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

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.


Results
The MgB 2 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 connectivity 28 . The PiCT technique allows in turn achieving a high density of the reacted MgB 2 phase and a high reproducibility of the sample properties 27 . 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 MgB 2 samples as reference, and ~ 50 C-doped samples, most of them with a nominal composition Mg(B 0.9 C 0.1 ) 2 . Indeed, x = 0.1 was proven to maximize H C2 in Mg(B 1−x C x ) 2 polycrystalline samples in the case of DWCNT 10 , 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 ours 29 , 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 behavior 30 . 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 MgB 2 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.
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 parameters 31 , as per the following quadratic polynomial: 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 0.0001 Å for all samples.

Sample ID
T c (K) ΔT c (K) μ 0 H irr @10 K (T) μ 0 H C2 @10 K (T) a (Å) c (Å) www.nature.com/scientificreports/ 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.
Equation (1) allowed us to identify the input-parameter combinations that maximize H C2 at 10 K. H C2 (T d ,H R,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 (µ 0 H 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.
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 (1)  www.nature.com/scientificreports/ partial texturing of the MgB 2 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 ⊥c C2 , regardless of the applied-field direction. Since H ⊥c C2 >H //c C2 , upon decreasing the applied field R(H) starts devting from its normal state value as soon as H ≤ H ⊥c C2 , 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 specimens 34 . In particular,     www.nature.com/scientificreports/ higher H Irr should be measured when H is perpendicular to the uniaxial-pressure direction 34 . 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 direction 34 .

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 rapidsynthesis 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 μ 0 H 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. . Data of Tables 1, 2 and 4 allow one to deduce that ~ 70% of the samples from this work have H Irr 0.7 H C2 . In the framework of the anisotropic Ginzburg-Landau theory, H Irr of untextured samples is described by: 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 from 10 are reported for comparison.
Scientific Reports | (2020) 10:17656 | https://doi.org/10.1038/s41598-020-74300-9 www.nature.com/scientificreports/ where γ=H ⊥c C2 /H //c C2 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 : 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 γ , p c can be calculated by Eq. (2) using for H ⊥c C2 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 γ 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 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 2.1 g/cm 3 , which is ~ 80% of the theoretical value for MgB 2 41 .
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 MgB 2 crystal structure, or remain interstitial within B rings 22,23 . Using the MgB 2 phase as a reference state, the enthalpy of formation at 0 K is negative in the case of C substitution, positive for interstitial C 23 . 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−x C x ) 2 samples, whilst c remains nearly constant 22,23 . In Mg(B 1−x C 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 [ [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 www.nature.com/scientificreports/ 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 MgB 2 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. Comparison between neutron-diffraction experiments and X-ray analyses has previously demonstrated that the actual level of C substitution in Mg(B 1-x C 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 rapidsynthesis 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 MgB 2 , and the two peaks at 193.5 eV and 190.5 eV to B 2 O 3 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 MgB 2 or Mg(B 1−x C 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 www.nature.com/scientificreports/ 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 = 0.1 52 . 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 MgB 2 . 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 MgB 2 lattice.
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 bands 24 . 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 MgB 2 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 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 MgB 2 structure also leads to a reduction of T C , which is steeper for a 3.06 Å as shown in Fig. 6b. Band filling due to electron doping is expected to lower T C 53 . In particular, a linear decrease of T C with x in Mg(B 1−x C 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 considered 53 . Our www.nature.com/scientificreports/ experimental observation that the slope of the T C vs a dependence changes for a 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 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 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 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 film 25 , and of a 0.75 μm-thick polycrystalline coated conductor deposited on SiC fibers that all showed µ 0 H C2 ~ 50 T 56 . 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 ⊥c C2 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):  www.nature.com/scientificreports/ 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, T 0 = 32.5 K ± 0.3 K, w 1 = 12.0 K ± 0.7 K, w 2 = 2.4 K ± 0.5 K, w 3 = 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 MgB 2 bulk samples in which disorder has been induced by doping, irradiation or synthesis conditions. At odds with what observed in bulk MgB 2 , 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 (μ 0 H C2 (4.2 K) > 40 T) measured in MgB 2 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 from 25 prepared by pulsed-laser deposition showed μ 0 H 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 band 15 . In this case, an upward curvature of H C2 vs T is expected. Contrary to what reported in 10 , 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−x C x ) 2 does not selectively increase the π-band intraband scattering rate, in agreement with theoretical expectations 24 . In view of the lower amount of substituted C in the samples from 10 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 in 10 and C should be more probably segregated outside the superconducting grains 45 . 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 axis 57 . No information about the type of texture is reported for the film 8 from 7 , which has μ 0 H C2 (4. 24 . 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 MgB 2 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 μ 0 H 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 μ 0 H C2 (4.2 K) ~ 50 T, while keeping T C above 25 K. www.nature.com/scientificreports/

Conclusions
We presented the results of a wide experimental campaign to investigate the role of carbon doping in the enhancement of H C2 in MgB 2 bulk samples. The main purpose of this work was understanding whether the very-high μ 0 H 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 MgB 2 in magnet technology. Based on theoretical and experimental evidences that structural disorder is the key to enhance H C2 , we produced samples by a rapidsynthesis 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, µ 0 H 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 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 MgB 2 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 µ 0 H 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 µ 0 H C2 in the 50 T range at 4.2 K.

Methods
Sample preparation. In-situ MgB 2 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 samples 41 . 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 in 29 . 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 mm 3 .
Screening DoE. We selected a 2 k−1 fractional factorial design for the screening DoE 59 . 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. 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 reproducibility 30 . 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 MgB 2 samples prepared with the same technique 29 . 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  Table 6.  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.

Sample ID Randomized order T (coded units) H (coded units) t (coded units) Q (coded units)
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/cm 2 . 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 Table 9. Synthesis conditions (in coded units) of all the samples prepared for the Response-Surface-Methodology DoE. Coded units refer to Table 8.  www.nature.com/scientificreports/ 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.
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 Suite 62 in order to evaluate the MgB 2 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 MgB 2 B 1s peak 50 . Sample measurement order was randomized and measurements performed on multiple sample positions over multiple experimental runs produced consistent results.