High Critical Current Density MgB2

1. Introduction

The highest critical transition temperature (T_c) among all the intermetallic superconductors in that was discovered MgB₂ has changed the previous approaches to the theory of superconductivity because the T_c limit in metallic superconductors had been believed to be ~ 30 K, which is predicted by the Bardeen-Cooper-Schrieffer (BCS) theory [1]. In the BCS theory of superconductivity [2, 3], the expression for T_c is derived as $$T_c=\theta {\text{e}}^{\left(-1/{\lambda }_{\text{eff}}\right)}$$, where $$\theta$$ is always equal to the MgB₂ Debye temperature, $${\theta }_D$$. λ_eff is defined as the electron and phonon coupling constant λ = N(E_F)× V. N(E_F) 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 T_c is limited to T_c ≈ 30 K. According to the BCS theory, an element or compound with larger N(E_F), V, and θ_D has high T_c value. However, θ_D of MgB₂ is comparable to those of other diborides and other light materials. Furthermore, the N(E_F) is relatively low because of the absence of d-electrons. Thus, the unusually high T_c 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 [4]. 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 E_2g mode. The unusually high T_c 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 [9]. 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 A_2u and B_1g. For the A_2u mode, both Mg and B move in opposite directions along c. For B_1g, 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 E_1u mode. Mg atoms are stationary in the E_2g 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 E_2g 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 (J_c) of pristine MgB₂ drops rapidly in high magnetic field due to the weak pinning forces and low upper critical field (H_c2). 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 (H_irr) have been greatly increased [14], 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, J_d, can be estimated from the Ginzburg-Landau (GL) formula:

$$J_d={\Phi }_0/\left[3/\left(\sqrt{3}\right)\pi {\mu }_0{\lambda }^2\left(T\right)\xi \left(T\right)\right],$$E1

where $${\Phi }_0$$ is the flux quantum, $${\mu }_0$$ the permeability of vacuum, $$\lambda$$ the penetration depth, and $$\xi$$ the coherence length [15]. However, it is not the theoretical maximum [16]. With the help of optimized pinning, about 15% of J_d 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. J_d 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% [16], 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 [18].

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 [21]. Insulating phases on the grain boundaries, such as MgO, boron oxides [22] or boron carbide [23], normal conducting phases [24], porosity, and cracks [25], 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 [26].

The concept of connectivity, A_con, has been introduced to quantify the effective cross-section (σ_eff) for supercurrents [21, 27]: A_con = σ_eff / σ₀, where σ₀ is the geometrical cross-section. The connectivity is estimated from the phonon contribution to the normal state resistivity through A_con ≈ Δρ_theo/Δρ_exp, with Δρ_theo ≈ 9 μΩ cm. This estimate is based on the assumption that σ_eff is reduced equivalently in the normal and superconducting state [18]. The supercurrents are limited by the smallest effective cross-section along the conductor. A single large transverse crack can reduce strongly J_c, while only slightly increases the resistivity of a long sample. Unreacted magnesium decreases Δρ_exp [28] and the effective paths for supercurrents. The thin insulating layers located on grain boundaries can decrease the effective connectivity A_con, inducing high Δρ_exp, although this kind of defect is transparent to supercurrents. The Δρ_theo values within the grains depend on the defects and strain in the grains. Sharma observed negative Δρ_exp in highly resistive samples [29]. Despite these objections, A_con 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 [27]. 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 [18].

The in-situ route is a practical technique to improve the H_c2 and J_c 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 H_c2. 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 T_c, and increases H_c2 [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 [57]. 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 [66], which influence the grain shape and size [67, 68], the secondary phases [69], MgB₂ density [70], and the element stoichiometry [71]. 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 [80], 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 [66]. 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 [91], stearic acid [92], and carbon nanotubes [76].

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 [94] 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 [99]. 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 H_c2 reduction [25]. 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 [18].

2. Thermal-strain-induced high J_c in high density SiC-MgB₂ bulk

The connectivity is considered to be a critical issue for improving the J_c 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 [103]. 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 J_c, H_irr, and H_c2 [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 J_c and T_c are improved by thermal strain on the interface between SiC and MgB₂ during the diffusion process [106] and hybrid physical-chemical vapor deposition (HPCVD) [107]. 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 J_c was derived from the height of the magnetization loop ΔM using the Bean model: J_c = 15ΔM/[π a³h)], where a and h are the radius and height of a cylindrical sample. The T_c 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].

Figure 1.

Rietveld refined XRD patterns of pure MgB₂ and 10wt% SiC doped MgB₂ samples made by the diffusion process at 850 ^oC for 10 h. The a-axis parameters are virtually the same for the two samples, whereas the c-axis parametar is stretched in the SiC-MgB₂ composite.

Figure 2.

(a) Thermal expansion coefficients (α) along the a-axis and c-axis for MgB₂ and SiC as a function of temperature. The averaged α(T) values for MgB₂ and SiC are plotted in the inset. (b) Plots of the normalized lattice changes for MgB₂ and SiC, and the thermal strain in the matrix during cooling from 1123 K to 0 K.

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 α_SiC decreases slightly from 5×10^-6/K at 1123 K to 2.5×10^-6/K at 0 K, whereas the ${\alpha }_{{\text{MgB}}_2}$ 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 [112], 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.

Figure 3.

(a) Bright field TEM image of the pure MgB₂ with indexed SAD pattern (inset). (b) High resolution electron microscope (HREM) image of interface in pure MgB₂ with FFT pattern along the [100] axis (inset). (c) Bright field TEM image of SiC-MgB₂. (d) HREM image of interface in SiC-MgB₂ and FFT patterns of SiC and MgB₂ from each side of the interface. The dashed line shows the interface of SiC and 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 [91], 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 [001] direction, while that for SiC is nearly isotropic, which is responsible for the enlarged c-axis of MgB₂.

Figure 4.

The magnetic J_c versus field at 5 K, 20 K, and 30 K for pure and nano-SiC doped samples. The inset shows the superconducting transitions of the two samples [103].

Based on the collective pinning model, [113], J_c is independent of the applied field in the single-vortex pinning regime (low magnetic field region: H < H_sb), where H_sb is the crossover field from single-vortex to small-bundle pinning. The J_c decreases exponentially in the small-bundle regime (high magnetic field: H_sb < H < H_irr). According to the dual model [14], the significant effect of SiC doping on J_c comes from the high level of C substitution on the B planes, which is responsible for the reduction of the self-field J_c[104, 105]. However, the SiC-MgB₂ composite sample shows not only an improved in-field J_c, but also no degradation in self-field J_c, as indicated in Figure 4. The approximate H_sb values are also indicated on the J_c curves for 20 K and 30 K, although H_sb 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 T_c 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 [107].

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 E_2g 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 [117]. 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 J_c 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 J_c values in pure the MgB₂ and the SiC-MgB₂ composite.

Figure 5.

Fitting and experimental results for the normalized ambient Raman spectrum of MgB₂ sintered at 850 ^oC for 10 h (a), and the cooling effect on the Raman spectrum (b) [103].

Figure 6.

Fitting and experimental results for the normalized ambient Raman spectrum of SiC-MgB₂ sintered at 850 ^oC for 10 h (a), and the cooling effect on the Raman spectrum (b) [103].

It should be noted that the broadened ω₂ peak in SiC-MgB₂ is a signal of strong electron-E_2g coupling, which is responsible for the high T_c in MgB₂. The electron-E_2g coupling constant is estimated from the Allen equation [118]: $${\Gamma }_2=2\pi {\lambda }_{E_{2g}}N(0){\omega }_{E_{2g}}^2$$, where Γ₂ is the ω₂ linewidth, $${\lambda }_{E_{2g}}$$ is the strength of the electron-E_2g 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), $${\lambda }_{E_{2g}}$$. The total density of states (DOS) at the Fermi energy, E_F, 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 [119]. 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 $${\lambda }_{E_{2g}}$$ values for the pure MgB₂ and SiC-MgB₂ are 2.327 and 2.706, respectively. The $${\lambda }_{E_{2g}}$$ of SiC-MgB₂ is just slightly higher than that of the pure MgB₂. However, the T_c 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 [120], as modified by Allen and Dynes:[121] $T_c=\frac{⟨{\omega }_{\log }⟩}{1.2}\exp \left(\frac{-1.04\left(1+\lambda \right)}{\lambda -{\mu }^{*}\left(1+0.62\lambda \right)}\right)$, where $⟨{\omega }_{\log }⟩=\left(390\times {\omega }_{E_{2g}}^2\times 690\right)$ is the averaged phonon frequency [122], with 390 and 690 cm^-1 being the phonon frequencies of the other modes in the MgB₂ system [123], μ^* is the Coulomb pseudopotential, taken as equal to 0.13 [124], 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 J_c and the T_c behaviors.

3. High connectivity MgB₂ wires fabricated by combined in-situ/ex-situ process

The self-field critical current density, J_c(0), of MgB₂ is much higher than for pure or doped samples processed with in-situ annealing [75, 81, 125-128]. J_c(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 J_c of bulk samples, the J_c(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 J_c 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 J_c was increased when the ex-situ powder ratio were 23% and 50% [133]. 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 J_c properties [134]. The home-made powder was fabricated through low temperature annealing, and the particle size was homogeneous and as small as ~200 nm. The J_c 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, T_c, 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 J_c 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. H_c2 and H_irr 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, A_F, was calculated based on the ρ(H, T) behavior.

Figure 7.

T_c dependence on sintering temperatures. The insets display the normalized magnetic moment dependence on measurement temperature for in-situ samples (upper) and for combined in-situ/ex-situ samples (lower) [134].

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 T_c values by 0.5 – 1 K compared with the in-situ/ex-situ ones. The T_c 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 T_c 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 8.

SEM images for the in-situ samples sintered at (a) 750 ^oC, (b) 850 ^oC, (c) 950 ^oC, and (d) 1050 ^oC. The crystal growth was improved gradually with increasing sintering temperature.

Figure 9.

SEM images for the combined in-situ/ex-situ samples sintered at (a) 750 ^oC, (b) 850 ^oC, (c) 950 ^oC, and (d) 1050 ^oC. The grains are inhomogeneous due to the ball milled MgB₂ precursor and the recrystallization effects.

Figure 10 compares the J_c dependence on applied magnetic field measured at 5 K and 20 K. The J_c value of 1050in is the worst because of the high MgO content. The J_c performances of 750in, 850in, 750inex, and 1050inex are quite similar over the whole field range. While the J_c deteriorated in 950in and 1050in under higher magnetic fields. 850inex and 950inex show amazingly high J_c properties. The J_c 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 J_c. The high crystal quality of 950in is of benefit for its high self-field J_c performance. 750inex and 850inex show low self-field J_c values. While 950inex and 1050inex display improved J_c, the values are lower than the self-field J_c of 950in due to the high MgO contents.

The J_c 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 [113]. The practicable J_c of MgB₂ falls into the single-vortex pinning region and the small-bundle pinning region in the phase diagram. The connectivity determines the J_c performance in the single-vortex pinning regime because the effective charge carrier density is responsible for the self-field J_c, while H_c2 and H_irr are responsible for the J_c performance in the small-bundle regime based on the disorder or defects.

Figure 10.

J_c dependence on applied magnetic field at 5 K and 20 K. The inset compares the self-field J_c behavior at 20 K [134].

A practical quantity to evaluate the connectivity is the active area fraction, A_F [21],

$$A_F=\frac{\Delta {\rho }_{\text{ideal}}}{\Delta \rho \left(300\text{K}\right)},$$E2

where,

$$\Delta {\rho }_{\text{ideal}}={\rho }_{\text{ideal}}\left(\text{300K}\right)–{\rho }_{\text{ideal}}\left(40\text{K}\right)\approx 9\mu \Omega \text{cm},$$E3

is the resistivity of fully connected MgB₂ without any disorder [18], and

$$\Delta \rho \left(300\right)=\rho \left(300K\right)–\rho \left(T_c\right).$$E4

Figure 11.

A_F dependence on sintering temperature. The short dashed line indicates A_F = 0.175. The upper left inset compares the resistivity of the in-situ samples. The upper right inset compares the resistivity of the combined in-situ/ex-situ samples [134].

Figure 11 compares the A_F of all the samples. All the samples show lower connectivity compared with those of ideal crystals, as indicated by the low A_F values. The A_F 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 A_F slightly, which indicates that the connectivity is easy to improve for in-situ samples. Although the MgO content is high in 1050in, its high A_F value is attributed to sufficient crystallization. However, its low magnetic field J_c was degraded by MgO. The combined in-situ/ex-situ samples show rather low A_F 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 A_F values for the in-situ/ex-situ samples due to the improved solid state reaction. The self-field J_c performances are also improved for 950inex and 1050inex.

The J_c 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 H_c2 and H_irr determine the strength of the pinning force, as shown in Figure 12. The high field J_c performance is in agreement with the H_c2 and H_irr behavior. The 1050in sample shows the lowest H_c2 and H_irr among all the samples. 850inex and 950inex show the highest H_c2 and H_irr, which are consistent with their high J_c values in the small-bundle region. The ex-situ powder induces strong disorder and proper crystallization, which are responsible for the high H_c2 and H_irr values.

Figure 12.

H_c2 and H_irr of (a) the in-situ samples and (b) the combined in-situ/ex-situ samples [134].

In summary, both connectivity and disorder show strong influences on the J_c properties of MgB₂. The connectivity is responsible for the high effective charge carrier density, which determines the self-field J_c performance. The strong flux pinning force induced by defects and disorder is responsible for the promising J_c in high magnetic field. The enhanced J_c performances of 850inex and 950inex are attributed to the optimized connectivity and disorder. The J_c values obtained in this work are still far below the J_d value, ~8.7×10⁸ A/cm² for pure MgB₂. The J_c 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 J_c of MgB₂ superconductors has been enhanced through many different kinds of dopants or additives [125], 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 J_c properties [140]. 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 J_c 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.

Figure 13.

XRD patterns of nano-SiC doped MgB₂/Fe wires fabricated by (a) the in-situ technique and (b) the in-situ/ex-situ technique, with sintering at 750, 850, 950, and 1050 ^oC, respectively. All patterns were indexed with MgB₂, MgO, Mg₂Si, SiC, and Mg [140].

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 (T_c) of the two batches of samples are compared in Figure 15. It is found that the T_c 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 T_c dependence on sintering temperature of the two batches of samples is exactly the same, which means that the T_c 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.

Figure 14.

SEM photos of (a) 850in and (b) 850inex [140].

Figure 15.

T_c dependence on sintering temperature. The insets show the normalized magnetic moment dependence on the measuring temperature for the in-situ samples (upper) and for the in-situ/ex-situ samples (lower) [140].

The J_c 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 J_c 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 J_c 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 J_c values but its J_c deteriorates with increasing magnetic field. The J_c properties of 950in, 850inex, and 950inex show outstandingly high J_c 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 J_c 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 J_c values in lower magnetic field. 1050in and 850inex show competitive self-field J_c. 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 J_c 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. ξ_ab(0) = 3.7 – 12 nm, and ξ_c(0) = 1.6 - 3.6 nm [110], 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 J_c performance in high magnetic fields. According to the collective pinning model [113], the J_c 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 J_c performance depends on the flux pinning force due to the increased high H_c2 and H_irr. The approximate H_sb values are also indicated on the J_c curves estimated at 20 K, as shown in the inset of Figure 16, where H_sb is the crossover field from single-vortex to small-bundle pinning based on the collective pinning model. However, H_sb has not been detected at 5 K due to the relatively high supercurrents [103].

Figure 16.

J_c at 5 K and 20 K. The inset indicates the J_c behavior in low magnetic field at 20 K. H_sb, the crossover field from single-vortex to small-bundle pinning, is indicated by the dashed-dotted line at its probable position on the J_c curves [140].

The strength of the pinning force can be reflected by the dependence of H_c2 and H_irr on the normalized temperature, as shown in Figure 17. Carbon substitution is one of the most effective methods to improve the H_c2 and H_irr because of the increased scattering by C doping, and the increased scattering can also contribute to decreased T_c and merging of the two gaps [144]. It should be noted that the H_c2 and H_irr for 750in, 850in, and 750inex have their highest values at low temperature, which means a strong flux pinning force. The poor J_c is of these samples attributed to the lower density of current carriers. Both 1050in and 1050inex show the lowest H_c2 and the lowest H_irr among all the samples. 850in, 950in, 850inex, and 950inex show competitive H_c2 and H_irr performances, which are responsible for their high J_c values under high magnetic field.

Figure 17.

Comparison of H_c2 (solid symbols) and H_irr (open symbols) for (a) MgB₂/Fe wires doped with nano-SiC and (b) MgB₂/Fe wires doped with nano-SiC, with sintering at 750, 850, 950, and 1050 ^oC, respectively [140].

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 J_c performance of nano-SiC doped MgB₂. The current carrier density determines the J_c behavior in the single-vortex regime, where J_c is independent of applied magnetic field. The flux pinning force determines the J_c 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.

5. Conclusions

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.

Acknowledgments

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.