High-field superconductivity in C-doped MgB₂ bulk samples prepared by a rapid synthesis route

Abstract

The upper critical field sets the thermodynamic limit to superconductivity. A big gap is present between the upper-critical-field values measured in MgB₂ polycrystalline bulk superconductors and those of thin films, where values as high as ~ 50 T have been achieved at 4.2 K. Filling this gap would unlock the potential of MgB₂ for magnet applications. This work presents the results of an extensive experimental campaign on MgB₂ bulk samples, which has been guided by a Design of Experiment. We modeled the dependence of the upper critical field on the main synthesis parameters and established a new record (~ 35 T at 4.2 K) preparing C-doped bulk samples by a non-conventional rapid-synthesis route. This value appears to be an upper boundary for the upper critical field in bulk samples. Structural disorder in films seems to act selectively on one of the two bands where superconductivity in MgB₂ takes place: this enhances the upper critical field while reducing the critical temperature only by few Kelvins. On the other hand, the critical temperature in bulk samples decreases monotonically when structural disorder increases, and this imposes a limit to the maximum achievable upper critical field.

Introduction

The MgB₂ superconductor has significant potential for practical applications. The main points of strength are its critical temperature close to 40 K, which may allow operating in cryogen-free environments, the low cost of precursor materials, and the ease of manufacture. However, today’s applications are limited to market niches, mainly constituted by low-field magnetic-resonance-imaging magnets and current leads^1,2,3,4,5,6. The upper critical field, H_C2, is well below 20 T at 4.2 K in polycrystalline binary MgB₂, whilst it can exceed 50 T in carbon-doped films^7,8. This value is about twice the H_C2 of Nb₃Sn, which is largely used in magnet applications and is considered one of the most promising candidates to realize next-generation particle-accelerator magnets⁹. In spite of the considerable efforts undertaken to reproduce the same results in polycrystalline bulk materials and wires, the highest µ₀H_C2 achieved so far is ~ 34 T at 4.2 K, as measured in a double-walled-carbon-nanotube (DWCNT) doped bulk sample¹⁰.

MgB₂ has a planar structure with honeycomb B layers separated by Mg atoms. Strong sp² hybrid σ bonding within the in-plane B atoms gives rise to the 2-dimensional σ band. Boron p_z orbitals lead to the 3-dimensional π band^11,12. Superconductivity takes place on the two bands with different energy gaps of ~ 2.2 meV (π band) and ~ 7.0 meV (σ band) at 0 K¹³. Both superconducting gaps vanish at the bulk critical temperature T_C¹⁴. Structural disorder can induce charge-carrier scattering on different channels: intraband scattering in each of the σ and π bands, and interband scattering between them¹⁵. Enhanced interband scattering leads to a decrease of T_C, whilst theoretical models predict that H_C2 can be significantly improved at low temperatures by selectively increasing the π-band intraband scattering^15,16. Nanoscale disorder can be tuned by chemical doping^8,10,17, irradiation^16,18, and preparation conditions^18,19. In the case of MgB₂, C proved to be the most effective way to enhance H_C2 by doping^10,20,21. C is not expected to have the same effect on the MgB₂ crystal structure in films and bulk samples^22,23, and this may lead to variations in the scattering rates^19,22,24. To date, the scenario that leads to record-high H_C2 in films is unclear. The out-of-equilibrium environment typical of film-growth processing may play a key role in enhancing H_C2. Indeed, µ₀H_C2(4.2 K) as high as ~ 44.5 T was measured in a binary film²⁵, indicating that C doping is not an exclusive way to achieve very high H_C2.

This work presents a systematic study of the effects of the synthesis conditions on H_C2 for C-doped bulk samples. We employed a rapid-synthesis route, which allowed us to explore ranges of variation of the synthesis conditions not achievable with traditional techniques. By means of a Design of Experiment, we defined the H_C2 response surface as a function of the main variables of the manufacturing route and thus determined the synthesis-parameter ranges that maximize H_C2. We found that, in spite of an enhanced substitution rate of C in Mg(B_1−xC_x)₂, µ₀H_C2 appears bounded to maximum values of ~ 26 T and ~ 35 T at 10 K and 4.2 K, respectively. These figures constitute new records for H_C2 in bulk samples but remain far below what is achievable by the material in film form. We show that in bulk samples, H_C2 and T_C correlate well with the lattice constant a. H_C2 values of bulk samples (from this work and from the literature) on which disorder has been introduced by different sources can be estimated with an uncertainty below ~  ± 20% by knowing T_C. The rapid-synthesis route allowed the production of samples with very-high irreversibility fields (H_Irr). This is an important result for applications because H_Irr defines the maximum field at which superconductors can be operated in magnets. The potential scalability for large volume productions of wires and bulk materials is another point of strength of this technique.

Results

The MgB₂ bulk sample manufacturing process consists of a combination of the Internal Magnesium Diffusion (IMD) and the Powder-in-Closed-Tube (PiCT) techniques^26,27. Samples prepared by IMD are typically characterized by high electrical connectivity²⁸. The PiCT technique allows in turn achieving a high density of the reacted MgB₂ phase and a high reproducibility of the sample properties²⁷. Samples were reacted using a laboratory-made induction furnace, which allowed us to heat with ramp rates as high as ~ 1000 °C/min and to quench the reaction process by injecting high-pressure Ar on the sample crucible. We prepared few binary MgB₂ samples as reference, and ~ 50 C-doped samples, most of them with a nominal composition Mg(B_0.9C_0.1)₂. Indeed, x = 0.1 was proven to maximize H_C2 in Mg(B_1−xC_x)₂ polycrystalline samples in the case of DWCNT¹⁰, which is the C-dopant used in this work. Details about the manufacturing process are reported in section Methods.

Based on a previous work of ours²⁹, we identified five synthesis parameters whose variation has a major effect on the samples’ superconducting properties, namely the heating ramp rate (HR), the dwell temperature (T_d), the dwell time (t), the pressure of the Ar-gas quenching jet (ArP), and the pressure applied to the precursors before synthesis (P). These variables reciprocally interact in the determination of H_C2, making the quest for the “best” synthesis conditions very challenging. The Design of Experiment (DoE) is a statistical tool used to determine the effects of experimental factors on a desired output in a system. It offers a set of advantages over the traditional one-variable-at-a-time approach since it can help to resolve parameter interactions and provide detailed maps of the system behavior³⁰. We used two types of DoE in this work. First, we carried out a screening DoE with the aim of identifying the area of the synthesis-parameter space where the highest H_C2 values are localized. We prepared 11 samples, which were characterized in terms of T_C, ΔT_C, H_C2, H_irr, and lattice parameters a and c. H_C2 and H_Irr were evaluated at 10 K, for two main reasons. The first one is conceptual, as the key interest in MgB₂ for magnet technology is for cryocooled systems operating above 4.2 K. The second one is practical and mainly related to the magnetic-field range (0–21 T) available at the University of Geneva, where most of the measurements were performed. The main properties of the samples prepared in the frame of the screening DoE are reported in Table 1. Based on this preliminary investigation, we performed a Response-Surface-Methodology (RSM) DoE, which is used to produce a detailed mathematical model of the process behavior as a function of the input variables^30,31. The RSM DoE required the preparation of 26 samples, which were all investigated at the University of Geneva. A sample selection was further characterized up to 35 T in an extended temperature range down to ~ 2.5 K at the LNCMI (Grenoble, France). The superconducting and structural properties of the samples prepared in the frame of the RSM DoE are reported in Table 2. Details on both DoEs, including comments on the experimental reproducibility, are reported in Methods.

Table 1 Main properties of the 11 samples of the screening Design of Experiment: critical temperature T_C, superconducting transition width ΔT_C, irreversibility field H_irr at 10 K, upper critical field H_C2 at 10 K, lattice parameters a and c. The estimated standard deviation for a and c is $\lesssim$ 0.0001 Å for all samples.

Table 2 Main properties of the 26 samples of the Response-Surface-Methodology Design of Experiment: critical temperature T_C, superconducting transition width ΔT_C, irreversibility field H_irr at 10 K, upper critical field H_C2 at 10 K, lattice parameters a and c. The estimated standard deviation for a and c is $\lesssim$ 0.0001 Å for all samples.

We used data from Table 2 to evaluate the H_C2 response surface as a function of the synthesis parameters. The best-fit surface was assessed with the software STATISTICA from StatSoft neglecting third-order interactions between the synthesis parameters³¹, as per the following quadratic polynomial:

$$\mu_{0} H_{C2} \left( {T_{d} ,HR,t,P} \right) = {\text{A}}_{0} + {\text{A}}_{1} T_{d} + {\text{A}}_{2} T_{d}^{2} + {\text{A}}_{3} HR + {\text{A}}_{4} HR^{2} + {\text{A}}_{5} t + {\text{A}}_{6} t^{2} + {\text{A}}_{7} P + {\text{A}}_{8} P^{2} + {\text{A}}_{9} T_{d} HR + {\text{A}}_{10} T_{d} t + {\text{A}}_{11} T_{d} P + {\text{A}}_{12} HR t + {\text{A}}_{13} HR P + {\text{A}}_{14} t P.$$

(1)

The best-fit values of the coefficients “${A}_{i}$” were evaluated by the least squares method and are listed in Table 3. For the RSM-DoE, we kept the Ar-jet pressure constant to its maximum value.

Table 3 Best-fit values of the coefficients ${\mathrm{A}}_{\mathrm{i}}$ of the $\it {{H}}_{{C}2}$ response surface at 10 K, as per Eq. (1).

Equation (1) allowed us to identify the input-parameter combinations that maximize H_C2 at 10 K. H_C2(T_d,HR,t,P) presents maxima in two synthesis-parameter regions characterized by: (1) high reaction temperature and low pressure (T_d > 900 °C and P < 250 MPa), (2) low reaction temperature and high pressure (T_d < 900 °C and P > 250 MPa). In order to allow the visualization the two maxima, panels (a) and (b) of Fig. 1 report two slices of the H_C2(T_d,HR,t,P) response surface performed at HR = 700 °C/min and P = 125 MPa, HR = 1000 °C/min and P = 375 MPa, respectively. The highest maximum (µ₀H_C2 ~ 26 T) is expected when combining high T_d and low P. Based on the predictions of the RSM DoE, we prepared further 7 samples selecting synthesis conditions favorable for H_C2. Details about the preparation conditions for this “post-DoE” batch are reported in Table 10 of Methods. Superconducting and lattice parameters are reported in Table 4.

Figure 1

H_C2(T_d,t) response surface at 10 K in the synthesis-parameter ranges where the DoE predicts the presence of relative maxima. Panel (a) is a slice of the H_C2(T_d,HR,t,P) surface for fixed values of HR (700 °C/min, “0” in coded units) and P (125 MPa, “α−” in coded units). From this plot it is deduced that at low pressure H_C2 is maximized for T_d $\gtrsim$ 950 °C and 60 min $\lesssim$ t $\lesssim$ 90 min. Panel (b) is a slice of H_C2(T_d,HR,t,P) for fixed values of HR (1000 °C/min, “+” in coded units) and P (375 MPa, “α+” in coded units). From this plot it is deduced that at high pressure H_C2 is maximized for T_d $\lesssim$ 950 °C and 60 min $\lesssim$ t $\lesssim$ 90 min.

Table 4 Superconductivity and lattice parameters of the “post-RSM-DoE” sample batch. Estimated standard deviation for a and c is $\lesssim$ 0.0001 Å for all samples.

The manufacturing process adopted in this work includes the flattening by uniaxial pressing of the SS tube filled with the precursors. It has been shown that non-hydrostatic cold-deformation processes may lead to a partial texturing of the MgB₂ crystallite c-axis along the applied-pressure direction^32,33. All samples from this study were investigated with the magnetic field perpendicular (⊥) to the uniaxial-pressure direction. Selected samples with the best in-field performances were further investigated for magnetic fields parallel (//) to the pressure direction. This short list includes samples from the RSM DoE (samples RSM_1, RSM_17, RSM_25) and from the post-RSM-DoE batch (samples M_1 and M_3). Obtained H_C2 and H_Irr, as evaluated at 10 K for the two orientations of the field, are reported in Table 5. In polycrystalline samples, R(H) measurements allow one to probe H_C2 for fields perpendicular to the crystallographic c axis, ${H}_{C2}^{\perp c}$, regardless of the applied-field direction. Since ${H}_{C2}^{\perp c}$>$H_{C2}^{//c}$, upon decreasing the applied field R(H) starts devting from its normal state value as soon as $H \le H_{C2}^{ \bot c}$, because grains with the c-axis perpendicular to the external-field direction become superconducting and do not contribute to the electrical resistance anymore. The variability between H_C2 values of Table 5 measured for the two field orientations is below ~ 5% and has to be considered as an experimental uncertainty inherent to the procedure adopted to determine H_C2. H_Irr is expected to be independent of the magnetic-field orientation in untextured samples, whilst a dependence is expected in textured or weakly-textured specimens³⁴. In particular, higher H_Irr should be measured when H is perpendicular to the uniaxial-pressure direction³⁴. In the case of the sample M_1, we measured H_Irr higher by ~ 15% in this orientation. This difference is above the measurement uncertainty of ~ 5% and is an indication of weak c-axis texturing along the applied-pressure direction³⁴.

Table 5 H_Irr and H_C2 values as measured in a selection of samples for two different orientations of the magnetic field, parallel and perpendicular to the uniaxial pressure direction.

Discussion

Record high H_C2 and H_Irr

Figure 2 presents the temperature (T) dependence of H_C2 (panel (a)) and H_Irr (panel (b)) for the best-performing samples from this work listed in Table 5. For comparison, we included in the plot the curves corresponding to the polycrystalline sample with the highest H_C2 from [10], which was prepared with a nominal C content x = 0.1 (hereafter, we refer to this sample as “[10]_0.1”). No texturing is expected for [10]_0.1 as it was reacted in the absence of any external pressure. We highlighted in the graph two zones associated with temperatures higher (white background) or lower (grey background) than 4 K. At T ≥ 4 K, the rapid-synthesis route allowed us to achieve H_C2 values comparable with the record-high H_C2 of [10]_0.1, for various combinations of the synthesis parameters. This is good news because it indicates that the synthesis conditions leading to μ₀H_C2 above 30 T at 4.2 K and above 23 T at 10 K can be adapted to consider specific manufacturing requirements. Furthermore, data from [10] represent the result of a single experiment that was never replicated. At T ≥ 4 K, H_C2 measured in the sample M_1 in parallel field overcomes that of the sample [10]_0.1 and thus set a new record for H_C2 in MgB₂ bulk samples. At T < 4 K, the H_C2(T) curves of samples from this work stay all below that of [10]_0.1. Furthermore, we do not observe any sudden increase of H_C2 at T $\lesssim$ 2 K. The temperature dependence of H_Irr is reported in Fig. 2b. Samples reacted by the rapid-synthesis route exhibit higher H_Irr with respect to [10]_0.1. The difference is up to ~ 8 T in the case of the sample M_1. This result is important for applications since H_Irr defines the operational limits in superconducting magnets. H_Irr seems to extrapolate linearly down to T = 0 K. In the case of untextured MgB₂ bulk samples prepared by standard synthesis routes, it is typically observed H_Irr ~ 0.5 H_C2^35,36. Data of Tables 1, 2 and 4 allow one to deduce that ~ 70% of the samples from this work have H_Irr $\gtrsim$ 0.7 H_C2. In the framework of the anisotropic Ginzburg–Landau theory, H_Irr of untextured samples is described by:

$$H_{Irr} = \frac{{H_{C2}^{ \bot c} }}{{\sqrt {\left( {\gamma^{2} - 1} \right)p_{c}^{2} + 1} }},$$

(2)

Figure 2

Temperature dependence of the upper critical field (a) and of the irreversibility field (b) for the best-performing samples from this work. Data of the record-high H_C2 bulk sample from¹⁰ are reported for comparison.

where γ=${H}_{C2}^{\perp c}/{H}_{C2}^{//c}$ is the upper-critical field anisotropy, and p_c is the percolation threshold, which represents the minimum superconducting-grain fraction for a continuous path through the superconductor^34,37. In granular superconductors, p_c depends on the coordination number, i.e., on the average number of grain first neighbors (that in turn depends on the grain packing density), but also on the presence of insulating spurious phases. p_c can thus be considered as an indicator of the electrical connectivity, K, in a superconductor: the higher p_c, the lower K. Data of Table 5 show that the difference in H_Irr measured for the two orientations of H is clearly above the experimental uncertainty only in the case of the sample M_1. For the other samples listed in Table 5, we can in a first approximation assume that texturing effects are negligible and make use of Eq. (2) to evaluate p_c. The low-temperature value of γ can be estimated from the sample critical temperature by the following empirical expression valid also for C-doped samples, which was derived in^18,38 comparing results obtained in a large set of samples with different T_C:

$$\gamma = \frac{{t_{c}^{2} + 16.7t_{c} \left( {1 - t_{c} } \right)}}{{3.88 - 3.724t_{c} }}.$$

(3)

Here ${t}_{c}$ = ${T}_{c}$/${T}_{c0}$ and ${T}_{c0}$ = 39.43 K is the T_C expectation for samples in the clean limit^18,38. Having estimated $\gamma$, ${p}_{c}$ can be calculated by Eq. (2) using for ${H}_{C2}^{\perp c}$ and H_Irr the experimental results reported in Table 5, averaging the values obtained in the two field orientations. Figure 3 reports ${p}_{c}$ and $\gamma$ evaluated at 10 K using Eqs. (2) and (3). Samples prepared by the rapid-synthesis route have ${p}_{c}$ smaller than that of sample [10]_0.1. Reported values are more generally low even when compared with further results from the literature for ${p}_{c}$, which is typically $\gtrsim$ 0.25 for C-doped samples^37,38,39,40. This result indicates that the higher H_Irr measured in the samples prepared with the rapid synthesis route has to be ascribed to a better electrical connectivity between the superconducting grains. This conclusion is further supported by the high values measured for the sample mass density. About 90% of the samples prepared in this work have mass density $\gtrsim$ 2.1 g/cm³, which is ~ 80% of the theoretical value for MgB₂⁴¹.

Figure 3

Low-temperature upper-critical-field anisotropy (γ) and percolation threshold p_C calculated as described in the text for the samples RSM_1, RSM_17, RSM_25, M_3 from this work and the record-high-H_C2 sample from¹⁰.

Effects of C doping on the crystal structure and electronic properties

We evaluated by X-ray powder diffraction experiments the a and c lattice parameters for all the samples produced in this campaign. Values are reported in Tables 1, 2 and 4. On the atomic scale, C can be substituted for B in the MgB₂ crystal structure, or remain interstitial within B rings^22,23. Using the MgB₂ phase as a reference state, the enthalpy of formation at 0 K is negative in the case of C substitution, positive for interstitial C²³. Therefore, substitutional insertion is energetically favored. C atoms can also be segregated outside the superconducting grains, thus impacting extrinsic superconducting properties such as grain connectivity or vortex pinning^21,42. Experimental and theoretical works have shown that lattice parameter a decreases upon augmenting x in bulk Mg(B_1−xC_x)₂ samples, whilst c remains nearly constant^22,23. In Mg(B_1−xC_x)₂ films, both a and c are observed to increase with the C-doping content^22,23. Figure 4a reports the experimental dependence of a on the nominal amount of C doping. We included in this chart a binary sample (BIN-STD_1) and three IMD bulk samples reacted in our laboratory using a conventional muffle furnace (STD_0.01, STD_0.025, STD_0.1), binary and DWCNT-doped samples from [10] ([10]_ BIN, [10]_0.01, [10]_0.025, [10]_0.05, [10]_0.1), and a selection of samples from the RSM DoE, that are representative of the a variability in our experiment. Samples reacted with conventional furnaces both from this work and from the literature show that a decreases upon increasing the nominal amount of C. However, a saturates at values approaching ~ 3.065 Å for x ~ 0.1, as also reported in^42,43,44,45. The rapid-synthesis route leads to a large variability of a in spite of the same nominal doping (x = 0.1). Very interestingly, most of the samples prepared with this process have a < 3.065 Å, indicating that the rapid-synthesis route allows for the substitution of a larger fraction of C into the B sites, at a same nominal doping. As a general trend, we found that low a values are typically associated with high T_d. The fact that high dwell temperatures are beneficial for C substitution in MgB₂ agrees with further results from the literature^44,46. A dedicated study would be needed to draw definitive conclusions about the microscopic mechanisms that lead to a more efficient C substitution when using the rapid-synthesis route. On the other hand, we can infer that the rapid heating and cooling (quench), which are unique characteristics of the employed route, play a certain role in enhancing the C-substitution efficiency with respect to conventional synthesis methods. In particular, it is possible that the C segregation out of the grains during a slow cooldown is hindered by the post-reaction quench.

Figure 4

Dependence of the a lattice parameter on the nominal C content (a) and with the effective substituted C content (b). The two panels include data from this work and from¹⁰. Lattice parameter uncertainty is smaller than the symbols’ size.

Comparison between neutron-diffraction experiments and X-ray analyses has previously demonstrated that the actual level of C substitution in Mg(B_1–xC_x)₂ can be estimated as x≈7.5⋅Δ(c/a), where Δ(c/a) is the change in c/a compared to a pure sample^21,44,47. The variation of a as a function of the actual substituted-C content is reported in Fig. 4b. Effective C substitution up to 90% of the nominal DWCNT content is reached with the rapid-synthesis route. The effectiveness of this route in substituting C for B is further confirmed by XPS, which we carried out on a batch of five samples prepared by the rapid-synthesis route (RSM_1, RSM_14, RSM_23, RSM_25, M_1) and four samples prepared using a standard furnace (STD_BIN, STD_0.01, STD_0.025, STD_0.1). Figure 5a shows the B 1s spectrum for the sample M_1, together with the result of least squares fitting of the spectrum considering pure Gaussian-line shapes, in agreement with previous reports^48,49. The B 1s spectrum is composed of three peaks centered at ~ 188.3 eV, ~ 190.5 eV and ~ 193.5 eV. In agreement with other XPS reports, we assign the main peak located at 188.3 eV to B in MgB₂, and the two peaks at 193.5 eV and 190.5 eV to B₂O₃ and other contaminants of B, respectively^11,48,50,51. The B 1s spectra of all measured samples are qualitatively similar and show only a single broad peak associated with MgB₂ or Mg(B_1−xC_x)₂. Interestingly, as shown in Fig. 5b, the binding energy of this peak increases as the a lattice constant contracts. Samples reacted with the rapid-synthesis route which have a < 3.065 Å exhibit a B 1s peak position located at up to ~ 0.2 eV higher binding energy compared to the samples with the larger a values. This binding-energy change can be attributed to the shift of the Fermi level due to the additional electrons doped into the system when substituting C for B^20,47. Therefore, Fig. 5b provides further evidence that substitutional C doping is higher for the samples prepared by the rapid-synthesis route. Since, as shown in Fig. 4, effective substitution in our samples is always less than 0.1, the maximum Fermi-level shift of ~ 0.2 eV is consistent with the value of ~ 0.3 eV theoretically predicted for effective x = 0.1⁵². The inset of Fig. 5b reports the best-fit values of the full width at half maximum (FWHM) as a function of the peak position for the B 1s peak attributed to MgB₂. The FWHM increases with the binding energy, which could be due to an increasing contribution from a peak component associated with B-C bonding, further confirming the increased substitution of C in the MgB₂ lattice.

Figure 5

(a) XPS B 1s spectrum of sample M_1 along with the Gaussian-peak best-fit curves. A linear background has been subtracted from the raw spectrum before fitting. (b) Variation of the lattice parameter a with the main-peak position of the B 1s spectrum. The inset shows the correlation between the main-peak position and its FWHM. Dashed straight lines are eye-guide lines.

Correlation between H_C2 and T_C with the a lattice parameter

Substituted, interstitial or intergranular C can affect differently the intraband and interband scattering rates. C substitution should primarily lead to an increase of the σ-band intraband scattering, whilst grain boundaries should affect the scattering rates on both σ and π bands²⁴. No study reports on the role of the energetically unfavorable interstitial C on the scattering rates. The lattice parameter a can be used as a sort of caliper to measure the C substitution in the MgB₂ lattice. Figure 6a shows the correlation between H_C2 at 10 K and a for all samples investigated in this study. Starting from the binary sample located at the bottom-right corner of the chart, one observes that H_C2 initially increases upon lowering a, it reaches a maximum when a ~ 3.06 Å and finally decreases for a $\lesssim$ 3.06 Å. The enhancement of H_C2 upon increasing the effective C doping has to be mainly ascribed to an increased intraband scattering, as further documented in the literature^15,16,20. The introduction of C atoms in the MgB₂ structure also leads to a reduction of T_C, which is steeper for a $\lesssim$ 3.06 Å as shown in Fig. 6b. Band filling due to electron doping is expected to lower T_C⁵³. In particular, a linear decrease of T_C with x in Mg(B_1−xC_x)₂ is theoretically predicted for doping levels up to x ~ 0.15, if changes in the bands and phonon spectrum due to the elemental doping are considered⁵³. Our experimental observation that the slope of the T_C vs a dependence changes for a $\lesssim$ 3.06 Å, which corresponds to x ~ 0.08, suggests that the effects of interband scattering cannot be ruled out in our series of samples, at least for those samples with a $\lesssim$ 3.06 Å. The enhancement of the interband scattering rate because of substitution of C for B has been further documented in the literature^15,53. In spite of the loss in condensation energy due to the lower T_C, moderate levels of lattice deformation characterized by a in the range ~ 3.06 Å to  ~ 3.08 Å lead to a net gain in terms of H_C2. A further decrease of the lattice parameter a (a $\lesssim$ 3.06 Å) results in a reduction of H_C2. Analogous conclusions are drawn when analyzing the evolution of H_C2 at 4.2 K with a.

Figure 6

Correlation between H_C2 at 10 K (a) and T_C (b) with the lattice parameter a. The highest H_C2 values are achieved when a is ~ 3.06 Å. For the same a value, a marked change in the slope of the T_C vs a dependence is observed. Dashed straight lines are eye-guide lines.

Figure 7 reports the correlation between H_C2(4.2 K) and T_C. We included in the chart all samples from this work investigated at T = 4.2 K and data available in the literature for C-doped^10,54 and irradiated bulk samples^16,55. A binary bulk prepared in our laboratory (STD_BIN) was added as a reference. In order to allow a comparison with the results obtained in films, we also included data of C-doped films^7,8, of a high-disorder binary film²⁵, and of a 0.75 μm-thick polycrystalline coated conductor deposited on SiC fibers that all showed µ₀H_C2 ~ 50 T⁵⁶. In the case of the films, we reported only H_C2 data measured with the field parallel to the surface (the highest values). Indeed, R(H) experiments carried out on polycrystalline samples provide an estimation of ${H}_{C2}^{\perp c}$³⁴. In bulk samples, H_C2 is maximized when T_C is in the range 34 K ± 2 K, regardless of the specific source of disorder (C doping, irradiation, synthesis conditions). All experimental H_C2(4.2 K) data of bulk samples can be predicted from T_C with an uncertainty below ~  ± 20% by an asymmetric 2-sigma function (dashed curve in Fig. 7):

$$\upmu _{0} H_{C2} \left( {{4}.{2} {\text{K}}} \right) = {\text{A}} + {\text{B}}*\left( {{1}/\left( {{1} + {\exp}\left( { - \left( {T_{C} - {\text{T}}_{0} + {\text{w}}_{{1}} /{2}} \right)/{\text{w}}_{{2}} } \right)} \right)} \right)*\left( {{1} - {1}/\left( {{1} + {\exp}\left( { - \left( {T_{C} - T_{0} - {\text{w}}_{{1}} /{2}} \right)/{\text{w}}_{{3}} } \right)} \right)} \right).$$

(4)

Figure 7

Correlation between H_C2 (4.2 K) and T_C for bulk samples from this work and from the literature. The dashed line represents the best fit curve of all experimental data obtained in bulk samples, as described in the text.

The parameters’ best-fit values and their statistical errors, as determined by least squares fitting, are A = 2.6 T ± 1.5 T, B = 31.2 T ± 2.4 T, T₀ = 32.5 K ± 0.3 K, w₁ = 12.0 K ± 0.7 K, w₂ = 2.4 K ± 0.5 K, w₃ = 1.1 K ± 0.2 K. The dashed curve of Fig. 7 can be considered as an empiric master curve for the H_C2 vs T_C dependence in MgB₂ bulk samples in which disorder has been induced by doping, irradiation or synthesis conditions. At odds with what observed in bulk MgB₂, a large variability of H_C2 is found for films with similar T_C. No correlation between H_C2 and the lattice constants a or c is reported for C-doped films^7,8. What is thus the origin of the very-high upper critical field (μ₀H_C2(4.2 K) > 40 T) measured in MgB₂ films? We found that only three samples out of all the C-doped films reported in the literature have measured H_C2(4 K) values distinctly above the bulk-sample master curve^7,8. C doping cannot be considered the only key to achieve high H_C2 because the binary film from²⁵ prepared by pulsed-laser deposition showed μ₀H_C2(4.2 K) ~ 44.5 T. Therefore, even if this and other studies prove that C doping can significantly enhance H_C2 with respect to binary samples, there has to be a specific “type of disorder” able to unlock much higher H_C2 values. The two-band theory suggests that H_C2 can be significantly enhanced at low temperatures if the π band is dirtier than the σ band¹⁵. In this case, an upward curvature of H_C2 vs T is expected. Contrary to what reported in¹⁰, our results do not give evidence of an upward curvature of H_C2(T) at low T. This indicates that C substitution in Mg(B_1−xC_x)₂ does not selectively increase the π-band intraband scattering rate, in agreement with theoretical expectations²⁴. In view of the lower amount of substituted C in the samples from¹⁰ with respect to those from this work, one could argue that interstitial C is at the origin of the observed upward curvature. However, the presence of interstitial C was not proven in¹⁰ and C should be more probably segregated outside the superconducting grains⁴⁵. We found that the films of Fig. 7 that have H_C2 above the bulk-sample master curve share a fiber texture, which is characterized by a rotational degree of freedom of the crystallographic a axis around the c axis⁵⁷. No information about the type of texture is reported for the film 8 from⁷, which has μ₀H_C2(4.2 K) ~ 52 T. Zhu et al. observed in their film a tilt at the nanometric scale of the c axis induced by C doping and pointed it out as the possible cause of the very-high H_C2²⁴. This kind of disorder may likely perturb the B p_z orbitals from which the π band arises^12,24, and possibly enhances selectively the intraband scattering in this band. It is possible that the same kind of lattice distortion is achieved in those of the C-doped films that present a fiber texture along the c axis. c-axis disorder due to nanometric inclusions was also reported for the binary MgB₂ film showing record-high H_C2^25,58. Further studies about the correlation between microstructural properties of films and H_C2 should be carried out to achieve a more complete understanding of the mechanisms responsible of the record H_C2 values. This study points out that in bulk samples the introduction of disorder by different sources enhances both the intraband and interband scattering rates leading to an upper limit for μ₀H_C2(4.2 K) of ~ 35 T. In the case of the thin films and of the coated conductor deposited on SiC fibers, it is another type of structural defect that leads to μ₀H_C2(4.2 K) ~ 50 T, while keeping T_C above 25 K.

Conclusions

We presented the results of a wide experimental campaign to investigate the role of carbon doping in the enhancement of H_C2 in MgB₂ bulk samples. The main purpose of this work was understanding whether the very-high μ₀H_C2 values of ~ 50 T at 4.2 K, as observed in disordered films, can also be achieved in polycrystalline samples. This would allow widening the application domain of MgB₂ in magnet technology. Based on theoretical and experimental evidences that structural disorder is the key to enhance H_C2, we produced samples by a rapid-synthesis route, which allowed us to explore ranges of variation of the synthesis conditions not achievable with traditional techniques. In particular, we quenched the synthesis process with the idea of freezing the system in out-of-equilibrium configurations. The study was guided by a Design of Experiment. This statistical tool allowed us to characterize the H_C2 surface response as a function of the synthesis parameters. We defined different regions of the synthesis-parameter space that maximize H_C2. Experimental H_C2 data reflected with good precision and reproducibility the predictions of the DoE. X-ray and XPS analyses demonstrated that the rapid-synthesis route allows us to reach levels of C substitution in the B sites not achievable with conventional manufacturing routes for bulk samples. Furthermore, we documented record-high H_Irr resulting from a good electrical connectivity between superconducting grains. This is an important result of this work, since H_Irr represents the maximum field at which a superconductor can be operated in superconducting magnets. In spite of the enhanced degree of C substitution, µ₀H_C2 appears to be bounded to maximum values of ~ 26 T and ~ 35 T at 10 K and 4.2 K, respectively. T_C and H_C2 correlate well with the contraction of the a lattice parameter. T_C decreases monotonously upon increasing the structural disorder but its variation becomes much steeper for a $\lesssim$ 3.06 Å. This value of the lattice parameter a corresponds to the maximum of H_C2, too. We also analyzed results reported in the literature for MgB₂ in the form of films and coated conductors. Contrary to the case of bulk samples, it is not possible to define a master curve that allows estimating H_C2 from T_C. The two-band theory for H_C2 demands for selective high scattering in the π band in order to achieve µ₀H_C2(4.2 K) as high as ~ 50 T. Our results indicate that C doping creates defects that act both as intraband and interband scattering centers, which respectively affect H_C2 and T_C. Furthermore, we did not observe any clear evidence of an upward curvature of H_C2 at low T, as expected in the case of a π band much dirtier than the σ band. The type of disorder present in the films that showed very-high H_C2 cannot be the same realized by C doping in bulk samples. Further investigations are needed to achieve a comprehensive understanding of this matter. Lattice deformations that produce a tilt of the c axis, which selectively affect the B p_z orbitals from which the π band arises, may be the key to achieve record-high µ₀H_C2 in the 50 T range at 4.2 K.

Methods

Sample preparation

In-situ MgB₂ bulk samples were prepared using amorphous 99+% purity B powders, 99.9+% purity Mg turnings and 90+% purity DWCNT as precursors. We added 50 wt.% Mg excess to the reagents with respect to the stoichiometric ratio Mg:B = 1:2, as we previously proved that this is beneficial to the electrical connectivity of the samples⁴¹. Precursors were handled in glove box under inert atmosphere (pure Ar) to prevent oxygen contaminations. B powders (mixed with the DWCNT in the case of doped samples) were sandwiched between two Mg pellets inside an AISI-316-L stainless (SS) tube. The tube ends were closed by hydraulic press and sealed by Tungsten-Inert-Gas welding. The central part of the tube was subsequently submitted to uniaxial pressure in order to further densify the precursors before the reaction. Samples were reacted using a laboratory-made induction furnace described in²⁹. The SS tube is inserted at the center of a water-cooled Cu coil, which is located inside a sealed chamber filled with Ar. The Cu coil induces currents in the SS crucible that acts as a susceptor. This allows reacting the precursors within the SS sheath with ramp rates as high as ~ 1000 °C/min. After the reaction dwell time, the synthesis process can be quenched by injecting Ar-gas at high pressure (up to 1.5 MPa). The sample temperature was recorded during the reaction using a pyrometer calibrated in the range 500–1200 °C. After reaction, we removed the SS sheath and cut samples of the desired size and shape by spark erosion. Typical dimensions of the samples used for electrical transport and structural characterizations are ~ 5 × 2 × 1 mm³.

Screening DoE

We selected a 2 ^k−1 fractional factorial design for the screening DoE⁵⁹. 2 is the number of levels for each factor (“−” and “ + ” in coded units) and k the number of factors or input variables. At this first stage, we let only 4 of the 5 input parameters vary, namely: heating ramp rate (HR), dwell temperature (T_d), dwell time (t), pressure of the Ar-gas quenching jet (ArP). 2 ^k−1 provides the number of experiments to be performed, which is 8 in our case. To this set of experiments, we added a “center point” (“0” in coded units) that represents the center value of all factors’ ranges. We replicated this run three times, preparing 11 samples in total. Tables 6 and 7 report the range of variation of the synthesis parameters and the specific samples’ synthesis conditions, respectively. Experiments were run in randomized order to guard against systematic biases.

Table 6 Range of variation of the synthesis parameters for the Screening DoE.

Table 7 Synthesis conditions (in coded units) of all the samples prepared for the Screening DoE. Coded units refer to Table 6.

Samples SCR_1, SCR_6 and SCR_11 are three replicas of the DoE “center point”. Replicating the center point in a DoE provides a measure of process stability and reproducibility³⁰. We found variations by ~ 1% for T_C, ~ 40% for ΔT_c., ~ 2% for H_Irr and ~ 4% for H_C2. The large spread of data found for ΔT_c does not seem to play a role in the variability of H_C2. The highest H_C2 values were found in samples whose synthesis process was quenched by injecting Ar gas at the highest pressure (1.5 MPa). This outcome agrees with results found in binary MgB₂ samples prepared with the same technique²⁹. Precursors did not react to form bulk samples when combining the lowest reaction temperature (850 °C) with the shortest dwell time (15 min). Therefore, it was not possible to characterize samples SCR_3 and SCR_5.

Response-surface-methodology DoE

We selected a 2^k full factorial design augmented with center points and axial points (denoted by α+ and α− in coded units). Axial points are outside the input-parameter hypercube defined by the “−” and “ + ” levels. They are fundamental to build a second-order polynomial for the determination of the response surface ^31,59. We used the following four input variables: heating ramp rate (HR), dwell temperature (T_d), dwell time (t), pressure applied to the precursors before synthesis (P). On the basis of the screening-DoE results, we fixed the Ar-jet pressure to 1.5 MPa. In view of the the low variability observed for H_C2 in the screening DoE, we performed only two replicas of the DoE center point (samples RSM_1 and RSM_7). Therefore, we performed 26 runs in total, composed by the 16 corners from the full factorial block (2⁴), 2 replications of the center point and 8 (2k) axial points. We run the DoE in randomized order. The two center-point replicas provided comparable values for all the investigated parameters but ΔT_C, which varies by ~ 3.2 K in the two samples. Tables 8 and 9 report the range of variation of the synthesis parameters for all samples prepared in the frame of this DoE. The RSM DoE allowed us to evaluate the response surface of H_C2(10 K) as a function of the four input variables. Based on the surface response predictions, we prepared further 7 samples with the aim of maximizing H_C2 and verifing the predictions of the DoE. Preparations conditions for samples belonging to this “post-RSM-DoE batch” are reported in Table 10. Samples M_1 and M_2 are localized in proximity of the high-T_d and low-P maximum of the synthesis-parameter space, samples M_3 and M_4 to the low-T_d and high-P one. The response surface extrapolates towards high µ₀H_C2 of ~ 25 T at pressures higher than the upper boundary of the explored range (P > 375 MPa) for T_d < 900 °C. Samples M_5, M_6 and M_7 were prepared following this indication for different combinations of HR and t. These samples resulted fragmented once extracted from the SS sheath, most probably because of the excessive stress exerted by the SS-crucible walls on the reacted MgB₂ bulk sample during the post-reaction quench. However, it was still possible to characterize them. We did not investigate H_C2 in an extrapolated high-T_d region above 1050 °C, which according to the DoE could lead to µ₀H_C2 ~ 27 T. Temperatures of ~ 1050 °C represent an upper limit for the mechanical strength of the 2-mm-thick-wall SS crucible, which has to withstand the internal overpressure from the Mg vapors. H_C2 values of samples M_1, M_2 and M_3 reproduce the RSM-DoE expectations with a precision within ~ 10%. Discrepancies up to ~ 25% are found for the three samples (M_5, M_6 and M_7) prepared at very high pressure, out of the DoE synthesis-parameter space. Synthesis conditions of M_4 vary from those of M_3 only for the dwell time (50 min in the place of 90 min). However, this sample showed a lower µ₀H_C2 (by ~ 3.5 T) and a higher T_C (by ~ 1.4 K) with respect to M_3, indicating that the dwell time plays an important role in enhancing the C-doping efficiency at low T_d < 900 °C.

Table 8 Range of variation of the synthesis parameters for the Response-Surface-Methodology DoE.

Table 9 Synthesis conditions (in coded units) of all the samples prepared for the Response-Surface-Methodology DoE. Coded units refer to Table 8.

Table 10 Synthesis conditions (in coded units) for the 7 samples of the post-RSM-DoE batch. Coded units refer to Table 8.

In-field electrical transport characterization

We investigated the samples’ electrical resistance (R) as a function of T and H by standard 4-wire measurements. Most of the samples were tested at the University of Geneva using a laboratory-made low-noise probe^60,61. We also designed and commissioned at the University of Geneva a dedicated low-noise probe to fit the 35 T magnet bore of the LNCMI facility in Grenoble. Both probes allow measuring up to 4 samples at the same time and choosing the samples’ orientation with respect to the H direction. Each sample was powered with excitation current in the range 1–10 mA in order to avoid heating effects. The voltage drop was amplified to increase the signal-to-noise ratio and measured with a nanovoltmeter. To determine the field dependence of the electrical resistance R(H), we swept the field at a constant rate of ~ 1 T/min. The probing current density was ~ 5 × 10^–2 A/cm². The R(H) dependence was investigated for fixed T values stabilized with a precision of ± 10 mK. The R(H) curves, as measured at different temperatures in the sample RSM_17, are reported in Fig. 8 for the sake of clarity. H_C2 and H_Irr were evaluated from the intersection of the linear fit of the superconducting transition with the normal-state R_N(H) and the R = 0 lines, respectively. The sample critical temperature (T_C) was evaluated from the R(T) curves acquired at H = 0. T_C is defined as the temperature at which the derivative dR/dT has a maximum. The width of the superconducting transition is defined as ΔT_C = T_90%–T_10%, where T_90% and T_10% are the temperatures at which R(T) is 90% and 10% of the normal state value just above the onset of the superconducting transition, respectively.

Figure 8

Resistance as a function of the magnetic field for fixed temperatures in the range 2.5–10 K. H_Irr(T) and H_C2(T) are defined as the magnetic field values corresponding to the intersection of the linear fit of the superconducting transition (dashed lines) with the superconducting −R=0− and the normal-state resistance −R_N(H)− lines, respectively.

Microstructural and electronic characterization

The samples’ microstructural properties were investigated by X-ray diffraction (XRD) and scanning electron microscopy (SEM) measurements. XRD patterns were collected on the PANalytical Empyrean powder diffractometer with the Bragg–Brentano geometry using the Cu Kα1 monochromatic radiation in the 2θ range between 20° and 120°. We performed a Rietveld refinement on the X-ray patterns by means of the FullProf Suite⁶² in order to evaluate the MgB₂ lattice parameters (a, c).

XPS measurements were performed using a Physical Electronics VersaProbe III system with a hemispherical analyser and monochromated Al Kα source. The energy scale linearity was calibrated with Au4f7/2 at 83.86 eV and Cu2p3/2 932.59 eV and data were referenced to the Ag3d5/2 peak at 368.36 eV. All data were measured at room temperature with a pass energy of 55 eV, at a take of angle of 45° and angular acceptance angle of +/− 20°. The samples were electrically grounded during measurement. The X-ray beam size on the sample was ~ 100 µm with a power of 25 W and chamber pressure was less than 1 × 10^–8 mbar. All samples were polished with sandpaper to remove a surface layer of at least ~ 200 μm in order to remove the layer of material at the surface resulting from spark-erosion cutting. Samples were sputter cleaned in-situ with 2 kV Argon ions for 18 min. Consistently with previous studies, we verified that sputter cleaning duration did not alter significantly the binding energy or FWHM of the MgB₂ B 1s peak⁵⁰. Sample measurement order was randomized and measurements performed on multiple sample positions over multiple experimental runs produced consistent results.