Two-band and pauli-limiting effects on the upper critical field of 112-type iron pnictide superconductors

The temperature dependence of upper critical field μ0Hc2 of Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 single crystals are investigated by measuring the resistivity for the inter-plane (H//c) and in-plane (H//ab) directions in magnetic fields up to 60 T. It is found that μ0Hc2(T) of both crystals for H//c presents a sublinear temperature dependence with decreasing temperature, whereas the curve of μ0Hc2(T) for H//ab has a convex curvature and gradually tends to saturate at low temperatures. μ0Hc2(T) in both crystals deviates from the conventional Werthamer-Helfand-Hohenberg (WHH) theoretical model without considering spin paramagnetic effect for H//c and H//ab directions. Detailed analyses show that the behavior of μ0Hc2(T) in 112-type Iron-based superconductors (IBSs) is similar to that of most IBSs. Two-band model is required to fully reproduce the behavior of μ0Hc2(T) for H//c, while the effect of spin paramagnetic effect is responsible for the behavior of μ0Hc2(T) for H//ab.

Scientific RepoRts | 7:45943 | DOI: 10.1038/srep45943 temperature region in this new type IBSs. Benefiting from the advanced technology of pulsed field measurement, in this work, we reported the temperature dependence of upper critical field μ 0 H c2 (T) of Ca 0.83 La 0.17 FeAs 2 and Ca 0.8 La 0.2 Fe 0.98 Co 0.02 As 2 single crystals by measuring the electrical resistivity in pulsed fields up to 60 T at Wuhan National High Magnetic Field Center. The behavior of μ 0 H c2 (T) and its anisotropy are systematically studied. Our results suggest that the two-band and spin-paramagnetic effects are shown to be responsible for the temperature-dependent behavior of μ 0 H c2,c (T) and μ 0 H c2,ab (T), respectively. Figure 1 presents the temperature dependence of the in-plane resistivity ρ(T) at zero field for (a) Ca 0.83 La 0.17 FeAs 2 and (b) Ca 0.8 La 0.2 Fe 0.98 Co 0.02 As 2 single crystals. The resistivity of both crystals monotonically decreases with decreasing temperature and shows no anomaly corresponding to the antiferromagnetic (AFM)/structural transition down to T c . The insets show the enlarged view near the superconducting transition. The transition temperature is estimated as T c 50% = 40.8 K for Ca 0.83 La 0.17 FeAs 2 and T c 50% = 38.8 K for Ca 0.8 La 0.2 Fe 0.98 Co 0.02 As 2. The transition width Δ T c , determined by adopting the criterion of 90%ρ n -10%ρ n , is 3.8 K for Ca 0.83 La 0.17 FeAs 2 , larger than the value of 1.1 K for Ca 0.8 La 0.2 Fe 0.98 Co 0.02 As 2 . The slightly wide superconducting transition for Ca 0.83 La 0.17 FeAs 2 seems to be a general feature in this compound, which may result from the inhomogeneity of La distribution. Upon a small amount of Co doping, single crystal quality can be improved significantly with sharp superconducting transition and large superconducting volume fraction [20][21][22][23] . Figure 2 shows the temperature dependence of the in-plane resistivity ρ(T) of Ca 0.83 La 0.17 FeAs 2 and Ca 0.8 La 0.2 Fe 0.98 Co 0.02 As 2 single crystals in low magnetic fields from 0 to 9 T for H//ab and H//c. With increasing fields, the resistivity transition width becomes broader and the onset of superconductivity shifts to lower temperatures gradually. It is worth noting that the superconducting transitions of Ca 0.83 La 0.17 FeAs 2 and Ca 0.8 La 0.2 Fe 0.98 Co 0.02 As 2 show different response to the increased magnetic field. For Ca 0.8 La 0.2 Fe 0.98 Co 0.02 As 2 , the field-induced broadening of superconducting transition is more pronounced for H//c, showing a tail structure at low temperatures, similar to the case in 1111-type IBSs and cuprates, which can be explained in terms of the formation of vortex-liquid phase 8,10,24,25 . In contrast, for Ca 0.83 La 0.17 FeAs 2 , the transition shows slightly field-induced broadening, indicating the vortex-liquid state region is very narrow, similar to the results of 122-type and 11-type In order to minimize the effects of superconducting fluctuation near 90%ρ n and vortex motion in the vortex-liquid region near 10%ρ n on the determination of μ 0 H c2 , we use the 50%ρ n criteria, which is widely accepted to be close to the real μ 0 H c2 , to define the μ 0 H c2 (T) values in the following 10,31 . The normal state resistivity ρ n was determined by linearly extrapolating the normal state resistivity into the superconducting state in ρ(T) and ρ(H) curves separately. μ 0 H c2 (T) of both crystals for H//ab and H//c directions along with the low magnetic field data up to 9 T were shown in Fig. 4. μ 0 H c2 (T) obtained from the pulsed field measurement follows well the curvature and values of the low filed ones. Data above 60 T were extracted by linear extrapolation of ρ(H) at μ 0 H < 60 T to ρ(H) = 0.5ρ n (T c , H) 18,32 . In several highly two-dimensional superconductors, the curvature of μ 0 H c2 (T) has been reported to vary depending on the criteria used to determine μ 0 H c2 8,14 . Thus, the μ 0 H c2 (T) defined by 80%ρ n and 20%ρ n were also presented in Supplementary Information (SI), Fig. S2. It is noted that the shape of μ 0 H c2 (T) curve does not change qualitatively when μ 0 H c2 (T) is defined by different criteria in this compound. In addition, a slight upward behavior near T c which commonly observed in some IBSs 14,33,34 , is also observed for both directions, might be due to the flux dynamics as is seen in cuprates 35 .

Discussion
Generally, two distinct mechanisms exist in superconductivity suppression under magnetic fields in type-II superconductors. One is the orbital pair-breaking effect, with opposite momenta acting on the paired electrons. In this case, the superconductivity is destroyed when the kinetic energy of the Cooper pairs exceeds the condensation energy. The other is attributed to the spin-paramagnetic pair-breaking effect, which comes from the Zeeman splitting of spin singlet cooper pairs. The superconductivity is also eliminated when the Pauli spin susceptibility energy is larger than the condensation energy. WHH theory, which could identify the contribution of each pair-breaking mechanism, was used to fit the μ 0 H c2 (T) curves, and the strength of the spin-paramagnetic effect and the spin-orbit effect were incorporated via the Maki parameter α and the spin-orbit interaction λ so , respectively 36 . According to the WHH theory, μ 0 H c2 (T) in the dirty limit can be described by the digamma function 37  In the absence of both spin-paramagnetic effect and spin-orbit interaction, α = 0 and λ so = 0, the orbital-limited upper critical field is expressed as In the weakly coupled BCS superconductors, the pauli limited field is given by 38 For conventional superconductors, μ 0 Hp (0) is usually much larger than μ 0 H c orb 2 (0), and therefore, their upper critical field is mainly restricted by the orbital pair breaking mechanism. While, the spin-paramagnetic effect may play an import role in pair breaking in some unconventional superconductors 10,12,[14][15][16][17][18][19] . In our case, we obtained µ H (0) Thus, it is likely that the upper critical field in both crystals is limited by the orbital effect for H//c, but is limited by the spin paramagnetic for H//ab.
As evidenced from Fig. 4, the experimental μ 0 H c2 (T) curves of both crystals deviate from the WHH model neglecting the spin paramagnetic effect (α = 0) and spin-orbit interaction (λ so = 0) in H//ab and H//c at low temperatures (dotted lines). For H//ab, the curve of μ 0 H c2 (T) falls below the WHH model (α = 0 and λ so = 0) and has a tendency to saturate at low temperatures, indicating the spin-paramagnetic effect should be considered, as we discussed above. The best fits (dashed lines) were obtained using Eq. (1) with α = 0.9 for Ca 0.83 La 0.17 FeAs 2 , and α = 1.9 for Ca 0.8 La 0.2 Fe 0.98 Co 0.02 As 2 . It is noteworthy here that the spin-orbit scattering is not necessary to have the best fit (λ so = 0). The negligible value of λ so compared to the other IBSs indicates the spin-orbit scattering is also rather weak in this compound 10,12,16,18,32 . According to Maki 36 , the paramagnetic limited field µ H (0)  In IBSs, the weak-coupling BCS formula usually underestimates the actual paramagnetic limit. This enhancement of pauli limited field seems to be a common feature in IBSs, presumably aroused by many-body correlation and the strong coupling effects in IBSs 12,16,18 .
For H//c, μ 0 H c2,c (T) shows an almost linear temperature dependence and tends to be saturated at low temperatures, similar to the results of 1111-type SmFeAs(O,F) single crystals 10 . At low temperatures, the μ 0 H c2,c (T) is slightly larger than the value predicted with Eq. (2). The sublinear increase and enhancement of μ 0 H c2,c (T) are generally observed in many multiband superconductors, e.g., MgB 2 and some IBSs [8][9][10]14,17,39,40 , which has been successfully explained by the two-band model 11 . The equation of μ 0 H c2 (T) for a two-band superconductor is given by:  10 . It should be noted that the results here are not sensitive to the choice of the coupling constants as discussed in previous report 9,14,41 . We find that the two-band model can also fit well, even if we change the value of λ properly. Although we can't give the exact coupling values and enable us to analysis the possible pairing scenarios in this novel IBSs, the fitting results by two-gap model can be seen to agree very well with our experimental data. In what follows, we discuss the different temperature dependence of μ 0 H c2 (T) for H//ab and H//c, which was generally observed in IBSs 10,13,14,16,17,42,43 . Why the two-band model is essential to explain the behavior of μ 0 H c2,c (T), but the effect of spin paramagnetic effect is mainly responsible for the behavior of μ 0 H c2,ab (T), is still unclear now. This calls for further investigations on this interesting question, both experimentally and theoretically. Previous studies have proposed that the cross section of the Fermi surface produces closed current loops that form vortices for H//c due to the quasi-two-dimensional Fermi-surface. Thus, the orbital pair-breaking mechanism plays a dominant role in destroying the superconductivity in high magnetic fields, therefore, the two-gap theory, taking into account the orbital pair-breaking effect. In contrast, for H//ab, closed loops cannot be easily formed because the cross-sectional area of the Fermi surface is almost fully open with negligible orbital effect, thus, the spin paramagnetic effect is a more dominant factor in μ 0 H c2,ab (T) 10,17 . This scenario seems also to be suitable for this 112-type IBSs since all Fermi-surfaces exhibit two-dimensional character except for the α band [3][4][5] . Here, we try to give another possible scenario relevant to the spin-locked superconductivity, which has been proposed in quasi-one-dimensional superconductor K 2 Cr 3 As 3 44 . The schematic diagram was shown in SI, Fig. S3. Under this physical scenario, the spins of Cooper pairs are predominantly aligned along the ab-plane in IBSs, thus, the behavior of μ 0 H c2,ab (T ) would be pauli-limited. For H//c, since there is little spin along the c-axis, μ 0 H c2,c (T) would be hardly effected by the pauli pair-breaking, and mainly restricted by the orbital effect instead. In this case, due to the multiband electronic structure, the enhancement of upper critical field μ 0 H c2,c (T) should be described by the multiband theory model.
, the superconducting coherence length ξ (0) can be estimated using the Ginzburg-Landau formula: H c2,c = Φ 0 /2π ξ (0)   (0) is the upper critical field for H//c, which is determined from fittting the experimental data using the two-band model. ξ ab (0) and ξ c (0) are the ab-plane and c-axis zero temperature coherence length calculated using μ 0 H c ab  nature and spin paramagnetism 10 . The decreasing γ with decreasing temperature in both crystals results from the enhanced μ 0 H c2,c (T) and the suppressed μ 0 H c2,ab (T).
In summary, we have investigated the temperature dependence of upper critical field of Ca 0.83 La 0.17 FeAs 2 and Ca 0.8 La 0.2 Fe 0.98 Co 0.02 As 2 single crystals under pulsed fields up to 60 T. Analysis based on the WHH model and two-band model indicates that, μ 0 H c2 (T) of this compound bears many similarities to most of IBSs, μ 0 H c2,ab (T) is clearly limited by the pauli limited effect at low temperatures, and the two-band model is required to describe the enhancement of the upper critical field μ 0 H c2,c (T). Our work clearly clarifies the behavior of the upper critical field of 112-type iron pnictide superconductors.

Method
Single crystal growth and basic characterizations. Single crystals of Ca 0.83 La 0.17 FeAs 2 and Ca 0.8 La 0.2 Fe 0.98 Co 0.02 As 2 were grown using the flux method as previous reports 20,22,23 . The single crystal x-ray diffraction (XRD) was performed using a Rigaku diffractometer with Cu Kα radiation (see SI, Fig. S1). Elemental analysis was performed by a scanning electron microscope equipped with an energy dispersive x-ray (EDX) spectroscopy probe.
Electrical resistivity measurements. In-plane electrical resistivity was performed by the standard four-probe method in low magnetic field up to 9 T in a Quantum Design Physical Property Measurement System (PPMS-9T) and in pulsed field up to 60 T at Wuhan National High Magnetic Field Center. Golden contacts were made by sputtering in order to provide a low contact resistance (less than 1 Ω.) in the pulsed field measurement.