Introduction

At present, conventional low-temperature superconductors such as Nb3Sn have been extensively applied in high-energy and nuclear physics, as well as in magnetic resonance imaging systems1. Inhomogeneity of Sn content is inevitable in practical Nb3Sn conductors, due to the high vapor pressure of Sn at the formation temperature of the A15 phase in a solid-state diffusion reaction2, 3. The tin variation in a conductor covers nearly the entire A15 phase field of binary Nb1−β Sn β with β = 0.18~0.2554. In a Nb3Sn conductor, the Sn gradient across the A15 layer has a remarkable impact on the local variation of the superconducting properties5. Experiments show that, the upper critical field B c2 varies almost linearly at ~5 T per at% between 19.5% and ~24 at% Sn, and the transition temperature T c versus tin concentration β also exhibits a linear relation within nearly the entire A15 phase field3. When β approaching stoichiometry of Nb1−β Sn β , β ≈ 24 at%, T c (or B c2) versus β no longer keeps the linear relationship due to the lattice softening (decreasing in phonon frequency)6, which means the varying Sn content leads to the cubic-tetragonal phase transformation of A15 lattices. The alloying addition (Ti and/or Ta), which is introduced in most modern high-field Nb3Sn conductors for increasing the electrical resistivity, suppressing the martensitic phase transformation and thus raising B c2, also becomes approximately linear with the Sn content β 7. As for other additives, the ZrO2 precipitates in Nb3Sn wires could refine Nb3Sn grain size such that change the pinning behavior8.

For Nb3Sn wires, the scaling law and the experiment show a disagreement in the field dependence of the pinning force at high reduced fields9. One of the reasons could be the inhomogeneity of microstructure and composition9. This also explains the observation that the scaling field lies below the experimental B c2 of Nb3Sn wires. In fact, the scaling field for the critical current reflects the average properties over the wire volume; it thus relates to the local variation of the critical field dependent on the composition gradient10. Cooley and the coauthors simulate the effect of Sn composition gradients on the superconducting properties of powder-in-tube (PIT) Nb3Sn strand, with an ideal structure modeled by concentric shells with varying Sn content7. They found that different Sn profiles have a pronounced effect on the irreversibility fields defined by the extrapolation of Kramer plots H K, and also that Sn gradients reduce the elementary pinning force, H K and the critical current density J c 7.

The temperature dependence of the upper critical field H c2(T) in inhomogeneous Nb3Sn conductors, as the field-temperature phase boundary, is comprehensively investigated by Godeke et al.2. It is concluded that, the complete field-temperature phase boundary can be described with the simplest form of the Maki-DeGennes (MDG) relation, and independent of compositional variation, measuring technique, criterion for the critical field and strain state2.

Various experiments11,12,13 and models14, 15 have recently been conducted to investigate the dependence of the superconductivity and magnetic properties of Nb3 Sn samples on Sn content and disorder. The underlying physics for the superconducting properties variation with the A15 composition in Nb3Sn is however still not very clear. The already-existing physical formulas for this dependence are insufficient for describing the state-of-art experimental results of binary Nb3Sn samples and practical Nb3Sn wires. In this paper, we develop a series of formulas based on GLAG theory; with respect to the previous formulas, the present ones systematically describe the major behaviors of the relevant experimental phenomena: the composition dependence of H c2, phase boundary and pinning behaviors. By this way, new physical insights into the importance of the composition inhomogeneity are provided. As for practical engineering significance, tin composition and possible additives are important for the very high J c now achieved in commercial strands16, 17. Therefore, describing the superconducting properties dependence on the A15 composition in theory will facilitate the understanding of the optimization for the critical current density J c. Based on the above considerations, it is of importance to investigate the physical mechanism of the superconducting properties dependences in inhomogeneous Nb3Sn samples.

In this paper, We model the variation of the upper critical field H c2 with Sn content in A15-type Nb1−β Sn β , within the GLAG theory frame. In this theory, the occurrence of H c2 is due to the breaking of orbital pair and the Pauli paramagnetic limiting. Two scattering mechanisms should be considered in type-II superconductors like Nb3Sn: electron-transport scattering by impurity (disorder) and spin-orbit scattering. As for A15 Nb3Sn, composition inhomogeneity and its deviation from stoichiometry may cause defect and site disorder in Nb3Sn lattices4, which contribute most to the scattering by impurity. Based on this physical picture, H c2 is correlated to the superconductivity parameters (the coherence length, the London penetration depth and etc.) as well as the scattering characteristics (the mean free path of electron transporting). As for the strong-coupling superconductor like Nb3Sn, one should include the correction for electron-phonon interaction to the weak-coupling BCS value. Since the microscopic parameters mentioned above cannot be determined directly, we correlate them to the transition temperature, normal-state resistivity and coefficient of electronic heat capacity. The three material parameters have been extensively measured as a function of tin content3, 4, 18, 19. In this manner, we can determine the superconducting properties variation with composition concentration. The following section will present the detail.

Model

Upper critical field at vicinity of superconducting transition temperature

The best quality Nb3Sn samples, with highest transition temperature T c’s and resistance ratios (RRR = ρ(300 K)/ρ(20 K)), have very narrow resistive transitions19. The transitions tend to broaden in high fields. Measured ternary PIT wire also exhibits a narrow transition at a wide range of fields and temperatures2. At the vicinity of T c, the upper critical fields H c2(T) are thus nearly the same, and almost independent on the selected critical-state criterion. We are then allowed to determine H c2 near T c uniquely. The resistivities ρ near T c are approximated as ρ(T c), measured at temperatures just above T c, and the coefficient of electronic heat capacity γ remains a constant at the vicinity of T c, satisfying the low-temperature heat capacity formula without undergoing a specific heat jump.

In the microscopic physical concept, the scattering of impurity (Supplementary Information A and Fig. A1) enters into the superconductivity of type II superconductors by changing the Ginzburg-Landau (GL) parameter κ at the vicinity of T c. We have κ in two limiting cases, κ clean(T) = κ(T c)χ 1(T) without scattering effect and κ dirty(T) = κ dirty(T c)χ 2(T) relevant to scattering, where κ clean(T c) ≈ 0.96δ L(0)/ξ 0 and κ dirty(T c) = 0.72δ L(0)/l (see Supplementary Information B for GLAG description of the superconductivity parameters). Here, δ L(0) is the London penetration depth of the magnetic field at 0 K. ξ 0 is the standard coherence length. l is the mean free path of electron transporting. κ clean and κ dirty refer to the clean limit (ξ 0 ≪ l) and dirty limit (ξ 0 ≫ l), respectively. χ 1(T) and χ 2(T) represent the temperature dependence of κ for the clean limit and dirty limit, respectively; calculations show that they vary little with T near T c, χ 1(T) ≈ χ 2(T) ≈ 120.

In light of the classic proposal19, 21, the superconductivity parameters δ L(0), ξ 0 and the scattering parameter l involved in the GL parameter κ can be linked to three independent material parameters, the transport scattering resistivity ρ, the low-temperature coefficient of electronic heat capacity γ and the superconducting transition temperature T c. Assuming a spherical Fermi surface and isotropic metal, we are allowed to use the electron conduction formula Eq. (A1) and the electron heat capacity relation Eq. (C1) in the GLAG description of the superconductivity parameters (Supplementary Information C), and then express l, ξ 0, δ L(0), κ clean and κ dirty as functions of ρ, γ and T c, Eqs. (C15)–(A19). Including the correction for the anisotropy in Nb3Sn (Supplementary Information D), one must consider the change of Fermi surface shape from the isotropic model, which leads to the corrected expressions, Eqs. (D3)–(D6).

Let us consider the dependences of T c, ρ and γ with Sn content β. Experiments3 show that T c variation with β is essentially linear up to ~24 at% ; while between 24 and 25 at% Sn content, T c versus β has a saturation, Fig. 1. The A15 type lattice undergoes a spontaneous cubic-tetragonal transformation in this range. The lattice softening, as one of the consequences, implies a decreasing in the lattice stiffness (ħω D = k B θ D) such that it reduces T c according to McMillan T c equation22 [Eq. (6)]. The normal-state resistivity ρ, as measured just above T c, decreases moderately with increase of β within 18~24 at%. Approaching β = 25 at%, ρ(β) exhibits an obviously stronger decrease. Below 24 at%, the coefficient of electronic heat capacity, γ, changes linearly with β by ~1.5 mJ · K−2 · mol−1 per at% Sn. Following the treatment to the experimental data in ref. 3, we use global fits to the experimental T c, γ and ρ versus β in the entire A15 range (Fig. 1 and Table 1),

$${T}_{{\rm{c}}},\gamma \,{\rm{or}}\,\rho =\{\begin{matrix}{a}_{1}\beta +{b}_{1}, & {\rm{linear}}\,\mathrm{fit},\\ {a}_{2}{\beta }^{3}+{b}_{2}{\beta }^{2}+{c}_{2}\beta +{d}_{2}, & {\rm{cubic}}\,{\rm{fit}}{\rm{.}}\end{matrix}$$
(1)
Figure 1
figure 1

Variations of material parameters T c, ρ, γ, θ D and 2Δ(0)/(k B T c) with Sn content β in the A15 range of binary Nb1−β Sn β . The experimental data are extracted from ref. 3. The solid lines represent the global fits to the experimental data in the A15 range.

Full size image
Table 1 Material parameters variation with Sn content β [at.%].
Full size table

We are now ready to calculate the dependence of the superconducting characteristic lengths δ L(0), ξ 0 and l with the Sn concentration β, by substituting Eq. (1) into Eqs. (D4), (D5) and (D6). The London penetration depth δ L(0) increases with Sn content, while the coherence length ξ 0 is reduced, Fig. 2. δ L(0) and ξ 0 are roughly with the same order of magnitude over the A15 range. By comparing the coherence length ξ 0 and the electronic mean free path l, one may distinguish the clean limit, the dirty limit and the intermediate state at any A15 composition. At lower Sn content, ξ 0 maintains rather high value compared to l; but this difference is mitigated as ξ 0 continues to decrease and l increase, for rising β until the phase transformation boundary. In the tetragonal phase range, ξ 0 is approximately equal to or even lower than l. This implies that, as Sn content gradually approaches the stoichiometry, Nb1−β Sn β undergoes a transition from the “dirty” limit (ξ 0 ≫ l) to the “clean” limit (ξ 0 ≪ l), and the phase transformation boundary may be taken as the boundary of this transition. In Fig. 2, we also present the variation of GL parameter κ with tin content, calculated by Eqs. (C19) and (D3). κ in the two limiting cases, κ clean and κ dirty, have the opposite change with Sn content; their change nearly counteract one another in the cubic phase range. While in the tetragonal phase range, κ dirty exhibits a more severe decrease compared to the increase in κ clean. Thus, κ = κ clean + κ dirty varies little in the cubic phase range, but decreases largely in the tetragonal phase.

Figure 2
figure 2

Composition dependence of the London penetration depth δ L(0), coherence length ξ 0 and electronic mean free path l (Top), and the GL parameter κ (Bottom).

Full size image

The magnetic properties of type II superconductors give \({H}_{{\rm{c2}}}=\sqrt{2}\kappa {H}_{{\rm{c}}}\) at the vicinity of T c, where H c is the thermodynamic critical field23. Combining this with the expressions κ and H c, Eqs. (C19), (C20) and (D3), renders the upper critical field H c2 in the clean limit and the dirty limit, Eqs. (C14) and (D7) (see Supplementary Information C and D for detail). Thus, H c2 at the intermediate ξ 0/l can be obtained by summing H c2,clean and H c2,dirty,

$${H}_{{\rm{c2}}}={C}_{1}{{T}_{{\rm{c}}}}^{2}(1-T/{T}_{{\rm{c}}}){\gamma }_{{\rm{mJ}}}^{2}+{C}_{2}{T}_{{\rm{c}}}(1-T/{T}_{{\rm{c}}}){\gamma }_{{\rm{mJ}}}{\rho }_{{\rm{\Omega }}{\rm{m}}}.$$
(2)

Here, \({C}_{1}=1.979\times {10}^{8}{\pi }^{1/3}c{e}^{-1}{k}_{{\rm{B}}}^{-2}\hslash {({n}^{2/3}S/{S}_{{\rm{F}}})}^{-2}\) and \({C}_{2}=1.356\times {{\rm{10}}}^{-6}{\pi }^{-1}ce{k}_{{\rm{B}}}^{-1}\) in which the physical constants are listed in Table A1. We have omitted multiplier R(λ tr), which is defined in the Gor’kov function19 to consider the effect of the ξ/l variations (ranging from ξ/l ≪ 1 to ξ/l ≫ 1), in the sum of H c2,clean and H c2,dirty [Eq. (2)]. This is reasonable since R(λ tr) changes little within the entire ξ/l range24. The involved three material parameters T c, γ mJ and ρ Ωm as well as their variations with the composition concentration in A15 Nb1−β Sn β have been widely measured3, Eq. (1) and Fig. 1. Thus, substituting Eq. (1) into Eq. (2), one finds the dependence of the upper critical field H c2 with Sn content. Note that the above formulas are valid at the vicinity of the superconducting transition temperature T c.

Corrections to upper critical field for electron-phonon interaction

A15 type Nb-Sn is a strong-coupling superconductor such that the corrections for electron-phonon (EP) interaction are required. We obtain the upper critical field, Eq. (2), based on the breaking and scattering of Cooper pairs in a weak-coupled interaction. Equation (2) is corrected for EP interaction in two ways: renormalizing the material parameters, and introducing correction parameters to superconductivity itself19, 22, 25.

We determine which material parameter should be renormalized in terms of Grimvall’s principle25. Both the density of states and the wave function are renormalized by EP interaction, and the electron mass in the absence of EP interactions is replaced by a renormalized electron mass m = m b(1 + λ ep), where b refers to band values. The electronic density of states υ(μ) at Fermi level is renormalized by an enhancement factor 1 + λ ep where λ ep is the EP interaction parameter, and thus the coefficient of the electronic heat capacity γ mJ is enhanced by (1 + λ ep)γ mJ. However, γ mJ is not changed for EP renormalization in Nb3Sn, since there are no EP renormalization effects in the change of the Fermi surface dimensions on alloy and the change in υ(μ) always depends on the Fermi level and follows almost rigidly any shift in Fermi energy. The electrical resistivity ρ Ωm [Eq. (A1)] is not renormalized, since the renormalization of the electron mass m exactly cancels against the renormalization of the scattering matrix element as it enters the averaged time τ between collisions. We find that, these are implicitly followed by Devantay et al.26, who do not take renormalizations on γ mJ and ρ Ωm. Thus, we take the only EP correction in Nb3Sn by multiplying a factor \({\eta }_{{H}_{{\rm{c2}}}}(T)\) for H c2,

$${H}_{{\rm{c2}}}={\eta }_{{H}_{{\rm{c2}}}}({T}_{{\rm{c}}})[{C}_{1}{T}_{{\rm{c}}}^{2}(1-T/{T}_{{\rm{c}}}){\gamma }_{{\rm{mJ}}}^{2}+{C}_{2}{T}_{{\rm{c}}}(1-T/{T}_{{\rm{c}}}){\gamma }_{{\rm{mJ}}}{\rho }_{{\rm{\Omega }}{\rm{m}}}].$$
(3)

Here, \({\eta }_{{H}_{{\rm{c2}}}}({T}_{{\rm{c}}})\) is the ratio of the strong-coupled magnetic pair-breaking parameter to the weak-coupled BCS value, and can be evaluated by the detailed EP spectrum24.

If T c and the energy gap Δ(0) at 0 K are determined experimentally, we have the strong-coupling correction η Δ(0) to Δ(0), η Δ(0) = (2Δ(0)/k B T c)meas/(2Δ(0)/k B T c)BCS = (2π/η)−1(2Δ(0)/k B T c)meas ≈ 0.283(2Δ(0)/k B T c)meas in which the BCS equation Δ(0) = (π/η)k B T c with η = 1.78 has been used for the second equal sign. The characteristic (equivalent Einstein) frequency ω 0 [erg] is then determined using19 (Supplementary Information E)

$${\eta }_{{\rm{\Delta }}(0)}=1+5.3{({\omega }_{0}{k}_{{\rm{B}}}^{-1}{T}_{{\rm{c}}}^{-1})}^{-2}\,\mathrm{ln}({\omega }_{0}{k}_{{\rm{B}}}^{-1}{T}_{{\rm{c}}}^{-1}).$$
(4)

The EP correction to H c2, \({\eta }_{{H}_{{\rm{c2}}}}({T}_{{\rm{c}}})\), can thus be obtained,

$${\eta }_{{H}_{{\rm{c2}}}}({T}_{{\rm{c}}})=1+{\pi }^{2}{({\omega }_{0}{k}_{{\rm{B}}}^{-1}{T}_{{\rm{c}}}^{-1})}^{-2}[0.6\,\mathrm{ln}({\omega }_{0}{k}_{{\rm{B}}}^{-1}{T}_{{\rm{c}}}^{-1})-0.26].$$
(5)

The another correction parameter (not involved in EP correction for Nb3Sn), i.e. the EP interaction parameter λ ep, is determined by the McMillan strong-coupled T c equation22

$${T}_{{\rm{c}}}=\frac{{\theta }_{{\rm{D}}}}{1.45}\exp (-\frac{1.04(1+{\lambda }_{{\rm{ep}}})}{{\lambda }_{{\rm{ep}}}-{\mu }^{\ast }(1+0.62{\lambda }_{{\rm{ep}}})}),$$
(6)

where θ D is the Debye temperature and μ * is the pseudo potential parameter for electron Coulomb repulsion (for Nb3Sn μ * ≈ 0.2). Note that to obtain a more accurate λ ep for Nb3Sn one should use the Allen-Dynes T c formula, which extends the application range 0 < λ ep < 1.5 of Eq. (6) to a large λ ep value. Experiments give λ ep ≈ 1.8 for Nb3Sn. However, the available experimental data is θ D and not the characteristic phonon frequency required in the Allen-Dynes T c formula. Thus we still apply Eq. (6) at the cost of some accuracy.

The corrections for EP interaction require the information of the Debye temperature θ D and the energy gap Δ(0) at 0 K, Eq. (3). Unfortunately, the available data are reported at very limited Sn content. We use linear fits to θ D and (2Δ(0)/k B T c) according to the identification3 of the gradual changes from strong coupling to weak coupling with decreasing β, Fig. 1.

Figure 3 presents the calculated EP correction parameters in terms of Eqs. (1), (5) and (6). The result of the strong-coupling correction \({\eta }_{{H}_{{\rm{c2}}}}\) to H c2 approximately equals to the calculated value19 of 1.17 at T c = 17.8 K. The calculated EP interaction parameter λ ep at the stoichiometry 25 at% Sn is slightly lower than the generally accepted value of 1.827. However, at larger derivation from the stoichiometry, the calculated λ ep differs relatively larger from the value of 1.8.

Figure 3
figure 3

EP correction parameters, the Debye temperature θ D and energy gap Δ(0), versus tin content.

Full size image

Upper critical field at temperature 0 K

In the following we will concern the behavior of H c2 at temperatures far from T c, where the GLAG theory does not apply. This will generalize the above results near T c [e.g. Eq. (3)] to a wide temperature range and up to 0 K. The temperature dependence of H c2 in the scaling law for Nb3Sn is determined by an approximate form of the Maki-de Gennes (MDG) relation28,

$${H}_{{\rm{c2}}}(T)={H}_{{\rm{c2}}}(0)[1-{(T/{T}_{{\rm{c}}})}^{\lambda }].$$
(7)

Recent measurements suggest λ ≈ 1.52; this value has a universe applicability to a wide range of off-stoichiometric samples and different methods of determining H c2 10. This value of λ is also the power determined from the MDG theory and Eliashberg theory28, 29. Godeke et al. demonstrate that the MDG description (7) is universal for H c2(T) relation of Nb1−β Sn β independent of the compositional variation2, 10. Recent experiment shows that this relation is applicable for Nb3Sn with or without undergoing the cubic-to-tetragonal transition11. We are then allowed to extrapolate H c2 at vicinity of T c to H c2(0) at temperature 0 K for any Sn content. If H c2(T 0) with T 0 → T c has been obtained using Eq. (3), then one can deduce H c2(0) as

$${H}_{{\rm{c2}}}(0)={H}_{{\rm{c2}}}({T}_{0})/[1-{({T}_{0}/{T}_{{\rm{c}}})}^{\lambda }].$$
(8)

We assume that EP interaction correction to H c2 far from T c is the same as that for H c2 near T c.

WHH (Werthamer-Helfand-Hohenberg) equation, derived from Gor’kov superconductivity theory (Green function method) and taking into account electron spin and spin-orbital scattering30, is capable of giving rather satisfactory descriptions for H c2 behavior of a wide range of commercial and experimental Nb3Sn wires. This equation can be written as a simple form

$${H}_{{\rm{c2}}}(0)=0.69{H}_{{\rm{c2}}}^{^{\prime} }{T}_{{\rm{c}}},$$
(9)

where \({H}_{{\rm{c2}}}^{^{\prime} }=-{(d{H}_{{\rm{c2}}}dT)}_{{T}_{c}}\)  13. Substituting the derivative of Eq. (3) into Eq. (9) and then comparing to Eq. (3), one finds H c2(0) = 0.69H c2(T 0)/(1 − T 0/T c). It is further shown that the only difference between WHH equation and MDG relation is the power of (T 0/T c); Since T 0/T c → 1 this difference has very limited impact on the H c2(0) values. This is consistent with the viewpoint30, 31 that MDG relation is a good approximation to Werthamer theory.

We now calculate the upper critical field B c2 near T c using Eqs. (1), (2) and (3), Fig. 4. The temperature T 0 near T c is designated as T 0 = 0.9T c. One finds the dependence of B c2(T 0) on Sn content β with/without the EP correction. The results of B c2(T 0) are then substituted into the MDG relation, Eq. (8), to obtain B c2 at 0 K, B c2(0). Since the WHH relation is equivalent to the MDG relation, we do not implement it redundantly. The calculation results are impressive (Fig. 4): B c2(0) obtained by B c2(T 0) with EP correction and then extrapolated by MDG relation (EP + MDG) is in good agreement with the experiments. This validates the above GLAG descriptions, and also the MDG description for the temperature dependence of the upper critical field at any composition over the A15 phase field.

Figure 4
figure 4

Upper critical field B c2 variation as a function of tin content β at: the vicinity of T c (Top) and 0K temperature (Bottom). MDG: extrapolation of H c2(T c) with MDG relation, Eqs. (1), (2) and (8); EP + MDG: extrapolation of EP corrected H c2(T c) with MDG, Eqs. (1), (2), (3) and (8); PPL: Pauli paramagnetic limit, Eqs. (1) and (10); EP + PPL: EP corrected Pauli paramagnetic limit, Eqs. (1) and (11); □: Experiment dataset-13; △: Experiment dataset-211.

Full size image

We find that, the cubic-tetragonal phase boundary at ~24 at% Sn separates the increasing B c2 versus β from the decreasing B c2 versus β. As shown both in the experimental B c2(0) versus tin content and the EP + MDG curve, the maximum ~29 T of B c2(0) appears at ~24 at% Sn, and at both sides of the peak, B c2(0) decreases as tin content deviates more from 24 at% Sn. In the tetragonal phase, B c2(0) has a more serious reduction; however, at the vicinity of 24 at% Sn, B c2(0) have nearly the same values at both sides of 24 at% Sn, namely in the cubic phase range and the tetragonal phase range. This is consistent with the experiment by Zhou et al.11, their tetragonal phase [B c2(0.3 K) = 29.1 T at β = 24.6 ± 0.2 at%] and cubic phase [B c2(0.3 K) = 29.0 T at β = 23.7 ± 0.4 at%] samples exhibiting almost identical B c2(0) ≈ 29 ± 0.2 T. We infer that, this coincidence occurs in a limited range, where the tin content deviates small from the phase transformation boundary; for a large deviation there is a stronger depression of B c2(0) in the tetragonal phase. This phenomenon is caused by the underlying relationship between the superconductivity of Nb3Sn and the related material parameters; physical properties of the latter is continually changed by the spontaneous cubic-tetragonal transformation.

We now consider the upper critical field B c2(0) decomposing into the component in the clean limit case [Eq. (D7)] and that in the dirty limit case [Eq. (C14)], B c2(0) = B c2,clean + B c2,dirty. One finds that B c2,clean increases as β raised, but for contrast the dirty one increases up to 24 at% and then deceases drastically, Fig. 5. As we already know, Nb1−β Sn β undergoes a transition from the “dirty” limit to “clean” limit as β increased. By comparing Figs 4 and 5, we find that, within 18 at%~24 at%, the component B c2,dirty dominates in B c2(0) and determines the trend of B c2(0) curve. Once crossing over the phase transformation boundary, B c2,clean takes over B c2(0) variation with β.

Figure 5
figure 5

Upper critical field B c2(0) in the clean and dirty limit case.

Full size image

Limit for upper critical field: Pauli paramagnetic limit

Preferential Pauli-paramagnetic lowering of normal-state free energy should place a limit on the orbital-pair-breaking H c2 of filamentary high-field superconductor32,

$${H}_{{\rm{c2}}}(0)\le {H}_{{\rm{p}}}(0)\equiv 1.84\times {10}^{4}{T}_{{\rm{c}}}[{\rm{Oe}}].$$
(10)

This equation is a rough estimation for the limit field H p(0), since the assumption that the zero temperature difference between superconducting and normal-state magnetizations be at least equal to the Pauli conduction-electron-spin magnetization, is somewhat contrary to experimental results33. The correction to H p(0) for EP interaction is given by

$${H}_{{\rm{p}}}(0)=1.84\times {10}^{4}{\eta }_{{H}_{{\rm{c}}}}(0){(1+{\lambda }_{{\rm{ep}}})}^{1/2}{T}_{{\rm{c}}}[{\rm{Oe}}],$$
(11)

where \({\eta }_{{H}_{c}}({T}_{c})=1+{\pi }^{2}{({\omega }_{0}{k}_{B}^{-1}{T}_{c}^{-1})}^{-2}[1.1\,\mathrm{ln}({\omega }_{0}{k}_{B}^{-1}{T}_{c}^{-1})+0.14]\) assuming \({\eta }_{{H}_{{\rm{c}}}}(T)\) is independent with temperature19.

In Fig. 4, PPL (Pauli paramagnetic limit), Eqs. (1) and (10), slightly lowers the EP + MDG curve after ~23 at% Sn, and it thus provides a good boundary with the experimental data before the phase transformation. This justifies that, PPL is independent with the EP interaction in Nb3Sn. In Fig. 4, EP corrected PPL [Eqs. (1) and (11)] is much higher than the experimental curve and has no restriction to B c2(0). This is consistent with the view by Orlando et al.19, who demonstrate that EP corrected PPL has nothing to do with Nb3Sn superconductivity, since the strong EP interaction in Nb3Sn increases largely the Pauli limiting field above its BCS value and the spin-orbit scattering is less involved. This explains why we only take the impurity scattering into account while exclude the spin-orbit scattering.

Results and Discussion

Temperature dependence of upper critical field

Now, we focus on the temperature dependence of B c2 at any A15 composition concentration, Eqs. (1), (3), (7) and (8). In Fig. 6, the temperature dependence B c2(T) increases at any temperature as raising the Sn content β within the cubic phase range. The tetragonal phase exhibits a reverse behavior, B c2 at 0 K and at most of other temperatures increasing with reduced Sn content. Orlando et al. show that in thin films B c2(0) is increased with ρ rising, however T c is suppressed2, 19. For ρ(T c) = 35 μΩ · cm, there exists B c2(0) = 29.5 T and T c = 16.0 K; while ρ(T c) = 9 μΩ · cm leads to B c2(0) = 26.3 T and T c = 17.4 K19. This corresponds to the calculated composition dependence in the tetragonal phase range, Figs 1 and 6. The measured bulk needle by Godeke et al. exhibits a similar behavior: ρ(T c) = 22 μΩ · cm at a 50% normal-state resistance criterion corresponds toB c2(0) = 27.4 T and T c = 16.5 K, and at a 90% criterion B c2(0) = 28.3 T and T c = 16.6 K2. A summary of the calculations and the comparisons to experiments is given in Table 2. Through the theoretical expressions, we also figure out the material parameter that is undetermined by experiments, Table 2.

Figure 6
figure 6

Temperature dependence B c2(T) at varying Sn content in the range of the cubic phase at increment of 1 at% from 18 at% to 24 at% Sn. The insert figure represents the situation in the tetragonal phase for 0.2 at% increment between 24.2 at% and 25.4 at% Sn.

Full size image
Table 2 Calculated H-T phase boundary and comparison to experiments.
Full size table

We also calculate B c2(T) for two rather highly homogenous Nb3Sn samples, Fig. 7. Their material parameters, reproduced from11, are listed in the insert of the figure. Note that in the absence of the raw experimental value of γ, we infer it from Eq. (1). Calculations show that, the dirtier sample (25 Sn-1800) with the larger resistivity (31.3 μΩ · cm) always exhibits a higher B c2 at any temperature while T c is suppressed. This follows the above observation by Orlando et al.19 and Godeke et al.2. The little discrepancy from the experiment may be attributed to the absence of the raw data of γ and the greater uncertainty in experimentally determining the composition concentration (fluctuation of ±0.7 at% Sn for the dirtier sample compared to ±0.2 at% for the cleaner one).

Figure 7
figure 7

Temperature dependence B c2(T) for two homogenous off-stoichiometric Nb3Sn samples, and comparison to experiments11.

Full size image

Flux pinning force and Kramer plot

We now extend the method for the upper critical field B c2 to account for the pinning behavior at different composition concentrations. Using the flux pinning model proposed by Kramer34, the pinning force per volume for Nb3Sn conductors, F p(B), is given by10

$${F}_{{\rm{p}}}(B)={J}_{{\rm{c}}}B=12.8{\kappa }^{-2}{B}_{{\rm{c2}}}^{2.5}{(B/{B}_{{\rm{c2}}})}^{0.5}{(1-B/{B}_{{\rm{c2}}})}^{2}\,[{\rm{GN}}\cdot {{\rm{m}}}^{-3}],$$
(12)

where J c is the critical current density. F p(B) is associated with Sn concentration through the composition dependences B c2(T,β) and κ(β), which are formulated by Eqs. (1), (3), (7), (8), (C19) and (D3). The Kramer function \({f}_{{\rm{K}}}(B)={J}_{{\rm{c}}}^{0.5}{B}^{0.25}\) is linear with the magnetic induction B and can identify B c2 at which f K(B) = 010,

$${f}_{{\rm{K}}}(B)={J}_{{\rm{c}}}^{0.5}{B}^{0.25}=1.1\times {10}^{5}{\kappa }^{-1}({B}_{{\rm{c2}}}-B)[{{\rm{A}}}^{0.5}{{\rm{m}}}^{-1}{{\rm{T}}}^{0.25}].$$
(13)

From Fig. 8 we observe a profound influence of the composition concentration on the field dependent pinning force F p(B). Within the cubic phase range, the increase of Sn content raises the F p(B) remarkably and shifts the peak in each F p(B) curve to the high field region. This is associated with the composition dependent superconducting properties. To be specific, the magnitude of B c2 determines the position of the peak in F p(B), while the height of F p(B) is related to both B c2 and κ, Eq. (12). B c2 in the cubic phase range increases as β rising (Fig. 4) while κ varies little (Fig. 2), thus resulting in the shift of the peak and the increase in the height of F p(B). This case differs from that for the tetragonal phase range, where rising β leads to a drastic decrease in both B c2 and κ. As in Fig. 8, the peak shifts to the lower field and the height is nearly the same in all curves (due to \({F}_{{\rm{p}}}(B)\propto {\kappa }^{-2}{B}_{{\rm{c2}}}^{2.5}\)). The experimental data of a Nb3Sn conductor locate roughly between the calculated F p(B) curves for different Sn contents. This implies that, to describe a real Nb3Sn conductor one may consider the composition gradient in the conductor which results in a weighted average of local homogenous composition properties. For Kramer plot, we find its variation consistent with F p(B) for the similar reasons (Fig. 8).

Figure 8
figure 8

Effect of composition concentration on the magnetic field dependence of the flux pinning force (Top) and Kramer plot (Bottom). The experiment data are extracted from refs 10 and 28.

Full size image

Composition gradient effect on practical Nb3Sn wires

In the following we will apply our descriptions for the superconducting properties variation with tin content in the concentric shells model proposed by Cooley et al.7, to arrive at the real composition dependent behavior in practical Nb3Sn conductors. In the Nb3Sn filament of PIT wires, the A15 layer exists between a central Sn-rich core and a coaxial Nb tube, Fig. 9. This structure is convenient for using a series of concentric shells with varying Sn concentration to simulate the composition inhomogeneity in the wire. The Sn content β varies with the position in the A15 layer7,

$$\beta =18+3.5[1-{r}^{N}+{(1-r)}^{1/N}]\,[\mathrm{at} \% ],$$
(14)

where N indicates the severity and steepness of the overall gradient and r is the normalized radius of the filament cross section. The radius r is set to be 0 at the A15/Sn-core interface and reaches its maximum 1 at the Nb/A15 interface. The actual position in this area is counted as R = 10 + 5r [μm], indicating that the inner radius of the A15 layer is R min = 10 μm and the outer radius is R max = 15 μm.

Figure 9
figure 9

Distribution of the Sn content β (Top left), pinning force F p (Top right) and magnetic moment m o (Bottom right) on the cross section of PIT wire filament (Bottom left), at temperature T = 4.2 K. N indicates the severity of the overall gradient.

Full size image

For PIT wires, the prefactor of the flux pinning force differs from the Kramer model [Eq. (13)], \({F}_{{\rm{p}}}(B)={J}_{{\rm{c}}}B=0.35{B}_{{\rm{c2}}}^{2}{(B/{B}_{{\rm{c2}}})}^{0.5}{(1-B/{B}_{{\rm{c2}}})}^{2}[{\rm{GN}}\cdot {{\rm{m}}}^{-3}]\) and \({f}_{{\rm{K}}}(B)=1.871\times {10}^{4}{B}_{{\rm{c2}}}^{-0.25}({B}_{{\rm{c2}}}-B)[{{\rm{A}}}^{0.5}{{\rm{m}}}^{-1}{{\rm{T}}}^{0.25}]\). We can then link B c2(T,β) and J c to the radius r, and the local magnetic moment is calculated as m o = πR 2 tLJ c [Am2], where t is the thickness of each of the 100 shells, i.e. t = (R max − R min)/100 and L is the sample length.

Figure 9 presents the Sn content variation with the radius r at different gradients N. It is found that, larger steepness of the overall gradient N has a more drastic decrease near the outer radius (closer to the real situation). The distribution of the magnetic moment m o along the radius r at different applied magnetic fields is also presented in Fig. 9. At any magnetic field B, the regions near the Nb/Nb3Sn (r = 1) and/or Nb3Sn/Sn-core (r = 0) interfaces appear no magnetic moment and thus loss of superconductivity. The larger region of the loss occurs for the higher magnetic field, with a suppression in the magnitude of magnetic moment. In fact, the magnetic moment is related to the local critical current density and thus the pinning force F p(B). One can find the Sn content dependence of F p(B) from Fig. 8; in the cubic phase range, F p(B) is raised with the Sn content increasing at any field, and F p for lower Sn content more probably turns to disappear at higher fields. These are the underlying reasons for the vanishing of the magnetic moment. The tetragonal phase has a similar corresponding relationship. So, the loss of superconductivity near the boundary of A15 layer is mainly associated with the change of the flux-pinning behavior due to the A15 composition variation.

Summing the contributions m o from each shell results in the total magnetic moment m ot, which can be measured by magnetometry. The critical current density J c of the wire is then expressed as \({J}_{{\rm{c}}}=3{m}_{{\rm{ot}}}/[\pi L({R}_{\max }^{3}-{R}_{\min }^{3})]\,[{A/m}^{{\rm{2}}}]\) in a longitudinal field7. In Fig. 10, we present the variation of J c as a function of the magnetic field B for different Sn gradients N. The homogenous sample results are presented to emphasize the effect of the tin gradient. The results of J c(B) allow one to depict the Kramer plot f k(B). The Kramer plot2 for the measurements on a SMI ternary PIT wire exhibits an anomalous curvature “tail” at the vicinity of B c2, Fig. 10. Two calculated homogenous samples with 21 at% and 23 at% appear no curvature, however, the inhomogeneous samples have a curved f k(B) when approaching B c2. An acceptable agreement with the experiment is found for the conductor with gradient severity N = 20. At this point, we reproduce the positive curvature in the Kramer plot of the PIT wire, emphasizing the importance of the composition inhomogeneity in the superconducting properties of practical Nb3Sn conductors. The composition dependent pinning force and Kramer function explain the disagreement between the results of the scaling laws and the experiments for practical inhomogeneous conductors.

Figure 10
figure 10

Effect of the composition gradient on the critical current density (Left) and Kramer function \({J}_{{\rm{c}}}^{0.5}{B}^{0.25}\) (Right) of PIT wires. The measurements on SMI ternary PIT wire are duplicated from ref. 2. The Kramer plot with a composition gradient N = 20 has a positive curvature approaching the experiment.

Full size image

Conclusions

Although Nb3Sn has been extensively used in fusion engineering area like ITER, the dependence of the superconducting properties with inhomogeneous composition is established incompletely in theory. Based on the GLAG theory frame, we derive a series of expressions for the superconductivity parameters [Eq. (1)] and the upper critical field as a function of the three material parameters [Eq. (2)]. These relations have a complete self-consistent theory basis describing the variation of superconductivity of Nb3Sn with Sn content. The corrections for EP interaction are included [Eq. (3)]. The theoretical estimation of H c2(0) variation with Sn content [Eqs. (3) and (8)], provided with the fits to the material parameters, shows an acceptable agreement with the experiments.

A15-type Nb-Sn undergoes a transition from “dirty” limit (ξ 0 ≫ l) to “clean” limit (ξ 0 ≪ l) as Sn content gradually approaches the stoichiometry [Eqs. (C19), (D4), (D5), (D6) and (D3)]. The change in superconductivity at the vicinity of the critical temperature and in the related material parameters determines the composition dependence of the upper critical field. The MDG description [Eq. (7)] is universal for describing H-T phase boundary over the A15 phase field.

In the cubic phase range, B c2(T) increases with raising Sn content [Eqs. (1), (3), (7) and (8)]. There appears an inverse B c2(T) behavior in the tetragonal phase range. A significant influence of the composition concentration on F p(B) [Eqs. (12) and (13)] curves is observed. Within the cubic phase range, the increase of Sn content raises the F p(B) remarkably and shifts the peak in each F p(B) curve to the right side. The peak shifts to the lower field and the shape is nearly the same in the tetragonal range. This can be explained by the composition dependencies B c2(T,β) and κ(β).

The effect of composition gradient on the superconducting properties of PIT wires is considered by applying the obtained formulas in Cooley’s concentric shells model [Eq. (14)]. The loss of superconductivity near the boundary of A15 layer is mainly associated with the change of the flux-pinning behavior due to the A15 composition variation. The inhomogeneous conductor with gradient severity N = 20 predicts well the curved “tail” approaching B c2 in the Kramer plot. This implies that the composition inhomogeneity is an important factor in the unusual phenomenon of the practical Nb3Sn conductors.

However, we cannot yet include the effect of the alloying addition (Ti and/or Ta) and the matrix material (bronze and/or copper) in a ternary Nb3Sn wire. The composition gradient is an important factor in the unusual phenomenon of practical Nb3Sn conductors but not the unique determinant. The most possible role of the alloying addition taken in Nb3Sn is the scattering impurity (Supplementary Information A), which changes the electrical resistivity deeply. We will include this effect in the future work to increase the practical value of the present theory.