p-wave superconductivity in iron-based superconductors

Abstract

The possibility of p-wave pairing in superconductors has been proposed more than five decades ago, but has not yet been convincingly demonstrated. One difficulty is that some p-wave states are thermodynamically indistinguishable from s-wave, while others are very similar to d-wave states. Here we studied the self-field critical current of NdFeAs(O,F) thin films in order to extract absolute values of the London penetration depth, the superconducting energy gap, and the relative jump in specific heat at the superconducting transition temperature, and find that all the deduced physical parameters strongly indicate that NdFeAs(O,F) is a bulk p-wave superconductor. Further investigation revealed that single atomic layer FeSe also shows p-wave pairing. In an attempt to generalize these findings, we re-examined the whole inventory of superfluid density measurements in iron-based superconductors and show quite generally that single-band weak-coupling p-wave superconductivity is exhibited in iron-based superconductors.

Introduction

The existence of p-wave superconductivity was hypothesized more than 50 years ago¹ and the fundamental mechanisms governing p-wave superconductivity are well developed in theory. There have however been problems finding a material that convincingly demonstrates p-wave superconductivity. The difficulties arise because some p-wave states are thermodynamically indistinguishable from s-wave states, whilst others would give very similar thermodynamic data to d-wave states². Sensitive probes for p-wave superconductivity must couple to either the odd parity or the spin part of the pairing. The vast majority of experimental works that have been reported to date concentrate on the latter. In spite of this a material that has bulk p-wave pairing remains to be found. Sr₂RuO₄ is one of the rare materials in which, for two decades now, p-wave superconductivity was thought to exist³, but recent experiments⁴ suggest that it is in all likelihood a d-wave superconductor. Thus, there is an on-going experimental search for p-wave pairing in new materials^5,6, including induced superconductivity in graphene-based systems⁷. The current status of the search for p-wave pairing was recently reviewed in^8,9.

Description of the Problem

One of the most robust ways of confidently detecting pairing type (i.e., s-, d-, or p-wave) in superconductors is the analysis of the temperature dependence of the superfluid density¹⁰:

$${\rho }_{s}(T)=\frac{1}{{\lambda }^{2}(T)}$$

(1)

where λ(T) is the London penetration depth. We note that this was the approach used by Hardy et al.¹¹ to demonstrate d-wave pairing in high-temperature superconducting cuprates. In Supplementary Fig. 1a we show the normalized superfluid densities, ρ_s(T/T_c) = ((λ(0)/λ(T/T_c))², for s-wave and d-wave superconductors and compare them with four possible scenarios of a weak-coupled p-wave superconductor in Supplementary Fig. 1b. The analysis of ρ_s(T) for p-wave pairing is much more complicated (in comparison with s- and d-wave) because in this case the gap function is given by^12,13,14:

$$\Delta (\hat{{\boldsymbol{k}}},T)=\Delta (T)f(\hat{{\boldsymbol{k}}},\hat{{\boldsymbol{l}}})$$

(2)

where Δ is the superconducting gap, k is the wave vector, and l is the gap axis. The electromagnetic response depends on the mutual orientation of the vector potential A and the gap axis which for an experiment is just the orientation of the crystallographic axes compared with the direction of the electric current. There are two different p-wave pairing states: “axial” where there are two point nodes, and “polar” where there is an equatorial line node. It can be seen from Supplementary Fig. 1 that the only p-wave case that is clearly distinguishable from s-wave and d-wave is polar A⊥l, which is the only case for which the second derivative of ρ_s(T/T_c) vs. T/T_c has opposite sign to all other scenarios for s-, d-, and p-wave pairing; that is, the temperature dependence of the superfluid density has positive curvature at all temperatures. The shapes of the superfluid densities for p-wave polar A||l and axial A⊥l cases are difficult to distinguish from their s-wave counterparts, and the p-wave axial A||l case is also difficult to distinguish from the dirty d-wave case.

In spite of these difficulties in the distinguishing of p-wave, s-wave and d-wave cases based on the shape of ρ_s(T), there is still the possibility to make this deduction based on the values of several superconducting parameters deduced from the ρ_s(T) analysis. For instance, Bardeen-Cooper-Schrieffer theory¹⁵ weak-coupling limits for these types of pairing are given in Supplementary Table I^{12,13,15,16,17}.

We note that, as mentioned by Gross-Alltag et al.¹³, only at very particular experimental conditions can the pure polar or pure axial cases of the p-wave superconductivity be observed. More likely, as was the case for heavy fermions^12,13, the hybrid cases will be observed in experiments.

In the case of iron-based superconductors (this class of unconventional superconductors includes more than 30 iron based superconductors discovered to date, which have 12 different crystallographic space groups^17,18), there is an obvious objection to them being p-wave superconductors, because Knight shift experiments showed that p-wave should be prohibited¹⁶. We note that consideration of the Knight shift in superconductors started in the early 1960s¹⁹ when it was believed that ferromagnetism is antagonistic to superconductivity. We suggest that a simple extrapolation of theoretical results in regards of the Knight shift obtained for classical BCS superconductors probably is not valid for the newly discovered class of iron-based superconductors.

We stress that there is an exceptional experimental condition under which p-wave superconductivity can be uniquely determined from the temperature dependence of the polar A⊥l case of ρ_s(T), and thus the lack of experimental studies for confidently detecting p-wave pairing is related not just to the fabrication of samples but also choosing an experimental technique for which the polar A⊥l orientation can be studied.

If we consider transport current flow in the basal plane c-axis oriented p-wave superconducting film then this is consistent with the case of polar A⊥l, which is equatorial line node mode with current flowing in the plane perpendicular to the gap axis. We note that the self-field critical current, J_c(sf,T), in thin superconducting films obeys the relation²⁰:

$${J}_{c}({\rm{sf}},T)=\frac{{\varphi }_{0}}{4\pi {\mu }_{0}}\cdot \frac{ln(\kappa )+0.5}{{\lambda }^{3}(T)}=\frac{{\varphi }_{0}}{4\pi {\mu }_{0}}\cdot (ln(\kappa )+0.5)\cdot {\rho }_{s}^{1.5}(T)$$

(3)

where ϕ₀ = 2.067 × 10⁻¹⁵ Wb is the magnetic flux quantum, µ₀ = 4π × 10⁻⁷ H/m is the magnetic permeability of free space, and κ = λ/ξ is the Ginzburg-Landau parameter, and thus J_c(sf,T) is proportional to ρ_s^1.5(T). In Supplementary Fig. 1c,d we show normalized plots of the temperature dependence of ρ_s^1.5(T/T_c) = ((λ(0)/λ(T/T_c))³ for s-, d-, and p-wave superconductors respectively, where λ(0) is the ground-state London penetration depth referring to the value in the limit T → 0 K.

In this paper, drawing upon previous work^20,21,22, we studied the self-field critical current density, J_c(sf,T), of NdFeAs(O,F) thin films with the aim of extracting the absolute values of the ground-state London penetration depth, λ(0), the ground-state superconducting energy gap, Δ(0), and the relative jump, ΔC/C, in specific heat at the superconducting transition temperature, T_c. Our initial purpose was to make an accurate determination of these superconducting parameters within a multiple s-wave gap scenario, due to this being the most widely accepted assumption regarding the superconducting pairing symmetry in iron-based superconductors^17,18.

However, the experimental J_c(sf,T) data as we show below was found to be incompatible with this scenario or even a multi-band d-wave scenario. Our analysis revealed that NdFeAs(O,F) is a single-band weak-coupling p-wave superconductor with

$$\frac{2\Delta (0)}{{k}_{B}{T}_{c}}=5.52\pm 0.06$$

(4)

where k_B = 1.381 × 10⁻²³ JK⁻¹ is the Boltzmann constant. This value is in good agreement with the majority of experimental data on direct measurements of 2Δ(0)/k_BT_c in iron-based superconductors, which is always reported to be in the range from 5 to 6^16,17,18.

To further prove our finding and explain why this pairing symmetry was not observed by other techniques, we re-examined available J_c(sf,T) data for thin films of other iron-based superconductors. All c-axis oriented thin films for which we re-analyse results herein demonstrate a single band p-wave polar A⊥l case as our own NdFeAs(O,F) film. These samples are:

• 1. Single atomic layer FeSe film with T_c > 100 K²³;

• 2. FeSe_0.5Te_0.5 thin film with T_c = 13 K²⁴;

• 3. (Li,Fe)OHFeSe thin film with T_c = 42.2 K²⁵;

• 4. Co-doped BaFe₂As₂ thin film with T_c = 21 K²⁶;

• 5. P-doped BaFe₂As₂ thin film with T_c = 29 K²⁷;

  We also analyse temperature dependent superfluid density, ρ_s(T), for several bulk superconductors:

• 6. LaFePO single crystal with T_c = 29 K²⁸;

• 7. (Li_0.84Fe_0.16)OHFe_0.98Se single crystal with T_c = 42.5 K²⁹;

• 8. Rb_0.77Fe_1.61Se₂ single crystal with T_c = 35.2 K³⁰;

• 9. K_0.74Fe_1.66Se₂ single crystal with T_c = 32.5 K³⁰;

• 10. Type-II Weyl semimetal T_d-MoTe₂ with T_c = 1.48–2.75 K³¹.

The latter is not iron-based superconductor, but we show that the formalism of single band p-wave superconductivity, we proposed herein, is equally well applied to this superconductor.

We thus found that p-wave gap symmetry indeed provides a consistent and reliable description of the whole variety of iron-based superconductors.

Results

The self-field critical current density of thin films

For a c-axis oriented film of an anisotropic superconductor having rectangular cross-section with width 2a and thickness 2b, the critical current density is given by the following equation³²:

$$\begin{array}{c}{J}_{c}(sf,T)\,=\,\frac{{\varphi }_{0}}{4\pi {\mu }_{0}}\cdot [\frac{ln({\kappa }_{c})+0.5}{{\lambda }_{ab}^{3}(T)}(\frac{{\lambda }_{ab}(T)}{b}\,\tanh (\frac{b}{{\lambda }_{ab}(T)}))\\ \,+\frac{ln(\gamma (T)\cdot {\kappa }_{c})+0.5}{\sqrt{\gamma (T)}\cdot {\lambda }_{ab}^{3}(T)}(\frac{{\lambda }_{ab}(T)}{a}\,\tanh (\frac{a}{{\lambda }_{ab}(T)}))]\end{array}$$

(5)

where λ_ab(T) and λ_c(T) are the in-plane and out-of-plane London penetration depths respectively, κ_c = λ_ab(T)/ξ_ab(T) and the electron mass anisotropy γ(T) = λ_c(T)/λ_ab(T). For isotropic superconductors $\gamma (T)\equiv 1$ and isotropic Ginzburg-Landau parameter, κ = λ(T)/ξ(T), replaces κ_c in Eq. 5.

Although it is well established³³ that γ(T) in iron-based superconductors is temperature dependent, in the case of thin films (b < λ_ab(0) ≪ a), Eq. 5 reduces to:

$${J}_{c}(sf,T)=\frac{{\varphi }_{0}}{4\pi {\mu }_{0}}\cdot [\frac{ln({\kappa }_{c})+0.5}{{\lambda }_{ab}^{3}(T)}+\frac{ln({\kappa }_{c})+ln(\gamma (T))+0.5}{\sqrt{\gamma (T)}\cdot {\lambda }_{ab}^{2}(T)\cdot a}]\cong \frac{{\varphi }_{0}}{4\pi {\mu }_{0}}\cdot [\frac{ln({\kappa }_{c})+0.5}{{\lambda }_{ab}^{3}(T)}]$$

(6)

which is independent of γ(T).

Based on this, in our analysis we use the ground state electron mass anisotropy γ(0) = λ_c(0)/λ_ab(0) which was taken from independent experimental reports and the basic equation for analysis of J_c(sf,T) was the following:

$$\begin{array}{c}{J}_{c}(sf,T)\,=\,\frac{{\varphi }_{0}}{4\pi {\mu }_{0}}\cdot [\frac{ln({\kappa }_{c})+0.5}{{\lambda }_{ab}^{3}(T)}(\frac{{\lambda }_{ab}(T)}{b}\,\tanh (\frac{b}{{\lambda }_{ab}(T)}))\\ \,+\,\frac{ln(\gamma (0)\cdot {\kappa }_{c})+0.5}{\sqrt{\gamma (0)}\cdot {\lambda }_{ab}^{3}(T)}(\frac{{\lambda }_{ab}(T)}{a}\,\tanh (\frac{a}{{\lambda }_{ab}(T)}))]\end{array}$$

(7)

NdFeAs(O,F) thin films

We have prepared thin films with two thicknesses of 2b = 30 and 90 nm. The Ginzburg-Landau parameter κ_c = 90 for NdFeAs(OF)^17,18,34,35 and its electron mass anisotropy γ = 5³⁶. Processed experimental J_c(sf,T) data for a NdFeAs(OF) thin film (bridge width 2a = 9 µm, film thickness 2b = 90 nm) is shown in Fig. 1(a) together with the absolute values of λ(T) calculated by numerical solution of Eq. 7.

Figure 1

(a) Experimental J_c(sf,T) and λ(T) calculated from Eq. 7 for a NdFeAs(OF) thin film; κ_c = 90 was used^17,18,34,35. Green data point indicates λ(0) = 195 nm deduced by Khasanov et al.³⁷ for NdFeAsO_0.85. (b) Scaling of ρ_s^1.5(T) for s-, d- and p-wave pairing to the experimental J_c(sf,T) data.

In Fig. 1(a) we also show the value of the ground-state London penetration depth λ(0) = 195 nm measured by µSR for NdFeAsO_0.85 as reported by Khasanov et al.³⁷. In Fig. 1(b) we have undertaken a manual scaling of ρ_s^1.5(T) to the experimental J_c(sf,T) data for weak coupled s-wave, d-wave, p-wave axial A||l, and p-wave polar A⊥l cases. It can be seen that only the latter provides a reasonable fit.

To deduce the fundamental superconducting parameters of the NdFeAs(O,F) thin film from the J_c(sf,T) data we employ the general approach of BCS theory¹⁵, in which the thermodynamic properties of a superconductor are derived from the superconducting energy gap, Δ(T). We used the temperature-dependent superconducting gap Δ(T) equation for the p-wave polar A⊥l case given by Gross-Alltag et al.^12,13 (which allows for variation in the coupling strength):

$$\Delta (T)=\Delta (0)\tanh (\frac{\pi {k}_{B}{T}_{c}}{\Delta (0)}\sqrt{\eta (\frac{\Delta C}{C})(\frac{{T}_{c}}{T}-1)})$$

(8)

with

$$\eta =\frac{2}{3}\frac{1}{{\int }_{0}^{1}{f}^{2}(x)dx}$$

(9)

where $f(x)=x$ for polar p-wave and

$$f(x)=\sqrt{1-{x}^{2}}\,for\,axial\,p-wave$$

(10)

and the equation for λ(T) also given by Gross-Alltag et al.^12,13:

$$\lambda (T)=\frac{\lambda (0)}{\sqrt{1-\frac{3}{4{k}_{B}T}{\int }_{0}^{1}\frac{1-{x}^{2}}{2}[{\int }_{0}^{\infty }\frac{d\varepsilon }{{\cosh }^{2}(\frac{\sqrt{{\varepsilon }^{2}+{\Delta }^{2}(T){f}^{2}(x)}}{2{k}_{B}{T}_{c}})}]dx}}$$

(11)

By substituting Eqs 8–11 in Eq. 7 for thick samples, or by using Eqs 3, 6 for thin samples for which the film thickness, 2b < λ(0), one can fit the experimental J_c(sf,T) data to the model and deduce λ(0), Δ(0), ΔC/C and T_c as free-fitting parameters. To help experimentalists use our model to infer λ(0), Δ(0), ΔC/C and T_c parameters from measured J_c(sf,T) data (which is not a trivial mathematical task), we have made our fitting code available online³⁸.

The result of the fit is shown in Fig. 2 and the parameters derived from the fit are found to be in good agreement with weak-coupling values predicted by BCS theory given by Gross-Alltag et al.^12,13. For instance, the deduced ΔC/C = 0.80 ± 0.01 and 2Δ(0)/k_BT_c = 5.52 ± 0.06 compare well with the predicted BCS weak-coupling values for polar orientation of 0.792 and 4.924, respectively (Supplementary Table I). In Fig. 1 we also show the value of the ground-state London penetration depth λ(0) = 195 nm measured by µSR for NdFeAsO_0.85 as reported by Khasanov et al.³⁷.

Figure 2

BCS fits to the experimental J_c(sf,T) data and λ(T) calculated from Eq. 7 for a NdFeAs(O,F) thin film assuming a p-wave polar A⊥l model (Eqs 7–11) and κ_c = 90^17,18,34,35. Green data point indicates λ(0) = 195 nm deduced by Khasanov et al.³⁷ for NdFeAsO_0.85. Derived parameters are: T_c = 40.5 ± 0.5 K, Δ(0) = 9.63 ± 0.03 meV, ΔC/C = 0.80 ± 0.01, λ(0) = 198.2 ± 0.1 nm, 2Δ(0)/k_BT_c = 5.52 ± 0.06. Fit quality is R = 0.99995.

A similar BCS ratio of 2Δ(0)/k_BT_c = 5.0–5.7 was found in the related compound Sm_1−xTh_xOFeAs reported by Kuzmicheva et al.³⁹. The weak-coupling scenario was also experimentally found in the related compound LaFeAsO_0.9F_0.1⁴⁰.

The deduced ground-state London penetration depth λ(0) = 198.2 ± 0.1 nm is also in very good agreement with independent measurements showing λ(0) = 195–200 nm³⁷. These results strongly support the conclusion that NdFeAs(O,F) is a p-wave superconductor.

FeSe single atomic layer film

To support our finding that some iron-based superconductors have p-wave pairing symmetry we performed a search for experimental J_c(sf,T) datasets for these materials. In Fig. 3 we show J_c(sf,T) and fit to Eq. 7 for the milestone report about FeSe single atomic layer sample with record transition temperature, ${T}_{c}\gtrsim 109\,K$, reported by Ge et al.²³.

Figure 3

BCS fits to the experimental J_c(sf,T) data²³ and λ(T) calculated from Eq. 7 for a single atomic layer FeSe film assuming a p-wave polar A⊥l model, and κ_c = 72. Derived parameters are: T_c = 116 ± 13 K, Δ(0) = 24.3 ± 1.5 meV, ΔC/C = 1.6 ± 1.6, λ(0) = 167 ± 2 nm, 2Δ(0)/k_BT_c = 4.9 ± 0.6. Fit quality is R = 0.8564.

To make this fit we made the assumption that the in-plane Ginzburg-Landau parameter κ_c = 72 does not change from its bulk⁴¹ and other single atomic layer film^42,43 values. The deduced λ_ab(0) = 167 nm is in good agreement with this assumption, taking into account that ξ_ab(0) = 2.4 nm⁴³. The fit to the p-wave model (Eqs 7–11) revealed that 2Δ(0)/k_BT_c = 4.9 ± 0.6 which is equal to the p-wave weak-coupling limit (Supplementary Table I) and more data are required to deduce ΔC/C with greater accuracy.

FeSe_0.5Te_0.5 thin film

The next example found in the literature is an FeSe_0.5Te_0.5 thin film (2a = 800 nm, 2b = 100 nm) where the raw J_c(sf,T) data from Nappi et al.²⁴ is shown in Fig. 4. To make a fit of J_c(sf,T) using Eqs 7–11, we used a Ginzburg-Landau parameter κ_c = 180^33,44 and electron mass anisotropy γ = 2.5^45,46.

Figure 4

BCS fits to the experimental J_c(sf,T) data²⁴ and λ(T) calculated from Eq. 7 for an FeSe_0.5Te_0.5 thin film (2a = 800 nm, 2b = 100 nm) assuming a p-wave polar A⊥l model, κ_c = 180 and γ = 2.5. Derived parameters are: T_c = 12.6 ± 0.4 K, Δ(0) = 2.96 ± 0.33 meV, ΔC/C = 1.1 ± 0.3, λ(0) = 970 ± 31 nm, 2Δ(0)/k_BT_c = 5.45 ± 0.6. Fit quality is R = 0.9711.

As can be seen, the fit matches excellently with the weak-coupling polar A⊥l p-wave case. We note that the derived λ(0) = 970 ± 31 nm is larger than the value reported by Bendele et al., λ(0) = 492 nm⁴⁵. We expect that this difference is related to some information mentioned by Nappi et al.²⁴, that during the preparation of the transport current bridge, the transition temperature of the film was reduced. We hypothesize that there was some minor damage caused to the current bridge edges. Based on this, the dissipation-free transport current is flowing along a narrower bridge, and thus the actual J_c(sf,T) will be higher than that calculated based on the nominal sample width 2a. Lower temperature data would of course be desirable to support our case for a p-wave scenario more strongly.

(Li,Fe)OHFeSe thin film

The next thin film presented here is (Li,Fe)OHFeSe (2a = 50 µm, 2b = 20 nm) where the raw J_c(sf,T) data reported by Huang et al.²⁵ is shown in Fig. 5. For a fit of J_c(sf,T) using Eqs 7–11, we take into account measurements of the in-plane coherence length ξ_ab(0) = 2.0 nm^25,47 and λ_ab(0) = 280 nm²⁹, which give the Ginzburg-Landau parameter as κ_c = 140. The electron mass anisotropy for this compound is γ = 10²⁹.

Figure 5

BCS fits to the experimental J_c(sf,T) data²⁵ and λ(T) calculated from Eqs 7–11 for a (Li,Fe)OHFeSe thin film (2a = 50 µm, 2b = 20 nm) assuming a p-wave polar A⊥l model, κ_c = 140 and γ = 10. Derived parameters are: T_c = 41.34 ± 0.08 K, Δ(0) = 11.2 ± 0.4 meV, ΔC/C = 1.87 ± 0.03, λ(0) = 360 ± 4 nm, 2Δ(0)/k_BT_c = 6.3 ± 0.2. Fit quality is R = 0.9997.

Despite the lack of low-temperature data points, the deduced λ(0) = 360 ± 4 nm is in reasonable agreement with the value λ(0) = 280 nm measured in a (Li_0.84Fe_0.16)OHFe_0.98Se single crystal by µSR experiments²⁹.

The deduced ratio of 2Δ(0)/k_BT_c = 6.3 ± 0.2 along with ΔC/C = 1.87 ± 0.03 together show that (Li,Fe)OHFeSe is likely a moderately strongly coupled p-wave superconductor. Analysis of the superfluid density measured by µSR on bulk (Li_0.84Fe_0.16)OHFe_0.98Se single crystals also reveals similar values of 2Δ(0)/k_BT_c and ΔC/C as those derived using J_c(sf,T) data. This analysis is presented in the Supplementary Information Section S1.

Co-doped BaFe₂As₂ thin film

Now we consider the most studied but perhaps least understood and most puzzling iron-based superconductor, BaFe₂As₂. This compound can be made to superconductor by substituting on different atomic sites. One of the most representative examples of the self-field critical current density in Co-doped BaFe₂As₂ was reported by Tarantini et al.²⁶. Raw J_c(sf,T) data²⁶ for the sample with 2a = 40 µm, 2b = 350 nm is shown in Figs 6 and 7. To fit the J_c(sf,T) dataset to Eqs 7–11, we take into account the Ginzburg-Landau parameter as κ_c = 66⁴⁸, and the electron mass anisotropy for this compound as γ = 1.5⁴⁹.

Figure 6

BCS fits to the experimental J_c(sf,T) data²⁶ and λ(T) calculated from Eq. 7 for a Co-doped BaFe₂As₂ thin film (2a = 40 µm, 2b = 350 nm) assuming a two-band s-wave model, κ_c = 66 and γ = 1.5. Derived parameters are: T_c1 = 21.24 ± 0.16 K, Δ₁(0) = 2.74 ± 0.05 meV, ΔC₁/C₁ = 0.93 ± 0.05, λ₁(0) = 234.8 ± 0.8 nm, 2Δ₁(0)/k_BT_c1 = 2.99 ± 0.05, T_c2 = 7.6 ± 0.2 K, Δ₂(0) = 0.92 ± 0.18 meV, ΔC₂/C₂ = 1.0 ± 0.2, λ₂(0) = 318 ± 23 nm, 2Δ₂(0)/k_BT_c2 = 2.8 ± 0.5. Fit quality is R = 0.9993. Green ball is λ(0) = 190 nm for Co-doped BaFe₂As₂⁴⁸.

Figure 7

BCS fits to the experimental J_c(sf,T) data²⁶ and λ(T) calculated from Eqs 7–11 for a Co-doped BaFe₂As₂ thin film (2a = 40 µm, 2b = 350 nm) assuming a p-wave polar A⊥l model. Derived parameters are: T_c = 20.8 ± 0.2 K, Δ(0) = 6.2 ± 0.2 meV, ΔC/C = 1.3 ± 0.1, λ(0) = 198.0 ± 0.8 nm, 2Δ(0)/k_BT_c = 6.9 ± 0.2. Fit quality is R = 0.9979. Green ball is λ(0) = 190 nm for Co-doped BaFe₂As₂⁴⁸.

There is a widely accepted view that this compound is a two-band s-wave superconductor^17,18. In the case of a two-band superconductor that has completely decoupled bands, J_c(sf,T) can be written in the form^21,22:

$${J}_{c}{(sf,T)}_{{\rm{total}}}={J}_{c}{(sf,T)}_{{\rm{band1}}}+{J}_{c}{(sf,T)}_{{\rm{band2}}}$$

(12)

where J_c(sf, T) for each band is as described by Eq. 3 with separate λ(0), Δ(0), ΔC/C and T_c values and all eight parameters may be used as free-fitting parameters. The raw J_c(sf,T) dataset measured by Tarantini et al.²⁶ was sufficiently rich that we were able to fit using all eight parameters. For s-wave superconductors the gap equation, Δ(T), is given by Eq. 8 with η = 2/3, and λ(T) is given by^12,13:

$$\lambda (T)=\frac{\lambda (0)}{\sqrt{1-\frac{1}{2\cdot {k}_{B}\cdot T}\cdot {\int }_{0}^{\infty }cos{h}^{-2}(\frac{\sqrt{{\varepsilon }^{2}+{\Delta }^{2}(T)}}{2\cdot {k}_{B}\cdot T})d\varepsilon }}$$

(13)

More details and examples of application of this s-wave weakly-coupled bands model can be found elsewhere^21,22.

The fit to this model is shown in Fig. 6. The fit quality is very high, R = 0.9993, and the deduced parameters for both bands agree well with other reports. The downside of this fit, as well as all previously applied two-band s-wave models, is that the deduced parameters are at times lower than the BCS weak-coupling limits.

For instance, $\frac{2\Delta (0)}{{k}_{B}{T}_{c}} < 3$ for both bands as compared with the BCS weak-coupling limit of $\frac{2\Delta (0)}{{k}_{B}{T}_{c}}=3.53$, and ${\frac{\Delta C}{C}|}_{T \sim {T}_{c}}\lesssim 1$ for both bands as compared with the BCS weak-coupling limit of ${\frac{\Delta C}{C}|}_{T \sim {T}_{c}}=1.43$.

The fit of the same J_c(sf,T) dataset to a single-band polar A⊥l p-wave model is presented in Fig. 7, where the deduced λ(0) = 198.0 ± 0.2 nm is in a good agreement with the reported λ(0) = 190 nm for cobalt-doped Ba-122 compounds⁴⁸. The other deduced parameters show that this compound has moderately strong coupling.

The significant advantage of this approach is that the fit has only four free-fitting parameters compared with eight for the two-band s-wave model. The additional four parameters for the two-band s-wave model give a remarkably insignificant improvement in the fit quality (R = 0.9993 compared to R = 0.9979), while dramatically increasing the mutual interdependency of the fit parameters. Similar arguments apply to the more exotic order parameter symmetries proposed, such as three-band models or s + is chiral symmetry models^16,17,18.

We consider that a good reason must be presented for requiring a more complex model than is needed to adequately explain the experimental data^50,51.

Also, it should be stressed that an unavoidable weakness of all multi-band models, ignoring the overwhelmingly large number of free-fitting parameters within these models, is that at least for one band (or, for two bands in the three-bands models) the ratio of the superconducting energy gap to the transition temperature is several times lower than the lowest value allowed within the most established theory of superconductivity, which is BCS¹⁵:

$$\frac{2\Delta (0)}{{k}_{B}{T}_{c}}\ll 3.53$$

(14)

Thus, in this paper, we present a model which is:

• 1. framed within the standard BCS single band theory.

• 2. provides superconducting parameters within weak-coupling BCS limits.

This means that our model is based on a minimal set of physical assumptions and provides values for several structurally different superconductors within the simplest weak-coupling BCS limits. In the next Section we make direct demonstration how experimental data can be processed within ${s}_{\pm }$ and p-wave models, for which we chose bulk T_d-MoTe₂ superconductor for which Guguchia et al.³¹ performed ρ_s(T) data fits to several conventional models, including s-wave, d-wave, and ${s}_{\pm }$ and ${s}_{++}$ models. And thus, this makes possible to compare our approach with ones proposed previously.

Bulk sample of Type-II Weyl semimetal Td-MoTe₂

Guguchia et al.³¹ reported the temperature dependent ρ_s(T) subjected to high pressure and performed data analysis for single band s-wave, d-wave, and ${s}_{\pm }$-wave models in their Fig. 4. In our Fig. 8 we show raw ρ_s(T) data fitted to polar A||l model. It can be seen that fits have very high quality.

Figure 8

BCS fits of ρ_s(T) data for bulk type-II Weyl semimetal T_d-MoTe₂³¹ measured at applied field of B = 20 mT to single-band p-wave polar A||l model (Eqs 8–11). (a) Fit quality is R = 0.9898; (b) R = 0.9650; (b) R = 0.9873.

In Fig. 9(a) we show evolution of $\frac{2{\Delta }_{i}(0)}{{k}_{B}{T}_{c}}$ ratios vs applied pressure deduced by Guguchia et al.³¹ within ${s}_{\pm }$-wave model. It can be seen that for Band 1 deduced ratio is in 2–3.5 times lower than the lowest value of 3.53 allowed by BCS theory for s-wave pairing symmetry.

Figure 9

Deduced $\frac{2\Delta (0)}{{k}_{B}{T}_{c}}$ values for bulk type-II Weyl semimetal T_d-MoTe₂ measured at applied field of B = 20 mT ³¹. (a) Two-band ${s}_{\pm }$-wave model. (b) Single-band p-wave polar A||l model.

In Fig. 9(b) we show $\frac{2\Delta (0)}{{k}_{B}{T}_{c}}$ evolution vs applied pressure for the value deduced by applying single-band p-wave polar A||l model. Deduced ratios for this model demonstrate lower uncertainties in comparison with ${s}_{\pm }$ model, and also ones show that T_d-MoTe₂ is moderately strong-coupling superconductor, for which coupling strength is linearly reducing towards weak-coupling limit of 4.06 (for this symmetry) while pressure is increased. This behavior is expected for this quasi-2D material as a direct consequence of the increase in interlayer coupling for 2D-nanosheets while applied pressure is increasing.

This example demonstrates that p-wave pairing symmetry perhaps is common feature for many unconventional superconductors.

P-doped BaFe₂As₂ thin film

Kurth et al.²⁷ reported the self-field critical current density for isovalently P-doped BaFe₂As₂ (Ba-122) single crystalline thin films deposited on MgO (001) substrates by molecular beam epitaxy. The film dimensions were 2a = 40 µm, and 2b = 107 nm. In Fig. 10 we show a fit of J_c(sf,T) to Eqs 7–11 using γ = 2.6^52,53,54 and κ = 93⁴⁸.

Figure 10

BCS fits to the experimental J_c(sf,T) data²⁷ and λ(T) calculated from Eqs 7–11 for a P-doped BaFe₂As₂ thin film (2a = 40 µm, 2b = 107 nm) assuming a p-wave polar A⊥l model, κ = 93 and γ = 2.6. T_c was fixed at 29 K. Derived parameters are: Δ(0) = 5.5 ± 0.6 meV, ΔC/C = 1.4 ± 1.0, λ(0) = 195 ± 5 nm, 2Δ(0)/k_BT_c = 4.4 ± 0.5. Fit quality is R = 0.486. Green ball is λ(0) = 200 nm for P-doped BaFe₂As₂⁴⁸.

Due to the experimental J_c(sf,T) dataset being limited to five data points, we fixed the transition temperature for the fit to T_c = 29 K. The deduced value of λ(0) = 195 ± 5 nm is in excellent agreement with the reported λ(0) = 200 nm for phosphorus-doped Ba-122 compounds⁴⁸. A richer experimental J_c(sf,T) dataset would be beneficial for more accurate determination of the other superconducting parameters.

LaFePO bulk single crystal: polar A⊥l

This was observed by a high resolution susceptometer based on a self-resonant tunnel diode circuit by Fletcher et al.²⁸. In Fig. 11 we show the raw data for their LaFePO Sample #1 with a fit to a p-wave ρ_s(T) polar A⊥l model. We fixed the T_c to the experimental value of 5.45 K. The fit is excellent across a very wide temperature range. All the deduced values are in excellent agreement with the weak-coupling limits of the p-wave polar A⊥l case.

Figure 11

BCS fits to the experimental ρ_s,ab(T) data²⁸ for a LaFePO sample assuming a p-wave polar A⊥l case. Derived parameters are: T_c = 5.45 K (fixed), Δ(0) = 1.186 ± 0.005 meV, ΔC/C = 0.84 ± 0.02, λ(0) = 248.9 ± 0.2 nm, 2Δ(0)/k_BT_c = 5.05 ± 0.02. Fit quality is R = 0.9996.

The temperature dependent superfluid density, ρ_s(T), in iron-based superconducting crystals has also been measured directly using muon-spin rotation (µSR) spectroscopy. For most iron-based superconductors reported in the literature, there is again the consistent observation that p-wave pairing symmetry exists in these materials. This analysis is given in the Supplementary Information.

As we show in this paper, experimental data for many iron-based superconductors clearly shows that p-wave superconductivity is surprisingly often observed in these materials.

Summary

Analysis of self-field critical current data and superfluid density data obtained on a wide variety of iron-based superconductors using p-wave models find superconducting parameters (ground-state penetration depth, superconducting gap polar A⊥l magnitude, and specific heat jump at the transition temperature) that are more consistent under a p-wave model compared with the generally-accepted s-wave model. Also, observation of the polar A⊥l model (where the shape is completely different to both s- and d-wave models) in both the self-field critical current data and superfluid density data strongly indicates the existence of p-wave pairing in these iron-based superconductors.

Methods

Superconducting NdFeAs(O,F) thin films were prepared by molecular beam epitaxy. First, the parent compound NdFeAsO was grown on MgO(001) at 800 °C, followed by the deposition of a NdOF over-layer at the same temperature, from which fluorine diffused into the NdFeAsO layer^48,49,53,54. Reflection high-energy electron diffraction confirmed the epitaxial growth of NdFeAsO as well as NdOF with smooth surfaces. Since NdOF is an isulator, the NdOF cap layer was removed by ion-beam etching for transport measurements. The NdFeAs(O,F) film was photolithographically patterned and ion-beam etched to fabricate bridges of 9 µm and 20 µm width and 1 and 2 mm in length.

For measurements of J_c(sf,T), a new system was built based on the Quantum Design Physical Property Measurement System. The new system adopts several parts of the system described elsewhere⁵⁵. A detailed account of the design and operational performance of the new system is given in ref.⁵⁶. The system is capable of supplying transport currents up to 30 A while maintaining a sample temperature of T = 2.0 ± 0.1 K, and currents up to 200 A at higher sample temperatures.