Limit to Self-Field Critical Current Density in Thin-Film, Type-II Superconductors

Abstract

In the last decade, the self-field critical current density J_c(s.f.) in Type-II superconductors has been considered fundamentally limited by a Silsbee-like criterion of J_c(s.f.) = H_c1/λ. We show that this universal limit to self-field critical current density J_c(s.f.) is not universally valid. We present several examples for this in YBa₂Cu₃O_7−δ-type and REBa₂Cu₃O_7−δ thin films and one for Nb thin films and show that calculated J_c(s.f.) using the Silsbee-like criterion using thermodynamic parameters has been substantially exceeded experimentally. We also show that J_c(s.f.) can be significantly improved by incorporation of artificial pinning centers (APCs), further implying that no such universal limit to J_c(s.f.) can exist because such an upper bound, J_c(s.f.) would have to be independent of APCs. These findings call for a revision of the accepted understanding of current-carrying limits in Type-II superconductors and reveal substantial potential for improving J_c in REBCO-based coated conductors through optimization of APCs for large-scale applications, including commercial nuclear fusion.

1. Introduction

Kilometer-long HTS wires based on heteroepitaxial deposition of high-temperature superconductors (HTS) on single-crystal-like substrates have generated tremendous excitement for many envisioned large-scale applications [1,2,3,4,5,6,7,8,9,10]. In particular, energy generation via commercial nuclear fusion is presently of great interest and has created a flurry of interest world-wide [1,2,3,4,5,6,7,8,9,10]. In addition, there are numerous large-scale applications of HTS materials for energy transmission, energy storage and use in electric power devices in the grid. All these large-scale HTS applications, the cost, or the price/performance metric of HTS wires dictates if an application will become commercially viable. A key route to achieve this goal is by significantly increasing the performance of the HTS wires without increasing the process cost. Hence, the question of the maximum performance or the critical current density, J_c, that can be achieved in a HTS wire is of paramount importance and of significant consequence to the field towards achieving conductor price/performance goals for large-scale HTS applications [11,12,13]. A well-known upper bound for the critical current density in any superconductor is the depairing current density J_d, the point at which Cooper pairs are destroyed and the material transitions to a resistive, non-superconducting state. J_d depends on key fundamental thermodynamic parameters such as the penetration depth, the coherence length, and on the measurement temperature. The depairing current density for Type-I superconductors is well-described by the Silsbee rule (proposed in 1917) which states that the depairing current density is the critical current density that, when flowing through the superconductor, induces a magnetic field equal to the critical field (H_c) of the Type-I superconductor causing a sharp breakdown of the superconducting state at the critical field [14], and so J_d = H_c/$\lambda$, where λ is the penetration depth. Unlike Type-I superconductors, in Type-II superconductors, the transition to the normal state is gradual, starting at the lower critical field, H_c1, where a mixed, superconducting and non-superconducting vortex state develops, and the fully resistive, non-superconducting state develops at the upper critical field, H_c2. For most technologically relevant Type-II superconductors, H_c2 is extremely large, whereas H_c1 is comparatively small. Silsbee’s criterion is therefore not applicable for determining the depairing current density in Type-II superconductors.

It has been argued that while the Silsbee criterion when considering the lower critical field, H_c1, does not result in the depairing current density (i.e., J_d is not equal to H_c1/$\lambda )$, it may predict a limit to attainable J_c(self-field), thereby providing an “upper limit” to possible J_c(self-field) in a Type-II superconductor. Substituting the standard textbook equation for H_c1 [15] in this formula, Talantsev and Tallon [16] obtain an expression for critical current density in thin-film, Type-II superconductors (as shown in Equation (4) of their paper):

$$J_c={\displaystyle \frac{{\varphi }_0}{4\pi {\mu }_0}}\times {\displaystyle \frac{\ln \left(\frac{\lambda \left(T\right)}{\xi \left(T\right)}\right)+0.5}{{\lambda }^3\left(T\right)}}$$

(1)

where ${\varphi }_0=2.06\times {10}^{-15} Wb$, is the flux quantum, ${\mu }_0=4\pi \times {10}^{-7}{\displaystyle \frac{Wb}{Am}}$, is the permeability of free space,$\lambda =$London penetration depth (nm) and $\xi =$coherence length (nm). The authors then examined published data in the literature on self-field J_c for various thin-film superconductors, as well as the reported thermodynamic parameters (penetration depth, $\lambda$, and coherence length, $\xi$), and then calculated the predicted J_c(s.f) obtained using Equation (1). As noted in their paper, they found no experimental case in which the measured self-field J_c exceeded the value predicted by Equation (1). They therefore proposed that the J_c obtained from Equation (1) represents an “upper limit” to the attainable self-field J_c in thin-film superconductors whose thickness is smaller than the penetration depth $\lambda$. As stated in their paper, the authors were surprised that even for Type-II superconductors, a similar criterion, J_c = H_c1/$\lambda$, provided the “upper limit” to attainable self-field critical current density, J_c(s.f.), and stated that this upper limit is valid for all Type-II superconductors and hence claimed that it was a “universal or fundamental limit” to attainable J_c(s.f.). In their paper, it is emphasized that this fundamental limit also applies to J_c for the second-generation high-temperature superconducting wires (also known as HTS 2G-wires or REBa₂Cu₃O_7−δ tapes or coated conductors) [17], which are the preferred option for high-field applications such as magnets for fusion projects [10]. Moreover, it is stated that the highest J_c(s.f., T) will be achieved with the lowest possible value of $\lambda$, and in REBa₂Cu₃O_7−δ this can only be achieved by full oxygenation (δ→0) and removing all impurity substitution.

2. Methods

In this paper we report the calculation of the predicted universal or fundamental limit to J_c(s.f.), referred to hereon as J_c(max, s.f.)_calc. by means of Talantsev and Tallon’s primary Equation (1) listed above [16], using reported experimental values of the fundamental thermodynamic material parameters (the London penetration depth and the coherence length) for cases approaching full oxygenation or for $\left(\delta \to 0\right)$ in REBa₂Cu₃O_7−δ and compare the obtained J_c(s.f.) values with experimentally reported J_c(s.f.) values. The J_c(s.f.) is also a reference value for the J_c at high magnetic fields [18]. Moreover, we address the fundamental question of whether the J_c(s.f.) can be affected by extrinsic artificial pinning centers.

First case: Using experimental values of thermodynamic parameters reported by Kiefl, R. F. et al. [19] for YBa₂Cu₃O_6.92, with the effective in-plane penetration depth λ_ab(T→0 K) ≈ 115.3 nm, and coherence length, ξ = 1.5 nm, this fundamental J_c(s.f.) limit calculated using Equation (1) is J_c(max, s.f., T →0 K)_calc. ≤ 42 MA/cm².

Second case: Using data from Sonier et al. [20], for an optimally doped sample of YBa₂Cu₃O_6.95 with T_c = 93 K and having λ_ab = 112 nm, we obtain the same value for this case as for the third case highlighted below.

Third case: Using measured thermodynamic parameters $\lambda$ = 112 nm and $\xi$ = 1.6 nm reported by Stangl et al. [21], for a nearly fully oxygenated thin-film of composition YBa₂Cu₃O_7−δ, the calculation for the fundamental limit or J_c(max, s.f.)_calc., is shown below:

$$\begin{gathered} J_c{\left(max,s.f.\right)}_{calc}=1.309·{10}^{-10}Am\times {\displaystyle \frac{\ln \left(\frac{112·{10}^{-9} m}{1.6·{10}^{-9} m}\right)+0.5}{{\left(112·{10}^{-9}\right)}^3 m^3}}=1.309·{10}^{-10} Am\times {\displaystyle \frac{4.25+0.5}{1.405·{10}^{-21} m^3}} \\ =1.309·{10}^{-10}\times 3.38·{10}^{21} \left({\displaystyle \frac{Am}{m^3}}\right)\approx 4.4·{10}^{11} {\displaystyle \frac{A}{m^2}}=4.4·{10}^7 {\displaystyle \frac{A}{{cm}^2}}=44 {\displaystyle \frac{MA}{{cm}^2}} \end{gathered}$$

3. Results and Discussion

Despite the values obtained starting from the reported Equation (1), Stangl et al. [21], report an experimentally measured J_c(s.f., 5 K) of ≈90 MA cm⁻² for this undoped YBa₂Cu₃O_7−δ (YBCO) thin-film approaching full oxygenation. The plot of J_c (H) at 5 K in field-range from self-field to 5 T, by Stangl [21] shown in Figure 1 clearly shows significantly higher experimentally obtained J_c than that predicted by Equation (1) using values for thermodynamic parameters λ and ξ measured and reported by them for this film [16]. This J_c(s.f., 5 K) is more than 100% higher than the value of 42 MA/cm² (which is 44 MA/cm² if λ and ξ for Stangl’s film shown above are considered) calculated using Talantsev and Tallon’s primary Equation (1). If Stangl et al.’s J_c(s.f.) measurements were made at lower temperatures, approaching 0 K, the J_c would have further increased manifold, even further exceeding J_c(max, s.f.)_calc. predicted by Equation (1).

The field and temperature dependences J_c (H, T) of a YBCO—1 vol%BaZrO₃ film with self-assembled columnar BZO nanorods fabricated using pulsed laser deposition (PLD), measured in transport by Xu [22,23] are shown in Figure 2. This Figure contains exactly the data presented in Figure 6.4 on page 104 and Figure 6.5 on page 105 of Xu’s thesis [22] and show that J_c for this film in external field at both 10 K and 5 K exceeds the theoretical limit in self-field (42 MA/cm²). The values of J_c(s.f.) are not reported because they exceed the current limit of their transport system, which for the specific bridge geometry and film thickness was 54.6 MA/cm². Even at this measurement limiting value of 54.6 MA/cm², the transport J_c(s.f.) is 30% higher than J_c(max, s.f.)_calc. = 44 MA/cm² for YBCO films at full oxygenation, predicted by Equation (1). In addition, if Xu et al.’s measurements were made at lower temperatures, approaching 0 K, the J_c would have increased manifold, even further exceeding this “limit” to self-field J_c. Lastly, since these YBCO films were not fully oxygenated (unlike Stangl’s films), the J_c(s.f.) is expected to further rise significantly upon oxygenation to produce an overdoped state as shown by Stangl.

Figure 1. J_c (H) for two films: (1) YBa₂Cu₃O_7−δ (YBCO) thin-film approaching full oxygenation from Stangl et al. [21], and (2) (Y,Gd)Ba₂Cu₃O_7−δ (YBCO) thin-film with BaHfO₃ additions from Miura et al. [24]. Shown in the figure in blue and green is the J_c(self-field) calculated using the “Universal or fundamental limit” per Equation (1) of 42 MA/cm² and 44 MA/cm² and corresponding to the first and third case in the article.

Figure 1. J_c (H) for two films: (1) YBa₂Cu₃O_7−δ (YBCO) thin-film approaching full oxygenation from Stangl et al. [21], and (2) (Y,Gd)Ba₂Cu₃O_7−δ (YBCO) thin-film with BaHfO₃ additions from Miura et al. [24]. Shown in the figure in blue and green is the J_c(self-field) calculated using the “Universal or fundamental limit” per Equation (1) of 42 MA/cm² and 44 MA/cm² and corresponding to the first and third case in the article.

Xu et al. [22,23], also reported an experimentally measured transport J_c(s.f., 4.2 K) ~55 MA/cm² at 4.2 K for H//c for a thin-film superconductor of composition 15 mol.% Zr-added (Gd, Y)-Ba-Cu-O. This J_c(s.f., 4.2 K) was ~30% higher than that predicted by Equation (1) of 44 MA/cm². Again, if Xu et al.’s measurements were made at lower temperatures, approaching 0 K, a further pronounced increase in the value of the J_c would have been obtained. Miura et al. [24] reported both a transport and magnetization J_c(s.f., 4.2 K) as well as calculated J_d(4.2 K) for (Y,Gd)Ba₂Cu₃O_7−δ (YBCO) thin films with BaHfO₃ additions as a function of carrier concentration. Figure 1 shows experimentally measured J_c(H) at 4.2 K in field-range from self-field to 7 T, by Miura et al. [24] showing significantly higher experimentally obtained J_c than that predicted by Equation (1) (nearly 200%) using values for thermodynamic parameters λ and ξ and as discussed above. Figure 3 shows J_c(s.f., 4.2 K) for three films with carrier concentration, p, equal to 0.18, 0.168 and 0.144. Only the film with p = 0.18 is overdoped and fully oxygenated. Miura et al. also provide experimentally measured values of the thermodynamic parameters, of $\lambda \mathrm{a}\mathrm{n}\mathrm{d} \xi$, for the films as a function of carrier concentration. Table 1 provides a summary of data from Miura along with the predicted J_c(max, s.f.)_calc. calculated using Equation (1).

Table 1. Thermodynamic parameters, $\lambda \mathrm{a}\mathrm{n}\mathrm{d} \xi$, J_c(s.f.) at 4.2 K, J_d (0 K) from Miura et al. [24], and J_c(max, s.f.)_calc. calculated using the equation, J_c(max, s.f.)_calc. = H_c1/$\lambda$ [16] for (Y,Gd)Ba₂Cu₃O_7−δ (YBCO) thin films with BaHfO₃ additions, at different carrier concentration, p.

Carrier Concentration, p	London Penetration Depth, λab (nm)	$\mathbf{Coherence} \mathbf{Length}, \mathit{\xi }$ab (nm)	Experimentally Measured Transport Jc(s.f., 4.2 K) (MA/cm2)	Estimated Jd(0) (MA/cm2)	$\mathbf{Calculated} {\mathbf{J}}_{\mathbf{c}}(\mathbf{max}, \mathbf{s}.\mathbf{f}.){\mathbf{calc}}_{.} = \mathbf{H}{\mathbf{c}}_1/\mathit{\lambda }$ Using Equation (1) from Reference [16]
0.18 (CC subs)	112	1.61	130	498.02	44.1
0.18 (SrTiO3)	112	1.61	150	498.02	44.1
0.168	122	1.84	90	367.68	33.8
0.144	143	2.14	62	230.57	21.0

Figure 2. J_c (H) for a PLD YBa₂Cu₃O_7−δ + 1vol%BaZrO₃ film measured by Xu [22]: (a) Selected data from Figure 6.4 of Xu showing J_c (H) at different temperatures, and (b) Selected data from Figure 6.5 from Xu showing J_c (T) at different applied fields. The dashed line in both figures corresponds to the J_c(max, s.f.)_calc. = H_c1/$\lambda$ for YBa₂Cu₃O_7−δ films, calculated by Equation (1).

Figure 2. J_c (H) for a PLD YBa₂Cu₃O_7−δ + 1vol%BaZrO₃ film measured by Xu [22]: (a) Selected data from Figure 6.4 of Xu showing J_c (H) at different temperatures, and (b) Selected data from Figure 6.5 from Xu showing J_c (T) at different applied fields. The dashed line in both figures corresponds to the J_c(max, s.f.)_calc. = H_c1/$\lambda$ for YBa₂Cu₃O_7−δ films, calculated by Equation (1).

Figure 3. J_c(s.f.) on the y-axis and J_d(s.f) on the x-axis at 4.2 K for (Y,Gd)Ba₂Cu₃O_7−δ (YBCO) thin-film with BaHfO₃ additions from Miura et al. as a function of carrier concentration, p [24], for p = 0.18, 0.168 and 0.144. The solid red line corresponds to J_c equal to 30% of estimated J_d. Also, shown in the plot is a dotted blue line (p = 0.144), a dotted green line (p = 0.168) and a dotted orange line (p = 0.18) corresponding to J_c(max, s.f.)_calc. = H_c1/$\lambda$. The highest J_c at p = 0.18 on a CC substrate was 130 MA/cm² and 150 MA/cm² for a film on SrTiO₃.

Figure 3. J_c(s.f.) on the y-axis and J_d(s.f) on the x-axis at 4.2 K for (Y,Gd)Ba₂Cu₃O_7−δ (YBCO) thin-film with BaHfO₃ additions from Miura et al. as a function of carrier concentration, p [24], for p = 0.18, 0.168 and 0.144. The solid red line corresponds to J_c equal to 30% of estimated J_d. Also, shown in the plot is a dotted blue line (p = 0.144), a dotted green line (p = 0.168) and a dotted orange line (p = 0.18) corresponding to J_c(max, s.f.)_calc. = H_c1/$\lambda$. The highest J_c at p = 0.18 on a CC substrate was 130 MA/cm² and 150 MA/cm² for a film on SrTiO₃.

As shown in Table 1, the J_c(s.f., 4.2 K) is 194% higher than the “limit” of 44.1 MA/cm² for the film on a coated conductor substrate. For the same film deposited on single-crystal, SrTiO₃ substrate, the J_c(s.f., 4.2 K) is 240% higher than this “limit”. For the incompletely oxygenated film corresponding to carrier concentration, p = 0.168, the J_c(s.f., 4.2 K) is 166% higher than the predicted limit of 33.8 MA/cm² for the film on a coated conductor substrate, calculated using the measured $\lambda \mathrm{a}\mathrm{n}\mathrm{d} \xi$ for this film. Lastly, for the incompletely oxygenated film corresponding to carrier concentration, p = 0.144, the J_c(s.f., 4.2 K) is 195% higher than the “limit” of 21.0 MA/cm² for the film on a coated conductor substrate, calculated using the measured thermodynamic parameters, $\lambda \mathrm{a}\mathrm{n}\mathrm{d} \xi$, for this film. Again, if Miura et al.’s measurements were made at lower temperatures, approaching 0 K, the J_c would have further increased by a significant margin, substantially exceeding the predicted limit to attainable self-field J_c. Even for films that were not fully oxygenated with p = 0.168 and 0.144, the J_c exceeded the predicted limit of 44.1 MA/cm² for a fully oxygenated film. In Figure 3, the solid red line corresponds to J_c equal to 30% of estimated J_d, the dotted blue line (p = 0.144), the dotted green line (p = 0.168) and the dotted orange line (p = 0.18) corresponds to the J_c calculated using the equation J_c(max, s.f.)_calc. = H_c1/$\lambda$. It can be seen that in all cases the experimentally obtained J_c significantly exceeds the J_c(max, s.f.)_calc. limit predicted by primary Equation (1). In fact, for all three values of the carrier concentration, the measured J_c exceeds the highest J_c calculated for the fully oxygenated, overdoped film shown by the orange dotted line. Figure 1 also shows J_c (H, 4.2 K) for the film on a coated conductor substrate which was overdoped in comparison to the Stangl’s films and the J_c (max, s.f.)_calc. It can be seen that the J_c(s.f.) limit predicted by Equation (1) has been greatly exceeded.

If one can find a way to lower the value of the penetration depth, $\lambda$, then the “fundamental limit” to J_c(s.f.) increases. Talantsev and Tallon [17] proposed doing so by using the lowest reported values for Gd_xY_1−xBa₂Cu₃O_6+y single-crystals by Perag-Barnea et al. [25] of λ_a = 103 ± 8 nm and λ_b = 80 ± 5 nm, giving an effective in-plane penetration depth of λ_eff = λ_ab = √λ_aλ_b = 91 nm. Perag-Barnea et al. obtained the magnetic penetration depth by relating the number of Gd ions exposed to the microwave magnetic field to the frequency-integrated intensity of the observed ESR transitions. Combined with a value ξ = 1.5 nm, Talantsev and Tallon calculated J_c(s.f., T →0 K) ≤ 78 MA/cm². However, this value of λ_eff has never been experimentally confirmed in another study on Gd_xY_1−xBa₂Cu₃O_6+y single-crystals or reported for any Gd_xY_1−xBa₂Cu₃O_6+y film on a coated conductor substrate or on a single-crystal substrate such as SrTiO₃. Regardless, even using this value of 78 MA/cm² as the fundamental limit to J_c(s.f.) for Gd_xY_1−xBa₂Cu₃O_6+y films, the J_c is exceeded by 67% for a film on a coated conductor substrate and as shown in Figure 3 and Table 1. Figure 3 shows a dotted black line at 78 MA/cm². The experimentally measured J_c exceeds this value by ~15% for p = 0.168, by ~67% for p = 0.18 for films on a coated conductor substrate and by ~92% for a film on a single-crystal substrate with p = 0.18.

In their paper [16], the authors argued that their universal self-field J_c limit applies only to films with thicknesses smaller than $\lambda$, since films thicker than $\lambda$ allow flux entry from the edges, implying that the proposed universal limit would no longer hold in such cases. However, this consideration becomes relevant only if the thickness dependence of J_c in REBCO films were such that J_c continued to increase for thicknesses exceeding $\lambda$. In contrast, experimental evidence shows that the self-field J_c increases rapidly at small film thicknesses [26,27,28], and in most cases grows exponentially for thicknesses below the penetration depth. This is clearly shown in Figure 1 of Matsui et al. [26], Figure 9 of McIntyre et al. [27] and Figure 3 in Foltyn et al. [28]. McIntyre et al. [27] reported on films made by chemical solution deposition with thicknesses ranging from 70 nm to 300 nm and Foltyn et al. [28], reported on films made using pulsed laser ablation with thicknesses ranging from 65 nm to 6.4 μm, and in both cases, the J_c(s.f.), increased exponentially at lower film thicknesses below 150 nm as shown in Figure 4 for in situ PLD films upto 4 μm in thickness. In fact, it is well appreciated that the high self-field J_c of very thin films is due to a dense array of misfit dislocations at the film/substrate interface, raising the self-field J_c of such very thin films [29,30,31], with the implication that these defects are causing the J_c(s.f.) to be increased. Given this trend of the thickness dependence of J_c(s.f.), it can be reasonably expected that J_c(s.f., 4.2 K) of films reported in Xu et al., Stangl et al. and Miura et al. will further increase quite substantially at lower film thicknesses, even more significantly surpassing the presumed universal or fundamental limit of self-field J_c.

Figure 4. J_c(s.f.) of thin films increases exponentially with decrease in thickness, for in situ PLD films [28] and also for ex situ films as shown in the inset [27]. In both cases, the J_c(s.f.) increases exponentially when film thickness is much smaller than the typical REBCO penetration depth.

For extreme Type-II superconductors such as REBa₂Cu₃O_7−δ (REBCO), where H_c2 is very high (150 T for optimally doped YBCO [32]) and H_c1 is very low (~34–35 mT for optimally doped YBCO [33]), the self-field generated by the current exceeds H_c1 even at low currents and the sample enters the mixed state where the J_c is then determined by vortex-pinning. Hence, it is not surprising that a self-field J_c calculated using the formula, J_c = H_c1/$\lambda$, was experimentally exceeded. In reference [16], the authors claim that self-field J_c is fully independent of pinning. They noted a single exception reported by Schindler et al. [34], where neutron irradiation increased the self-field J_c from 19.4 to 32 MA/cm². However, this result was ruled out on the grounds that Schindler et al. employed a high electric-field criterion. It is correct that artificial pinning centers (APCs) generated via irradiation after film growth can in most cases lower J_c (s.f.) due to a loss of effective cross-section area and/or damage to the two-dimensional CuO₂ planes. However, there are many additional studies in the literature that show the J_c (s.f.) to increase very substantially with introduction of APCs. In contrast to post-growth irradiation, where APC formation is decoupled from the film-growth process, non-irradiative methods for introducing APCs directly influence film growth, morphology, microstructure, crystallographic texture, and epitaxy, and in most cases lead to a significant enhancement of the J_c (s.f.). Here, we cite only a few studies showing clear increase in J_c (s.f.) via introduction of APCs—see Figure 4 of Miura et al. [24], Figure 4 of Diez-Sierra et al. [35], Figure 3 of Vaimala et al. [36], Figure 2 of Jha et al. [37] and Figure 4 of Jha et al. [38]. As mentioned previously, this can also potentially be a secondary effect due to improved film microstructure, morphology, texture, etc., with the introduction of APCs. In Wee et al. [39], defect engineering resulted in a massive increase in transport J_c(s.f., 77 K), from 1.4 MA/cm² to 2.5 MA/cm² (~78% increase in J_c(s.f.)) for a film of NdBa₂Cu₃O_7−δ with the incorporation of engineered, non-superconducting, nanoscale APCs compared to a NdBa₂Cu₃O_7−δ with no engineered APCs incorporated. Figure 5 shows that for this film, 50% of the film was nanostructured such that the BaZrO₃ (BZO) nanorods were aligned perpendicular to the film and for the other 50% of the film, the BZO nanorods were aligned parallel to the c-axis of YBCO [39]. This film had a combined thickness of 0.86 μm.

Figure 5. Hybrid NdBa₂Cu₃O_x + BaZrO₃ (BZO) films, with a total thickness of 0.86 μm, deposited using PLD on a coated conductor substrate [39]. The bottom half of the film has BZO nanorods aligned parallel to the substrate surface or parallel to the ab-plane of NdBa₂Cu₃O_x, and the top half of the film with BZO nanorods aligned perpendicular to the substrate surface or parallel to the c-axis of NdBa₂Cu₃O_x. (a) Angular dependence of I_c at 77 K, 1 T, showing that pinning corresponds to the orientation of BZO. (b) Schematic of film cross-section showing alignment of BZO in film and (c) high-resolution transmission electron microscopy image showing alignment of BZO nanostructures. The dashed white line separates the two halves of the film with the blue and red arrows pointing to BZO nanostructures in bottom and top halves of the film.

While the available data for HTS materials other than REBCO is limited, we further show that this universal or fundamental limit to J_c(s.f.) also does not apply to other Type-II superconductors. We consider Nb, a classical Type-II superconductor. Dinner et al. reported achievement of extremely high critical current density in Nb thin films via through-thickness arrays of APCs via ion-milling through anodized aluminum oxide thin- film templates [40]. They reported experimentally measured transport J_c (H) in Nb thin films with optimized APCs (in terms of size and spacing) to be 50 times greater than Nb film with no APCs at self-field [40]. In fact, Figure 6 shows J_c (H) at 5.0 K for a Nb thin-film with and without APCs from Dinner et al. [40] with the field applied perpendicular to the film surface. The black points and line correspond to J_c for the film with no APCs. The red points and line correspond to J_c for the film with nanoscale APCs. The transport J_c(s.f., 5 K) of the Nb film with no APCs or pinning centers was ~0.6 MA/cm² and the transport J_c (s.f., 5 K) of the Nb film with optimized, through thickness APCs was 30 MA/cm². This further clearly demonstrates that in Type-II superconductors, J_c(s.f.) is also dependent on pinning. Therefore, our analysis shows that (Y,Gd)Ba₂Cu₃O_7−δ (YBCO) thin films with BaHfO₃ and through-thickness arrays of APCs via ion-milling through anodized aluminium oxide thin-film templates in Nb thin films were both extremely effective in increasing self-field J_c.

Figure 6. J_c (H) at 5K for a Nb thin-film with and without APCs from Dinner et al. [40]. The black points and line correspond to J_c for the film with no APCs. The red points and line correspond to J_c for the film with nanoscale APCs. Also shown in the figure with a blue and green asterisk are J_c values calculated using Equation (1) for coherence length, ξ, of 23.1 nm and 40 nm, respectively. Inset: a magnification in the range 0 T and 0.1 T is shown to better visualize the higher values of the experimentally obtained self-field J_c for the film with APCs (red points) than the ones calculated using the so-called predicted maximum attainable self-field J_c using expression, J_c(max, s.f.)_calc. = H_c1/$\lambda$.

The penetration depth, $\lambda$, for these films was 120 nm [40,41] and the coherence length, ξ(5 K) = 40 nm [41]. However, since, from Ginzburg-Landau theory, ${\mathsf{\xi }\left(T\right)=\mathsf{\xi }}_0{\left(1-{\displaystyle \frac{T}{T_c}}\right)}^{-1/2}$, with T_c = 7.5 K (from Dinner et al., page 4), it results ξ₀ = 23.1 nm at T = 0 K. Based on these thermodynamic parameters, Table 2 shows the calculated values of J_c(max, s.f.)_calc. using the primary equation mentioned above corresponding to J_c(max, s.f.)_calc. = H_c1/$\lambda$.

Table 2. Thermodynamic parameters, $\lambda \mathrm{a}\mathrm{n}\mathrm{d} \xi$ from Dinner et al. [40], and J_c(max, s.f.)_calc. calculated using the equation, J_c(max, s.f.)_calc. = H_c1/$\lambda$ [16] for Nb thin films.

λ(0) (nm)	ξ (nm)	Measured Jc(s.f., 5 K) (MA/cm2)	Jc(max, s.f.)calc. (MA/cm2)
120	40 (@ 5 K)	30	12.1 (@ 5 K)
120	23.1 (@ 0 K)	30	16.3

In Figure 6, the blue and green asterisk correspond to these J_c values calculated using the equation reported in reference [16], for coherence length, ξ, of 23.1 nm (@ 0 K) and 40 nm (@ 5 K), respectively. As shown in Table 2 above, the experimentally obtained transport J_c(s.f.) of 30 MA/cm² is ~100% higher than that predicted by [16]. The inset of Figure 6 clearly shows the experimentally obtained self-field J_c compared to that calculated using the so-called predicted maximum attainable self-field J_c using expression, J_c (max, s.f.)_calc. = H_c1/$\lambda$. The transport measurements reported in Dinner et al. [40] were made on 12 μm wide bridges, wide enough for the geometrical issues to be not relevant. A validation for this stems from the fact that for same bridge width, the Nb film with no APCs only had a transport J_c(s.f.) of ~0.6 MA/cm².

Although Xu [22], published in 2012, reported transport J_c of YBCO thin films on CC substrates, Xu et al. [23], published in 2014, reported transport J_c of (Y,Gd)BCO thin films on CC substrates, and Dinner et al. [40], published in 2011, reported transport J_c of Nb thin films, these references were not considered in reference [16]. In addition, significant enhancements to transport J_c(s.f.) of YBCO thin films on CC substrates via incorporation of APCs have been shown recently in many reports [21,22,23,24,35,36,37,38,39,40]. These references now cast strong doubt on the premise of a universal or fundamental limit to J_c(s.f.) for Type-II superconductors given by the expression, J_c(max, s.f.)_calc = H_c1/$\lambda$, as their primary equation.

In short, we have shown the following:

(1)

Experimentally obtained, J_c(s.f.), significantly higher than that calculated using Talantsev and Tallon’s primary Equation (1) [16] have been reported (by Xu et al. [22,23], Stangl et al. [21] and Miura et al. [24]) for YBa₂Cu₃O_7−δ-type and REBa₂Cu₃O_7−δ thin films and by Dinner et al. [40] for Nb films, rendering predictions of J_c(s.f.) by a Silsbee-like criterion to be incorrect. A summary of the samples, materials, parameters, and self-field critical current densities considered in this work is reported in Table 3:

Table 3. Summary of samples and self-field critical current density results.

Material/Sample	APCs Type and Morphology	λ (nm)	ξ (nm)	T (K)	Calculated Jc(max,s.f.) (MA/cm2)	Experimental Jc(s.f.) (MA/cm2)	Increment Relative to the Theoretical Limit
YBCO (Stangl)	Pure with no APC’s, near full oxygenation	112	1.6	5	≈44	≈90	+104%
YBCO + 1% BZO (Xu)	Columnar BaZrO3 nanorods	112	1.6	5, 10	≈44	≥54.6	≥+24%
15 mol.% Zr-added (Gd, Y)-Ba-Cu-O (Xu)	Columnar BaZrO3 nanorods	112	1.6	4.2	≈44	55	+25%
(Y,Gd)BCO + BHO (Miura, p = 0.18, CC)	Columnar BaHfO3 nanorods	112	1.61	4.2	44.1	130	+194%
(Y,Gd)BCO + BHO (Miura, p = 0.18, STO)	Columnar BaHfO3 nanorods	112	1.61	4.2	44.1	150	+240%
(Y,Gd)BCO + BHO (Miura, p = 0.168)	Columnar BaHfO3 nanorods	122	1.84	4.2	33.8	90	+166%
(Y,Gd)BCO + BHO (Miura, p = 0.144)	Columnar BaHfO3 nanorods	143	2.14	4.2	21.0	62	+195%
Nb thin-film (Dinner)	Columnar nanoholes	120 120	40 (@ 5 K) 23.1 (@ 0 K)	5 5	12.1 16.3	30 30	+148% +84%

(2)

Due to the established, rapid decrease in the critical current density with thickness, for films thinner than the films reported in point (1) above, we expect an even higher J_c(s.f.) further exceeding the J_c (max, s.f.)_calc. limit predictions, and

(3)

J_c(s.f.) can be significantly increased with introduction of extrinsic artificial pinning centers [35,36,37,38,39,40], further rendering the J_c(max, s.f.)_calc. limit using a Silsbee-like criterion invalid.

In an alternative analysis, Mastushita and Kiuchi estimated the upper limit of achievable critical current density via optimization of APCs under the influence of kinetic energy using Ginzburg-Landau theory [42]. They estimated the upper limit of the achievable J_c to be 67% of the depairing current density in the London limit [42]. They argue that the strong flux pinning by self-assembled nanorods when optimized, could provide such high J_c approaching this upper limit of 67% of J_d [42]. Self-assembled nanorods or nanocolumns consisting of perovskite (BaMO₃) and double perovskite (Ba₂RETaO₆ and Ba₂RETaO₆) [43,44,45,46,47,48,49,50,51,52,53,54,55,56] nanorods have been shown to form controllable self-assembled microstructures with greatly enhanced J_c and flux-pinning. Very high-J_c has also been achieved in microstructures comprising randomly oriented nanoparticles of Y₂BaCuO_x [57], BaZrO₃ [58,59,60,61] and RE₂O₃ [62,63]. A critical-current-by-design paradigm to tailor and enhance J_c using time-dependent Ginzburg-Landau simulations coupled with Monte-Carlo and molecular dynamics has been proposed to further optimize J_c [11]. The data collectively demonstrate that J_c(s.f.) is a highly tunable quantity governed by microstructural design, suggesting that optimization strategies rather than fixed limits should guide practical materials development, and that further opportunities are present for theorists to develop deeper physical insights to more accurately predict a limiting J_c(s.f.) in thin-film Type-II superconducting films.

4. Conclusions

It was shown that the “universal or fundamental limit” to self-field J_c calculated using a Silsbee-like criterion of J_c(max, s.f.)_calc. = H_c1/$\lambda$, for Type-II superconductors is not valid. This was established via detailed calculations for several YBa₂Cu₃O_7−δ-type and REBa₂Cu₃O_7−δ thin films and then comparing the obtained J_c(max, s.f.)_calc. values to experimental J_c(s.f.) values. This was also shown to be the case for another Type-II superconductor, Nb, where experimental J_c(s.f.) was found to be significantly higher than the predicted fundamental limit calculated via the equation, J_c(max, s.f.)_calc. = H_c1/$\lambda$, using reported thermodynamic parameters for the film. In addition, it was established that in Type-II superconductors, the J_c(s.f.) also depends on extrinsic pinning centers, a key fundamental question in the field. The fact that J_c(s.f.), in superconductors with different substrates, can be significantly improved by incorporation of artificial pinning centers (APC’s), further implies that no universal limit to J_c(s.f.) can exist because for such a universal and fundamental limit to exist, J_c(max, s.f.)_calc. would have to be independent of extrinsic pinning centers and different pinning mechanisms.

This leads to an important conclusion that the performance of REBa₂Cu₃O_7−δ based thin-film coated conductors have not reached their limit to attainable J_c(s.f.). Since the depairing current density in (Y,Gd)Ba₂Cu₃O_7−δ (YBCO) thin films with BaHfO₃ is still a factor of three to four higher than the highest experimentally achieved J_c, the potential headroom for increasing J_c is still enormous. There remains substantial room for enhancement of J_c of commercial coated conductors which could significantly improve the price-to-performance ratio of high-temperature superconducting wires. Such progress would directly accelerate the deployment of superconductors across large-scale technologies, including energy generation (commercial nuclear fusion), transmission (superconducting ac and dc cables), and storage (superconducting magnetic energy storage systems, SMES), as well as in energy-efficient devices such as motors, generators, and fault-current limiters. Given the recent excitement generated due to the possibility of commercial nuclear fusion enabled by coated conductor HTS wires, this is an extremely important and key point to be considered in scaling-up fabrication of HTS wires.