Mechanism of enhanced critical fields and critical current densities of MgB₂ wires with C/Dy₂O₃ co-additions

Abstract

A series of monofilamentary powder-in-tube MgB₂ wires were fabricated with 2 mol. % C doping and co-additions of 0–3 wt. % Dy₂O₃. Irreversibility fields (μ₀H_irr), upper critical fields (μ₀H_c2), and transport critical currents were measured, and from these quantities, anisotropies $(\gamma )$ and electronic diffusivities $(D_{\pi ,\sigma })$ were estimated. The addition of 1 wt. % Dy₂O₃ to already optimally C-doped MgB₂ wires produced higher H_c2//ab, H_c2//c, and H_irr values at 4.2 K. In addition, the critical current density, J_c, increased with Dy₂O₃ concentration up to 1 wt. % where non-barrier J_c reached 4.35 × 10⁴ A/cm² at 4.2 K, 10 T. At higher temperatures, for example, 20 K and 5 T, co-additions of 2 mol. % C and 2 wt. % Dy₂O₃ improved non-barrier J_c by 40% and 93% compared to 2 and 3 mol. % C doping, respectively. On the other hand, measurements of T_c showed that C/Dy₂O₃ co-additions increase interband scattering rates at a lower rate than C doping does (assuming C doping levels giving similar levels of low-T μ₀H_c2 increase as co-addition). Comparisons to a two-band model for μ₀H_c2 in MgB₂ allowed us to conclude that the increases in H_c2//ab, H_c2//c, and H_irr (as well as concomitant increases in high-field J_c) with Dy₂O₃ addition are consistent with increases primarily in intraband scattering. This suggests C/Dy₂O₃ co-addition to be a more promising candidate for improving non-barrier J_c of MgB₂ at temperatures above 20 K.

I. INTRODUCTION

Investigation into the in-field performance of MgB₂ is complicated by its multi-band (gap) structure.¹ Two σ bands are derived from the in-plane, sp2 hybridized orbitals of B; these σ bands are two-dimensionally distributed within the B layers. In addition, two π bands are derived from the π-bonding and antibonding of the out-of-plane p_z orbitals; these π bands are three-dimensionally distributed throughout the lattice.^2,3 For an MgB₂ single crystal, this multi-band mechanism results in two upper critical fields μ₀H_c2//c and μ₀H_c2//ab and a resultant anisotropy γ = H_c2//ab/H_c2//c, where //c and //ab indicate two magnetic field directions—parallel to the c-axis and the ab-plane, respectively.⁴ Of course, the superconducting properties of an MgB₂ polycrystalline sample or a wire are also affected by the multi-band mechanism, although the manifestation is slightly different. For a polycrystalline wire, e.g., μ₀H_c2 and μ₀H_irr can be extracted from a ρ vs μ₀H (ρ vs T) curve, taking μ₀H_c2 as 0.9ρ_n and μ₀H_irr as 0.1ρ_n, where ρ_n is the normal state resistivity. If each grain in a polycrystalline MgB₂ sample is assumed to be a single-crystal superconducting domain having constant H_c2//c and H_c2//ab, the H_c2 of the polycrystalline MgB₂ will be equivalent to H_c2//ab, indicating the critical field where the last (first) MgB₂ grain exhibits superconducting-normal state transition in increasing (decreasing) magnetic field. Thus, in this work, H_c2 always corresponds to H_c2//ab unless otherwise noted. On the other hand, H_irr is tied to H_c2//c.

One of the main reasons to pursue higher critical fields is for the superconductor in question to be able to exhibit supercurrents in higher magnetic fields. However, in addition to higher critical fields, sufficient flux pinning and MgB₂ connectivity are required. In terms of direct attempts to increase connectivity, a number of mechanical pressing methods have been applied to PIT in-situ MgB₂ wires in order to improve their grain connectivity. Two typical mechanical pressing methods investigated have been hot isostatic pressing (HIP)^5,6 and cold-high-pressure-densification (CHPD).^7–9 Gajda et al. showed that HIP induced the formation of high-density and high-uniformity MgB₂ and increased J_cs three times at both 4.2 and 20 K.⁵ Hossain et al. showed that transport J_cs and H_irrs of 10 wt. % C₄H₆O₅-added MgB₂ conductors cold-densified at 1.48 GPa were significantly enhanced at 20 K.⁷

On the other hand, dozens of dopants and additives have been investigated. Carbon doping is the most commonly used method to improve the high field transport J_cs of MgB₂ conductors, which it does mainly through increasing its upper critical fields. Carbon can be added in many forms, including SiC,^10–15 nanocarbon,^16–18 C₆₀,¹⁹ carbohydrate,^20,21 as well as by the use of C-predoped amorphous B powder.^22,23 Significant J_c improvement at high temperature was achieved by adding rare earth (RE) oxides into undoped MgB₂ or C-doped MgB₂.^24–29 It has been seen that adding RE oxides into MgB₂ can significantly increase flux pinning and therefore increase J_c(μ₀H) due to the formation of REB_y (y = 4 or 6) and MgO nanoinclusions within the MgB₂ matrix. For example, Chen et al. reported that in-field J_cs were increased while lattice parameters and T_cs were unchanged for MgB₂ bulks pinned with DyB₄ inclusions for 0.5–5.0 wt. % additions of Dy₂O₃.²⁷ This is because these nano-inclusions can act both directly as flux pinning centers as well as acting to increase the density of grain boundaries.^27,30 However, Dy₂O₃ also has been seen to increase H_irr, indicating that Dy₂O₃ addition also decrease the electronic diffusivities, since H_irr is tied to H_c2//c. Yang et al. showed nano-sized Dy-based precipitates distributed in the grains of MgB₂ conductors doped with 0.5 wt. % Dy₂O₃ did not alter T_cs, but led to uniform 1T increase in H_irr over a wide range of temperatures.³¹ These studies point out clearly that Dy₂O₃ reacts during heat treatment to form intragrain and intergrain nanoscale DyB₄ precipitates but does not substitute into MgB₂ lattice; therefore, the increased H_irr seems to be not related to the atomic substitution into the Mg or B sublattice. To date, however, there is not complete set of magnetic and transport data taken on the same series of MgB₂ strands with various Dy₂O₃ compositions, and hence, the effects of Dy₂O₃ addition on transport J_c(μ₀H), H_irr(T), and H_c2(T) behavior of MgB₂ have not yet been fully explored.

In this work, a systematic study was conducted on J_cs and critical fields (H_c2//ab, H_c2//c, and H_irr) for a series of Dy₂O₃/C co-added powder-in-tube in situ MgB₂ wires, which were then compared to the results for a C-doped-only wire. The first aim is to fully investigate the temperature dependence of H_irr and H_c2 for Dy₂O₃/C co-added MgB₂ wires over a wide temperature range (4.2–30 K). In addition, C/Dy₂O₃ co-addition is compared with C doping, and it is shown that the co-addition can improve MgB₂ performance over a wide temperature range. The results, with the aid of a two-band model for H_c2 in MgB₂, are interpreted in terms of intra- and inter-band scattering with C/Dy₂O₃ co-additions.

II. EXPERIMENTAL DETAILS

A series of monofilamentary MgB₂ wires with C and Dy₂O₃ additions were fabricated by Hyper Tech Research Inc. (HTR). The wires were fabricated using mixed powders of Mg (<40 μm, 99%), “2 mol. % C-pre-doped” B (10–100 nm in size provided by Specialty Materials Inc. SMI), and Dy₂O₃ (99.9%, 100 nm, Sigma Aldrich) filled into a Nb tube surrounded by a Monel tube. After full reaction between Mg, pre-doped B, and Dy₂O₃, the wires have actual C-doping level of 2.10 mol. %³² and Dy₂O₃ concentrations of 0.0, 1.0, 2.0, and 3.0 wt. % with respect to MgB₂. The wires have a 0.84 mm outer diameter and were heat-treated (HT) either at 650 °C/1h or at 675 °C/2h and were then furnace cooled to room temperature.

Resistivity, R(T), was measured from 15 to 45 K in transverse magnetic fields ranging from 0.005 to 12 T. These R(T) measurements were conducted on 10 mm segments of the 0.0 and 1.0 wt.  % Dy₂O₃-added samples using a Quantum Design Model 6000 Physical Property Measuring System (PPMS) as described in Ref. 33. Values of μ₀H_c2 and μ₀H_irr were extracted from R(T) curves at 0.9R_n and 0.1R_n, respectively. μ₀H_c2 and μ₀H_irr data at 4.2 K for 0.0 and 1.0 wt. % Dy₂O₃-added samples were obtained from measurements of resistivity as a function of magnetic field, R(μ₀H), measured in liquid helium in fields up to 30 T in a resistive magnet at the National High Magnetic Field Laboratory. Transport 4.2 K I_c measurements were performed at OSU at pool boiling liquid helium as a function of transverse magnetic field, H, up to 12.5 T on 5 cm-long-segments of selected wires using a four-point-probe method. Transport I_cs were extracted from the V–I curves using a gauge length of 5 mm and an electric field criterion of 1.0 μV cm⁻¹. Non-barrier J_cs of these PIT MgB₂ wires were calculated by dividing the conductor I_c value by the MgB₂ filamentary area (summed over the whole strand), and this is the same as the area inside the Nb chemical barrier.

III. RESULTS

A. Critical current densities

Figure 1 shows the 4.2 K transport non-barrier J_cs as a function of transverse magnetic field for samples with 2 mol. % C doping and Dy₂O₃ compositions from 0 to 3 wt. % (all additions in this work are with respect to the final MgB₂). The non-barrier J_cs of 2 mol. % C-doped-only samples (Dy0.0-C2-650 and Dy0.0-C2-675) reported in Refs. ³⁴ and ³⁵ are included in Figs. 1(a) and 1(b), respectively, for comparison. Also shown in Figs. 1(a) and 1(b) are non-barrier J_cs of a 2 mol. % C/0.5 wt. % Dy₂O₃ co-added MgB₂ wire HT at 675 °C (Dy0.5-C2-675).³¹ At both HT conditions, the 1.0 wt. % Dy₂O₃ addition (with baseline 2 mol. % C doping) achieved the highest transport J_cs (e.g., 3.08 × 10⁴ A/cm² for Dy1.0-C2-650 and 4.35 × 10⁴ A/cm² for Dy1.0-C2-675 at 4.2 K, 10 T). Increasing Dy₂O₃ concentration from 1.0 wt. % to higher values (2.0 and 3.0 wt. %) reduced J_c slightly from this maximum value to 3.28 × 10⁴ and 2.96 × 10⁴ A/cm², respectively. We will argue below that the main contribution to this behavior has to do with H_c2 increases associated with additional intraband scattering with increasing Dy₂O₃ additions, although there may also be a contribution-enhanced pinning associated with DyB₄ and MgO nanoinclusions^36,37 that have collected at the grain boundaries.

FIG. 1.

Field dependence of transport non-barrier J_cs for the samples HT with (a) 650 °C/1h and (b) 675 °C/2h. Non-barrier J_cs of samples Dy0.0-C2-650 from Refs. ³⁴ and ³⁵, Dy0.0-C2-675 from Refs. ³⁴ and ³⁵, and Dy0.5-C2-675 from Ref. 31 are included for comparison.

From Figs. 1(a) and 1(b), it can be seen that the most important difference with 1.0 wt. % Dy₂O₃ addition is at high fields. This suggests that while in principle some pinning from nanoprecipitates might be expected, the high-field J_cs are mainly driven by increased H_irr or H_c2. Of course, we could perhaps increase H_c2 with further C additions. We can then compare the J_c–μ₀H data for higher C-doped samples with the samples that had both Dy₂O₃ and 2 mol. % C both at 4.2 K as well as at higher temperatures. To this end, Fig. 2 compares the transport non-barrier J_c(μ₀H) of sample Dy2.0-C2-650 in this study and Dy0.5-C2-675 from Ref. 31 with the MgB₂ samples with 2.0 and 3.0 mol. % C doping from Ref. 38, which are labeled as Pure-C2-675 and Pure-C3-700 from 4.2 to 25 K. At 20 K/5 T, sample Dy0.5-C2-675 with co-addition of 0.5 wt. % Dy₂O₃ generated transport J_c of 1.80 × 10⁴ A/cm² from Ref. 31 and sample Dy2.0-C2-675 with co-addition of 2.0 wt. % Dy₂O₃ generated 9.65 × 10³ A/cm² compared to 6.92 × 10³ A/cm² for 2.0 mol. % C doping and 5.00 × 10³ A/cm² for 3.0 mol. % C doping from Ref. 38.

FIG. 2.

Comparison for transport non-barrier J_c at 4.2, 15, 20, and 25 K between 2.0 mol. % pure-C-doped sample Pure-C2-675 from Ref. 38, 3.0 mol. % pure-C-doped sample Pure-C3-700 from Ref. 38, 2 mol. % C/2.0 wt. % Dy₂O₃ co-added sample Dy2.0-C2-650 in this study, and 2.0 mol. % C/0.5 wt. % Dy₂O₃ co-added sample Dy0.5-C2-675 from Ref. 31.

Compared to sample Pure-C2-675, although sample Pure-C3-700 has higher non-barrier J_cs at 15 K, Pure-C3-700 has similar 20 K J_cs and obtained an even smaller 25 K J_cs at all applied fields, with respect to the 2 mol. % C sample. Thus, for C-doped samples, 3.0 mol. % C doping is best below 20 K, and 2.0 mol. % C doping is best above 20 K. Bringing in now Dy2.0-C2-650 to the comparison, it meets the performance of the Pure-C3-700 sample at 15 K and outperforms both C only doped chemistries at 20 and 25 K. Finally, Dy0.5-C2-675 outperforms all other samples at 20 and 25 K. These results indicate that Dy₂O₃ co-doping with 2 mol. % C co-addition is a more effective way to improve J_c than heavier C doping over a wide temperature range. At higher temperatures, the lighter Dy₂O₃ doping is better.

B. Upper critical fields and irreversibility fields

Figure 3(a) shows examples of R(μ₀H) curves from which values of μ₀H_c2 and μ₀H_irr were extracted at 4.2 K for Dy0.0-C2-675 and Dy1.0-C2-675. For each sample, two R(μ₀H) curves were obtained by first increasing μ₀H from 0 to 30 T and then decreasing it from 30 to 0 T. μ₀H_c2(4.2 K) is the average value of μ₀H_c2s extracted from two R(μ₀H) curves, and same averaging was applied to μ₀H_irr(4.2 K). Figure 3(b) displays μ₀H_c2 and μ₀H_irr vs T for Dy0.0-C2-675 and Dy1.0-C2-675. Little difference is seen at 15 K and above, but μ₀H_c2(4.2 K) and μ₀H_irr(4.2 K) are indeed increased by 2.29 and 2.85 T by the 1.0 wt. % Dy₂O₃ addition, respectively. Here, we have used a resistive method to obtain μ₀H_irr. However, in some studies, the irreversibility field is defined at the point where J_c reaches 10³ A/cm², we have denoted the irreversibility field extracted in this alternative way μ₀H_irr*. Table I gives the values for μ₀H_irr* for Dy2.0-C2-650, Pure-C2-675, and Pure-C3-700. It can be seen that H_irr* is suppressed at higher temperatures by increasing the C concentration from 2 to 3 mol. %. However, the C/Dy₂O₃ co-addition generates a larger H_irr* than 2 mol. % C doping at all temperatures, even though the enhancements are more pronounced at lower temperatures.

FIG. 3.

(a) R-μ₀H curves of Dy0.0-C2-675 and Dy1.0-C2-675 at 4.2 K. (b) μ₀H_c2 and μ₀H_irr as a function of temperature for samples Dy0.0-C2-675 and Dy1.0-C2-675.

TABLE I.

Comparison of μ₀H_irr* (Unit: T) between Pure-C3-700, Pure-C2-675, and Dy2.0-C2-650.

Temperature (K)	Dy2.0-C2-650	Pure-C3-700a	Pure-C2-675a
4.2	19.1	21.6	15.9
15	12.2	11.1	9.70
20	8.01	6.92	7.00
25	5.01	3.28	4.00

^a μ₀H_irr*s of Pure-C3-700 and Pure-C2-675 are extracted from non-barrier J_c(μ₀H) curves from Ref. 38.

C. Anisotropy and H_c2//c

The above results indicate that 4.2 K H_c2 (=H_c2//ab) is increased by Dy₂O₃/C co-addition, while no increase is seen at higher temperatures. On the other hand, H_irr (or H_irr*), which is correlated to H_c2//c, is increased over a larger range of temperatures, even though the largest increases are seen at low fields. Eisterer's percolation model for MgB₂ allows us to extract the values of H_c2//c from H_irr using the following relationship connecting measured H_c2 (=H_c2//ab), γ (and thus H_c2//c), and an effective percolation threshold p_c* with a fixed number:^39–41

$$\mathrm{\Delta }T={\displaystyle \frac{1-\sqrt{({\gamma }^2-1)p_c^{\ast 2}+1}}{d{H}_{c2}/dT}H.}$$ (1)

This indicates that H_c2//c, while not directly measurable in polycrystalline samples, is directly proportional to H_irr. That is, H_irr and H_c2//c are directly tied together, with H_irr being slightly different with H_c2//c by an amount related to a percolation effect. The essence of the model is that a certain number of grains, which are, in principle, randomly oriented with respect to the applied field, are required to be superconducting in order for a superconducting path to be continuous. As field or temperature is decreased from the normal state, more and more lengths along a given path are contiguous, until at H_irr, a complete path exists. The percolation model describes not only the nature of resistive transition, but also influences the form of J_c vs μ₀H, given as^39–41

$$J_c(\alpha )=F_0{\displaystyle \frac{{[1-H/H_{c2}(\alpha )]}^2}{{\mu }_0\sqrt{H{H}_{c2}(\alpha )}},}$$ (2)

where J_c(α) is the microscopic J_c dominated by grain-boundary flux pinning, F₀ is the pinning strength, and α represents the angle between the magnetic field and the c-axis of MgB₂ grain. Then, macroscopic J_c of polycrystalline MgB₂ can be given as^39–41

$$\begin{array}{rl}{J}_c& =\underset{0}{\overset{J_c^{max}({\mu }_{0}H)}{\int }}{\left({\displaystyle \frac{p_{Mg{B}_2}p(J)-p_c}{1-p_c}}\right)}^{1.79}dJ\\ & =\underset{0}{\overset{J_c^{max}({\mu }_{0}H)}{\int }}{\left({\displaystyle \frac{p_{Mg{B}_2}-p_c}{1-p_c}}\right)}^t{\left({\displaystyle \frac{p(J)-p_c/p_{Mg{B}_2}}{1-p_c/p_{Mg{B}_2}}}\right)}^{1.79}dJ\\ & =\underset{0}{\overset{J_c^{max}({\mu }_{0}H)}{\int }}{A}_{con}{\left({\displaystyle \frac{p(J)-p_c^{\ast }}{1-p_c^{\ast }}}\right)}^{1.79}dJ,\end{array}$$ (3)

where p(J) is the fraction of superconducting grains whose J_c(α) is larger than J, p(MgB₂) is the fraction of MgB₂ grains in the polycrystalline sample, p_c = p(MgB₂) × p_c* is percolation threshold, $J_c^{max}({\mu }_{0}H)$ is defined as $p[J_c^{max}({\mu }_{0}H)]$ that is equal to p_c, and the values of p_c and t are determined to be 0.17–0.31 and 1.79, respectively.³⁹ Thus, four parameters (F_m, μ₀H_c2, γ, and p_c*) are required for obtaining best fitting of transport J_c using the percolation model and F_m = F₀ × A_con is the effective flux pinning strength. For Dy0.0-C2-675 and Dy1.0-C2-675, the field dependence of 4.2 K non-barrier J_c has been fit by the percolation model; results are shown in Fig. 4. The 4.2 K values of μ₀H_c2 (=μ₀H_c2//ab) for Dy0.0-C2-675 and Dy1.0-C2-675 were measured and were used in the fitting of Fig. 4 and listed in Table II. Table II also gives the values of other fitting parameters, and the calculated values for μ₀H_c2//c. p_c* are assumed to be fixed at 0.2 for simplicity. In some cases, the samples transitioned by quench, but this was interpreted as J_c for present purposes (previous work has shown these values to be very close to the critical surface). The effective flux pinning strength of 1.0 wt. % reaches 4.12 × 10⁶ A T cm⁻², which is 51% higher than that of 0.0 wt. % wire. This result confirms previous reports that Dy₂O₃ additions increase flux pinning.²⁷ More interesting, however, is that Dy₂O₃ addition decreases the anisotropy γ. The 4.2 K γ of Dy0.0-C2-675 is 2.52, but 1 wt. % addition of Dy₂O₃ decreases γ to 2.14. The 4.2 K μ₀H_c2//c for Dy0.0-C2-675 and Dy1.0-C2-675 are calculated to be 9.34 T and 12.30 T, respectively. Consequently, the Dy₂O₃ addition can further increase both H_c2//c and H_c2//ab for C-doped MgB₂ at 4.2 K, and provides a stronger enhancement of H_c2//c than H_c2//ab.

FIG. 4.

Experimental transport non-barrier J_cs (solid dots), quench Js (cross dots), and modeling J_cs (solid lines) for Dy0.0-C2-675 and Dy1.0-C2-675. Transport non-barrier J_cs of Dy0.0-C2-675 are from Ref. 34 and 35. Quench Js of Dy0.0-C2-675 are from Ref. 35.

TABLE II.

List of fitting parameters for the percolation model and calculated μ₀H_c2//c at 4.2 K.

Fitting Parameter	Dy0.0-C2-675	Dy1.0-C2-675
Fm (A T cm−2)	2.73 × 106	4.12 × 106
Anisotropy, γ	2.52	2.14
μ0Hc2 (=μ0Hc2//ab) (T)	23.5	26.3
μ0Hc2//c (T)	9.34	12.3

The 1.0 wt. % Dy₂O₃ addition tends to decrease anisotropy γ at temperatures ≤20 K and correspondingly enhances H_c2//c, Fig. 5. The values of γ are calculated for both Dy1.0-C2-675 and Dy0.0-C2-675 using Eq. (1). The γ value of 3.8 at 23.4 K for Dy0.0-C2-675 is similar to that of C-doped MgB₂ reported by Eisterer,⁴² for which γ = 4 at 21 K as the T_c of C-doped MgB₂ is 35 K. Anisotropy γ grows with temperature for both samples due to anisotropic σ bands determining γ at high temperatures while isotropic π bands define γ at low temperatures.⁴³ It has been reported that, without H_c2//ab enhancement, adding 0.5 wt. % Dy₂O₃ enhances H_irr for C-doped MgB₂.³¹ Based on present results, the H_irr enhancement can be considered as a result of H_c2//c enhancement or decrease in γ caused by Dy₂O₃ addition. Nevertheless, the increases in H_c2//ab and H_c2//c are not pronounced at high temperatures. In any case, it is of interest to explore further the H_c2//ab(T) and H_c2//c(T) behaviors as influenced by Dy₂O₃ addition. To do this, we will model H_c2//ab and H_c2//c values extracted above using the two band, anisotropic model for MgB₂.

FIG. 5.

Temperature dependence of (a) anisotropy γ and (b) μ₀H_c2//c for Dy0.0-C2-675 and Dy1.0-C2-675.

IV. DISCUSSION

According to Gurevich's discussion,^1,43 H_c2 is related to electronic diffusivities (D_σ and D_π) associated with σ and π bands, respectively. Associated with the anisotropy of MgB₂, D_σ and D_π also have an angular dependence, which can be given as

$$D_n(\alpha )=\sqrt{{(D_n^{ab})}^2\mathrm{co}{\mathrm{s}}^2(\alpha )+(D_n^{ab})(D_n^c)\mathrm{si}{\mathrm{n}}^2(\alpha )},$$ (4)

where α represents the angle between the direction of the applied field and the c-axis of the MgB₂ lattice and $D_n^{ab}$ and $D_n^c$ are electronic diffusivities in the ab-plane and c-axis (again index n corresponds to the π or σ band), respectively. Thus, we have given an expression for the effective diffusivity in a plane perpendicular to α, using the defined electronic diffusivities for each band along a given direction within the crystal. In this case, the limiting cases of D_n(π/2) and D_n(0) can be expressed as

$$D_n(0)=D_n^{ab}$$ (5)

and

$$D_n(\pi /2)=\sqrt{D_n^{ab}{D}_n^c}.$$ (6)

The electronic diffusivities, D_σ and D_π, represent the degree of intraband scattering for MgB₂ conductor and decreased D_σ and D_π are the reflection of increased intraband scattering rates caused by impurity doping/addition. While the H_c2(T) behavior of MgB₂ is affected by both intraband scattering rates as well as interband scattering rates, parametrized by Γ_πσ and Γ_σπ, and T_c is also correlated with interband scattering. In Gurevich's two-gap dirty limit theory, the effects of intra- and interband scattering on the H_c2 of a two-gap MgB₂ having a σ band and a π band can be described by linearized Usadel equations,^1,44

$$\omega f_{\sigma }-{\displaystyle \frac{D_{\sigma }}{2}{\mathrm{\Pi }}^2{f}_{\sigma }={\mathrm{\Delta }}_{\sigma }+(f_{\pi }-f_{\sigma }){\mathrm{\Gamma }}_{\sigma \pi },}$$ (7)

$$\omega f_{\pi }-{\displaystyle \frac{D_{\pi }}{2}{\mathrm{\Pi }}^2{f}_{\pi }={\mathrm{\Delta }}_{\pi }+(f_{\sigma }-f_{\pi }){\mathrm{\Gamma }}_{\pi \sigma },}$$ (8)

where f_σ,π is the Usadel green function associated with the σ or π band, ω is the Matsubara frequency, D_σ,π is the intraband diffusivity due to intraband scattering in either σ or π band, Γ_σπ and Γ_πσ are interband scattering rates, and Δ_σ,π is the σ or π bandgap. Then, for a two-gap MgB₂ conductor, the solution to the linearized Usadel equations for the condition T = T_c can be expressed as^1,43

$$U\left({\displaystyle \frac{g}{t_c}}\right)=\psi \left({\displaystyle \frac{1}{2}+{\displaystyle \frac{g}{t_c}}}\right)-\psi \left({\displaystyle \frac{1}{2}}\right)=-{\displaystyle \frac{\ln t_c(w\ln t_c+{\lambda }_0)}{w\ln t_c+p},}$$ (9)

where t_c = T_c/T_c0, $g=({\mathrm{\Gamma }}_{\sigma \pi }+{\mathrm{\Gamma }}_{\pi \sigma })\hslash /(2\pi k_B{T}_{c0})$ is called the interband scattering parameter, T_c is the critical temperature of MgB₂ due to interband scattering, and T_c0 is the critical temperature of MgB₂ without interband scattering (Γ_σπ = Γ_πσ = 0). Here, $\psi (x)$ is the di-gamma function, while W, λ₀, and p are appropriate functions of interband scattering rates and BSC coupling constants of λ_σσ = 0.810, λ_σπ = 0.119, λ_πσ = 0.09, and λ_ππ = 0.285, which is given by^1,44–46

$$w={\lambda }_{\sigma \sigma }{\lambda }_{\pi \pi }-{\lambda }_{\sigma \pi }{\lambda }_{\pi \sigma },$$ (10)

$${\lambda }_{-}={\lambda }_{\sigma \sigma }-{\lambda }_{\pi \pi },$$ (11)

$${\lambda }_0={({\lambda }_{-}^2+4{\lambda }_{\sigma \pi }{\lambda }_{\pi \sigma })}^{1/2},$$ (12)

$${\mathrm{\Gamma }}_{\pm }={\mathrm{\Gamma }}_{\sigma \pi }\pm {\mathrm{\Gamma }}_{\pi \sigma },$$ (13)

$$2p={\lambda }_0+[{\mathrm{\Gamma }}_{-}{\lambda }_{-}-2{\lambda }_{\sigma \pi }{\mathrm{\Gamma }}_{\pi \sigma }-2{\lambda }_{\pi \sigma }{\mathrm{\Gamma }}_{\sigma \pi }]/{\mathrm{\Gamma }}_{+}.$$ (14)

Equation (9) demonstrates the relationship between g, T_c, and interband scattering rates. On the other hand, for the condition of H = H_c2, the solution to the linearized Usadel equations can be expressed as¹

$$\begin{array}{rl}& 2w(\ln t+U_{+})(\ln t+U_{-})+({\lambda }_0+{\lambda }_i)(\ln t+U_{+})\\ & +({\lambda }_0-{\lambda }_i)(\ln t+U_{-})=0,\end{array}$$ (15)

where $t=T/T_{c0}$, $U_{\pm }=\psi (1/2+\hslash {\mathrm{\Omega }}_{\pm }/(2\pi k_{B}T))-\psi (1/2)$, ${\mathrm{\Omega }}_{\pm }$ are the function of D_π,σ, Γ_σπ, Γ_πσ, and H_c2 and can be expressed as

$$2{\mathrm{\Omega }}_{\pm }={\omega }_{+}+{\mathrm{\Gamma }}_{+}\pm {\mathrm{\Omega }}_0,$$ (16)

$${\mathrm{\Omega }}_0={({\omega }_{-}^2+{\mathrm{\Gamma }}_{+}^2+2{\mathrm{\Gamma }}_{-}{\omega }_{-})}^{1/2},$$ (17)

$${\omega }_{\pm }=(D_{\sigma }\pm D_{\pi })\pi {\mu }_0{H}_{c2}/{\text{Ø}}_0,$$ (18)

$${\lambda }_i=[({\omega }_{-}+{\mathrm{\Gamma }}_{-}){\lambda }_{-}-2{\lambda }_{\sigma \pi }{\mathrm{\Gamma }}_{\pi \sigma }-2{\lambda }_{\pi \sigma }{\mathrm{\Gamma }}_{\sigma \pi }]/{\mathrm{\Omega }}_0,$$ (19)

where ${\text{Ø}}_0$ is the flux quantum. Therefore, Gurevich's two-gap dirty limit theory builds a quantitative relationship between H_c2, T_c, and intra- and interband scattering for two-gap MgB₂.

Pure MgB₂ has relatively low critical fields, which leads to J_c values that drop off relatively quickly with increasing applied magnetic field. Up until now, the most effective way to increase H_c2s of MgB₂ at 4.2–20 K is through the C substitution into the B sublattice, which leads to the increase in intraband scattering rates (smaller D_π,σ). Gurevich's two-gap dirty limit theory indicates that the π bands are much dirtier than σ bands for C-doped MgB₂ conductors $(D_{\pi }\ll D_{\sigma })$.^1,44,45 However, the H_c2 of MgB₂ is decreased at relatively high temperatures (≥20 K) if heavy C doping is used; this leads to a corresponding decrease in J_cs at high temperatures. For example, although the J_cs of powder-in-tube in situ MgB₂ wires are improved at 4.2 K by increasing C doping from 2.0 to 3.0 mol. %, the J_cs are suppressed at 25 K.^32,38 These results are induced by the fact that heavy C doping increases interband scattering rates, which is reflected in a decrease in T_c.

The T_cs of Dy0.0-C2-675 and Dy1.0-C2-675 are measured to be 36.4 and 34.9 K, respectively, and the g values of the samples are solved to be 0.054 and 0.081 using Eq. (11). Once the values of g are known, the values of interband scattering rates, Γ_σπ and Γ_πσ, can be calculated by solving the simultaneous equations of $N_{\sigma }{\mathrm{\Gamma }}_{\sigma \pi }=N_{\pi }{\mathrm{\Gamma }}_{\pi \sigma }$, $N_{\pi }\approx 1.3N_{\sigma }$, and $g=({\mathrm{\Gamma }}_{\sigma \pi }+{\mathrm{\Gamma }}_{\pi \sigma })\hslash /(2\pi k_B{T}_{c0})$. Then, by inserting the values of Γ_σπ and Γ_πσ into Eqs. (15)–(19), μ₀H_c2 can be treated as the solution to Eq. (15) for certain values of T, D_σ, and D_π. In other words, the curve of μ₀H_c2 vs T can be modelled by setting the values of D_σ and D_π if the values of Γ_σπ and Γ_πσ are known. Figure 6 shows μ₀H_c2//ab and μ₀H_c2//c vs T for Dy0.0-C2-675 and Dy1.0-C2-675. Also shown is a fit based on Gurevich's two-gap dirty limit theory and the fitting parameters are listed in Table III. The fitting parameters of ${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}$ were used to calculate H_c2//ab, and H_c2//c is associated with the values of $D_{\sigma }^{ab}$. Since the π band is more isotropic than the σ band, $D_{\pi }^{ab}/D_{\pi }^c\sim 1$ and $D_{\sigma }^c\ll D_{\sigma }^{ab}$,^1,47,48 the fitting parameter of $D_{\pi }$ representing ${(D_{\pi }^{ab}{D}_{\pi }^c)}^{1/2}$, $D_{\pi }^{ab}$, and $D_{\pi }^c(D_{\pi }=D_{\pi }^{ab}=D_{\pi }^c={(D_{\pi }^{ab}{D}_{\pi }^c)}^{1/2})$ was used to fit the data of μ₀H_c2//ab(T) and μ₀H_c2//c(T). At higher temperatures, the experimental values of both H_c2//ab and H_c2//c are lower than the theoretical curves for both samples. This could be due to an increase in p_c with temperature (due to C or Dy₂O₃ accumulation at the grain boundaries and related “inhomogeneities”) as suggested by Eisterer.³⁹ According to Eq. (1), the increase in p_c with temperature induces ΔT_c broadening of ρ–T curves in superconductors at high temperatures. Moreover, ΔT_c broadening at high temperature can also be caused by thermal fluctuations.³⁹ Both H_c2//ab(T) and H_irr(T) extracted from ρ–T curves with ΔT_c broadening are smaller than those extracted from ρ–T curves without ΔT_c broadening, as they correspond to 90% and 10% of the normal state, respectively. Since H_c2//c is associated with H_irr, H_c2//c is also underestimated due to ΔT_c broadening at high temperatures. The difference between our H_c2 fitting parameters, D_σs, with those obtained by Refs. ⁴⁹ and ⁵⁰ is within 40%. Since our H_c2 fitting follows the rule that π bands are isotropic $(D_{\pi }=D_{\pi }^{ab}=D_{\pi }^c)$, our D_π value is comparable to only one of D_πs reported by Refs. ⁴⁹ and ⁵⁰.

FIG. 6.

Temperature dependence of μ₀H_c2//ab and μ₀H_c2//c and fitting curves for (a) Dy0.0-C2-675 and (b) Dy1.0-C2-675.

TABLE III.

Parameters for fitting μ₀H_c2//ab(T) and μ₀H_c2//c(T) curves and calculated $D_{\sigma }^{ab}$ and $D_{\sigma }^c$.

Fitting parameter	Dy0.0-C2-675	Dy1.0-C2-675
g	0.054	0.081
${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}(\mathrm{c}{\mathrm{m}}^2/\mathrm{s})$	1.73	1.59
$D_{\sigma }^{ab}(\mathrm{c}{\mathrm{m}}^2/\mathrm{s})$	5.45	4.50
$D_{\pi }=D_{\pi }^{ab}=D_{\pi }^c(\mathrm{c}{\mathrm{m}}^2/\mathrm{s})$	0.0172	0.0100

In the first place, it can be seen that Dy₂O₃ addition do increase g and also decrease ${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}$, $D_{\sigma }^{ab}$, and $D_{\pi }$ further for already C-doped MgB₂. In other words, both intraband and interband scattering rates of C-doped MgB₂ are increased by adding Dy₂O₃. We note that Dy₂O₃ additions, unlike C and Al doping, are unable to induce B or Mg-site substitution within the MgB₂ lattice. It is also known that secondary phases that collect within the grain boundaries can increase μ₀H_c2//ab.⁵¹ However, given the fact that DyB₄ nanoinclusions are known to form within the grains and that we see increases in both μ₀H_c2//ab and μ₀H_irr, the most probable mechanism for the increase in band scattering is that DyB₄ nanoinclusions^27,31 have short-range localized strain in the MgB₂ matrix. The effects of Dy₂O₃ additions can be separated into two states. First, the properties [ ${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}=1.73\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$, $D_{\pi }=0.0172\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$, g = 0.054] of Dy0.0-C2-675 are set as starting points, which is called state 1. Then, the influence of 1.0 wt. % Dy₂O₃ addition on intraband scattering is accounted for, which leads to a transition from state 1 to state 2 by decreasing the values of ${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}$ and $D_{\pi }$ to 1.59 and 0.0100 cm²/s, respectively. Next, increasing the value of g to 0.081 accounts for the influence of 1.0 wt. % Dy₂O₃ addition on the interband scattering effect, leading to a transition from state 2 to state 3. Figure 7 shows the change of μ₀H_c2//ab(T) caused by the whole set of transitions from state 1 to state 2 to state 3. Decreasing ${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}$ from 1.73 to 1.59 cm²/s and $D_{\pi }$ from 0.0172 to 0.0100 cm²/s induces H_c2//ab enhancements, leading to a Δμ₀H_1 → 2 of +2.357 T at 4.2 K, +1.245 T at 15 K, +0.949 T at 20 K, and +0.663 T at 25 K, as shown in Table IV. By keeping ${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}=1.59\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$ and $D_{\pi }=0.0100\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$ unchanged, increasing g from 0.054 to 0.081 induces H_c2//ab suppression, leading to a Δμ₀H_2 → 3 of −0.727 T at 4.2 K, −0.431 T at 15 K, −0.663 T at 20 K, and −0.843 T at 25 K. Therefore, adding 1.0 wt. % Dy₂O₃ into already 2 mol. % C-doped MgB₂ causes the change of state 1 → state 2 → state 3, generating a Δμ₀H (=Δμ₀H_1 → 2+ Δμ₀H_2 → 3) of +1.630 T at 4.2 K, +0.814 T at 15 K, +0.287 T at 20 K, and −0.180 T at 25 K, Table IV. Thus, while overall the rate of interband to intraband scattering is lower for Dy₂O₃ than C, allowing Dy₂O₃/C co-doping to lead to higher H_c2//c and H_c2//ab at elevated temperatures, Dy₂O₃ does have some interband scattering. Thus, heavier 1.0 wt. % Dy₂O₃ addition leads to the small changes in H_c2//ab at relatively higher temperatures.

FIG. 7.

Effects of ${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}$, $D_{\pi }$, and g on μ₀H_c2//ab(T) behavior of MgB₂. State 1 represents the performance of ${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}=1.73\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$, $D_{\pi }=0.0172\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$, and g = 0.054. State 2 represents the performance of ${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}=1.59\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$, ${\mathit{D}}_{\mathit{\pi }}$ = 0.0100 cm²/s, and g = 0.054. State 3 represents the performance of ${(D_{\sigma }^{ab}{D}_{\sigma }^c)}^{1/2}=1.59\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$, $D_{\pi }=0.0100\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$, and g = 0.081.

TABLE IV.

Comparison of μ₀H_c2 (unit: T) for state 1, state 2, and state 3 in Fig. 7. Δμ₀H_1 → 2 and Δμ₀H_2 → 3 represent the transitions from state 1 to state 2 and from state 2 to state 3, respectively. Δμ₀H is the sum of Δμ₀H_1 → 2 and Δμ₀H_2 → 3.

T (K)	μ0Hc2 in state 1	μ0Hc2 in state 2	Δμ0H1 → 2	μ0Hc2 in state 3	Δμ0H2 → 3	Δμ0H
4.2	23.54	25.89	+2.357	25.17	−0.727	+1.630
15	13.80	15.05	+1.245	14.61	−0.431	+0.814
20	10.59	11.54	+0.949	10.88	−0.663	+0.287
25	7.427	8.090	+0.663	7.247	−0.843	−0.180

Increased values of g tend to reduce H_c2, H_irr, and H_irr* at higher temperatures (≥20 K). Such an increase in g can be reflected by a decrease in T_c. In this study, a parameter T_c* is introduced as a surrogate for T_c. It is defined as the temperature where H_irr* vs T curve extrapolates to zero, Fig. 8(a). When μ₀H_irr*–T data are not available, similar effective T_c*s can be obtained by linearly extrapolating μ₀H_irr vs T curves to zero. The influence of C doping alone as well as C/Dy₂O₃ co-addition on T_c* can be clarified by setting 2 mol. % C concentration as a reference, as is shown in Fig. 8(b). Here, a concentration difference, Δχ, is defined and obtained by subtracting the 2 mol. % from doping or co-addition. Therefore, for samples that only have C doping, Δχ = X − 2 mol. % corresponds to X mol. % C doping. Since the concentration of 2 mol. % C/Y wt. % Dy₂O₃ co-addition can be expressed as (2 mol. % + Y wt. %), the Δχ of this co-addition is Y wt. %. For example, the Δχ of 2.0 wt. % corresponds to 2.0 mol. % C/2.0 wt. % Dy₂O₃ co-addition. As shown in Fig. 8(b), as compared to C/Dy₂O₃ co-addition, C doping decreases T_c* at a much faster rate, producing a stronger interband scattering and hence greater suppressed H_c2/H_irr at high temperatures. The overall results indicate that Dy₂O₃/C co-addition is more promising than C doping for seeking better performances of MgB₂ over a wide temperature range.

FIG. 8.

Comparison of (a) μ₀H_irr*s and T_c* for Dy2.0-C2-650, Pure-C2-675, and Pure-C3-700 and (b) T_c* between C/Dy₂O₃ co-addition and C doping. The H_irr*s of Pure-C2-675 and Pure-C3-700 are extracted from the J_c–μ₀H data from Ref. 38. The T_c*s are extracted from the μ₀H_irr*–T data from Ref. 38 and μ₀H_irr–T data from Ref. 32 for 0–4 mol. % C doping. The T_c* of 0.5 wt. % Dy₂O₃ are extracted from the μ₀H_irr–T data reported in Ref. 31.

V. CONCLUSIONS

This study confirms that Dy₂O₃ additions can further increase H_c2 and H_irr of already optimally C-doped MgB₂ at 4.2 K. However, this study goes further, systematically showing the effect of Dy₂O₃ addition on J_cs, μ₀H_c2s, μ₀H_irrs, critical field anisotropies of C-doped PIT MgB₂ wires fabricated with 2 mol. % C doping and co-addition of 0–3 wt. % Dy₂O₃. Irreversibility fields (μ₀H_irr), upper critical fields (μ₀H_c2), and transport critical currents were measured, and from these quantities, the anisotropies (γ) and intraband diffusivities (D_π,σ) were estimated for the first time for wires with C and Dy₂O₃ co-additions. The results were compared to previous results for wires with C doping only, and further band scattering, primarily intra-band scattering, was observed due to Dy₂O₃ additions. The highest 4.2 K, 10 T non-barrier J_c of 4.35 × 10⁴ A/cm² was achieved by an MgB₂ wire with 1.0 wt. % Dy₂O₃ addition (to 2.0 mol. % C-doped strands) indicating 1.0 wt. % is the optimal Dy₂O₃ concentration (for co-doping with 2.0 mol. % C) for maximizing the non-barrier J_cs of the MgB₂ wire at 4.2–15 K. Above 15 K, we found that 0.5 wt. % Dy₂O₃ doping with 2.0 mol. % C performs better, but in general, Dy₂O₃ + C doping outperforms C-doped-only samples comparing our results in this present work to previous studies. This is because the increases in intraband scattering induced by Dy₂O₃ addition generates lower electronic diffusivities and thus leads to further increased values of H_irr*, H_irr, H_c2//ab, and H_c2//c for C-doped MgB₂. The μ₀H_c2 modeling based on the two-band theory shows that the increases in H_irr, H_c2//ab, and H_c2//c primarily resulted from the increases in intraband scattering with Dy₂O₃ additions. Thus, C/Dy₂O₃ as co-additives have the potential to obtain higher H_irrs than C dopant as C/Dy₂O₃ co-addition increases interband scattering at a lower rate than C doping does, suggesting that C/Dy₂O₃ co-additions are a better approach for improving non-barrier J_c of MgB₂ at higher temperatures.

NOMENCLATURE

D_σ = 

Electronic diffusivity associated with σ band

D_π = 

Electronic diffusivity associated with π band

D_σ(π/2) = 

D_σ for ab-plane parallel to the magnetic field

D_σ(0) = 

D_σ for c-axis parallel to the magnetic field

D_σ,_π = 

Electronic diffusivities due to intraband scattering of σ or π band

$D_{\sigma }^{ab}$ = 

D_σ along the direction of ab-plane

$D_{\sigma }^c$ = 

D_σ along the direction of c-axis

f_σ,π = 

Usadel green function associated with σ or π band

F₀ = 

Flux pinning strength

F_m = 

Effective pinning strength

g = 

Interband scattering parameter

J_c = 

Critical current density

J_c(α) = 

Microscopic critical current density of MgB₂ grain

k_B = 

Boltzmann constant

p_c = 

Percolation threshold

p_c* = 

Effective percolation threshold that is equal to $p_c/p_{Mg{B}_2}$

p(J) = 

Fraction of superconducting grains whose J_c is larger than J

$p_{Mg{B}_2}$ = 

Fraction of MgB₂ grains in conductor (number of MgB₂ grains over the total number of MgB₂ grains and voids)

R = 

Resistance of MgB₂ wire

R_n = 

Normal state resistance of MgB₂ wire

t = 

Normalized temperature that is defined as T/T_c0

t_c = 

Normalized critical temperature that is defined as T_c/T_c0

T_c = 

Critical temperature of MgB₂ wire

T_c* = 

Critical temperature that is extrapolated from data of μ₀H_irr vs T or μ₀H_irr* vs T

T_c0 = 

Critical temperature of MgB₂ without interband scatterings

U(x) = 

The function that is defined as $\psi (1/2+x)-\psi (x)$

μ₀H_c2 = 

Upper critical field of MgB₂ wire

μ₀H_c2//ab = 

Upper critical field when the direction of external field is parallel with ab-plane of MgB₂ lattice

μ₀H_c2//c = 

Upper critical field when the direction of external field is parallel with c-axis of MgB₂ lattice

γ = 

H_c2 anisotropy that is equal to H_c2//ab/H_c2//c

μ₀H = 

External magnetic field

ρ = 

Resistivity of MgB₂ wire

ρ_n = 

Normal state resistivity of MgB₂ wire

μ₀H_irr = 

Irreversibility field that is extrapolated from data of R vs μ₀H or R vs T

μ₀H_irr* = 

Irreversibility field that is extrapolated from J_c vs μ₀H

α = 

Angle between magnetic field direction and c-axis

Γ_πσ and Γ_σπ = 

Interband scattering rates

ω = 

Matsubara frequency

Δ_σ,π = 

σ or π bandgap

$\psi (x)$ = 

Digamma function

λ_σσ and λ_ππ = 

BSC constants that describe intraband coupling

λ_σπ and λ_πσ = 

BSC constants that describe interband coupling

W, λ₀, λ₋, λ_i and p = 

Appropriate functions of interband scattering rates and BSC coupling constants

${\mathrm{\Omega }}_{\pm }$ and ${\mathrm{\Omega }}_0$ = 

Appropriate functions of intraband scattering rates, interband scattering rates, and upper critical fields

${\mathrm{\varnothing }}_0$ = 

Flux quantum

ℏ = 

Reduced Planck constant

Δμ₀H_1 → 2 = 

μ₀H_c2//ab of state 2 minus μ₀H_c2//ab of state 1

Δμ₀H_2 → 3 = 

μ₀H_c2//ab of state 3 minus μ₀H_c2//ab of state 2

Δμ₀H = 

Sum of Δμ₀H_1 → 2 and Δμ₀H_2 → 3

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

F. Wan: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (lead); Writing – original draft (lead); Writing – review & editing (equal). M. D. Sumption: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Supervision (equal); Writing – review & editing (equal). E. W. Collings: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.