldfarb, 1989).

**3.2 Sample characterization**

**4. Results and discussion 4.1 Morphology and density**

crystallinity and small grains.

Fig. 2. Microstructure of Mg diffusion processed samples via various sintering conditions including (a) 1100 ºC 0.5 h, (b) 1100 ºC 1 h, (c) 1100 ºC 3 h, (d) 660 ºC 24 h, (e) 660 ºC 48 h, (f) 660 ºC 96 h, (g) 1100 ºC 0.1h - 660 ºC 3 h, (h) 1100 ºC 0.1 h - 660 ºC 6 h, (i) 1100 ºC 0.1h - 660 ºC 12 h, (k) 660 ºC 12 h - 1100 ºC 0.25 h, and (l) 660 ºC 12 h - 1100 ºC 0.5 h. (j) Microstructure of compacted B powder before sintering.

Structural Characteristic and Superconducting

diffusion process could not avoid MgO formation.

Radiation source CuK

Temperature (K) 298

and the calculated one was evaluated by goodness of fit *s*.

(º) 0.01

Sintering

2

**4.3 Influence on** *T***c**

Performance of MgB2 Fabricated by Mg Diffusion Process 297

level of disorder and small grains in the high-density MgB2 samples, which are essential for enhancing the flux pinning effect. The MgO nanoparticles are also well known to act as pinning sites in MgB2 thin films (Eom et al., 2001; Singh et al., 2008), whereas it is difficult to control the size in polycrystalline MgB2 samples and large MgO grains could impede supercurrent flow (Maeda et al., 2011a). In this respect, it is necessary to eliminate the MgO impurity from polycrystalline MgB2 materials. However, the MgO secondary phase of the four MgB2 samples is not sensitive to sintering conditions. This indicates that even Mg

condition 1100ºC1h 660ºC48h 1100ºC0.1h - 660ºC6h 660ºC12h - 1100ºC0.5h X-ray powder diffraction data

Goodness of fit *s* 1.32 1.27 1.23 1.23 Lattice parameters *a* (Å) 3.0864 (2) 3.0850 (2) 3.0859 (2) 3.0868 (2) *c* (Å) 3.5210 (1) 3.5210 (1) 3.5210 (1) 3.5211 (2) Unit cell volume *V* (Å3) 29.047 (2) 29.020 (3) 29.038 (3) 29.054 (3) Reflection of MgB2 (101) FWHM(º) 0.15 0.31 0.29 0.10 Weight fraction MgB2 (wt%) 95.9 95.8 96.3 96.1 MgO (wt%) 4.1 4.2 3.7 3.9

Table 1. Results of Rietveld refinement on XRD data for Mg diffusion processed samples via four major sintering conditions that are 1100 ºC for 1 hour, 660 ºC for 48 hours, 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 6 hours, and 660 ºC for 12 hours followed by second sintering at 1100 ºC for 0.5 hour. The fitting quality between the observed pattern

Figure 3 shows the real and imaginary components of the ac susceptibility for four MgB2 samples sintered via four major conditions. It was found that the onset of diamagnetism at the dc field of 0 T for the MgB2 sample sintered at 660 ºC for 48 hours is ~ 38 K and 1 K lower than those for the other samples. The decrease in *T*c is most likely due to lattice disorder caused by the reduction of unit cell volume, as indicated by the refinement result listed in Table 1. It is noteworthy that although the other three samples show ~ 39 K of the onset diamagnetism at the dc field of 0 T, the superconducting transition curves are different. Specifically, the transition width of the MgB2 sample first sintered at 1100 ºC followed by second sintering at 660 ºC is ~ 1.5 K and approximately two times broader than that of the MgB2 sample sintered by the one-step process at either 1100 ºC or 660 ºC. It was

(Å) 1.54060 (CuKα1), 1.54443 (CuKα2)

However, the Mg diffusion processed samples at 660 ºC were brittle and the mass densities are ~ 1.5 g/cm3. The value is almost half of the theoretical density, even though the SEM images of the corresponding samples seem to be dense. The porous structure caused by the low density may intensively limit grain connectivity and hence supercurrent flow. On the other hand, the two-step process first sintered at 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 3 -12 hours was found to form well-developed crystalline grains, as can be seen in Fig. 2(g), (h), and (i). The average grain size is below 1 m and smaller than that of Mg diffusion processed samples via the one-step sintering at 1100 ºC. However, unlike Mg diffusion processed samples via the one-step sintering at 660 ºC which have small grains, the mass densities are ~ 2.4 -2.5 g/cm3 and those are above 90 % of the theoretical density. The two-step sintering process achieves both high density and small grains, which arise from the natures of high-temperature sintering and low-temperature sintering, respectively. In contrast to the highly dense structure related to the well-connected small grains, the two-step process first sintered at 660 ºC for 12 hours followed by second sintering at 1100 ºC for 0.25 - 0.5 hour was found to result in loosely connected grains, as can be seen in Fig. 2(k) and (l). The mass densities are ~ 1.7 g/cm3 and this result also confirm its porous structure. The corresponding samples were observed to have well-developed crystalline and relatively large grains. The two-step sintering process yields low-density structure and large crystal grains due to low-temperature sintering and high-temperature sintering, respectively.

#### **4.2 Phase composition and lattice structure**

All Mg diffusion processed samples were observed to have the same phase composition, which consists of MgB2 with a trace amount of MgO from XRD -2 scans. Hereafter the samples will be referred as MgB2 samples. In order to examine deeper insight of microstructure, structural parameters of four MgB2 samples sintered via four major conditions were refined and its result is listed in Table 1. It was found that the *a*-lattice parameter is slightly changed owing to various sintering effects, while leaving the *c*-lattice parameter the same. The *a*-lattice parameter of the MgB2 sample sintered at 660 ºC for 48 hours shows 3. 0850 Å, and it is the lowest value of the four samples. The decrease in the *a*lattice parameter reduces the unit cell volume of MgB2. The reduction may introduce structural disorder in the lattice and decrease the DOS at the *E*F responsible for a considerable change in *T*c and thus *B*c2. The XRD full width at half maximum (FWHM) is well known to be closely related to crystallite size and structural disorder and it is also listed in Table 1. The one-step sintering process at 1100 ºC and the two-step process first sintered at 660 ºC followed by second sintering at 1100 ºC were found to obtain relatively narrow FWHM reflecting well-crystallized and large grain. This is consistent with the SEM result, as can be seen in Fig. 2. On the other hand, the FWHM values of the rest of MgB2 samples were observed to be much broader than those of the other MgB2 samples. The broad FWHM of the MgB2 sample sintered at the one-step process at 660 ºC is caused by poorly crystallized and small grains, as also indicated by the SEM observation. However, it should be noted that the FWHM of the MgB2 sample first sintered at 1100 ºC followed by second sintering at 660 ºC is comparable to that of the sample sintered at 660 ºC for 48 hours, although the corresponding sample seems to have well crystallized grains which are distinctly lager than that of MgB2 samples sintered at the one-step process at 660 ºC. This indicates that thanks to the second sintering at the low temperature, the two-step process could achieve both a high

However, the Mg diffusion processed samples at 660 ºC were brittle and the mass densities are ~ 1.5 g/cm3. The value is almost half of the theoretical density, even though the SEM images of the corresponding samples seem to be dense. The porous structure caused by the low density may intensively limit grain connectivity and hence supercurrent flow. On the other hand, the two-step process first sintered at 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 3 -12 hours was found to form well-developed crystalline grains, as can be seen in Fig. 2(g), (h), and (i). The average grain size is below 1 m and smaller than that of Mg diffusion processed samples via the one-step sintering at 1100 ºC. However, unlike Mg diffusion processed samples via the one-step sintering at 660 ºC which have small grains, the mass densities are ~ 2.4 -2.5 g/cm3 and those are above 90 % of the theoretical density. The two-step sintering process achieves both high density and small grains, which arise from the natures of high-temperature sintering and low-temperature sintering, respectively. In contrast to the highly dense structure related to the well-connected small grains, the two-step process first sintered at 660 ºC for 12 hours followed by second sintering at 1100 ºC for 0.25 - 0.5 hour was found to result in loosely connected grains, as can be seen in Fig. 2(k) and (l). The mass densities are ~ 1.7 g/cm3 and this result also confirm its porous structure. The corresponding samples were observed to have well-developed crystalline and relatively large grains. The two-step sintering process yields low-density structure and large crystal grains due to low-temperature sintering and high-temperature sintering,

All Mg diffusion processed samples were observed to have the same phase composition,

samples will be referred as MgB2 samples. In order to examine deeper insight of microstructure, structural parameters of four MgB2 samples sintered via four major conditions were refined and its result is listed in Table 1. It was found that the *a*-lattice parameter is slightly changed owing to various sintering effects, while leaving the *c*-lattice parameter the same. The *a*-lattice parameter of the MgB2 sample sintered at 660 ºC for 48 hours shows 3. 0850 Å, and it is the lowest value of the four samples. The decrease in the *a*lattice parameter reduces the unit cell volume of MgB2. The reduction may introduce structural disorder in the lattice and decrease the DOS at the *E*F responsible for a considerable change in *T*c and thus *B*c2. The XRD full width at half maximum (FWHM) is well known to be closely related to crystallite size and structural disorder and it is also listed in Table 1. The one-step sintering process at 1100 ºC and the two-step process first sintered at 660 ºC followed by second sintering at 1100 ºC were found to obtain relatively narrow FWHM reflecting well-crystallized and large grain. This is consistent with the SEM result, as can be seen in Fig. 2. On the other hand, the FWHM values of the rest of MgB2 samples were observed to be much broader than those of the other MgB2 samples. The broad FWHM of the MgB2 sample sintered at the one-step process at 660 ºC is caused by poorly crystallized and small grains, as also indicated by the SEM observation. However, it should be noted that the FWHM of the MgB2 sample first sintered at 1100 ºC followed by second sintering at 660 ºC is comparable to that of the sample sintered at 660 ºC for 48 hours, although the corresponding sample seems to have well crystallized grains which are distinctly lager than that of MgB2 samples sintered at the one-step process at 660 ºC. This indicates that thanks to the second sintering at the low temperature, the two-step process could achieve both a high

scans. Hereafter the

respectively.

**4.2 Phase composition and lattice structure**

which consists of MgB2 with a trace amount of MgO from XRD

level of disorder and small grains in the high-density MgB2 samples, which are essential for enhancing the flux pinning effect. The MgO nanoparticles are also well known to act as pinning sites in MgB2 thin films (Eom et al., 2001; Singh et al., 2008), whereas it is difficult to control the size in polycrystalline MgB2 samples and large MgO grains could impede supercurrent flow (Maeda et al., 2011a). In this respect, it is necessary to eliminate the MgO impurity from polycrystalline MgB2 materials. However, the MgO secondary phase of the four MgB2 samples is not sensitive to sintering conditions. This indicates that even Mg diffusion process could not avoid MgO formation.

Table 1. Results of Rietveld refinement on XRD data for Mg diffusion processed samples via four major sintering conditions that are 1100 ºC for 1 hour, 660 ºC for 48 hours, 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 6 hours, and 660 ºC for 12 hours followed by second sintering at 1100 ºC for 0.5 hour. The fitting quality between the observed pattern and the calculated one was evaluated by goodness of fit *s*.

## **4.3 Influence on** *T***c**

Figure 3 shows the real and imaginary components of the ac susceptibility for four MgB2 samples sintered via four major conditions. It was found that the onset of diamagnetism at the dc field of 0 T for the MgB2 sample sintered at 660 ºC for 48 hours is ~ 38 K and 1 K lower than those for the other samples. The decrease in *T*c is most likely due to lattice disorder caused by the reduction of unit cell volume, as indicated by the refinement result listed in Table 1. It is noteworthy that although the other three samples show ~ 39 K of the onset diamagnetism at the dc field of 0 T, the superconducting transition curves are different. Specifically, the transition width of the MgB2 sample first sintered at 1100 ºC followed by second sintering at 660 ºC is ~ 1.5 K and approximately two times broader than that of the MgB2 sample sintered by the one-step process at either 1100 ºC or 660 ºC. It was

Structural Characteristic and Superconducting

**4.4 Influence on** *B***c2**

conditions.

conditions.

Performance of MgB2 Fabricated by Mg Diffusion Process 299

Figure 4 shows the temperature dependence of *B*c2 for MgB2 samples sintered via various

Fig. 4. The temperature dependence of *B*c2 derived from ac susceptibility with *f* = 76.97 Hz, 0*H*ac = 10 T, and 0*H*dc = 0, 0.2, 0.5, 1, 2, 3, 4, and 5 T for MgB2 samples sintered via various

It was found that although all MgB2 samples has positive curvatures of *B*c2 at *T* ~ *T*c reflecting suppression of the diffusivity in the *σ* bands compared to that in the *π* bands (Gurevich, 2003), the behaviours of curves at lower temperature are considerably different. The *B*c2 slopes of the MgB2 samples first sintered at 1100 ºC followed by second sintering at 660 ºC or sintered only at 660 ºC are steeper in comparison with those of the MgB2 samples sintered at 1100 ºC. The structural disorder of both samples may be responsible for this. By contrast, the *B*c2 slopes of the MgB2 samples sintered at 660 ºC followed by second sintering

reported that the broad transition width arises from the nature of two kinds of superconducting MgB2 phases, which are good ordering phase and a relatively lower level of ordering phase owing to the two-step sintering process (Maeda et al., 2008). The lower level of ordering phase could be a cause for the broad FWHM responsible for structural disorder, as observed in the XRD pattern (Table 1). Contrary to this, the transition width of the MgB2 sample first sintered at 660 ºC followed by second sintering at 1100 ºC is ~ 0.3 K, suggesting that the sample consists of nearly perfect crystallite MgB2 grains without any lattice disorder.

Fig. 3. The real and imaginary components of ac susceptibility with *f* = 76.97 Hz, 0*H*ac = 10 T, and 0*H*dc = 0 T for Mg diffusion processed samples via four major sintering conditions that are 1100 ºC for 1 hour, 660 ºC for 48 hours, 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 6 hours, and 660 ºC for 12 hours followed by second sintering at 1100 ºC for 0.5 hour.

## **4.4 Influence on** *B***c2**

298 Superconductors – Properties, Technology, and Applications

reported that the broad transition width arises from the nature of two kinds of superconducting MgB2 phases, which are good ordering phase and a relatively lower level of ordering phase owing to the two-step sintering process (Maeda et al., 2008). The lower level of ordering phase could be a cause for the broad FWHM responsible for structural disorder, as observed in the XRD pattern (Table 1). Contrary to this, the transition width of the MgB2 sample first sintered at 660 ºC followed by second sintering at 1100 ºC is ~ 0.3 K, suggesting that the sample consists of nearly perfect crystallite MgB2 grains without any

Fig. 3. The real and imaginary components of ac susceptibility with *f* = 76.97 Hz, 0*H*ac = 10 T, and 0*H*dc = 0 T for Mg diffusion processed samples via four major sintering conditions that are 1100 ºC for 1 hour, 660 ºC for 48 hours, 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 6 hours, and 660 ºC for 12 hours followed by second sintering at

lattice disorder.

1100 ºC for 0.5 hour.

Figure 4 shows the temperature dependence of *B*c2 for MgB2 samples sintered via various conditions.

Fig. 4. The temperature dependence of *B*c2 derived from ac susceptibility with *f* = 76.97 Hz, 0*H*ac = 10 T, and 0*H*dc = 0, 0.2, 0.5, 1, 2, 3, 4, and 5 T for MgB2 samples sintered via various conditions.

It was found that although all MgB2 samples has positive curvatures of *B*c2 at *T* ~ *T*c reflecting suppression of the diffusivity in the *σ* bands compared to that in the *π* bands (Gurevich, 2003), the behaviours of curves at lower temperature are considerably different. The *B*c2 slopes of the MgB2 samples first sintered at 1100 ºC followed by second sintering at 660 ºC or sintered only at 660 ºC are steeper in comparison with those of the MgB2 samples sintered at 1100 ºC. The structural disorder of both samples may be responsible for this. By contrast, the *B*c2 slopes of the MgB2 samples sintered at 660 ºC followed by second sintering

Structural Characteristic and Superconducting

samples.

conditions.

**6. References**

**5. Conclusion**

Performance of MgB2 Fabricated by Mg Diffusion Process 301

second sintering at 660 ºC is significantly higher than those of the other MgB2 samples. The enhancement of *J*c at not only low field but also high field is clearly due to the twostep sintering process which yields well-connected small grains with a high level of disorder in highly dense MgB2 samples. It is also interesting to note that even though the *J*c of the MgB2 sample sintered at 1100 ºC for 1 hour is nearly the same as that of the MgB2 sample sintered at 660 ºC for 48 hours, the behaviours of the curves are somewhat different. Specifically, the low-field *J*c of the MgB2 sample sintered at 1100 ºC for 1 hour is slightly higher than that of the MgB2 sample sintered at 660 ºC for 48 hours, whereas the corresponding behaviours of high-field *J*c are distinctly opposite. This results from high density and lack of pinning sites caused by good crystallinity, which are responsible for the increase in self-field *J*c and the decrease in high-field *J*c, respectively. Owing to the low density and nearly perfect crystallization, the MgB2 samples sintered at 660 ºC followed by second sintering at 1100 ºC show the lowest values of in-field *J*c in all MgB2

In summary, this chapter has introduced structural characteristic and superconducting performance of pure MgB2 polycrystalline samples fabricated by Mg diffusion processes. The Mg diffusion processes easily yield MgB2 phase and a trace amount of MgO phase in polycrystalline samples. The microstructure and mass density of the MgB2 samples can be controlled by Mg diffusion processes via various sintering conditions. The two-step sintering process, namely, the sintering for a short period of 0.1 hour at 1100 ºC first, followed by the second sintering at 660 ºC, produces highly dense MgB2 samples characterized by well-connected small grains with a high level of disorder. The unique feature results in considerable improvement of *B*c2 and in-field *J*c. Therefore, Mg diffusion process is critical for controlling the structure of MgB2 polycrystalline samples, and there is great potential for further enhancement of *J*c by careful optimisation of sintering

An, J. M. & Pickett, W. E. (2001). Superconductivity of MgB2: Covalent bonds driven

Avdeev, M.; Jorgensen, J. D.; Ribeiro, R. A.; Bud'ko, S. L. & Canfield, P. C. (2003). Crystal

Bengtson, A. K.; Bark, C. W.; Giencke, J.; Dai, W. Q.; Xi, X. X.; Eom, C. B. & Morgan, D.

Braccini, V.; Gurevich, A.; Giencke, J. E.; Jewell, M. C.; Eom, C. B.; Larbalestier, D. C.;

chemistry of carbon-substituted MgB2. *Physica C - Superconductivity and Its*

(2010). Impact of substitutional and interstitial carbon defects on lattice parameters

Pogrebnyakov, A.; Cui, Y.; Liu, B. T.; Hu, Y. F.; Redwing, J. M.; Li, Q.; Xi, X. X.; Singh, R. K.; Gandikota, R.; Kim, J.; Wilkens, B.; Newman, N.; Rowell, J.; Moeckly, B.; Ferrando, V.; Tarantini, C.; Marre, D.; Putti, M.; Ferdeghini, C.; Vaglio, R. & Haanappel, E. (2005). High-field superconductivity in alloyed MgB2 thin films.

metallic. *Physical Review Letters*, 86(19): 4366-4369, ISSN 0031-9007.

in MgB2. *Journal of Applied Physics*, 107(2): 023902, ISSN 0021-8979.

*Applications*, 387(3-4): 301-306, ISSN 0921-4534.

*Physical Review B*, 71(1): 012504, ISSN 1098-0121.

at 1100 ºC are intensively shelving compared to those of the other MgB2 samples. This indicates that the MgB2 samples do not have any lattice defects, as suggested by the result of the narrow superconducting transition (Figure 3).

#### **4.5 Influence on** *J***c**

Figure 5 shows the magnetic field dependence of *J*c at 20 K for MgB2 samples sintered via various conditions. It was found that the *J*c curves are highly sensitive to sintering conditions which lead to considerably differences in MgB2 structure, as shown in the results of SEM observation, XRD analysis, and ac susceptibility. The most noticeable feature shown in this figure is that the in-field *J*c of the MgB2 samples sintered at 1100 ºC followed by

Fig. 5. Magnetic field dependence of *J*c at 20 K for MgB2 samples sintered via various conditions.

second sintering at 660 ºC is significantly higher than those of the other MgB2 samples. The enhancement of *J*c at not only low field but also high field is clearly due to the twostep sintering process which yields well-connected small grains with a high level of disorder in highly dense MgB2 samples. It is also interesting to note that even though the *J*c of the MgB2 sample sintered at 1100 ºC for 1 hour is nearly the same as that of the MgB2 sample sintered at 660 ºC for 48 hours, the behaviours of the curves are somewhat different. Specifically, the low-field *J*c of the MgB2 sample sintered at 1100 ºC for 1 hour is slightly higher than that of the MgB2 sample sintered at 660 ºC for 48 hours, whereas the corresponding behaviours of high-field *J*c are distinctly opposite. This results from high density and lack of pinning sites caused by good crystallinity, which are responsible for the increase in self-field *J*c and the decrease in high-field *J*c, respectively. Owing to the low density and nearly perfect crystallization, the MgB2 samples sintered at 660 ºC followed by second sintering at 1100 ºC show the lowest values of in-field *J*c in all MgB2 samples.

## **5. Conclusion**

300 Superconductors – Properties, Technology, and Applications

at 1100 ºC are intensively shelving compared to those of the other MgB2 samples. This indicates that the MgB2 samples do not have any lattice defects, as suggested by the result of

Figure 5 shows the magnetic field dependence of *J*c at 20 K for MgB2 samples sintered via various conditions. It was found that the *J*c curves are highly sensitive to sintering conditions which lead to considerably differences in MgB2 structure, as shown in the results of SEM observation, XRD analysis, and ac susceptibility. The most noticeable feature shown in this figure is that the in-field *J*c of the MgB2 samples sintered at 1100 ºC followed by

Fig. 5. Magnetic field dependence of *J*c at 20 K for MgB2 samples sintered via various

the narrow superconducting transition (Figure 3).

**4.5 Influence on** *J***c**

conditions.

In summary, this chapter has introduced structural characteristic and superconducting performance of pure MgB2 polycrystalline samples fabricated by Mg diffusion processes. The Mg diffusion processes easily yield MgB2 phase and a trace amount of MgO phase in polycrystalline samples. The microstructure and mass density of the MgB2 samples can be controlled by Mg diffusion processes via various sintering conditions. The two-step sintering process, namely, the sintering for a short period of 0.1 hour at 1100 ºC first, followed by the second sintering at 660 ºC, produces highly dense MgB2 samples characterized by well-connected small grains with a high level of disorder. The unique feature results in considerable improvement of *B*c2 and in-field *J*c. Therefore, Mg diffusion process is critical for controlling the structure of MgB2 polycrystalline samples, and there is great potential for further enhancement of *J*c by careful optimisation of sintering conditions.

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**14**

İbrahim Karaca

 *Turkey*

**Measurement of Levitation Forces in**

**High Temperature Superconductors**

Magnetic Levitation (MagLev) has become a popular of new technological applications since the discovery of the new superconductor materials, in 1987. MagLev transportation systems have been ignited the human's imagination about levitated motion. MagLev systems have become very important in the transportation and other applications in the future life in the world is generally based on the superconductor technology. These applications are ranging from in an energy efficient prototype of a cryogen transfer line to in space energy storage

The superconducting behaviors of materials originated from two basic physical properties. One of them is disappearance of the electrical resistance below the transition temperature ( ) *TC* in the superconducting materials. The another of them is special magnetic properties in an applied magnetic field ( ) *H* including perfect diamagnetism at low fields (low temperature superconductors (LTS) and high temperature superconductors (HTSs)) and the penetration of quantized magnetic flux (vortices or flux lines) at higher fields regions (HTSs). The electrons in the superconducting state can be described by a macroscopic wave function which is undisturbed by scattering and, in the absence of applied fields and currents, uniform in space over macroscopic distances. This coherence of the superconducting state is preserved even in the presence of weak currents and magnetic fields (below the upper critical field 2 ( ) *HC* , which is a necessary condition for the practical use of the high temperatures superconductors (HTSs) in strong magnetic fields. The additional condition for new applications in these superconductors materials have been used to avoid any motion of flux lines. This can be achieved through the introduction of pinning centers in the superconductor, interacting, in most cases, with the normal conducting core of the flux lines. A flux line is composed of a normal conducting core regions and a surrounding circulating supercurrents area where the magnetic field and supercurrents fall off within penetration depth. The critical current density ( ) *CJ* increases with the applied magnetic field ( ) *H* increases and penetration depth ( )

decreases. The force-distance (permanent magnets and superconductor) hysteresis loops during the descending and ascending process expanded with critical current density

**1. Introduction**

systems.

increases.

*Department of Physics, Niğde University, Niğde*

Zhu, Y.; Matsumoto, A.; Senkowicz, B. J.; Kumakura, H.; Kitaguchi, H.; Jewell, M. C.; Hellstrom, E. E.; Larbalestier, D. C. & Voyles, P. M. (2007b). Microstructures of SiC nanoparticle-doped MgB2/Fe tapes. *Journal of Applied Physics*, 102(1): 013913, ISSN 0021-8979.

## **Measurement of Levitation Forces in High Temperature Superconductors**

## İbrahim Karaca

*Department of Physics, Niğde University, Niğde Turkey*

## **1. Introduction**

306 Superconductors – Properties, Technology, and Applications

Zhu, Y.; Matsumoto, A.; Senkowicz, B. J.; Kumakura, H.; Kitaguchi, H.; Jewell, M. C.;

0021-8979.

Hellstrom, E. E.; Larbalestier, D. C. & Voyles, P. M. (2007b). Microstructures of SiC nanoparticle-doped MgB2/Fe tapes. *Journal of Applied Physics*, 102(1): 013913, ISSN

> Magnetic Levitation (MagLev) has become a popular of new technological applications since the discovery of the new superconductor materials, in 1987. MagLev transportation systems have been ignited the human's imagination about levitated motion. MagLev systems have become very important in the transportation and other applications in the future life in the world is generally based on the superconductor technology. These applications are ranging from in an energy efficient prototype of a cryogen transfer line to in space energy storage systems.

> The superconducting behaviors of materials originated from two basic physical properties. One of them is disappearance of the electrical resistance below the transition temperature ( ) *TC* in the superconducting materials. The another of them is special magnetic properties in an applied magnetic field ( ) *H* including perfect diamagnetism at low fields (low temperature superconductors (LTS) and high temperature superconductors (HTSs)) and the penetration of quantized magnetic flux (vortices or flux lines) at higher fields regions (HTSs). The electrons in the superconducting state can be described by a macroscopic wave function which is undisturbed by scattering and, in the absence of applied fields and currents, uniform in space over macroscopic distances. This coherence of the superconducting state is preserved even in the presence of weak currents and magnetic fields (below the upper critical field 2 ( ) *HC* , which is a necessary condition for the practical use of the high temperatures superconductors (HTSs) in strong magnetic fields. The additional condition for new applications in these superconductors materials have been used to avoid any motion of flux lines. This can be achieved through the introduction of pinning centers in the superconductor, interacting, in most cases, with the normal conducting core of the flux lines. A flux line is composed of a normal conducting core regions and a surrounding circulating supercurrents area where the magnetic field and supercurrents fall off within penetration depth. The critical current density ( ) *CJ* increases with the applied magnetic field ( ) *H* increases and penetration depth ( ) decreases. The force-distance (permanent magnets and superconductor) hysteresis loops during the descending and ascending process expanded with critical current density increases.

Measurement of Levitation Forces in High Temperature Superconductors

critical current ( ) *CJ* as constant.

behavior of the HTSs.

materials;

ii. History of applied field; iii. The initial state of the system; iv. The boundary conditions.

penetrated (Johansen, *et al.*, 1994; Riise, *et al.,* 1994).

order to study the influence of orientation, material, or shape of superconducting samples, comparing, sometimes, thin films with bulk samples (Navau & Sanchez, 1998). On the other hand, a few models have been developed, most of them studying the interaction between a superconductor and a permanent magnet. Hellman *et al.* and Yang presented models based on total flux exclusion of the samples that thus describe the behavior of LTSs or HTSs materials with higher critical current densities (Navau & Sanchez, 1998). Schönhuber and Moon, Chan *et al.*, and Torng and Chen have taken into account the penetration of supercurrents, the latter with an applied field provided by a pair of oppositely wound coils (Schonhuber & Moon, 1994; Chan, *et al.,* 1994; Torng & Chen, 1992). Some papers have considered granular structure for the superconductors when grains are completely

All over the models mentioned above assume that a superconducting sample is small enough to consider the magnetic field gradient constant along it. Demagnetization field of the sample due to the finite dimensions of the HTSs has been neglected by the same time. Demagnetization effects have been take into only account in theoretical calculation so far Tsuchimoto *et al.* have calculated the dependence of levitation force upon different geometrical parameters of a PM-HTSs system by using the an axisymmetric boundary element analysis (Tsuchimoto, *et al.,*1994). Portabella *et al.* have been used finite element calculations for the levitation force-distance ( ) *F x* hysteresis loop in order to estimate the value of critical current of a HTSs. Navau and Sanchez calculated the dependence of maximum vertical force upon the length of superconductor, considering both the demagnetization effect and the nonuniformity of the field gradient. In spite of these efforts, a systematic treatment of vertical levitation force for which the applied magnetic field can not be assumed to have a constant gradient has not been developed yet. In the scope of this chapter, we have been observed vertical levitation force as a function of distance between PM and HTSs. Moreover, most of the developments consider the superconductor to be in the critical state and describe it by means of Bean's critical-state model considering the

The most HTSs present a dependence of their critical current density ( ) *CJ* upon the internal magnetic field. Some analytical expressions have been proposed and successfully applied to describe the magnetic properties of superconductors (Navau & Sanchez, 1998). Therefore it is necessary to include such dependence in the study of the levitation

Kim's model dependence on the critical current density was developed by Tsuchimoto *et al.*  with numerical (Navau & Sanchez, 1998). But, physical properties dependence on critical

i. Character of response to applied field, equations of dependence on properties of

current density ( ) *CJ* in levitation behavior has not been studied in general so far.

To obtain critical current density ( ) *CJ* , the consideration includes:

309

The mechanism of superconductivity in high temperature superconductor (HTS) is still controversially discussed. The Bardeen-Cooper-Schrieffer (BCS) concept of a macroscopic quantum state of Cooper pairs remains the common basis in almost all theories proposed for HTS. However, there is no consensus about the origin of the pairing so far. The lack of an isotope effect in HTS of highest *TC* would favor a nonphonon pairing mechanism as, for instance, d-wave pairing, whereas its presence in several HTS with lower*TC* suggests a phonon-mediated s-wave pairing. The most experimental features favor d-wave pairing. A direct test of the phonon mechanism for pairing in HTSs is to check the sign change in the energy gap ( ( )) *E k g* as function of momentum ( ) *k* . A d-state energy gap is composed of two pairs of positive and negative leaves forming a four-leaf clover. In contrast, the magnitude of an s-state may be anisotropic but always remains positive.

## **2. Levitation with superconductors**

Superconducting levitation by using the permanent magnet (PM) is a fascinating property of the magnetic behavior of these materials. It can be mainly defined in two regimes for HTSs. In the first regime, the magnetic induction applied an external magnetic field to samples are expelled from the superconductor and do not penetrate inside of the superconductor materials, which is defined as Meissner effect. This effect is possible only if the applied magnetic field does not exceed the first critical field *(*so called *HC*1 *)* of the HTS. To obtain experimentally levitation in the Meissner state, it is convenient to close the magnet very carefully, after cooling the superconductor. In the second regime of the levitation, i.e., the mixed state, a vortex penetration through for HTSs, occurs only for fields between the first ( *HC*1 ) and the second critical magnetic field ( *HC*2 ) of the HTSs. This is so because magnetic lines could be pinned if they penetrate in the sample and the superconductor would be switched to the mixed state. It is clearly that the levitation measurement is finding in the mixed state acceptable. Magnetic levitation effects are seen Meissner states as the same in HTSs and LTSs. But, magnetic levitation effects originated from flux lines for only HTSs.

When a superconductor is in the presence of a changing applied magnetic field, supercurrents ( *CI* ) are induced inside the sample producing a diamagnetic response of the material. The interaction of the supercurrents with an inhomogeneous magnetic field can produce stable levitation. The flux lines in the small currents densities ( *C J J* for a certain critical current *CJ* ) for the real HTSs can be pinned by inhomogenities. Under this condition, when an external applied field increases or decreases, the magnetic flux moves within the superconductor towards its interior or exterior until a critical slope is reached in the flux profile. The current density in this critical state attains its maximum value *CJ* . The critical slope behavior has been modeled by Bean's critical-state model and its extensions, phenomenological. Brandt showed qualitatively how from the pinning of flux lines some properties of levitation of HTSs can be explained in detail (Brandt, 1998).

Moon *et al.* measured the vertical forces on a levitating superconductor showing some of the main properties (Moon, 1988). This measurement has been extended by several groups in

The mechanism of superconductivity in high temperature superconductor (HTS) is still controversially discussed. The Bardeen-Cooper-Schrieffer (BCS) concept of a macroscopic quantum state of Cooper pairs remains the common basis in almost all theories proposed for HTS. However, there is no consensus about the origin of the pairing so far. The lack of an isotope effect in HTS of highest *TC* would favor a nonphonon pairing mechanism as, for instance, d-wave pairing, whereas its presence in several HTS with lower*TC* suggests a phonon-mediated s-wave pairing. The most experimental features favor d-wave pairing. A direct test of the phonon mechanism for pairing in HTSs is to check the sign change in the energy gap ( ( )) *E k g* as function of momentum ( ) *k* . A d-state energy gap is composed of two pairs of positive and negative leaves forming a four-leaf clover. In contrast, the magnitude of an s-state may be

Superconducting levitation by using the permanent magnet (PM) is a fascinating property of the magnetic behavior of these materials. It can be mainly defined in two regimes for HTSs. In the first regime, the magnetic induction applied an external magnetic field to samples are expelled from the superconductor and do not penetrate inside of the superconductor materials, which is defined as Meissner effect. This effect is possible only if the applied magnetic field does not exceed the first critical field *(*so called *HC*1 *)* of the HTS. To obtain experimentally levitation in the Meissner state, it is convenient to close the magnet very carefully, after cooling the superconductor. In the second regime of the levitation, i.e., the mixed state, a vortex penetration through for HTSs, occurs only for fields between the first ( *HC*1 ) and the second critical magnetic field ( *HC*2 ) of the HTSs. This is so because magnetic lines could be pinned if they penetrate in the sample and the superconductor would be switched to the mixed state. It is clearly that the levitation measurement is finding in the mixed state acceptable. Magnetic levitation effects are seen Meissner states as the same in HTSs and LTSs. But, magnetic levitation effects originated

When a superconductor is in the presence of a changing applied magnetic field, supercurrents ( *CI* ) are induced inside the sample producing a diamagnetic response of the material. The interaction of the supercurrents with an inhomogeneous magnetic field can produce stable levitation. The flux lines in the small currents densities ( *C J J* for a certain critical current *CJ* ) for the real HTSs can be pinned by inhomogenities. Under this condition, when an external applied field increases or decreases, the magnetic flux moves within the superconductor towards its interior or exterior until a critical slope is reached in the flux profile. The current density in this critical state attains its maximum value *CJ* . The critical slope behavior has been modeled by Bean's critical-state model and its extensions, phenomenological. Brandt showed qualitatively how from the pinning of flux lines some

Moon *et al.* measured the vertical forces on a levitating superconductor showing some of the main properties (Moon, 1988). This measurement has been extended by several groups in

properties of levitation of HTSs can be explained in detail (Brandt, 1998).

anisotropic but always remains positive.

**2. Levitation with superconductors**

from flux lines for only HTSs.

309

order to study the influence of orientation, material, or shape of superconducting samples, comparing, sometimes, thin films with bulk samples (Navau & Sanchez, 1998). On the other hand, a few models have been developed, most of them studying the interaction between a superconductor and a permanent magnet. Hellman *et al.* and Yang presented models based on total flux exclusion of the samples that thus describe the behavior of LTSs or HTSs materials with higher critical current densities (Navau & Sanchez, 1998). Schönhuber and Moon, Chan *et al.*, and Torng and Chen have taken into account the penetration of supercurrents, the latter with an applied field provided by a pair of oppositely wound coils (Schonhuber & Moon, 1994; Chan, *et al.,* 1994; Torng & Chen, 1992). Some papers have considered granular structure for the superconductors when grains are completely penetrated (Johansen, *et al.*, 1994; Riise, *et al.,* 1994).

All over the models mentioned above assume that a superconducting sample is small enough to consider the magnetic field gradient constant along it. Demagnetization field of the sample due to the finite dimensions of the HTSs has been neglected by the same time. Demagnetization effects have been take into only account in theoretical calculation so far Tsuchimoto *et al.* have calculated the dependence of levitation force upon different geometrical parameters of a PM-HTSs system by using the an axisymmetric boundary element analysis (Tsuchimoto, *et al.,*1994). Portabella *et al.* have been used finite element calculations for the levitation force-distance ( ) *F x* hysteresis loop in order to estimate the value of critical current of a HTSs. Navau and Sanchez calculated the dependence of maximum vertical force upon the length of superconductor, considering both the demagnetization effect and the nonuniformity of the field gradient. In spite of these efforts, a systematic treatment of vertical levitation force for which the applied magnetic field can not be assumed to have a constant gradient has not been developed yet. In the scope of this chapter, we have been observed vertical levitation force as a function of distance between PM and HTSs. Moreover, most of the developments consider the superconductor to be in the critical state and describe it by means of Bean's critical-state model considering the critical current ( ) *CJ* as constant.

The most HTSs present a dependence of their critical current density ( ) *CJ* upon the internal magnetic field. Some analytical expressions have been proposed and successfully applied to describe the magnetic properties of superconductors (Navau & Sanchez, 1998). Therefore it is necessary to include such dependence in the study of the levitation behavior of the HTSs.

Kim's model dependence on the critical current density was developed by Tsuchimoto *et al.*  with numerical (Navau & Sanchez, 1998). But, physical properties dependence on critical current density ( ) *CJ* in levitation behavior has not been studied in general so far.

To obtain critical current density ( ) *CJ* , the consideration includes:

Measurement of Levitation Forces in High Temperature Superconductors

from Eq. (1-4) as the next form,

using the Eqs. (1) and (2), respectively if the

where

*H*0 /

identical.

1994),

Where

the

*et al*., 1994).

**d. Yin model,**

A general critical-state model was developed for the critical current density. It is obtained

( ) ( ,)

condition *H H i* / 1 0 is satisfied in eq. (5) and the *K* and *n* are set to 0 *J T C* () H

, respectively (Xu, et al., 1990). In the conditions of *H H i* / 1 0 and

*J T J HT*

*C i*

with the preceding conditions and letting *HH x i* / 0 , it has

0 0

>

(Moon, 1990; Sanchez & Navau, 2001; Brandt, 1990; De Medeiros, *et al.* 1998),

i. Biot–Savart's law: Calculating *BPM* with Biot–Savart's law.

0 ,*U*0 , ( ) *W P E J HP V f*

term in the *WV* corresponds to 0

 *HH H C n* 2

coefficient (with 2

*x x*

lim 1 lim 1

 *C C*

0

1 / () *C*

is a dimensionless constant. Eq. (5) can be converted to Bean and Kim models by

*H HT*

*i*

**H**0, where **H**0 is a finite value with dimension of field, and taking the limit of ( ) *C i J H*

( ) ( ) lim ( , ) ( )exp( / )

It can be seen that Eq. (6) has the same form as Eq. (4) and that their physical meanings are

Material's equation, known as Yin model, has been developed in the form as (Yin *et al*.,

0 0 ( ) 2 exp sinh *U WV WL EJ H kT kT*

velocity, the pinning potential, the viscous dissipation term of flux motion with viscosity

of the volume of the moving flux bundle and the range of the force action, respectively. In

Yin model includes pinning barrier term, viscous dissipation term and direction dependent. It provides a unified description which covers all regimes of flux motion: thermally activated flux flow, flux creep, critical state and flux flow. Yin model contain Anderson–Kim model, Bardeen–Steven model and critical state model ( ) *FJH P C* as its specific forms (Yin

There are different ways of calculation of magnetic field ( ) *HPM* of permanent magnets

*J T J T J HT JT H H*

*C i xH H C i*

311

and

1 , with

(5)

0 and 1. Eq. (3) is obtained when the

0 (1/ )( / ) 0

*x x* 0 (6)

(7)

*f* ), the energy due to Lorentz driving force, a product

exp ( ) *U J kT* . *U* represents the activation barrier.

, *W JHP L* and *P* are a prefactor with dimension of

Models on magnetization of the HTSs as can be seen in below.

#### **a. Bean model,**

Bean proposed a so-called critical-state model in which the hysteresis loop is plugged in a macroscopic parameter critical current density *CJ* (Bean, 1964). According to the Bean model, critical current density *CJ* presumes a constant and given as

$$\mathbf{J}\_{\mathbb{C}}(\mathcal{H}\_{i'}T) = \mathbf{J}\_{\mathbb{C}}(T) \tag{1}$$

where *Hi* and *T* are the local magnetic field and the temperature with respectively. In the model also pointed out that *CJ* was directly determined by the microstructure of the superconductors. The relationship of *CJ* versus temperature can be expressed as 0 0 ( ) ( )/( ) *C CC C JT J T T T T* (Berger, *et al.* 2007), where *TC* is the critical temperature and *C*0 *J* the critical current density according to the reference temperature 0 ( ) *T* .

#### **b. Kim model,**

Anderson and Kim modified the Bean model (Anderson & Kim, 1964; Kim, et al. 1962) and *CJ* was suggested that it should vary with the local magnetic field and should have the form

$$f\_{\mathbb{C}}(H\_i, T) = \frac{f\_{\mathbb{C}}(T)}{1 + H\_i \, / \, H\_0} \tag{2}$$

where *H*0 is a macroscopic materials parameter with the dimension of field. The model was found to fit the experimental results for some conventional superconductors for a solid cylinder shaped superconductors.

According to Kim model the *CJ* depend on the term *H H i* / 0 , which may vary considerably among different materials. Watson showed that, for some systems, the condition *H H i* 0 is fulfilled. Eq. (2) shows a simple linear field dependence of *CJ* , ( ) *Ci i J H A CH* , where *A* and *C* are constants (Hindley & Watson, 1969 ).

#### **c. Power law model,**

Irie and Yamafuji are focus on the specific pinning mechanisms, a power-law field dependence of *CJ* was suggested (Irie & Yamafuji, 1967),

$$f\_{\mathcal{C}}(H\_i, T) = \frac{K(T)}{H\_i^n} \tag{3}$$

where *K* is a materials parameter and *n* directly reflects pinning strength. An empirical formula based on experimental results developed by Fietz et al. (Fietz, *et al.,* 1964),

$$\mathcal{J}\_{\mathbb{C}}(H\_{i'}T) = \mathcal{J}\_{\mathbb{C}}(T) \exp\left(-H\_i \,/\, H\_0\right). \tag{4}$$

Bean proposed a so-called critical-state model in which the hysteresis loop is plugged in a macroscopic parameter critical current density *CJ* (Bean, 1964). According to the Bean

where *Hi* and *T* are the local magnetic field and the temperature with respectively. In the model also pointed out that *CJ* was directly determined by the microstructure of the superconductors. The relationship of *CJ* versus temperature can be expressed as 0 0 ( ) ( )/( ) *C CC C JT J T T T T* (Berger, *et al.* 2007), where *TC* is the critical temperature and

Anderson and Kim modified the Bean model (Anderson & Kim, 1964; Kim, et al. 1962) and *CJ* was suggested that it should vary with the local magnetic field and should have the

( ) ( ,) 1 /

where *H*0 is a macroscopic materials parameter with the dimension of field. The model was found to fit the experimental results for some conventional superconductors for a solid

According to Kim model the *CJ* depend on the term *H H i* / 0 , which may vary considerably among different materials. Watson showed that, for some systems, the condition *H H i* 0 is fulfilled. Eq. (2) shows a simple linear field dependence of *CJ* , ( ) *Ci i J H A CH* , where *A*

Irie and Yamafuji are focus on the specific pinning mechanisms, a power-law field

( ) ( ,) *C i n*

where *K* is a materials parameter and *n* directly reflects pinning strength. An empirical

formula based on experimental results developed by Fietz et al. (Fietz, *et al.,* 1964),

*i*

*K T J HT H* (3)

0 ( , ) ( )exp ( / ) *Ci C i J HT J T H H* . (4)

*J T J HT H H*

*C*

*i*

0

( ,) () *Ci C J HT J T* (1)

(2)

Models on magnetization of the HTSs as can be seen in below.

model, critical current density *CJ* presumes a constant and given as

*C*0 *J* the critical current density according to the reference temperature 0 ( ) *T* .

*C i*

**a. Bean model,**

**b. Kim model,**

cylinder shaped superconductors.

**c. Power law model,**

and *C* are constants (Hindley & Watson, 1969 ).

dependence of *CJ* was suggested (Irie & Yamafuji, 1967),

form

A general critical-state model was developed for the critical current density. It is obtained from Eq. (1-4) as the next form,

$$J\_{\mathbb{C}}(H\_{i'}T) = \frac{J\_{\mathbb{C}}(T)}{\left[1 + H\_i \,/\, H\_0(T)\right]^\beta} \tag{5}$$

where is a dimensionless constant. Eq. (5) can be converted to Bean and Kim models by using the Eqs. (1) and (2), respectively if the 0 and 1. Eq. (3) is obtained when the condition *H H i* / 1 0 is satisfied in eq. (5) and the *K* and *n* are set to 0 *J T C* () H and , respectively (Xu, et al., 1990). In the conditions of *H H i* / 1 0 and 1 , with *H*0 / **H**0, where **H**0 is a finite value with dimension of field, and taking the limit of ( ) *C i J H* with the preceding conditions and letting *HH x i* / 0 , it has

$$\lim\_{i} I\_{\mathbb{C}}(H\_{i},T) = \frac{I\_{\mathbb{C}}(T)}{\lim\_{\mathbf{x} \to \mathbf{0}} \mathbb{E} \mathbf{1} + \mathbf{x} \mathbf{J}^{\mathcal{S}}} = \frac{I\_{\mathbb{C}}(T)}{\lim\_{\mathbf{x} \to \mathbf{0}} \mathbb{E} \mathbf{1} + \mathbf{x} \mathbf{J}^{(1/\mathbf{x})(H\_{0}/H\_{0})}} = I\_{\mathbb{C}}(T) \exp(-H\_{i} \mid H\_{0}) \tag{6}$$

It can be seen that Eq. (6) has the same form as Eq. (4) and that their physical meanings are identical.

#### **d. Yin model,**

Material's equation, known as Yin model, has been developed in the form as (Yin *et al*., 1994),

$$E(f) = 2\nu\_0 H \exp\left[-\left(\frac{\mathcal{U}\_0 + \mathcal{W}\_V}{kT}\right)\right] \sinh\left[\frac{\mathcal{W}\_L}{kT}\right] \tag{7}$$

Where0 ,*U*0 , ( ) *W P E J HP V f* , *W JHP L* and *P* are a prefactor with dimension of velocity, the pinning potential, the viscous dissipation term of flux motion with viscosity coefficient (with 2 *HH H C n* 2 *f* ), the energy due to Lorentz driving force, a product of the volume of the moving flux bundle and the range of the force action, respectively. In the term in the *WV* corresponds to 0 exp ( ) *U J kT* . *U* represents the activation barrier. Yin model includes pinning barrier term, viscous dissipation term and direction dependent. It provides a unified description which covers all regimes of flux motion: thermally activated flux flow, flux creep, critical state and flux flow. Yin model contain Anderson–Kim model, Bardeen–Steven model and critical state model ( ) *FJH P C* as its specific forms (Yin *et al*., 1994).

There are different ways of calculation of magnetic field ( ) *HPM* of permanent magnets (Moon, 1990; Sanchez & Navau, 2001; Brandt, 1990; De Medeiros, *et al.* 1998),

i. Biot–Savart's law: Calculating *BPM* with Biot–Savart's law.

Measurement of Levitation Forces in High Temperature Superconductors

Levitation force (N)

0,0

0,5

1,0

1,5

2,0

Airgap(mm) 0 5 10 15 20 25

1. The HTS's are of HTS with a strongly irreversible magnetic behavior, i.e., their

2. The levitation phenomenon between HTS and PM materials has been studied for magnetic granularity or polycrystalline bulk materials in general. The magnetic

During the past years, many research groups have done extensive studies of levitation forces using HTS bulk materials of various kinds (Takao *et al.* 2011; Lu *et al.,* 2011; Krabbes *et al.*, 2000). The higher levitation force and stable levitation or suspension between a PM and a bulk superconductor, makes superconducting bulks possible for various applications such as superconducting magnetic bearings, flywheel, magnetic levitation transportation

The levitation forces measure between permanent magnet and superconducting high temperature, bulk or film in general. The critical current has the crucial role in the levitation force measuring system. The levitation force between HTS and PM originated from by the interaction between the induced electric current in the HTS and the magnetic field from the PM. The critical current density ( ) *CJ H* is closely related with the flux pinning center density and flux pinning force of the sample, and also closely related with the flux pinning, depinning process and the magnetic flux creep rate during the levitation force measurement. The ( ) *CJ H* is also a fixed parameter (if we consider the process based on

Fig. 1. The levitation forces evolution of the distance of between HTS and PM with

magnetization depends on the prehistory of the external magnetic fields.

levitation force studies are very poor for single crystalline materials.

experimental and different theoretical models (Chun & Lee, 2002).

3. A finite levitation force is required non uniform magnetic field.

systems, generators or motors and permanent magnets.

**3. Levitation experimental system and data analaysis**

Experimental data Lorentz equation Maxwell stress tensor Virtual work method

313

Field expression in cylindrical coordinates (John David Jackson, 1998), is

$$
\vec{H}(\vec{r}) = \vec{\nabla} \times \vec{A} = \frac{\mu\_0}{4\pi} \int \int\_V \frac{\vec{J}(\vec{r}) \times (\vec{r} - \vec{r}')}{|\vec{r} - \vec{r}'|} dv' \tag{8}
$$

Numerical methods have been developed towards to solve the ( ) *CJ H* and the levitation force. Analysis and simulation can be carried out based on above models and various parameters.

The levitation forces evaluation of distance between HTS and PM (or air gap) are seen in Fig. 1. As can be seen from this figure, levitation forces in a magnetic media for experimental and theoretical calculation decreases when distance between HTS and PM increases.

There are several methods for the calculation of levitation force in magnetic field. Some of these methods are Lorentz force equation (Chun &Lee, 2002), Maxwell's stress tensor (Amrhein & Krein, 2009) and virtual work methods (Vandevelde& Melkebeek, 2001). The Lorentz force equation method is important to know the exact distribution of the critical current density and the magnetic field. The Maxwell's stress tensor method is calculated by the integration of the Maxwell's stress tensor on an arbitrary surface surrounding the object.

The magnetic force as the virtual work method is given by the derivation of the co-energy with respect to a virtual displacement.

The magnetic levitation forces evolution of the distance of between HTS and PM with experimental and different theoretical models are seen in Fig. 1.

From this figure, levitation force nearly exponentially increase while distance of between HTS and PM decreases. But, the experimental results nearly agree with the Lorentz force equation methods. In other words, from the comparison, the result by the Lorentz force methods is in strict correspondence with the experimental results quantitatively according to the others simulation methods. Agreement between the Lorentz force methods with the experimental results originated perhaps from the penetration of the applied field into HTSs. Because of the penetration of the applied field into the HTS occurs where the vortex state is. This field passes through the HTSs produce a flux tubes. This states in this type studied so called known repulsive magnetic field in general. Finally, compatible Lorentz force method with the experimental result is not surprising.

Since the discovery of high temperature superconductors (HTSs) this interaction has achieved considerable attention, mainly due to its capability of producing stable noncontacting conditions for such (HTS-PM) systems. The magnetic levitation forces can be demonstrated very easily in experimentally. But, their theoretical models are far more difficult. The reason for these difficulties are the below.

 0 () ' ( ) ' 4 ' *V Jr r r Hr A dv*

Numerical methods have been developed towards to solve the ( ) *CJ H* and the levitation force. Analysis and simulation can be carried out based on above models and various

The levitation forces evaluation of distance between HTS and PM (or air gap) are seen in Fig. 1. As can be seen from this figure, levitation forces in a magnetic media for experimental

There are several methods for the calculation of levitation force in magnetic field. Some of these methods are Lorentz force equation (Chun &Lee, 2002), Maxwell's stress tensor (Amrhein & Krein, 2009) and virtual work methods (Vandevelde& Melkebeek, 2001). The Lorentz force equation method is important to know the exact distribution of the critical current density and the magnetic field. The Maxwell's stress tensor method is calculated by the integration of the Maxwell's stress tensor on an arbitrary surface surrounding the

The magnetic force as the virtual work method is given by the derivation of the co-energy

The magnetic levitation forces evolution of the distance of between HTS and PM with

From this figure, levitation force nearly exponentially increase while distance of between HTS and PM decreases. But, the experimental results nearly agree with the Lorentz force equation methods. In other words, from the comparison, the result by the Lorentz force methods is in strict correspondence with the experimental results quantitatively according to the others simulation methods. Agreement between the Lorentz force methods with the experimental results originated perhaps from the penetration of the applied field into HTSs. Because of the penetration of the applied field into the HTS occurs where the vortex state is. This field passes through the HTSs produce a flux tubes. This states in this type studied so called known repulsive magnetic field in general. Finally, compatible Lorentz force method

Since the discovery of high temperature superconductors (HTSs) this interaction has achieved considerable attention, mainly due to its capability of producing stable noncontacting conditions for such (HTS-PM) systems. The magnetic levitation forces can be demonstrated very easily in experimentally. But, their theoretical models are far more

and theoretical calculation decreases when distance between HTS and PM increases.

iii. Vector potential method: Calculating vector potential *A PM*( ), *H* is the curl of *A* .

Field expression in cylindrical coordinates (John David Jackson, 1998), is

(8)

*r r*

in current-free area, *HPM* is the

ii. Scalar potential method: Calculating scalar potential

gradient of

parameters.

object.

.

with respect to a virtual displacement.

with the experimental result is not surprising.

difficult. The reason for these difficulties are the below.

experimental and different theoretical models are seen in Fig. 1.

Fig. 1. The levitation forces evolution of the distance of between HTS and PM with experimental and different theoretical models (Chun & Lee, 2002).

During the past years, many research groups have done extensive studies of levitation forces using HTS bulk materials of various kinds (Takao *et al.* 2011; Lu *et al.,* 2011; Krabbes *et al.*, 2000). The higher levitation force and stable levitation or suspension between a PM and a bulk superconductor, makes superconducting bulks possible for various applications such as superconducting magnetic bearings, flywheel, magnetic levitation transportation systems, generators or motors and permanent magnets.

## **3. Levitation experimental system and data analaysis**

The levitation forces measure between permanent magnet and superconducting high temperature, bulk or film in general. The critical current has the crucial role in the levitation force measuring system. The levitation force between HTS and PM originated from by the interaction between the induced electric current in the HTS and the magnetic field from the PM. The critical current density ( ) *CJ H* is closely related with the flux pinning center density and flux pinning force of the sample, and also closely related with the flux pinning, depinning process and the magnetic flux creep rate during the levitation force measurement. The ( ) *CJ H* is also a fixed parameter (if we consider the process based on

Measurement of Levitation Forces in High Temperature Superconductors

Fig. 2. Schematic representation of the preparing of HTS sample.

Fig. 3. Schematic representation of the annealing process of the HTS disk.

Magnetic properties were measured by using the magnetic levitation force-distance (between HTS and PM) system at the temperature 77 K in Fig. 4. Measurements were performed in the zero field cooled (ZFC) regimes in applied magnetic field 0.15 *H T r* .

315

Bean's model) because the sample is the same, so the levitation force is only related with the total induced electric current in the sample which is depending on magnet moving speed, this is the only variable during the measurement. Superconductors is different from perfect conductor, that if a superconductor is cooled below certain temperature (is called critical temperature, *TC* ) and an external magnetic field is applied then the superconductor repel the magnetic field because of the supercurrent. This super-current is made of the inside the superconductor via of the induced of the external field as the same Lenz law. This situation, the superconductor has the supercurrent as defined superconductor. Firstly grains effected and a current circulate interior of the grain, which is called internor current, *CI* . Secondly all of the interior current sums to surface of the superconductor and they called screening current, *SI* ; which is repulsive and screening to external magnetic field.

The screening currents produce a mutually repulsing behavior, so the external magnetic field do not penetrate interior of the superconductor, which is called Meissner effect in the superconductor. The German physicists Meissner and Ochsenfeld discovered the phenomenon in 1933 by measuring the magnetic field distribution outside superconducting thin and lead samples. These samples, in the presence of an applied magnetic field, were cooled below what is called their superconducting transition temperature. Below the transition temperature the samples canceled nearly all magnetic fields inside. They detected this effect only indirectly; because the magnetic flux is conserved by a superconductor, when the interior field decreased the exterior field increased. The experiment demonstrated for the first time that superconductors were more than just perfect conductors and provided a uniquely defining property of the superconducting state.

Macroscopic transport properties mechanism of the grain boundary effect on HTS is not entirely understood yet, though on this mechanism several theoretical models have been suggested (Luine & Kresin, 1998; Coble, 2009). In particular, it is assumed that the grain boundary effects are describing as that to following factors: (i) a decrease of electron free path in vicinities of grain boundaries; (ii) stress-field-induced structural disorder and pinnig of magnetron vortices at grain boundaries; (iii) formation of the antiferromagnetic phase within grain cores; (iv) faceting of grain boundaries and d-symmetry of the superconducting order parameter, (v) deviations from bulk stoichiometry in vicinities of grain boundaries.

Figures 2-3 shows a schematic representation of the preparing of the HTS sample with a wet-technique (or ammonium nitrate method) and annealing process of the HTS disk, respectively. The starting material was the powder of nominal Bi1.84Pb0.34Sr1.91Ca2.03Cu3.06O10 compound, which was synthesized by a wet-technique using high-purity (>99.99%) Bi2O3, PbO, SrCO3, CaCO3 and CuO, We produce our powder by using this method at room temperature. Then, the powder was calcined at 700 °C for 10h, twice as much, and pressed into pellets with 225 MPa. The pellets were placed in a preheated furnace at 850 °C for 100 h then directly cooled to room temperature for quench. After this sintering process pellets regrinding and the powders at various pressures and time were pressed in to cylindrical superconductor disk (CSD: Bi1.84Pb0.34Sr1.91Ca2.03Cu3.06O10**)** with diameter ( 2*r )* of 13 mm and height ( *h* ) of 13.52 mm. The CSD was placed in a preheated furnace at 815 °C at a rate of 0.1 °C/min for 100 h then directly cooled to room temperature. The details of producing of HTS material was given in previously our work (Karaca, 2009).

Bean's model) because the sample is the same, so the levitation force is only related with the total induced electric current in the sample which is depending on magnet moving speed, this is the only variable during the measurement. Superconductors is different from perfect conductor, that if a superconductor is cooled below certain temperature (is called critical temperature, *TC* ) and an external magnetic field is applied then the superconductor repel the magnetic field because of the supercurrent. This super-current is made of the inside the superconductor via of the induced of the external field as the same Lenz law. This situation, the superconductor has the supercurrent as defined superconductor. Firstly grains effected and a current circulate interior of the grain, which is called internor current, *CI* . Secondly all of the interior current sums to surface of the superconductor and they called screening