In-plane p-wave coherence length in iron-based superconductors

High-temperature superconductivity in iron-based layered compounds discovered by Hosono group (Kamihara et al 2006 J. Am. Chem. Soc. 128 10012) is fascinating physical phenomenon which still has many unanswered questions. One of these questions is the superconducting gap symmetry in iron-based superconductors (IBS), for which the most agreed concept is multiple-band $s$-wave symmetry. Recently, an alternative concept of single-band $p$-wave symmetry has been proposed. To disprove/reaffirm the latter concept, in this paper we analyse temperature dependent in-plane coherence length in FeSe, FeSe1-xTex, Ba(Fe1-x(Co,Ni)x)2As2 and Ca10(Pt4As8)((Fe,Pt)2As2)5 in order to extract the gap-to-critical-temperature ratio, 2$\Delta$(0)/$k$$_B$$T$$_c$, and the specific-heat-jump ratio, $\Delta$C/C,in these compounds. In the result, we report that deduced ratios are in a good agreement with the concept of single-band $p$-wave superconductivity in these materials.


I. Introduction
The discovery of superconductivity in Fe-based compound of LaOFeP with superconducting transition near boiling point of liquid helium by Hosono's group [1] demonstrated that there is more deep link between the superconductivity and the ferromagnetism which both for decades considered as mutually exclusive phenomena.
Further studies revealed more than dozen iron-based superconductors (IBS) families [2][3][4][5], including several systems which exhibit transition temperatures well above 21 K (i.e., the boiling point of hydrogen, which can be considered as a natural border from low-and hightemperature superconductors).
Superconducting gap symmetry in IBS is one of central question in understanding of the phenomenon in these compounds. Widely accepted view is that IBS exhibits multiple-band swave gap symmetry, in particular ± -wave [6][7][8][9][10][11][12]. It should be noted, that this model has been proposed at very early stage of IBS studies, when precise experimental data for majority of IBS compounds has not been reported, and ± -wave model is mainly originated from firstprinciples calculations, rather than from analysis of experimental data. However, recent thorough analysis of experimental temperature dependent superfluid density, s(T), for several IBS systems [13] showed that single-gap p-wave model describes experimental data remarkably well, which is, in addition, required only 4 free-fitting parameters (i.e., transition temperature, Tc, ground state London penetration depth, (0), ground state superconducting energy gap, (0), and relative jump in specific heat at transition temperature, C/С), while ± -wave model (or any other multiple-gaps model) requires more than twice free-fitting parameters (because, in addition to doubled number of mentioned above parameters, the model requires inter-band coupling constants). Due to a large number of free-fitting parameters, there is unlikely that all parameters are mutual independent, and, thus, an overfitting problem can be prominent when ± -wave model is applied for the analysis of experimental data.
In this paper we attempt to reaffirm or disprove single-band p-wave superconductivity model in IBS compounds by the analysis of temperature dependent c-axis upper critical field, Bc2,c(T) (i.e., when external applied magnetic field applied parallel to [001] direction of single crystal). This field is also called in-plane upper critical field and it is defined as [14]: where 0 = 2.068 • 10 −15 Wb is magnetic flux quantum, and ( ) is in-plane coherence length.
In the result, we conclude that all analysed IBS materials are single-band p-wave superconductors.

II. Description of the approach
To analyse Bc2,c(T) data we use an approach which has been proposed in our previous works [15][16][17], and which is based on utilization a general relation between in-plane London penetration depth, ab(T), and in-plane the coherence length, ab(T): where ( ) is in-plane Ginzburg-Landau parameter [14]. Thus, Bc2,c(T) can be expressed in term of in-plane superfluid density, , ( ): where Rnorm is normal state resistance [18][19][20][21][22]. Despite of this, hereafter, we will use the designation of Bc2(T) for the field at which initial stage of dissipation has been registered in experiment. It should be noted that not for all IBS materials and not all measurements (and, especially, for measurements at high pulsed magnetic field [23]) the criterion of R = 0  can be used because of experimental challenges, in these cases the lowest possible value of ( , ) ( ) ≲ 0.1 will be applied to define Bc2(T) from R(T,B) curves as described below.
In our previous work [13], we showed that the self-field critical current density, Jc(sf,T), in thin (when the film thickness, 2b, is less than c-axis London penetration depth, c(0)) caxis oriented films of IBS described by: where , ,⊥ ( ) ≡ It can be seen (Fig. 3,b) that Bc2,c(T) data for s-wave superconductor V3Si is only slightly deviate from the calculated weak-coupling 2 ( )⋅ , ( ) 2 (0)⋅ , (0) curve, due to V3Si is moderately strong coupling superconductor (with the gap-to-critical-temperature ratio of 2⋅Δ(0) ⋅ = 3.8 [41]), which is slightly larger than weak-coupling value of 2⋅Δ(0) ⋅ = 3.53 for which the curve was calculated. In section 3.1 we perform numerical fit of this dataset (rather than manual scaling) which confirms moderately-strong coupling electron-phonon interaction in V3Si superconductor.
It should be also stressed that manual scaling (Fig. 3,b) Bc2,c(T) data for Ba(Fe1-xCox)2As2 [39] is nicely match weak-coupling T/T c While our primarily finding is shown in Fig. 3 where kB is the Boltzmann constant, and the amplitude of temperature dependent superconducting gap, (T), is given by [29,30]: where ΔC/C is the relative jump in electronic specific heat at Tc, and  = 2/3 for s-wave superconductors.
In following sections, where Bc2,c(T) data of IBS is analyzed, we use fitting equation: where subscripts p and ⊥ designate polar perpendicular case of p-wave symmetry for which general gap function is given by [29,30]: where, (T) is the superconducting gap amplitude, k is the wave vector, and l is the gap axis.
9,10 to fit the same dataset. The fit result is shown in Fig. 4.

FeSe single crystal
FeSe is iron-based superconductor which has the simplest crystalline structure (which whole IBS family) and simultaneously exhibiting the most intriguing property to be thinning down to atomic thickness, its critical temperature shows unprecedented rise in Tc [47][48][49][50]. In this paper we analyse Bc2,c(T) data for bulk FeSe crystals which exhibit Tc ~ 8-9 K.
In Fig. 8 we show Bc2,c(T) data reported by Vedeneev et al [51] (defined by zero resistance criterion of ( , ) = 0.0 , see Fig. 6(a) [51]) and fit this dataset to Eqs.  gap of (0) = 2.3 ± 0.3 meV, which is in a good agreement with independent measurement the amplitude of the superconducting gap in FeSe [52]. The ratio of C/C is within the uncertainty is in a good agreement with independent measurements of 1.55 [53].
To perform fit we fixed Tc to its experimental value of 10.1 K. It can be seen in Fig. 10 that deduced parameters are in good agreement with weak-coupling limit of p-wave gap symmetry. It should be also mentioned that this first reported Bc2(T) data already showed that neither s-, nor d-wave superconducting energy gap cannot describe experimental data (please see Fig. 3).

V. Conclusions
In this paper we describe a general approach to deduce primary parameters of superconductors, i.e. ab(0), Δ(0), ΔC/C, Tc, and 2(0)/kBTc by the analysis of experimental upper critical field data, Bc2(T), which is defined by strict criterion of ( , ) → 0, and which is usually referred as the irreversibility field, Birr(T).
We should stress that Bc2(T) data in iron-based superconductors cannot be fitted in the assumption of s-or d-wave symmetry. We demonstrate that all analysed Bc2(T) data in ironbased superconductors can be perfectly fitted to p-wave superconducting gap symmetry.