Classifying superconductivity in an infinite-layer nickelate Nd$_{0.8}$Sr$_{0.2}$NiO$_2$

Recently Li et al (2019 Nature 572 624) discovered a new type of oxide superconductor Nd0.8Sr0.2NiO2 with T$_c$ = 14 K. To classify superconductivity in this infinite-layer nickelate experimental upper critical field, B$_{c2}$(T), and the self-field critical current densities, J$_c$(sf,T), reported by Li et al (2019 Nature 572 624), are analysed in assumption of s-, d-, and p-wave pairing symmetries and single- and multiple-band superconductivity. Based on deduced the ground-state superconducting energy gap, $\Delta$(0), the London penetration depth, $\lambda$(0), the relative jump in electronic specific heat at Tc, $\Delta$C/C, and the ratio of 2$\Delta$(0)/k$_B$T$_c$, we conclude that Nd0.8Sr0.2NiO2 is type-II high-$\kappa$ weak-coupled single-band s-wave superconductor.

In this paper, to classify superconductivity in this new class of oxide superconductors the temperature-dependent upper critical field, Bc2(T), and the self-field critical current density, Jc(sf,T), are analysed within s-, d-, and p-pairing symmetries. In result, it is shown that infinitelayer Nd0.8Sr0.2NiO2 nickelate is weak-coupled single band s-wave superconductor.

II. Models description
The Ginzburg-Landau theory [14] has two fundamental lengths, one is the coherence length, (T), and the second is London penetration depth, (T). The ground state coherence length, (0), is given by [14,15]: ]. (2) The second model was proposed by Baumgartner et al [20] (B-WHH): And the third model was proposed recently in our recent report [21]: .07•( ) 4 1.77 where kB is Boltzmann constant, and (T) is the temperature-dependent superconducting gap, for which analytical expression was given by Gross et al [22]: where (0) is the ground state energy gap amplitude, ΔC/C is the relative jump in electronic specific heat at Tc,  = 2/3 for s-wave superconductors [22].
Thus, (0) and Tc can be obtained by fitting experimental Bc2(T) data to Eqs. 2-4. In addition, C/C, (0) and, thus, the ratio of 2Δ(0) , can be deduced as free-fitting parameters by fitting experimental Bc2(T) data to Eq. 4. More details about the procedures can be found elsewhere [23].
There is an alternative way to deduce (0), C/C, Tc and 2Δ(0) by the fit of experimental self-field critical current density, Jc(sf,T), to universal equation, which is for thin-film superconductors reduced to simple form [23,24]: 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 the London penetration depth, (T), is given by: for s-wave superconductors, where (T) is given by Eq. 5 [22,25].
for d-wave superconductors, where the superconducting energy gap, (T,), is given by [22,25]: where m(T) is the is the maximum amplitude of the k-dependent d-wave gap given by Eq. 5,  is the angle around the Fermi surface subtended at (, ) in the Brillouin zone (details can be found elsewhere [22,25,26]). In Eq. 9 the value of  = 7/5 [22,25,26].
3. And p-wave symmetry [22,25], which only recently was tested to fit critical current densities in superconductors [22,25]: where subscripts p, a, ⊥, and ∥ designate polar, axial, perpendicular and parallel cases respectively. For this symmetry, the gap function is given by [22,25]: where, (T) is the superconducting gap amplitude, k is the wave vector, and l is the gap axis.
Thus, temperature dependence of (T) is determined by mutual orientation of the vector potential, A, and the gap axis, l, which is for transport current experiment just the orientation of the crystallographic axes of the film compared with the direction of the electric current.

III. Bc2(T) analysis
There are several criteria to define Bc2(T) from experimental R(T) curves. In this paper ti define Bc2(T) we use the criterion of 3% of normal state resistance, Rnorm(T), for R(T) curves of Nd0.8Sr0.2NiO2 presented in Fig. 4(a) by Li et al [3]. The fits of Bc2(T) data to three models are shown in Fig. 1. It can be seen that (0) values deduced by three models are close to each other and following analysis of Jc(sf,T) will be utilized an average value of:  Deduced values by the fit to Eq. 4: are, within uncertainties, equal to BCS [34] weak-coupling limits of 3.53 and 1.43 respectively, and the former deduced value is equal to recently deduced value of: for s-wave oxide superconductor of Ba0.51K0.49BiO3 [35].
It should be noted that there is no sign in experimental Bc2(T) data that Nd0.8Sr0.2NiO2 exhibits two superconducting band state, which can be seen as sharp enhancement in amplitude of Bc2(T) at critical temperature of the second superconducting band opening (see for details Ref. 36).

IV. Jc(sf,T) analysis
The critical current density, Jc, is defined as the lowest, detectable in experiment, value of electric power dissipation in a superconductor on electric current flow. For available E(I) curves presented by Li et al [3] in their Fig. 3(f), the critical current density at self-field condition (when no external magnetic field is applied), Jc(sf,T), can be defined at the lowest value of electric field of Ec = 3 V/cm. Experimental Jc(sf,T) deduced by this Ec criterion and the fit to single band s-wave model (i.e., Eqs. 6,7 for which (0) = 5.7 nm was fixed) are shown in Fig. 2(a). It can be seen that the fit is excellent, and deduced superconducting parameters ( Fig. 2(a) and Table 1) are within BCS weak-coupling limits.  Table I. Deduced (0) = 740 ± 3 nm is similar to (0) = 690-850 nm measured for samples possessing maximal Tc values for cuprate counterpart La1-xSrxCuO2 [37].

Pairing symmetry and experiment geometry
Deduced values and these symmetries can be excluded from further consideration.
The cases of polar Al and axial Al gap symmetries are still hypothetically possible (Table 1), and Jc(sf,T) fit to these models are shown in Figs. 2(b,c) respectively, however, for given experimental conditions (i.e. epitaxial c-axis oriented thin film) expected geometry is polar Al [21].
It should be also noted that there is no sign for two-band superconductivity in Nd0.8Sr0.2NiO2 which usually can be detected by a sharp enhancement in Jc(sf,T) at critical temperature of the second band opening [24,36].
By taking in account a good agreement between 2Δ(0) and Δ values deduced for s-wave symmetry from Bc2(T) and Jc(sf,T) analyses (Eqs. 17, 18 and Table 1, respectively), which are, in addition, within BCS weak-coupling limits for this symmetry, and a fact that s-wave pairing symmetry is the most conventional one, we can conclude that Nd0.8Sr0.2NiO2 nickelate is weak-coupling single band high- s-wave superconductor.

V. Conclusions
Recently discovered [3] an infinite-layer nickelate Nd0.8Sr0.2NiO2 superconductor is a new member of bulk oxide superconductors for which experimental Bc2(T) and Jc(sf,T) data are analysed in this paper.
In result, it is found that an infinite-layer nickelate Nd0.8Sr0.2NiO2 is weak-coupling single band high- s-wave superconductor.