Introduction

Recently, the newly discovered 112-type Iron-based superconductors (IBSs) Ca1−xRExFeAs2 (RE = rare-earth elements) have attracted tremendous interest due to their peculiar properties compared with other IBSs1,2. It crystallizes in a monoclinic crystal structure with a space group of P21, consisting of alternating stacking of FeAs and As zigzag bond layers. The FeAs layers have been proven to be responsible for the High-Tc superconductivity, while the unique As zigzag bond layers could generate anisotropic Dirac cones, and an additional three-dimensional (3 D) hole pocket which is absent in other IBSs3,4,5, and may also be possible for realizing the Majorana related physics6. However, so far the pair mechanism and many basic physical properties both in the superconducting and normal states are still not well understood in this compound. Upper critical field μ0Hc2, is one of the most important superconducting parameters for gathering an understanding of unconventional superconductivity including coherence length, coupling strength, and insights into the pair-breaking mechanism. Moreover, the anisotropy of μ0Hc2, which is related to the dimensionality and the topology of the underlying electronic structure, also becomes important for potential applications as well as for understanding multiband effects. Many efforts have been made to investigate the μ0Hc2(T) ever since the discovery of IBSs7. For instance, in 1111-type IBSs, μ0Hc2,c(T) shows a significant upturn behavior in (La, Nd)FeAs(O,F)8,9, and linearly increases with decreasing temperature but tends to be saturated at low temperatures in SmFeAs(O,F)10. Both behaviors can be accounted by the two-band theory11. μ0Hc2,ab(T) exhibits a downturn and flattening curvature with decreasing temperature, which is mainly ascribed to the spin-paramagnetic effect10,12. For 122-type and 111-type IBSs, μ0Hc2,c(T) exhibits a linear increase down to the lowest temperature, and μ0Hc2,ab(T) also shows a downturn curvature with decreasing temperature. These behaviors can also be interpreted using the two-band theory and spin-paramagnetic effect, respectively13,14,15,16,17. Furthermore, spin-paramagnetic effect dominates in both of μ0Hc2,ab(T) and μ0Hc2,c(T) in the 11-type IBSs(e.g., Fe1+yTe1−xSex)18,19. Nevertheless, the behavior of μ0Hc2(T) at high fields and low temperatures is not clear yet in this newly discovered 112-type IBSs. Therefore, it is interesting to check whether Ca1−xRExFeAs2 may represent a new type of IBSs, because the multiband electronic structure and peculiar properties in Ca1−xRExFeAs2 may contribute to the behavior of upper critical filed.

Previous studies of μ0Hc2(T) were just performed near Tc in low magnetic field20,21. Thus, measurement in higher magnetic field is essential to clarify the behavior of upper critical field and its anisotropy γ at low 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 μ0Hc2(T) of Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 single crystals by measuring the electrical resistivity in pulsed fields up to 60 T at Wuhan National High Magnetic Field Center. The behavior of μ0Hc2(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 μ0Hc2,c(T) and μ0Hc2,ab(T), respectively.

Results

Figure 1 presents the temperature dependence of the in-plane resistivity ρ(T) at zero field for (a) Ca0.83La0.17FeAs2 and (b) Ca0.8La0.2Fe0.98Co0.02As2 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 Tc. The insets show the enlarged view near the superconducting transition. The transition temperature is estimated as  = 40.8 K for Ca0.83La0.17FeAs2 and  = 38.8 K for Ca0.8La0.2Fe0.98Co0.02As2. The transition width ΔTc, determined by adopting the criterion of 90%ρn–10%ρn, is 3.8 K for Ca0.83La0.17FeAs2, larger than the value of 1.1 K for Ca0.8La0.2Fe0.98Co0.02As2. The slightly wide superconducting transition for Ca0.83La0.17FeAs2 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 fraction20,21,22,23.

Figure 1
figure 1

Temperature dependence of the in-plane electrical resistivity ρ(T) for (a) Ca0.83La0.17FeAs2 and (b) Ca0.8La0.2Fe0.98Co0.02As2 single crystals at zero field. The insets of (a) and (b) show an enlarged view of resistivity near the superconducting transition, Tc was determind by the 50% normal state resistivity ρn.

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Figure 2 shows the temperature dependence of the in-plane resistivity ρ(T) of Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 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 Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 show different response to the increased magnetic field. For Ca0.8La0.2Fe0.98Co0.02As2, 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 phase8,10,24,25. In contrast, for Ca0.83La0.17FeAs2, 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 IBSs18,26,27,28,29. This discrepancy seems to be consistent with the different vortex dynamics in both samples. Indeed, a small amount of Co doping distinctly influences the vortex dynamics and flux pinning, in comparison with Ca0.83La0.17FeAs2, the second magnetization peak was clearly observed and critical current density enhanced significantly through introducing more effective pinning center in Ca0.8La0.2Fe0.98Co0.02As223,30.

Figure 2
figure 2

Temperature dependence of in-plane resistivity (nomalized to 45 K) of Ca0.83La0.17FeAs2 for (a) H//ab and (b) H//c, and of Ca0.8La0.2Fe0.98Co0.02As2 for (c) H//ab and (d) H//c at the various magnetic fields from 0 to 9 T(0, 0.5, 1, 2, 3, 5, 7 and 9 T).

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The magnetic field dependence of resistivity ρ(H) of Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 single crystals for H//ab and H//c are presented in Fig. 3. Since there is no hysteresis for ρ(H) under μ0H sweep up and down processes, only data collected during the down sweep of the magnet are shown. Obviously, the superconducting transitions are suppressed upon increasing magnetic field for both directions. In addition, the superconductivity of Ca0.8La0.2Fe0.98Co0.02 is more robust against magnetic field compared with Ca0.83La0.17FeAs2, indicating Ca0.8La0.2Fe0.98Co0.02As2 has a higher μ0Hc2(0) than Ca0.83La0.17FeAs2.

Figure 3
figure 3

Magnetic field dependence of in-plane resistivity ρ(H) of Ca0.83La0.17FeAs2 for (a) H//ab and (b) H//c, and of Ca0.8La0.2Fe0.98Co0.02As2 for (c) H//ab and (d) H//c measured at fixed temperatures in pulsed fields up to 60 T.

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In order to minimize the effects of superconducting fluctuation near 90%ρn and vortex motion in the vortex-liquid region near 10%ρn on the determination of μ0Hc2, we use the 50%ρn criteria, which is widely accepted to be close to the real μ0Hc2, to define the μ0Hc2(T) values in the following10,31. The normal state resistivity ρn was determined by linearly extrapolating the normal state resistivity into the superconducting state in ρ(T) and ρ(H) curves separately. μ0Hc2(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. μ0Hc2(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 μ0H < 60 T to ρ(H) = 0.5ρn(Tc, H)18,32. In several highly two-dimensional superconductors, the curvature of μ0Hc2(T) has been reported to vary depending on the criteria used to determine μ0Hc28,14. Thus, the μ0Hc2(T) defined by 80%ρn and 20%ρn were also presented in Supplementary Information (SI), Fig. S2. It is noted that the shape of μ0Hc2(T) curve does not change qualitatively when μ0Hc2(T) is defined by different criteria in this compound. In addition, a slight upward behavior near Tc which commonly observed in some IBSs14,33,34, is also observed for both directions, might be due to the flux dynamics as is seen in cuprates35.

Figure 4
figure 4

The upper critical field μ0Hc2(T) versus temperature for (a) Ca0.83La0.17FeAs2 and (b) Ca0.8La0.2Fe0.98Co0.02As2 single crystals. Symbols of the open circles and filled circles represent the data obtained in the low magnetic field and pulsed field, respectively. The dotted lines are the WHH fits neglecting the spin paramagnetic effect for H//ab and H//c, respectively, while the dashed lines are the best fits to the experimental data by WHH model with the pauli-limiting effect considered for H//ab. The solid lines for H//c are fitting to the data using the two-band model.

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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 μ0Hc2(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 λso, respectively36. According to the WHH theory, μ0Hc2(T) in the dirty limit can be described by the digamma function37

where , and .

In the absence of both spin-paramagnetic effect and spin-orbit interaction, α = 0 and λso = 0, the orbital-limited upper critical field is expressed as

In the weakly coupled BCS superconductors, the pauli limited field is given by38

For conventional superconductors, μ0Hp (0) is usually much larger than μ0 (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 superconductors10,12,14,15,16,17,18,19. In our case, we obtained  = 75 T for Ca0.83La0.17FeAs2 and  = 71.4 T for Ca0.8La0.2Fe0.98Co0.02As2. The slope values of dμ0Hc2/dT near Tc is 3.98 T/K (9.22 T/K) and 1.23 T/K (2.26 T/K) for H//ab and H//c in Ca0.83La0.17FeAs2 (Ca0.8La0.2Fe0.98Co0.02As2), respectively. Using the WHH model, i.e. Eq. (2), one can estimate the orbital limited upper critical fields, which gives  = 112.5 T (248 T) and  = 34.9 T (60.9 T) for Ca0.83La0.17FeAs2 (Ca0.8La0.2Fe0.98Co0.02As2). In both crystals, the values of are much smaller than the corresponding μ0HBCSp(0). In contrast, is much larger than . 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 μ0Hc2(T) curves of both crystals deviate from the WHH model neglecting the spin paramagnetic effect (α = 0) and spin-orbit interaction (λso = 0) in H//ab and H//c at low temperatures (dotted lines). For H//ab, the curve of μ0Hc2(T) falls below the WHH model (α = 0 and λso = 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 Ca0.83La0.17FeAs2, and α = 1.9 for Ca0.8La0.2Fe0.98Co0.02As2. It is noteworthy here that the spin-orbit scattering is not necessary to have the best fit (λso = 0). The negligible value of λso compared to the other IBSs indicates the spin-orbit scattering is also rather weak in this compound10,12,16,18,32. According to Maki36, the paramagnetic limited field expresses as , where the Maki parameter α, is given by , μ0Hp(0) is the zero temperature pauli limited field. The values of μ0Hp,ab(0) using α obtained from WHH model fitting for Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 are 176.75 T and 184.56 T, respectively, larger than the corresponding in both crystals. 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 IBSs12,16,18.

For H//c, μ0Hc2,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 crystals10. At low temperatures, the μ0Hc2,c(T) is slightly larger than the value predicted with Eq. (2). The sublinear increase and enhancement of μ0Hc2,c(T) are generally observed in many multiband superconductors, e.g., MgB2 and some IBSs8,9,10,14,17,39,40, which has been successfully explained by the two-band model11. The equation of μ0Hc2(T) for a two-band superconductor is given by:

where t = T/Tc, a0 = 2(λ11λ22 − λ12λ21)/λ0, a1 = 1 + (λ11 − λ22)/λ0, a2 = 1 − (λ11 − λ22)/1/2 λ0, λ0 = ((λ11 − λ22)2 + 4λ12λ21)1/2, h = Hc2D1/(2Φ0/T), η = D2/D1 and U(x) = Ψ(1/2 + x) − Ψ(1/2). Ψ(x) is the digamma function, D1 and D2 are the diffusivity of each band, λ11, λ22 denote the intra-band coupling constants, and λ12, λ21 are the inter-band coupling constants. Due to the lack of microscopic theory of pairing mechanism, we choose the values of λ referring to previous reports in SmFeAs(O,F) from ref. 10. Here, we assume the intra-band coupling dominant the μ0Hc2,c(T) and take the inter-band coupling value λ12 = λ21 = 0.17 (λ12 = λ21 = 0.13), and the intra-band coupling value λ11 = 0.7, λ22 = 0.3 (λ11 = 0.78, λ22 = 0.3) for Ca0.83La0.17FeAs2(Ca0.8La0.2Fe0.98Co0.02As2). The two-band model can fit the experimental data well as shown in Fig. 4 with η = 7.5 for Ca0.83La0.17FeAs2 and η = 8 for Ca0.8La0.2Fe0.98Co0.02As2. The values of η are close to that of SmFeAs(O,F) (η = 9)10. It should be noted that the results here are not sensitive to the choice of the coupling constants as discussed in previous report9,14,41. 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 μ0Hc2(T) for H//ab and H//c, which was generally observed in IBSs10,13,14,16,17,42,43. Why the two-band model is essential to explain the behavior of μ0Hc2,c(T), but the effect of spin paramagnetic effect is mainly responsible for the behavior of μ0Hc2,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 μ0Hc2,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 band3,4,5. Here, we try to give another possible scenario relevant to the spin-locked superconductivity, which has been proposed in quasi-one-dimensional superconductor K2Cr3As344. 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 μ0Hc2,ab(T) would be pauli-limited. For H//c, since there is little spin along the c-axis, μ0Hc2,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 μ0Hc2,c(T) should be described by the multiband theory model.

Superconducting parameters of Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 obtained from analysis above are summarized in Table 1. Using and , the superconducting coherence length ξ (0) can be estimated using the Ginzburg-Landau formula: Hc2,c = Φ0/2π and Hc2,ab = Φ0/2πξab(0) ξc(0), where Φ0 = 2.07 × 10−15 T m2 is the flux quantum. We calculated ξab(0) = 2.65 nm and ξc(0) = 1.47 nm for Ca0.83La0.17FeAs2, ξab(0) = 1.93 nm and ξc(0) = 1.44 nm for Ca0.8La0.2Fe0.98Co0.02As2, respectively. ξc(0) of both crystals is larger than the distance d (~1.035 nm) between the adjacent FeAs conducting layers, indicating a 3D superconductivity in this novel superconductor, despite the layered nature of their crystal structure.

Table 1 Superconducting parameters of Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 from the analysis of μ0Hc2(T).
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The temperature dependence of anisotropy of μ0Hc2(T) is shown in Fig. 5 as a function of reduced temperature t = T/Tc for Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 single crystals. The values of γ reside in the range 1.2~3.2 (2.3 – 5.4) for Ca0.83La0.17FeAs2 (Ca0.8La0.2Fe0.98Co0.02As2) in the temperature region of T = (0.51 ~ 1) Tc. The anisotropy γ values of both crystals show nonmonotonic temperature-dependent, which first increase and then decrease with decreasing temperature, in consistence with our earlier studies23. This strong temperature dependence of γ is similar to that of SmFeAs(O, F), which may originate from the combined effect of two-band nature and spin paramagnetism10. The decreasing γ with decreasing temperature in both crystals results from the enhanced μ0Hc2,c(T) and the suppressed μ0Hc2,ab(T).

Figure 5
figure 5

The anisotropic parameter γ versus the reduced temperature T/Tc for Ca0.83La0.17FeAs2 (red circles) and Ca0.8La0.2Fe0.98Co0.02As2 (blue circles) single crystals.

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In summary, we have investigated the temperature dependence of upper critical field of Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 single crystals under pulsed fields up to 60 T. Analysis based on the WHH model and two-band model indicates that, μ0Hc2(T) of this compound bears many similarities to most of IBSs, μ0Hc2,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 μ0Hc2,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 Ca0.83La0.17FeAs2 and Ca0.8La0.2Fe0.98Co0.02As2 were grown using the flux method as previous reports20,22,23. 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.

Additional Information

How to cite this article: Xing, X. et al. Two-band and pauli-limiting effects on the upper critical field of 112-type iron pnictide superconductors. Sci. Rep. 7, 45943; doi: 10.1038/srep45943 (2017).

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