The highest critical transition temperature (Tc) among all the intermetallic superconductors in that was discovered MgB2 has changed the previous approaches to the theory of superconductivity because the Tc limit in metallic superconductors had been believed to be ~ 30 K, which is predicted by the Bardeen-Cooper-Schrieffer (BCS) theory . In the BCS theory of superconductivity [2, 3], the expression for Tc is derived as , where is always equal to the MgB2 Debye temperature, . λeff is defined as the electron and phonon coupling constant λ = N(EF)× V. N(EF) is the normal state electron number density the Fermi surface, and V is the average electron interaction matrix element corresponding to the attraction. A weak coupling, λ << 1, is assumed to exist between the electrons and phonons in the initial BCS theory. In this case, the value of Tc is limited to Tc ≈ 30 K. According to the BCS theory, an element or compound with larger N(EF), V, and θD has high Tc value. However, θD of MgB2 is comparable to those of other diborides and other light materials. Furthermore, the N(EF) is relatively low because of the absence of d-electrons. Thus, the unusually high Tc in MgB2 has confused researchers with respect to the origin of its superconductivity. Considerable theoretical and experimental work has been conducted to explore the superconducting mechanism in MgB2.
MgB2 is the first superconductor to be proved to have two distinct superconducting gaps in its superconducting state . Initially, an unconventional exotic superconducting mechanism was suggested for the material [5, 6]. Then, other researchers proposed hole superconductivity, which is similar to what occurs in high temperature superconductors (HTS), based on the fact that holes are the dominant charge carriers in the normal state [7, 8]. MgB2 has now been accepted as a phonon-mediated BCS type superconductor. The superconductivity is attributed to selective coupling between specific electronic states and specific phonons, such as the E2g mode. The unusually high Tc value arises from the strong phonon anharmonicity.
Choi et al. have calculated the phonon frequencies and electron-phonon interactions from frozen phonon calculations at all the symmetry points of the Brillouin zone . Six non-acoustic modes at the Brillouin zone centre Γ are divided into four distinct phonon modes based on the point symmetry of the lattice. The two vibrations along the c-axis are singly degenerate modes of A2u and B1g. For the A2u mode, both Mg and B move in opposite directions along c. For B1g, the B atoms move in opposite directions, while the Mg is stationary. The other two modes along the x or y directions involving only in-plane motions are doubly degenerate. The vibration of Mg and B planes in opposite directions along the x or y directions is E1u mode. Mg atoms are stationary in the E2g mode, while B atoms exhibit a breathing vibration in the x or y directions. This mode is highly anharmonic. The theoretical vibration energy of the E2g mode is around 75 meV [10, 11], which is in agreement with the results from Raman measurements [12, 13].
MgB2 is easy to make into bulk, wire, tape, and thin film forms. However, the critical current density (Jc) of pristine MgB2 drops rapidly in high magnetic field due to the weak pinning forces and low upper critical field (Hc2). Many techniques have been employed to improve the application potential of MgB2, such as chemical doping, irradiation, thermo-mechanical processing, and magnetic shielding. Although the critical current density, upper critical field, and irreversibility field (Hirr) have been greatly increased , many difficulties need to be overcome for further application. The origin of the flux pinning force and relevant fabrication techniques will be discussed in this work. Lattice distortion is found to be one of the most effective sources of flux pinning centers in pure MgB2. Then, the combined effects of connectivity and lattice disorder on the flux pinning force are investigated based on nanosize SiC doped MgB2.
The depairing current density, Jd, can be estimated from the Ginzburg-Landau (GL) formula:
where is the flux quantum, the permeability of vacuum, the penetration depth, and the coherence length . However, it is not the theoretical maximum . With the help of optimized pinning, about 15% of Jd can be obtained at low magnetic fields in superconductors [17, 18]. The high field values for pure MgB2 are λ = 80 nm and ξ = 12 nm, respectively. Jd at zero K is estimated as ~1.3 × 108 A.cm−2. The contribution from π-band charge carriers to the depairing current density is quite low, only about 10% , and the interaction energy induces the difference from the high field value. The depairing current is reduced in samples with defects because of the increased λ values .
The grain boundaries in MgB2 do not show the weak link effect, and clean grain boundaries are not obstacle to supercurrents [19, 20]. On the other hand, dirty grain boundaries do potentially reduce the critical current . Insulating phases on the grain boundaries, such as MgO, boron oxides  or boron carbide , normal conducting phases , porosity, and cracks , can further reduce the cross-section effective of supercurrents. The high porosity in in-situ prepared MgB2 is responsible for its low density, only about half (or less) of its theoretical value .
The concept of connectivity, Acon, has been introduced to quantify the effective cross-section (σeff) for supercurrents [21, 27]: Acon = σeff / σ0, where σ0 is the geometrical cross-section. The connectivity is estimated from the phonon contribution to the normal state resistivity through Acon ≈ Δρtheo/Δρexp, with Δρtheo ≈ 9 μΩ cm. This estimate is based on the assumption that σeff is reduced equivalently in the normal and superconducting state . The supercurrents are limited by the smallest effective cross-section along the conductor. A single large transverse crack can reduce strongly Jc, while only slightly increases the resistivity of a long sample. Unreacted magnesium decreases Δρexp  and the effective paths for supercurrents. The thin insulating layers located on grain boundaries can decrease the effective connectivity Acon, inducing high Δρexp, although this kind of defect is transparent to supercurrents. The Δρtheo values within the grains depend on the defects and strain in the grains. Sharma observed negative Δρexp in highly resistive samples . Despite these objections, Acon is very useful for estimating the connectivity of samples if the resistivity is not too high. A clear correlation between the resistivity and the critical current was found in thin films . It should be noted that this procedure is not always reliable, although it offers a possibility for understanding the influences of the connectivity on the supercurrent .
The in-situ route is a practical technique to improve the Hc2 and Jc performance of MgB2 using magnesium or MgH2 as the precursor material [30-39], which reacts with boron after mixing and compacting. Low annealing temperature generates MgB2 samples with small grains [25, 40-52] due to the poor crystallinity, and the great amount of grain boundaries result in strong pinning and high Hc2. Magnesium deficient samples can be fabricated by adjusting the stoichiometry of the precursor materials. This method can generate strong lattice strain in MgB2, which decreases Tc, and increases Hc2 [44, 53-56]. On the other hand, an excess magnesium ratio in the raw materials can compensate for the evaporation loss of Mg due to its low melting temperature and the reaction loss due to the reaction with oxygen or the sheath material. The morphology and particle size of the precursor magnesium powders are crucial for the superconductivity and superconducting performance of the final MgB2 . The grain size of the initial boron powder also has a significant influence on the MgB2 samples. Some researchers have employed ball milling and mechanical alloying of the magnesium and boron mixture, reducing the grain size and enhancing the critical supercurrent [55, 58-65].
The reaction kinetics between magnesium and boron can be modified by chemical or compound dopants , which influence the grain shape and size [67, 68], the secondary phases , MgB2 density , and the element stoichiometry . Carbon doping is one of the most promising methods to improve the superconducting performance of MgB2. The carbon sources include B4C [72, 73], carbon [52, 66, 67, 74], carbon nanotubes [75-78], nanodiamonds [78, 79], NbC , SiC [41, 51, 57, 66, 72, 81-89], and organic compounds [39, 47, 90]. SiC is one of the most promising dopants because it can react with magnesium and boron to form C doped MgB2 at quite low temperatures (600 oC), based on the dual reaction model . Higher processing temperatures are necessary for most of the other carbon sources, leading to grain growth and worse pinning. Comparable results to those with SiC have also obtained, however, with nanoscale carbon powder , stearic acid , and carbon nanotubes .
The in-situ technique also suffers from its disadvantage of low mass density, which is originates from the annealing process, because the precursor powder has a lower density than MgB2. High pressure synthesis increases the density [93, 94]. On the other hand, the density of ex-situ materials is usually close to the real density of MgB2  and can be further improved by hot isostatic pressing (HIP) [91, 95, 96]. However, the grain size of ex-situ produced materials is comparatively large and inhomogeneous due to the post annealing [97, 98]. Small grains (~ 100–200 nm) have also been reported . A high temperature heat treatment is necessary for the ex-situ process to improve the connectivity [99-102]. This heat treatment leads to recrystallization and Hc2 reduction . It is concluded that disorder induced by the low temperature processing is insufficient for high superconducting performance of the ex-situ MgB2. Furthermore, thermally stable defects, such as dopants, are necessary for enhancing the high field performance .
2. Thermal-strain-induced high Jc in high density SiC-MgB2 bulk
The connectivity is considered to be a critical issue for improving the Jc of MgB2 based superconductors. Furthermore, an efficient flux pinning force is crucial for high magnetic field application. Here, we employed an in-situ diffusion process to make high density MgB2 bulks and investigate the efficiency of the high flux pinning force induced by lattice thermal strain . Microstructural analysis and Raman scattering measurements were employed to investigate the origins of the huge flux pinning force. Doping nano-SiC particles into MgB2 has been proven to be particularly effective in significantly enhancing Jc, Hirr, and Hc2 [14, 104, 105]. In contrast to chemical doping effects, tensile stress is believed to act as a source of strong flux pinning centers when there is no reaction between SiC and MgB2. Both the Jc and Tc are improved by thermal strain on the interface between SiC and MgB2 during the diffusion process  and hybrid physical-chemical vapor deposition (HPCVD) . The influences of the stress field on the flux pinning force and the electron-phonon coupling constant are discussed to clarify the superconducting performance of high density SiC-MgB2 composite fabricated through the diffusion method.
Crystalline B with 99.999% purity was pressed into pellets or mixed with 10wt% SiC particles and then pressed into pellets. The pellets were sealed in iron tubes and padded with 99.8% Mg powder. The Mg to B atomic ratio was 1.15:2.0. The diffusion process is time dependente. The sintering condition were 1123 K for 10 h under a flow of high purity argon gas to achieve fully reacted MgB2 bulks. Then the samples were cooled down to room temperature. X-ray diffraction (XRD) was employed to characterize the phases, and the results were refined to determine the a-axis and c-axis lattice parameters and the lattice distortion. Microstructure observations were performed with a transmission electron microscope (TEM). The magnetic Jc was derived from the height of the magnetization loop ΔM using the Bean model: Jc = 15ΔM/[π a3h)], where a and h are the radius and height of a cylindrical sample. The Tc value could be deduced from the temperature dependence curve of the magnetic susceptibility M(T). To observe the temperature effects of lattice strain, Raman spectra were collected using a confocal laser Raman spectrometer (Renishaw inVia plus) under a 100× microscope. The excitation laser is Ar+ 514.5 nm.
The density of the pure MgB2 sample is about 1.86 g/cm3, which is about 80% of the theoretical density. This value is much higher than those of the samples made by the in-situ process, which were less than 50%. The SiC-MgB2 composite shows an even higher density of 1.91 g/cm3 due to the SiC addition. Figure 1 shows the Rietveld refinement XRD patterns of the pure MgB2 and the 10wt% SiC doped MgB2 samples. In this case, SiC did not react with magnesium and boron to form C doped MgB2 and Mg2Si. The product is a SiC-MgB2 composite. The Rietveld refinement analysis indicates that the unreacted SiC content was about 9.3wt%, similar to the SiC content in the precursor. This is consistent with the absence of Mg2Si in the XRD pattern as shown in Figure 1. The result is different when the SiC-doped MgB2 prepared by the in-situ technique [14, 41, 84], in which only a very small amount of SiC remaines, while Mg2Si is always present due to the reaction of Mg with SiC. The a- and c-axis lattice parameters were 3.0850 Å and 3.5230 Å for pure MgB2, and 3.0840 Å and 3.5282 Å for SiC doped MgB2, respectively. The a-axis parameters are virtually equivalent for the two samples, whereas the c-axis one is slightly stretched in the SiC-MgB2 composite. The phenomenon is different from the shortening of the a-axis parameter in in-situ processed SiC doped MgB2, while the c-axis parameter remains unchanged [14, 89].
To explain the abnormal c-axis enlargement of the SiC-MgB2 composite, the thermal expansion coefficients, α, of MgB2 and SiC are considered. It is reasonable to assume that both the MgB2 and the SiC are in a stress-free state at the sintering temperature of 1123 K due to the relatively high sintering temperature over a long period of time. However, the lattice parameters are determined by the thermal strain during the cooling process. The temperature dependences of the α values for MgB2 and SiC are especially different. Figure 2(a) plots the α(T) for MgB2 and SiC along the a- and c-axes, based on the data of References [108-111]. It clearly shows the weak temperature dependence ofα(T) for SiC in both directions, whereas the changes are great for MgB2 and are characterized by high anisotropy. The averaged α(T) is also huge in MgB2 as shown in the inset of Figure 2(a). The αSiC decreases slightly from 5×10-6/K at 1123 K to 2.5×10-6/K at 0 K, whereas the drops quickly from 1.7×10-5/K at 1123 K to zero at 0 K. Based on α(T), the normalized lattice change and lattice strain in the MgB2 matrix of SiC-MgB2 composite during cooling from 1123 K to 0 K can be derived, as shown in Figure 2(b). An assumption of Figure 2(b) is that the two phases are strongly connected and the volume shrinkage of MgB2 is confined by the relatively stable SiC. The normalized lattice strain is estimated to be -0.55% in SiC-MgB2 along the c-axis at room temperature. The negative value corresponds to tensile strain in the MgB2. The large c-axis strain in the doped MgB2 results in an enlargement in the c-axis by 0.15 % in comparison with pure MgB2. As estimated from the Williamson-Hall model , the lattice strain is 0.208 and 0.306 along the a-axis, and 0.292 and 1.13 along the c-axis for pure and SiC-MgB2, respectively. The lattice strain along the c-axis in the SiC-MgB2 has increased from that of pure MgB2 by a factor of 4, which is attributed to the high anisotropy in the thermal expansion coefficient of MgB2.
The unreacted SiC buried in the MgB2 matrix is believed to be one of the most effective sources of strain, and the strongly connected interfaces of SiC and MgB2 are the most effective flux pinning centers. The micro morphologies can be detected using TEM to explore the defects and grain boundaries both in the pure MgB2 and in the SiC-MgB2. Figures 3(a) shows a bright field image of pure MgB2. A high density of defects, such as dislocations and lattice distortion, is observed in the MgB2 phase, and the grain size is about 100 nm,as estimated from the grain boundaries. In contrast to with the highly porous structure in the MgB2 samples , the samples made by the diffusion process are well connected with high density. The indexed selected area diffraction (SAD) image shows very pure polycrystalline MgB2. A high resolution grain boundary image is shown in Figure 3(b). The interface is very clean and well connected. The indexed fast Fourier transform (FFT) pattern indicates that the right part parallels the (1 1 0) plane. The micro structure of SiC-MgB2 is similar with that of pure MgB2 with high density of defects. Furthermore, nanosize SiC particle are detected in the MgB2 matrix as indicated in Figure 3(c). Figure 3(d) shows the interface of SiC and MgB2. Based on the FFT analysis, the interface is marked with a dashed line on the image. The left side is a SiC grain paralleling the (1 0 1) plane and the right side is a MgB2 grain paralleling the (0 0 1) plane. This kind of interface will impose tensile stress along the c-axis in MgB2 because the thermal expansion coefficient for MgB2 is highly anisotropic in the  direction, while that for SiC is nearly isotropic, which is responsible for the enlarged c-axis of MgB2.
Based on the collective pinning model, , Jc is independent of the applied field in the single-vortex pinning regime (low magnetic field region: H < Hsb), where Hsb is the crossover field from single-vortex to small-bundle pinning. The Jc decreases exponentially in the small-bundle regime (high magnetic field: Hsb < H < Hirr). According to the dual model , the significant effect of SiC doping on Jc comes from the high level of C substitution on the B planes, which is responsible for the reduction of the self-field Jc[104, 105]. However, the SiC-MgB2 composite sample shows not only an improved in-field Jc, but also no degradation in self-field Jc, as indicated in Figure 4. The approximate Hsb values are also indicated on the Jc curves for 20 K and 30 K, although Hsb has not been detected at 5 K due to the relatively high supercurrents. The in-situ processed SiC doped MgB2 normally shows a decrease in Tc of 1.5 to 2 K, [14, 104, 105], but this present SiC-MgB2 composite sample features a small drop of 0.6 K, as shown in the inset of Figure 4. This phenomenon is attributed to the absence of any reaction between Mg and SiC, as well as the stretched MgB2 lattice, as indicated by the XRD pattern .
To investigate whether the lattice strain is significant in SiC-MgB2 during low temperature measurements to obtain M(H) and M(T) curves, Raman spectra were collected before and after the measurements. The Raman spectra for the pure MgB2 are shown in Figure 5(a) and (b) to compare the cooling effects on the matrix. Both the spectra have been fitted with three peaks: ω1, ω2, and ω3 [114-116]. Based on the previous results, ω2 is the reflection of the E2g mode at the Г point of the Brillouin zone in the simple hexagonal MgB2 structure (space group: P6/mmm), while ω1 and ω3 come from the lattice distortion. The effects of ω1 are not discussed in the following analysis because of its small influence on the spectra. As indicated by the fitting parameters that are shown in Figure 5, both the peak centers and the full width at half maximum (FWHM) values show negligible differences before and after the low temperature measurements because of synchronic volume fluctuation. The weak temperature dependence of the Raman spectra for pure MgB2 is in agreement with the results of Shi et al . The ω2 peak of the Raman spectrum of SiC-MgB2 before the low temperature measurement has shifted to the low frequency of 585 cm-1, as shown in Figure 6(a). The FWHM of the ω2 peak increases from ~200 cm-1 to 210 cm-1. Furthermore, the FWHM of the ω3 peak increases from ~93 cm-1 to 125 cm-1. The variations in both the Raman shift and the FWHM indicate strong lattice strain in the SiC-MgB2 composite. Figure 6(b) shows the cooling effect on the Raman spectrum of SiC-MgB2. The FWHM of the ω2 peak further increases to 228 cm-1, and the frequency of the ω3 peak shifts to 770 cm-1. These results suggest that the stress field is very strong during the low temperature measurements in the SiC-MgB2 composite. Considering the stable defect structures in the sample at room temperature and the measurement temperatures, the high Jc performance is attributed to the thermal strain. Although the interface or grain boundaries themselves are effective flux pinning centers, the thermal strain provides more efficient flux pinning force, based on the comparison of the Jc values in pure the MgB2 and the SiC-MgB2 composite.
It should be noted that the broadened ω2 peak in SiC-MgB2 is a signal of strong electron-E2g coupling, which is responsible for the high Tc in MgB2. The electron-E2g coupling constant is estimated from the Allen equation : , where Γ2 is the ω2 linewidth, is the strength of the electron-E2g coupling, and N(0) is the density of states on the Fermi surface. The experimental phonon frequency and linewidth are simply and directly related to the electron-phonon coupling constant (EPC), . The total density of states (DOS) at the Fermi energy, EF, of pure MgB2 is taken as 0.354 states/eV/cell/spin. The σ band contribution is 0.15 and the π band is 0.204 . N(0) is assumed to be constant for the small changes of electrons and holes in MgB2 and SiC-MgB2. Taking the fitting values of the ω2 peaks with cooling effects, the values for the pure MgB2 and SiC-MgB2 are 2.327 and 2.706, respectively. The of SiC-MgB2 is just slightly higher than that of the pure MgB2. However, the Tc of SiC-MgB2 is slightly decreased compared to that of the pure MgB2. The total EPC constants are degraded by the scattering effects of SiC impurities in the MgB2 matrix, which can be estimated with the McMillan formula , as modified by Allen and Dynes: , where is the averaged phonon frequency , with 390 and 690 cm-1 being the phonon frequencies of the other modes in the MgB2 system , μ* is the Coulomb pseudopotential, taken as equal to 0.13 , and λ is the total EPC constant. Taking these values, λ is calculated as 0.888 in pure MgB2 and 0.886 in SiC-MgB2, respectively. Although the values are very similar, the λ of MgB2 is a little higher because of its low impurity scattering effects. It should be noted that the λ values were overestimated using the McMillan formula for the two-gap nature of MgB2. However, the overestimations do not change the dependence of λ on the strain effect due to the main contribution of the σ-band. The residual resistivity of SiC-MgB2 is 16 μΩ cm, but it is just 12 μΩ cm for pure MgB2, due to the weak impurity scattering effects.
In summary, the thermal strain originating from the interface of SiC and MgB2 is one of the most effective sources of flux pinning centers to improve the supercurrent critical density. The weak temperature dependence of the thermal expansion coefficient of SiC stretches the MgB2 lattice as the temperature decreases. The thermal strain supplies much more effective flux pinning force than the interfaces and grain boundaries themselves. The low temperature effects on Raman spectra include very strong lattice stretching at the application temperature of MgB2, which has benefits from both the Jc and the Tc behaviors.
3. High connectivity MgB2 wires fabricated by combined in-situ/ex-situ process
The self-field critical current density, Jc(0), of MgB2 is much higher than for pure or doped samples processed with in-situ annealing [75, 81, 125-128]. Jc(0) values have been reported as high as 3.5×107 A/cm2 at 4.2 K and 1.6×108 A/cm2 at 2 K in highly connected thin films made by HPCVD [129, 130]. The connectivity is much lower than the theoretical value in in-situ MgB2 because the technique involves a liquid-solid phase reaction process with considerable shrinkage due to the high density of MgB2 compared to the powder mixture of Mg and B [104, 115]. Although the diffusion process could increase the Jc of bulk samples, the Jc(0) and disorder of wires and tapes cannot be improved using similar methods. Several research groups have reported possible techniques to increase the connectivity of in-situ filamentary tapes. High pressure sintering and cold high pressure densification (CHPD) can make it possible to fabricate high density wire and tape samples [96, 131, 132]. Nevertheless, neither HIP nor CHPD are suitable for long MgB2 wires and tapes, and the Jc values are still lower than needed for practical application. Kováč et al. reported a mixed in-situ/ex-situ process to make MgB2 wires, and it was found that the Jc was increased when the ex-situ powder ratio were 23% and 50% . They used commercial MgB2 powder from Alfa Aesar in this process with a very wide grain size distribution ranging from submicrometer size up to 100 μm. In this work, high quality home-made ex-situ powder was used to repeat the mixed in-situ/ex-situ process to develop MgB2/Fe wires with high connectivity and strong disorder to increase both the low and high field Jc properties . The home-made powder was fabricated through low temperature annealing, and the particle size was homogeneous and as small as ~200 nm. The Jc dependence on microstructure, connectivity, and disorder in MgB2 wires was analyzed based on the microstructure observations.
MgB2 wires were fabricated by the powder-in-tube (PIT) process using a ball-milled mixture of Mg (99%) and amorphous B (99%). The in-situ wires were sintered at 750, 850, 950, and 1050 °C for 30 min in high purity Ar and marked as 750in, 850in, 950in, and 1050in, respectively. The in-situ MgB2 powder was sintered at 650 °C for 30 min in high purity Ar flow and then ball-milled. Then the PIT method was employed to fabricate in-situ/ex-situ combined MgB2/Fe wires using 1:3 ratio of precursor powders to form a mixture of Mg and B powders. All the green wires were annealed at 750, 850, 950, and 1050 °C for 30 min in high purity Ar and marked as 750inex, 850inex, 950inex, and 1050inex, respectively.
The phases and microstructure were characterized by XRD (D/max-2200) and field emission gun scanning electron microscopy (FEG-SEM: JSM-6700F) at room temperature. The superconducting properties were detected from 5 K to 305 K using a Physical Properties Measurement System (PPMS: Quantum Design). The critical superconducting transition temperature, Tc, is defined as the onset point on the magnetic moment vs. temperature curve, M(T), measured in a magnetic field of 25 Oe. The magnetic Jc was derived from the Bean model. The resistivity dependence on applied magnetic field and temperature, ρ(H, T), was measured using the four-probe method with H from 0 T to 13 T applied perpendicularly to the current direction, from 5 K to 305 K. Hc2 and Hirr were defined as the magnetic field values at 90% and 10% on the superconducting transition on the ρ(H, T) curve, respectively. The active connectivity factor, AF, was calculated based on the ρ(H, T) behavior.
According to the indexed XRD patterns, the samples contain a small amount of MgO. The MgO contents are high in 950in, 1050in, and 1050exin compared with the other samples. The broad transition widths from the normal state to the superconducting state of these samples confirm the high impurity contents, as shown in the insets of Figure 7. The 1050in transition width is ~7 K compared with the width of ~2 K for the other samples, which is attributed to the degraded connectivity of the magnetic flux due to the high MgO content. The in-situ sintered samples show higher Tc values by 0.5 – 1 K compared with the in-situ/ex-situ ones. The Tc of 1050in is similar to that of 1050inex because of the high amount of impurity phase. Ball milling of the raw materials induces a great amount of defects in the in-situ/ex-situ samples, which display lower Tc compared with the in-situ samples [135, 136]. The connectivity of the in-situ/ex-situ samples is worse than those of the in-situ samples judging from their big transition widths. The low amount of Mg evaporation may be responsible for the narrow transition width of 1050inex compared with that of 1050in because the stable precursor MgB2 and low quantity of Mg in the raw materials reduce the magnesium loss during the sintering at 1050 oC.
Typical SEM images of the in-situ samples are shown in Figure 8. The grain size of 750in is about 300 nm, and the grains show an isolated distribution due to independent growth. 850in, 950in, and 1050in have big clusters of grains because the increasing sintering temperatures have extended the crystal growth time. Some clusters are as big as 1 μm in 1050in. The crystallinity is enhanced at higher sintering temperatures. A high sintering temperature induces raw magnesium evaporation and MgO formation, which can be observed as small white particles without contrast under SEM in 1050in.
The crystal shapes for the in-situ/ex-situ samples are irregular with a dispersed distribution of grain sizea, as shown in Figure 9. The grain size of 750inex is small because of the low sintering temperature. The recrystallization effect is triggered for 850inex, 950inex, and 1050inex, and big grains as large as 1 μm are observed. 1050in and 1050inex show similar microstructures because the high sintering temperature provides enough energy and a long time for the crystal growth.
Figure 10 compares the Jc dependence on applied magnetic field measured at 5 K and 20 K. The Jc value of 1050in is the worst because of the high MgO content. The Jc performances of 750in, 850in, 750inex, and 1050inex are quite similar over the whole field range. While the Jc deteriorated in 950in and 1050in under higher magnetic fields. 850inex and 950inex show amazingly high Jc properties. The Jc values at 5 K under 8 T magnetic field are around five times higher than those of the best in-situ samples. The inset of Figure 10 compares the low field performances measured at 20 K. The in-situ samples, except for 1050in, show competitive low field Jc. The high crystal quality of 950in is of benefit for its high self-field Jc performance. 750inex and 850inex show low self-field Jc values. While 950inex and 1050inex display improved Jc, the values are lower than the self-field Jc of 950in due to the high MgO contents.
The Jc performance depends on the flux pinning mechanism under different magnetic field intensities. The collective pinning model classifies the disorder-induced spatial fluctuations in the vortex lattice into three regimes based on the strength the magnetic field: e.g. single-vortex, small-bundle, large-bundle, and charge-density-wave type relaxation of the vortex lattice . The practicable Jc of MgB2 falls into the single-vortex pinning region and the small-bundle pinning region in the phase diagram. The connectivity determines the Jc performance in the single-vortex pinning regime because the effective charge carrier density is responsible for the self-field Jc, while Hc2 and Hirr are responsible for the Jc performance in the small-bundle regime based on the disorder or defects.
A practical quantity to evaluate the connectivity is the active area fraction, AF ,
is the resistivity of fully connected MgB2 without any disorder , and
Figure 11 compares the AF of all the samples. All the samples show lower connectivity compared with those of ideal crystals, as indicated by the low AF values. The AF value of 750in is just 0.169, and high sintering temperature enhances the connectivity. It is ~ 0.26 for 850in. Higher sintering temperatures than 850 oC can improve the AF slightly, which indicates that the connectivity is easy to improve for in-situ samples. Although the MgO content is high in 1050in, its high AF value is attributed to sufficient crystallization. However, its low magnetic field Jc was degraded by MgO. The combined in-situ/ex-situ samples show rather low AF values compared with the in-situ samples. The phenomenon is consistent with the high resistivity of the in-situ/ex-situ samples, as shown in the insets of Figure 11. The gradually improved connectivity improves the AF values for the in-situ/ex-situ samples due to the improved solid state reaction. The self-field Jc performances are also improved for 950inex and 1050inex.
The Jc performance in the small-bundle region depends on a strong flux pinning force. The flux pinning centers could be lattice distortion, most types of defects, and grain boundaries [110, 137]. The temperature dependences of Hc2 and Hirr determine the strength of the pinning force, as shown in Figure 12. The high field Jc performance is in agreement with the Hc2 and Hirr behavior. The 1050in sample shows the lowest Hc2 and Hirr among all the samples. 850inex and 950inex show the highest Hc2 and Hirr, which are consistent with their high Jc values in the small-bundle region. The ex-situ powder induces strong disorder and proper crystallization, which are responsible for the high Hc2 and Hirr values.
In summary, both connectivity and disorder show strong influences on the Jc properties of MgB2. The connectivity is responsible for the high effective charge carrier density, which determines the self-field Jc performance. The strong flux pinning force induced by defects and disorder is responsible for the promising Jc in high magnetic field. The enhanced Jc performances of 850inex and 950inex are attributed to the optimized connectivity and disorder. The Jc values obtained in this work are still far below the Jd value, ~8.7×108 A/cm2 for pure MgB2. The Jc improvement in MgB2 should be explored based on the chemical doping effects and combined in-situ/ex-situ process.
4. Nano-SiC doped MgB2 wires made by combined In-Situ/Ex-Situ process
The combined in-situ/ex-situ process has proved to be a promising technique for the fabrication of practical MgB2 wires. The Jc of MgB2 superconductors has been enhanced through many different kinds of dopants or additives , especially different carbon sources [75, 81, 126-128, 138, 139]. In this work, a mixed in-situ/ex-situ technique was employed to develop nano-SiC doped MgB2 wires with high connectivity and strong flux pinning force to increase both the low and high field Jc properties . The SiC particle size is another critical issue for introducing strong flux pinning forces into MgB2. The size of SiC used in this work is smaller than the sizes used in previous research, and the Jc dependence on sintering temperature also shows a very different trend [81, 104, 141, 142].
The powder-in-tube (PIT) process was employed to make practical MgB2 wires from a ball-milled mixture of Mg (99%), B (99%, amorphous), and SiC (< 15 nm).]The sample fabrication and characterization are similar to the techniques mentioned for the pure samples in the last section.
Figure13 shows the XRD patterns of the two batches of samples. According to the indexed XRD patterns, all samples show quite high purity of MgB2, with only small amounts of MgO and un-reacted Mg and SiC. The un-reacted Mg can be detected because of the high content of SiC in the raw materials [104, 115, 143]. The most interesting phase change relates to the change in the Mg2Si content with sintering temperature. 750in shows very high Mg2Si content, which decreases with increasing sintering temperature and becomes a trace peak in 1050in. However, more than a trace of Mg2Si can only be found in samples sintered at lower temperature using the combined in-situ/ex-situ method, 750inex and 850inex. The variation of Mg2Si content is an important signal of the C and Si distributions in the MgB2 matrix. Figure 14 shows SEM images of 850in and 850inex. The 850in sample contains large slits between MgB2 clusters due to the volume contraction during the in-situ sintering of Mg and B powders. The 850inex sample shows hard-packed MgB2 clusters because the ex-situ precursor is a course of nucleating centers and releases the strain of the in-situ MgB2.
The critical transition temperatures (Tc) of the two batches of samples are compared in Figure 15. It is found that the Tc values of the in-situ sintered samples are always slightly lower than those for the samples from the other batch, except for 1050in, and the Tc dependence on sintering temperature of the two batches of samples is exactly the same, which means that the Tc depends greatly on the sintering temperature, but not on the different techniques. However, the transition widths from the normal state to the superconducting state are quite different for the two batches of samples, as shown in the inset of Figure 15. The transition widths of the in-situ sintered samples are quite broad compared with those fabricated by the combined technique. The transition width is about 4 K for 750in and becomes 3 K for the 850in and 950in samples sintered at higher temperature due to the high crystallinity. It should be noted that the transition of 1050in shows a two-step behaviour, which may be attributed to the inhomogeneous carbon substitution effect or the inhomogeneous SiC distribution in the raw materials. The transition widths of all the samples made by the in-situ/ex-situ combined technique are 2.5 K, showing behaviour that is independent of the sintering temperature. This means that the crystallinity is increased through the in-situ/ex-situ combined technique because the precursor MgB2 powder is a source of high quality nucleating centres for the newly formed MgB2 during the solid-liquid reaction between the magnesium and the boron.
The Jc dependence on the applied field is shown in Figure 16 for typical samples, which were measured at 5 K and 20 K, respectively. It should be noted that the Jc dependence on sintering temperature in this work is totally different from previously reported results, because the solid-liquid reaction dynamics is different due to the small SiC particle size, less than 15 nm, which is much smaller than the particle sizes used before. The Jc benefits from the high sintering temperature. 750in, 850in, and 750inex display non-competitive performance over the whole field range. 1050inex has very high low field Jc values but its Jc deteriorates with increasing magnetic field. The Jc properties of 950in, 850inex, and 950inex show outstandingly high Jc performances among all the samples. It is concluded that the in-situ/ex-situ combined technique only requires a lower sintering temperature to achieve high quality MgB2 wires, which is very important for industrial application in terms of energy saving and equipment simplification. The Jc values of 750inex are double those of 750in at 5 K and 20 K over the measured magnetic field range. The inset of Figure 16 displays the low field performances at 20 K to avoid the influence of the flux jumping effect. 750in and 750inex show quite low Jc values in lower magnetic field. 1050in and 850inex show competitive self-field Jc. The ball-milling process used to produce the ex-situ MgB2 powder destroyes the porous structure and enhances the density of MgB2 fabricated by the in-situ/ex-situ combined process. This is because of the small particle size of SiC used in this work, which induces different reaction dynamics during the in-situ or in-situ/ex-situ processing [81, 104, 141, 142], so that the present Jc dependence on sintering temperature is quite different from what has been previously reported. It is proposed that the liquid Mg reacts with SiC first to form Mg2Si and releases free C at low sintering temperature. Then the Mg2Si reacts with B to form MgB2 and releases free Si at high sintering temperatures. Both C and Si have very small sizes and cannot be detected by XRD. The coherence length, ξ, of MgB2 is anisotropic. ξab(0) = 3.7 – 12 nm, and ξc(0) = 1.6 - 3.6 nm , which is shorter than the particle size of Mg2Si. The Mg2Si particles cannot be effective flux pinning centers, but are rather useless impurities in the MgB2 matrix, which decrease the density of current carriers. However, the free C and Si can be very strong flux pinning centers because of their small sizes, which are responsible for the high Jc performance in high magnetic fields. According to the collective pinning model , the Jc performance in the low magnetic field region depends on the density of current carriers due to its weak field dependence, while the high magnetic field Jc performance depends on the flux pinning force due to the increased high Hc2 and Hirr. The approximate Hsb values are also indicated on the Jc curves estimated at 20 K, as shown in the inset of Figure 16, where Hsb is the crossover field from single-vortex to small-bundle pinning based on the collective pinning model. However, Hsb has not been detected at 5 K due to the relatively high supercurrents .
The strength of the pinning force can be reflected by the dependence of Hc2 and Hirr on the normalized temperature, as shown in Figure 17. Carbon substitution is one of the most effective methods to improve the Hc2 and Hirr because of the increased scattering by C doping, and the increased scattering can also contribute to decreased Tc and merging of the two gaps . It should be noted that the Hc2 and Hirr for 750in, 850in, and 750inex have their highest values at low temperature, which means a strong flux pinning force. The poor Jc is of these samples attributed to the lower density of current carriers. Both 1050in and 1050inex show the lowest Hc2 and the lowest Hirr among all the samples. 850in, 950in, 850inex, and 950inex show competitive Hc2 and Hirr performances, which are responsible for their high Jc values under high magnetic field.
In conclusion, high sintering temperature can improve the critical current density of small-particle-size SiC doped MgB2. The two-step reactions between Mg, SiC, and B release free C and Si to form strong flux pinning centers. The current carrier density and flux pinning force are important factors in the improvement of the Jc performance of nano-SiC doped MgB2. The current carrier density determines the Jc behavior in the single-vortex regime, where Jc is independent of applied magnetic field. The flux pinning force determines the Jc performance in the small-bundle pinning regime, where the doping induced defects are believed to act as flux pinning centers. Nano-SiC doped MgB2/Fe wires fabricated by the combined process are worth ongoing research to develop optimized processing parameters for practical purposes.
The diffusion method can greatly improve the critical current density compared with the normal technique, which indicates that the critical current density greatly depends on the connectivity of MgB2 grains. The combined process improves the connectivity of MgB2 grains and the compactness of the superconducting core in wires, which induces high critical current density in zero field. The flux pinning force can also be improved by dopants for magnetic field application. Further research could focus on parameter optimization of the combined process to fabricate high quality MgB2 wires.
The authors thank Dr. T. Silver for her useful discussions. This work is supported by the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, the Australian Research Council (project ID: DP0770205), and Hyper Tech Research Inc.