Iron chalcogenide superconductors at high magnetic fields

Thermal fluctuations in FeCh-11 superconductors can result in the thermally activated flux flow (TAFF) behavior where the vortex bundles hop between neighboring pinning centers. According to the TAFF theory, the resistivity in TAFF region can be expressed as [44–46]

$$\begin{equation} \rho =(2\nu _{0}LB/J)\exp(-J_{{\rm c}0}BVL/T)\sinh(JBVL/T), \end{equation} \tag{ 3 }$$

where ν₀ is an attempt frequency for a flux bundle hopping, L is the hopping distance, B is the magnetic induction, J is the applied current density, J_c0 is the critical current density in the absence of flux creep, V is the bundle volume and T is the temperature. If J is small enough and JBV L/T ≪ 1, we obtain

$$\begin{equation} \rho =(2\rho _{{\rm c}}U/T)\exp(-U/T)=\rho _{0f}\exp(-U/T), \end{equation} \tag{ 4 }$$

where U = J_c0BV L is the thermal activation energy and ρ_c = ν₀LB/J_c0, which is usually considered to be temperature independent. For cuprate superconductors, the prefactor 2ρ_cU/T is usually assumed as a constant ρ_0f [45], therefore, ln ρ(T,μ₀H) = ln ρ_0f − U(T,μ₀H)/T. On the other hand, according to the condensation model [46], U(T,μ₀H) = (μ₀H_c(t))²ξⁿ(t), where μ₀H_c is the thermodynamic critical field, ξ is the coherence length, t = T/T_c, and n depends on the dimensionality of the vortex system with the range from 0 to 3. Since μ₀H_c∝1 − t, and ξ∝(1 − t)^−1/2 near T_c [47], it is obtained that U(T,μ₀H) = U₀(μ₀H)(1 − t)^q, where q = 2 − n/2. It is generally assumed that U(T,μ₀H) = U₀(μ₀H)(1 − t), i.e. n = 2, and the ln ρ − 1/T becomes Arrhenius relation

$$\begin{equation} \ln\rho (T,\mu _{0}H)=\ln\rho _{0}(\mu _{0}H)-U_{0}(\mu _{0}H)/T, \end{equation} \tag{ 5 }$$

where ln ρ₀(μ₀H) = ln ρ_0f + U₀(μ₀H)/T_c is a temperature-independent term and U₀(μ₀H) is the apparent activation energy. Furthermore, it can be concluded that −∂ ln ρ(T,μ₀H)/∂T⁻¹ = U₀(μ₀H). Hence, the ln ρ versus 1/T should be linear in the TAFF region. The slope is U₀(μ₀H) and its y-intercept represents ln ρ₀(μ₀H).

As shown in figures 1(a) and (b), the experimental data can be fitted using the Arrhenius relation very well for both FeSe and Fe_1.03Te_0.55Se_0.45 (Se-45) single crystals [48, 49]. The good linear behavior indicates that the temperature dependence of U(T,μ₀H) is approximately linear, i.e. U(T,μ₀H) = U₀(μ₀H)(1 − t) [45, 46]. The log ρ(T,μ₀H) lines for compounds extrapolate to the same temperature T_cross, which should be equal to T_c [50]. For FeSe single crystal, the extrapolated temperatures are about 11.1 K for both H∥(101) and H⊥(101). This behavior has also been observed in NdFeAsO_0.7F_0.3 single crystals for H∥c, however, it is not the case for H∥ab [51]. For FeSe single crystal, the obtained U₀ are similar for H∥(101) and H⊥(101) (inset of figure 1(a)) and much smaller than in Fe(Te,Se) [49, 52]. Moreover, ln ρ₀(μ₀H) − U₀(μ₀H) is linear for both field directions (inset of figure 1(a)). Fits using ln ρ₀(μ₀H) = ln ρ_0f + U₀(μ₀H)/T_c yielded values of ρ_0f and T_c as 37(1) mΩ cm and 11.2(1) K for H∥(101) and 39(2) mΩ cm and 11.2(1) K for H⊥(101). The T_c values are consistent with the values of T_cross within the error bars. As shown in figures 1(c) and (d), the field dependence of U₀(H) exhibits a power-law behavior for both FeSe and Se-45, i.e. U₀(μ₀H) ∼ (μ₀H)^−α. There are two types of power-law behavior at low and high fields. For FeSe, α ∼0.25 for μ₀H < 3 T and α ∼ 0.7 for μ₀H > 3 T. Se-45 exhibits similar behavior to FeSe (α ∼ 0.25 for μ₀H < 2 T and α ∼ 0.55 for μ₀H > 2 T). The weak power-law dependence of U₀(μ₀H) at low fields implies that single-vortex pinning dominates in this region [44], followed by a quicker decrease of U₀(μ₀H) at high fields where the vortex spacing becomes significantly smaller than penetration depth in higher fields and a crossover to a collective-pinning regime occurs. In this region, the U₀(μ₀H) becomes strongly dependent on the field [53]. A similar crossover has been observed in Nd(O,F)FeAs single crystals [51]. Because α = 0.5 and 1 correspond to a planar-defect pinning and a point-defect pinning, respectively [54], for FeSe, the fitted α values vary between 0.5 and 1, suggesting that the pinning centers may be mixed with point and planar defects. On the other hand, for Se-45, the fitted α values are close to 0.5, implying that the vortices are mainly pinned by the collective point defects in the high-field region. 

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Large fitting errors occur for Fe_1.14Te_0.91S_0.09 (S-09) single crystal when using the Arrhenius relation in the TAFF region (solid lines), especially for H∥ab (figures 2(a) and (b)). Deviation from the Arrhenius relation indicates that the assumptions of constant ρ_0f and linear temperature dependence of U(T,μ₀H) could be invalid in S-09. If U(T,μ₀H) = U₀(μ₀H)(1 − t) and ρ_0f = const, this will result in −∂ ln ρ(T,μ₀H)/∂T⁻¹ = U₀(μ₀H), and U₀(μ₀H) should be a set of horizontal lines as shown in figures 2(c) and (d). The limited lengths of horizontal lines correspond to the temperature interval used for estimating U₀(μ₀H) in the Arrhenius relation. It can be seen that −∂ ln ρ(T,μ₀H)/∂T⁻¹ increases sharply with decreasing temperature, which was also observed in Bi-2212 thin films [55]. The center of each U₀(μ₀H) horizontal line approximately intersects the −∂ ln ρ(T,μ₀H)/∂T⁻¹ curve, and therefore each U₀(μ₀H) is only the average value of its −∂ ln ρ(T,μ₀H)/∂T⁻¹ in the fitting temperature region. Hence, the U(T,μ₀H) determined using the Arrhenius relation does not reflect the true evolution of thermal activation energy with temperature in S-09, particularly for H∥ab.

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If ρ_0f = 2ρ_cU(T,μ₀H)/T is temperature dependent due to the term U(T,μ₀H) = U₀(μ₀H)(1 − t)^q and q can be different from 1, then it can be shown from equation (4) [56] that

$$\begin{eqnarray} \ln\rho &=&\ln(2\rho _{{\rm c}}U_{0}(\mu _{0}H))+q\ln (1-t)\nonumber\\ &&-\ln T-U_{0}(\mu _{0}H)(1-t)^{q}/T \end{eqnarray} \tag{ 6 }$$

and thus

$$\begin{equation} -\partial \ln\rho /\partial T^{-1}=[U_{0}(\mu _{0}H)(1-t)^{q}-T][1+qt/(1-t)], \end{equation} \tag{ 7 }$$

where q, ρ_c and U₀(μ₀H) are temperature-independent free parameters, and T_c derived from the Arrhenius relation is used for fitting. Figures 2(a) and (b) reveal that all fitting curves (dashed lines) agree well with experimental data and the results are better than the Arrhenius relation, especially for H∥ab. Figures 2(c) and (d) clearly show the advantage of equation (7) over the Arrhenius relation. It can effectively capture the upturn trend of −∂ ln ρ/∂T⁻¹ with decreasing temperature. As shown in figure 3(a), the U₀(μ₀H) values obtained using equation (6) are comparable to that in FeSe but smaller than in Fe(Te,Se) [48, 49, 52]. Similar to FeSe and Fe(Te,Se), there are also two ranges in the field dependence of U₀(H) curves with different α for both field directions in S-09. For H∥ab, α = 0.12(2) for μ₀H < 5 T and α = 1.7(3) for μ₀H > 5 T; For H∥c, α = 0.21(3) for μ₀H < 5 T and α = 1.3(2) for μ₀H > 5 T. In the low-field region, the small α suggests that single-vortex pinning dominates, similar to FeSe and Fe(Te,Se). At high fields α is larger than 1, implying that the material enters a collective pinning regime and the pinning centers are point defects [54]. On the other hand, the values of q change from about 1 for H∥c to 2 for H∥ab, independent on the field intensity for both directions (figure 3(b)). The value of q = 2 has also been observed in many cuprates superconductors [55–57]. The nonlinear temperature dependence of U₀(T,μ₀H) and non-constant prefactor ρ_0f are also common to other iron-based superconductors [58, 59].

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It should be noted that the low volume fraction of superconductivity in Fe(Te,S) should not significantly affect the analysis of TAFF. Because of the inhomogeneity of crystals, the conductivity can be expressed as σ = σ_sc + σ_normal, where σ_sc is the conductivity of superconducting fraction and σ_normal is the conductivity of the normal fraction of the crystal. When T < T_c,zero, σ = σ_sc = ∞, i.e. ρ = 0 and the normal-state part of sample is short-circuited. On the other hand, the resistivity in the TAFF regime is about one to three orders of magnitude less than in the normal state [45, 46]. It means that although the conductivity σ_sc in this range is finite, it is still much larger than σ_normal, and we can approximate σ ≈ σ_sc.

In K_0.64Fe_1.44Se₂ and Tl_0.58Rb_0.42Fe_1.72Se₂ single crystals, as shown in figure 4, the good linear behavior of ln ρ(T,μ₀H) versus 1/T in a wide temperature range for both field directions indicates that the temperature dependence of U(T,μ₀H) is approximately linear [60], i.e. q = 1, and the ρ_0f is constant, similar to FeSe and Fe(Te,Se) [48, 49]. For K_0.64 Fe_1.44Se₂, fitting by the function ln ρ₀(μ₀H) = ln ρ_0f + U₀(μ₀H)/T_c yields ρ_0f = 5.3(1) and 12.4(1) Ω cm and T_c = 31.8(3) and 32.3(2) K for H∥ab and H∥c, respectively (inset of figure 4(a)). On the other hand, the log ρ(T,μ₀H) lines for different fields extrapolate to one temperature T_cross ∼ 32 K for both field directions, consistent with the values of T_c within the error bars. The U₀(μ₀H) values obtained for both superconductors are much larger than in FeCh-11 materials; they are comparable to that in polycrystalline NdFeAsO_0.7F_0.3 [51], but are still much smaller than in polycrystalline SmFeAsO_0.9F_0.1 [58]. The power-law fitting yields α = 0.78(2) and 0.84(2) for H∥ab and H∥c in K_0.64Fe_1.44Se₂ and α = 0.27(2) for H∥ab and 0.22(2) for H∥c in Tl_0.58Rb_0.42Fe_1.72Se₂ (insets of figures 4(b)–(d)). For both materials, the field dependencies of U₀(μ₀H) are similar for both field directions, indicating that the pinning forces weakly depend on orientation [60]. But the values of α are rather different. For K_0.64Fe_1.44Se₂, the values of α are similar to those in FeSe, suggesting that both point and planar defects act as the pinning centers. On the other hand, the much smaller values of α in Tl_0.58Rb_0.42Fe_1.72Se₂ imply that the single-vortex pinning mechanism is dominant in fields up to 15 T.

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Figure 6 shows the J_c of FeSe, FeTe_0.61Se_0.39 and FeTe_0.9S_0.1 single crystals at various fields and temperatures [48, 63, 64]. At T ∼ 5 K, the J_c of FeSe is about 2.4 × 10³ A cm⁻² for H⊥(101) at zero field. This is lower than J_c in FeTe_0.61Se_0.39 and FeTe_0.9S_0.1, which is about 2 × 10⁵ and 3 × 10⁵ A cm⁻² for H∥c, respectively. FeTe_0.61Se_0.39 shows relative high J_c and it remains above 7 × 10⁴ A cm⁻² at μ₀H = 5 T. Moreover, the T_c of optimally doped Fe(Te,Se) (Se ∼ 0.4 − 0.5) is about 14 K, which is higher than in Fe(Te,S) and FeSe. It implies that the optimally doped Fe(Te,Se) might be suitable for applications. On the other hand, the anisotropy γ_Jc of J_c (γ_Jc = J^H∥(101)_c/J^H⊥(101)_c or J^H∥ab_c/J^H∥c_c) is about 4–5 for FeSe and FeTe_0.9S_0.1 [48, 64].

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When compared to FeCh-11 superconductors, the much higher T_c and μ₀H_c2 of AFeSe-122 superconductors immediately raise a question whether the J_c values of these materials are more promising for practical applications. As shown in figure 7(a), however, the early studies indicate that J_c of K_xFe_2−ySe₂ is rather low (∼(0.7–2.6) ×10³ A cm⁻² at 5 K) [65, 66], even lower than that of FeSe [48]. Therefore, it is important to explore pathways for the enhancement of J_c in AFeCh-122 compounds.

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It was soon found that sulfur doping can remarkably enhance the J_c of K_xFe_2−ySe_2−zS_z [67]. As shown in figure 7(a), the J^ab_c(μ₀H) (H∥c) exhibits a small increase in the high-field region for z = 0.32 when compared to the z = 0 sample. On the other hand, the enhancement is about one order of magnitude for z = 0.99 (J^ab_c(0) = 7.4 × 10³ A cm⁻² at 1.8 K) in the entire magnetic-field range. For higher S content, the J^ab_c(0) is still much larger than in z = 0 sample, but the J^ab_c(μ₀H) at high fields is smaller. It should be noted that the T_c decreases significantly when z > 0.32. When compared to z = 0 (T_c, onset = 33.0 K), the z = 0.99 crystal has T_c,onset = 24.6 K, whereas z = 1.04 corresponds to T_c,onset = 18.2 K [68]. Therefore, the sample with z = 0.99 exhibits the best performance among sulfur-doped samples investigated. Moreover, the ratio of J^c_c(μ₀H)/J^ab_c(μ₀H) is approximately 1 and is smaller than in FeCh-11 superconductors. Although the J_c of z = 0 sample is much larger than in K_xFe_2−ySe₂ and also larger than in FeSe, it is still smaller than in FeTe_0.61Se_0.39 and FeTe_0.9S_0.1.

Metallic and superconducting state was induced by post-annealing and quenching treatment in as-grown and insulating K_xFe_2−ySe₂ crystals, which is a notable improvement in superconducting properties [69]. Therefore, some effects of this process on the J_c are also expected [70]. Post-annealing and quenching process can further increase J^ab_c(0) to 7.4 × 10³ A cm⁻² at 1.8 K which is about 50 times larger than that of the as-grown sample (figure 4(a)). This cannot be simply ascribed to the improvement of the superconducting volume fraction, because the volume fraction of the quenched crystal is only about four times larger than that of the as-grown crystal. It could also be related to the enhanced flux pinning force due to the post-annealing and quenching treatment. The J_c in the quenched crystal is also slightly higher than that in K_xFe_2−ySe₂ crystals grown using the one-step technique [71] and are the highest known J_c values among AFeCh-122 type materials until now. The quenched sample also exhibits a good performance at high fields. The J^ab_c(μ₀H = 4.8 T) for the quenched sample is still larger than 10⁴ A cm⁻² at 1.8 K, whereas for the as-grown sample, it has already decreased about one order of magnitude.

As shown in figure 7(b), detailed studies on the temperature and field dependences of the vortex pinning force Fp = μ0HJc show that it follows the Dew-Huges [72] scaling law: fp∝hp(1 − h)q, where fp = Fp/Fmaxp is the normalized vortex pinning force, Fmaxp corresponds to the maximum pinning force and h = H/Hirr is the reduced field. The results indicate that there is a dominant pinning mechanism. The obtained p and q is 0.86(1) and 1.83(2), respectively and the value of hmaxfit = p/(p + q) ≈ 0.32 is consistent with the peak positions (hmaxexp ≈ 0.33) of the experimental curves at different temperatures. Those values are close to the expected values for core normal point-like pinning (p = 1, q = 2 and hmaxfit = 0.33) [69]. These point-like pinning centers might come from the random distribution of iron vacancies after quenching, similar to FeAs-122 type materials [73, 74]. On the other hand, the Fmaxp can be fitted using Fmaxp∝(μ0Hirr)α with α = 1.67(1) (inset of figure 4(b)), close to the theoretical value (α = 2) for the core normal point-like pinning [72]. Moreover, as shown in figure 7(c), the temperature dependence of μ0Hirr can be fitted with equation μ0Hirr(T) = μ0Hirr(0)(1 − t)β, and we obtained β = 1.21(1), close to the characteristic value of 3D giant flux creep (β = 1.5) [53]. For type-II superconductors, vortices interact with pinning centers either via the spatial variations in the Tc ('δTc pinning') or by scattering of charge carriers with reduced mean free path l near defects ('δl pinning') [44]. These two pinning types result in the different temperature dependences of Jc(t) in the single vortex pinning regime i.e. low-field and zero-field regions. For δTc pinning, JδTcc,H=0(t) = Jc,H=0(0)(1 − t2)7/6(1 + t2)5/6 while for δl pinning, Jδlc,H=0(t) = Jc,H=0(0)(1 − t2)5/2(1 + t2)−1/2 [77]. As shown in figure 3(c), the Jc,H=0(t) is between the two curves corresponding to δTc and δl pinnings, respectively, but much closer and similar in shape to the δTc-pinning curve. Using Jc,H=0(t) = xJδTcc,H=0(t) + (1 − x)Jδlc,H=0(t), the experimental data can be fitted very well with x = 0.74(2), suggesting that both δTc and δl pinnings play roles in the quenched KxFe2−ySe2 single crystals, but the former mechanism is dominant. The dominant effect of δTc pinning also implies that the main pinning centers lead to the distribution of Tc in their vicinity or might even be non-superconducting, like Y2O3 and Y–Cu–O precipitates in YBa2 Cu3O7−x thin films [78]. Although the Jabc of quenched KxFe2−ySe2 single crystals is enhanced significantly when compared to the as-grown samples, it is still smaller than in FeTe0.61Se0.39 and other iron pnictide superconductors [73, 74]. There might be more room to improve the current-carrying capacity for AFeCh-122 superconductors in the future using methods such as A site ionic disorder and nanometer-scale second phases.

It should be noted that if the phase separation scale is microns or even larger, the insulating region could have some effect on the calculation of the absolute values of J_c. This is because the volume of insulating phase has been used to calculate the J_c from M − H curve, i.e., the real J_c values might be even larger than the calculated ones. However, the phase separation should have minor effects on the field and temperature dependences of J_c, i.e. the pinning mechanism should not be affected by this type of phase separation. If the phase separation is in the nanometer range, as it has been observed in K_xFe_2−ySe₂ by transmission electron microscopy [75], the effects on J_c are similar to what has been discussed above regarding the pinning mechanism. The insulating second phase can be considered as the pinning center, similar to nanometer-scale BaZrO₃ inclusions in YBCO coatings on metals [76].

For practical applications, especially for fabrication of superconducting wires and tapes, superconductors are usually made in the polycrystalline bulk or thin-film form. Therefore, besides enhancement of J_c by various artificial pinning centers in the grains, i.e. enhancement of intragrain properties, the GBs properties are also of great interest for the applications of iron-based superconductors.

There are two studies so far on the Ba(Fe,Co)₂As₂ epitaxial thin films deposited on bicrystal substrates (figure 8) [79, 80]. Both show that the J_c across bicrystal grain boundary (BGB) has an exponential dependence on GB misorientation angle θ, indicating a weak-link effect for high-angle BGBs. However, the critical GB misorientation angles θ_c are different: about 6° and about 10° [79, 80]. According to the study of Katase et al, the slopes of the exponential decay are much smaller than those for YBCO BGBs, and the critical angle is approximately 9–10°, which is substantially larger than that of YBCO (∼3–5°, inset of figure 8(a)) [80, 81]. Larger critical angle suggests smaller constraints for the in-plane orientation for substrate or buffer layers to obtain high-J_c iron-based superconducting films on flexible metal substrates. This observation will certainly be advantageous for tape applications. Similar study is lacking in iron selenide superconductors, and therefore related studies on FeCh-11 or AFeCh-122 films will be of interest to understand the GB properties.

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After the discovery of iron-based superconductors, the trials for fabrication of iron-based superconducting wires were immediately carried out [82–84]. There are two methods to prepare iron-based superconducting wires and tapes, power-in-tube (PIT) and coated conductor techniques.

PIT technique has been used widely in Bi-based cuprate oxide wires (first-generation high-temperature superconductor (HTS) wires) and MgB₂ [85, 86]. Ozaki et al [87] fabricated Fe(Se,Te) wires by an ex situ PIT method with the transport J_c(0) of about 64 A cm⁻² at 4.2 K. Recently, chemical-transformation PIT method was used to fabricate FeSe wires with a transport J_c(0) of 588 A cm² at 4.2 K for a three-core wire (figure 9(a)) [88]. Although the J_c is enhanced considerably, it is much smaller than that of iron arsenic superconducting wires prepared using PIT method. For example in (Ba,K)Fe₂As₂ the J_c(0) reaches 1 × 10⁴ A cm⁻² at 4.2 K [89]. But it should be noted that the J_c for both iron chalcogenide and iron arsenic superconducting wires fabricated using PIT method are much smaller than those of single crystals [48, 63, 74], implying that the texturing of grains in wires is important due to the GB weak-link effect.

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Coated conductor technique has been successfully developed for the YBCO coated conductors (second-generation HTS wires) to overcome the weak-link effect [61]. The FeSe_0.5Te_0.5 thin films deposited on Hastelloy substrates, on which an MgO buffer layer has been formed by ion beam assisted deposition (IBAD), exhibit a nearly isotropic J_c(0) exceeding 2 × 10⁵ A cm⁻² at 4.2 K and still higher than 1 × 10⁴ A cm⁻² at a magnetic field as high as 25 T (figure 9(b)) [90]. This gives the superior property in high fields (⩾20 T) when compared to Nb₃Sn or Ni–Ti wires. The fitting of field dependency of the F_p using Dew-Huges scaling law yields q ∼ 2 and p ∼ 1 (figure 9(c)), similar to K_xFe_2−y(Se,S)₂ and quenched K_xFe_2−ySe₂ [67, 70]. This suggests that the point-defect core pinning is dominant due to the inhomogeneous distribution of Se and Te [90]. In the core-pinning regime, F_p is a product of the individual F_p, ind and the pinning center density n. This implies that the J_c of FeSe_0.5Te_0.5 thin films can still be enhanced by simply adding more defects to act as pinning centers [90]. Moreover, the weak-link effect of FeSe_0.5Te_0.5 may not be as severe as that of YBCO. Although the IBAD substrates have many low-angle GBs in the textured MgO template, the J_c(0) of FeSe_0.5Te_0.5 thin films on IBAD-MgO with in-plane misorientation angle Δϕ = 4.5° is just a little lower than that of thin films deposited on LaAlO₃ single-crystal substrate [90]. These results should be confirmed with further studies of the thin films grown on controlled bicrystal substrates. On the other hand, depositing FeSe_0.5Te_0.5 thin films directly on textured metal tapes may be possible because FeSe_0.5Te_0.5 thin films are prepared in vacuum unlike YBCO-coated conductors. This considerably simplifies the synthesis procedure and reduces production costs [62]. A similar study on the Ba(Fe,Co)₂As₂ thin films deposited on IBAD-MgO-buffered Hastelloy substrates shows that the J_c(0) is (1.2–3.6) × 10⁶ A cm⁻² at 2 K with Δϕ ∼3°, regardless of the larger Δϕ (5.5–7.3°) of the MgO buffer layers [91]. It is much larger than that of Ba(Fe,Co)₂As₂ wire prepared by PIT method and comparable to that on MgO single crystals and remains at 1 × 10⁶ A cm⁻² at T ⩽ 10 K [89, 91]. This observation is consistent with the fact that the Δϕ of Ba(Fe,Co)₂As₂ is much smaller than the θ_c = 9° [91]. These results demonstrate that the coated conductors technique is a promising method to fabricate iron-based superconducting tapes for high field applications.

The s wave Cooper pair is formed by two electrons that attract together with opposite spins and momenta. External magnetic fields contributes to depairing via two primary mechanisms (figure 10(a)) [92]. One is the orbital pair breaking mechanism. The Lorentz force acts on paired electrons with opposite momenta and the superconductivity is destroyed when the kinetic energy exceeds the condensation energy of the Cooper pairs. Another mechanism is the Pauli paramagnetic pair breaking. When the Pauli spin susceptibility energy (Zeeman energy) exceeds the condensation energy as shown in figure 10(b), the singlet pair with opposite spins is broken into unbound triplet along the field direction (Zeeman effect, also called spin-paramagnetic effect).

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Through the Maki parameters α (see below) and λ_so [93], the effects of Pauli spin paramagnetism and spin-orbital scattering have been included in the Werthamer–Helfand–Hohenberg (WHH) theory for a single-band s wave weak-coupled type-II superconductor in the dirty limit [94]. The μ₀H_c2(T) is given by

$$\begin{eqnarray} \displaystyle\ln \frac{1}{t}&=&\left(\frac{1}{2}+\frac{\mathrm{i}\lambda_{\mathrm{so}}}{4\gamma}\right)\psi\left( \frac{1}{2}+\frac{h+\lambda _{\mathrm{so}}/2+\mathrm{i}\gamma}{2t}\right) \nonumber \\ & &\displaystyle+\left(\frac{1}{2}-\frac{i\lambda_{\mathrm{so}}}{4\gamma}\right)\psi \left(\frac{1}{2}+ \frac{h+\lambda_{{\rm so}}/2-{\rm i}\gamma }{2t}\right)-\psi\left(\frac{1}{2}\right),\nonumber\\ \end{eqnarray} \tag{ 8 }$$

where ψ(x) is digamma function, γ ≡ [(αh)² − (λ_so/2)²]^1/2 and

$$\begin{equation} h=\frac{4\mu _{0}H_{\mathrm{c}2}(T)}{\pi ^{2}T_{\mathrm{c}}(-\mathrm{d}\mu _{0}H_{\mathrm{c}2}(T)/\mathrm{d}T)_{T=T_{\mathrm{c}}}} \end{equation} \tag{ 9 }$$

If the orbital effect is dominant (α = 0) and the spin-orbital scattering is negligible (λ_so = 0), the equation can be simplified as

$$\begin{equation} \ln \frac{1}{t}=\psi \left(\frac{1}{2}+\frac{h}{2t}\right)-\psi \left(\frac{1}{2}\right), \end{equation} \tag{ 10 }$$

which describes the temperature dependence of orbital limited upper critical field μ₀H*_c2(T). At zero temperature it becomes

$$\begin{equation} \mu _{0}H_{\mathrm{c}2}^{\ast }(0)=-0.693T_{\mathrm{c}}(\mathrm{d}\mu _{0}H_{\mathrm{c}2}(T)/\mathrm{d}T)_{T=T_{\mathrm{c}}}. \end{equation} \tag{ 11 }$$

On the other hand, if we only consider the spin-paramagnetic effect alone, the zero-temperature Pauli limiting field μ₀H_p(0) (Chandrasekhar–Clogston limit field for a weakly coupled superconductor) is [95, 96]

$$\begin{equation} \mu _{0}H_{\mathrm{p}}(0)[T]=\Delta /\sqrt{2}\mu _{\mathrm{B}}=1.86T_{\mathrm{c}}, \end{equation} \tag{ 12 }$$

where Δ is the s wave superconducting gap. Including strong-coupling corrections in the BCS theory due to electron–phonon and electron–electron interactions gives [92, 97]

$$\begin{equation} \mu _{0}H_{\mathrm{p}}(0)[T]=1.86(1+\lambda _{\mathrm{ep}})^{\varepsilon }\eta _{\Delta }\eta _{\mathrm{ib}}(1-I). \end{equation} \tag{ 13 }$$

Here η_Δ describes the strong coupling intraband correction for the gap, I is the Stoner factor I = N(E_F)J, N(E_F) is the electronic density of states (DOS) per spin at the Fermi energy E_F, J is an effective exchange integral, η_ib is introduced to describe phenomenologically the effect of the gap anisotropy, λ_ep is electron–phonon coupling constant and ε = 0.5 or 1 [92, 97, 98].

When compared to the orbital pair breaking effect, the relative contribution of the Pauli paramagnetic pair breaking effect is quantified by the Maki parameter $\alpha =\sqrt {2}H_{{\rm c}2}^{\ast }(0)$/H_p(0). When α > 1, the spin-paramagnetic effect becomes essential [93] and the actual upper critical field μ₀H_c2(0) is given by

$$\begin{equation} \mu _{0}H_{\mathrm{c}2}(0)=\mu _{0}H_{\mathrm{c}2}^{\ast }(0)/\sqrt{1+\alpha ^{2}}. \end{equation} \tag{ 14 }$$

In the single-band clean limit, the Maki parameter α is given by [40]

$$\begin{equation} \alpha =\frac{\pi ^{2}\Delta m}{4E_{\mathrm{F}}m_{0}},H{\parallel} c, \end{equation} \tag{ 15 }$$

$$\begin{equation} \alpha =\frac{\pi ^{2}\Delta mv_{ab}}{4E_{\mathrm{F}}m_{0}v_{{\rm c}}},H{\parallel}ab, \end{equation} \tag{ 16 }$$

where v_ab and v_c are the Fermi velocities in the ab plane and along the c-axis, respectively, m is the electron effective mass and m₀ is free electron mass. For conventional single-band BCS superconductors, because of m ∼ m₀, v_ab ∼ v_c, and low T_c, the α is usually small, i.e. the orbital pair breaking mechanism is dominant. On the other hand, for some exotic superconductors, such as heavy fermion superconductors with m ≫ m₀ or organic superconductors with large ratio of v_ab/v_c, the condition of α > 1 can be easily satisfied and the spin-paramagnetic effect is dominant. When α is large enough and superconductors are in the clean limit, the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state may appear, resulting in a non-zero momentum of the Cooper pairs and the spatial oscillations of the superconducting order parameter [99, 100].

On the other hand, for the superconductors with several disconnected Fermi surfaces, the multiband effects on the μ₀H_c2 should be considered. The most famous example is MgB₂ where the μ₀H_c2 can be described successfully using the two-band BCS model in the dirty limit [101, 102].

In the dirty limit, taking into account both orbital and spin-paramagnetic effect with interband and intraband scattering contributions, the μ₀H_c2(T) is given by

$$\begin{eqnarray} & &(\lambda _{0}+\lambda _{i})(\ln t+U_{+})+(\lambda _{0}-\lambda _{i})(\ln t+U_{-}) \nonumber \\ & &+2w(\ln t+U_{+})(\ln t+U_{-})=0, \end{eqnarray} \tag{ 17 }$$

where t = T/T_c0, and

$$\begin{equation} \lambda _{0}=(\lambda _{-}^{2}\pm 4\lambda _{12}\lambda _{21})^{1/2}, \end{equation} \tag{ 18 }$$

$$\begin{equation} \lambda _{i}=[(\omega _{-}+\gamma _{-})\lambda _{-}-2\lambda _{12}\gamma _{21}{-}2\lambda _{21}\gamma _{12}]/\Omega _{0}, \end{equation} \tag{ 19 }$$

$$\begin{equation} \lambda _{\pm }=\lambda _{11}\pm \lambda _{22}, \end{equation} \tag{ 20 }$$

$$\begin{equation} \omega _{\pm }=(D_{1}\pm D_{2})\pi \mu _{0}H_{{\rm c}2}/\Phi _{0}, \end{equation} \tag{ 21 }$$

$$\begin{equation} \gamma _{\pm }=\gamma _{12}\pm \gamma _{21}, \end{equation} \tag{ 22 }$$

$$\begin{equation} \Omega _{0}=[(\omega _{-}+\gamma _{-})^{2}+4\gamma _{12}\gamma _{21}]^{1/2}, \end{equation} \tag{ 23 }$$

$$\begin{equation} w=\lambda _{11}\lambda _{22}-\lambda _{12}\lambda _{21}, \end{equation} \tag{ 24 }$$

$$\begin{equation} U_{\pm }=\mathrm{Re}\left[\psi \left(\frac{1}{2}+\frac{\Omega _{\pm }+i\mu _{{\rm B}}H_{{\rm c}2}}{2\pi T} \right)-\psi \left(\frac{1}{2}\right)\right], \end{equation} \tag{ 25 }$$

$$\begin{equation} 2\Omega _{\pm }=\omega _{+}+\gamma _{+}\pm \Omega _{0}. \end{equation} \tag{ 26 }$$

D₁ and D₂ are intraband diffusivities of the bands 1 and 2; λ₁₁ and λ₂₂ are the intraband couplings in the bands 1 and 2, and λ₁₂ and λ₂₁ describe the interband couplings between bands 1 and 2; γ₁₂ and γ₂₁ are interband scatting rates between band 1 and 2. If neglecting interband scattering (γ₁₂ = γ₂₁ = 0), we obtain

$$\begin{equation} \begin{array}{c} (\lambda _{0}+\lambda _{-})(\ln t+U[(k+\mathrm{i})h])+(\lambda _{0}-\lambda _{-})[\ln t+U[(\eta k+\mathrm{i})h])\!\!\!\!\\[4pt] +2w(\ln t+U[(k+\mathrm{i})h])(\ln t+U[(\eta k+\mathrm{i})h])=0, \end{array} \end{equation} \tag{ 27 }$$

where $U[x]=\mathrm {Re}\left[\psi \left(\frac {1}{2}+x\right)-\psi \left(\frac {1}{2}\right)\right]$, h = μ₀H_c2D₀/(2Φ₀T), D₀ = μ_BΦ₀/π = ℏ²/2m is quantum diffusivity, k = D₁/D₀, η = D₂/D₁. When the spin-paramagnetic effect is negligible (D₀ ≪ D_1,2), H_c2 can be simplified to

$$\begin{equation} \begin{array}{rcl} (\lambda _{0}+\lambda _{-})(\ln t+U[kh])+(\lambda _{0}-\lambda _{-})(\ln t+U[\eta kh]) & & \\ +2w(\ln t+U[kh])(\ln t+U[\eta kh])=0. & & \end{array} \end{equation} \tag{ 28 }$$

Moreover, for η = 1, equation (28) can be further reduced to equation (10) [93, 94].

All parameters of equations (15) and (16) that might lead to a large α exist in iron-based superconductors [40], therefore the spin-paramagnetic effect needs to be considered. Furthermore, there are several disconnected electron and hole sheets at the FS that originate from the hybridized d-orbitals of iron [29]. This suggests that the multiband effect on the μ₀H_c2(T) should also be taken into account. For iron arsenic superconductors, due to the different contributions of multiband and spin-paramagnetic effects, there are various temperature dependences of μ₀H_c2(T) for different systems (see the review paper about this topic [103] and references therein). Figure 11 shows several typical μ₀H_c2(T) dependences in iron arsenic superconductors. FePn-1111 system (LnFeAs(O,F), figure 11(a)) shows a pronounced upturn curvature in μ₀H_c2,c(T) at low temperatures. In contrast, μ₀H_c2,ab(T) exhibits a downturn curvature with decreasing temperature [51, 104]. Both can be explained within a two-band theory with a high diffusivity ratio of electron band to hole band, but the spin-paramagnetic effect also needs to be considered, especially for H∥ab [51, 104, 105]. For the FeAs-122 system (A(Fe,Co)₂As₂, figure 11(b)), the upturn curvature of μ₀H_c2,c(T) present in FeAs-1111 system does not appear, but it still shows a positive curvature at temperatures far below T_c without saturation. On the other hand μ₀H_c2,ab(T) tends to saturate with decreasing temperature [106, 107]. This can also be interpreted using a two-band theory with a larger diffusivity ratio of two bands when compared to FePn-1111 system [106, 107]. Similar to FePn-1111 system, the spin-paramagnetic effect may also have some effect on the μ₀H_c2,ab(T) [107]. For KFe₂As₂ and LiFeAs (figures 11(c) and (d)), both μ₀H_c2,ab(T) and μ₀H_c2,c(T) show saturation trends at low temperatures with different negative curvatures. The former can be ascribed to the spin-paramagnetic effect and the latter is mainly determined by the orbital-limited field in the one-band scenario [108, 109].

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Both multiband and spin-paramagnetic effects contribute to μ₀H_c2(T) in iron chalcogenide superconductors. However, the multiband effect is much weaker when compared to iron arsenides, and the spin-paramagnetic effect is usually the dominant factor.

In Fe(Te,Se) (as shown on examples of Fe_1.02Te_0.61Se_0.39 (Se-39) and Fe_1.05Te_0.89Se_0.11 (Se-11) in figures 12(a) and (b)) both μ₀H_c2(T) for H∥ab and H∥c can be explained well using the WHH model with spin-paramagnetic effect when neglecting spin-orbital scattering. It indicates that the spin-paramagnetic effect is the dominant pair-breaking mechanism for both H∥ab and H∥c [110, 111, 114]. For the Fe(Te,Se) in the clean limit (grown by Bridgman–Stockbarger technique) the μ₀H_c2(T) is Pauli limited. This may suggest the emergence of the FFLO state at low temperatures [114]. The dominance of spin-paramagnetic effect in Fe(Te,Se) may be due to the disorder induced by Te(Se) substitution/vacancies and excess Fe in Fe(2) site [92, 115]. For Se-39, more Se doping introduces more disorder than in Se-11. This effect could contribute to the larger α_H∥ab of Se-39 when compared to Se-11. However, it cannot explain the inverse trend of α_H∥c. Therefore, another effect must compete with disorder. It may be the effect of excess Fe in Fe(2) position, which is the unique feature of 11-system, different from other Fe pnictide superconductors. The Fe(2) has larger local magnetic moment than Fe(1) in Fe-(Te,Se) layers. The Fe(2) moment is present even if the SDW antiferromagnetic ordering of the Fe plane is suppressed by doping or pressure, contributing to N(E_F) [25]. According to equation (13), μ₀H_p(0) can be decreased if the Stoner factor increases via enhancement of J or N(E_F). Excess Fe in Fe(2) site with local magnetic moment could interact with itinerant electron in Fe layer, resulting in exchange-enhanced Pauli paramagnetism or Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, thus enhancing J. Hence, higher content of excess Fe in Se-11, could lead to larger α_H∥c than in Se-39. Another possibility may be that the N(E_F) is decreased with increasing the content of Se [116]. This trend will also enhance the Pauli-limited field, i.e. suppress the spin-paramagnetic effect.

As shown in figure 12(c), spin-paramagnetic effect is also dominant in μ₀H_c2(T) of S-09 with larger α and obvious spin-orbital scattering [117], when compared to Se-39 and Se-11. The spin-paramagnetic effect should have the same origin as in Se-39 and Se-11 due to disorder. Moreover, since the N(E_F) of FeS is larger than that of FeSe, [116], it is likely that the N(E_F) of S-09 is larger than that of Fe(Te,Se) with the same doping content. This will lead to the smaller μ₀H_p(0), i.e. larger α. On the other hand, non-zero λ_so can also be explained via increasing Kondo-like scattering from excess Fe in Fe(2) site, consistent with the definition of λ_so, which is proportional to the spin-flip scattering rate [93, 94]. It is also consistent with the Kondo behavior of S-09 in the normal state, related to the excess Fe in Fe(2) that act as the Kondo-like impurities [117]. The presence of Kondo-type interactions in binary iron chalcogenides, first inferred from the normal-state scattering and μ₀H_c2(T) behavior, has later been confirmed by the neutron scattering measurements [118].

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Furthermore, the evaluated mean free path of S-09, l = 1.35 nm using the Drude model l = ℏ(3π²)^1/3/(e²ρ₀n^2/3), suggests that S-09 is a dirty-limit superconductor since l/ξ(0) = 0.396 [117]. Therefore, the FFLO state at high fields is unlikely in S-09 because the short mean free path will remove any momentum anisotropy [99, 100].

The μ₀H_c2,H∥(101)(T) of FeSe single crystal at low temperatures is larger than the value evaluated from the WHH theory with α = 0 and λ_so = 0 (figure 12(d)) [119], which is distinctively different from Fe(Te,Se) and Fe(Te,S) where the WHH curve is far above the experimental data. The enhancement of the μ₀H_c2(T) at low temperatures and high fields implies that the multiband effect is not negligible. By using the coupling constants, determined from muon spin resonance (μSR) experiment with very small interband coupling [120], the μ₀H_c2,H∥(101)(T) data from both rf and resistivity measurements can be very well explained using a two-band model with η = 0.40. It is similar to the value of FeAs-122 but much larger than that of other two-band iron-based superconductors, such as FeAs-1111 [51, 104]. Large η leads to the absence of the upturn of μ₀H_c2(T) at low temperatures, which is observed in FeAs-1111 superconductors [51, 104], but not in FeAs-122 compounds [106]. We have also performed simulations for different values of coupling constants: (i) dominant intraband coupling, w > 0 and (ii) dominant interband coupling, w < 0. The different sets of fitting parameters results in almost identical result, fitting the experimental data well (figure 12(d)). The derived η is in the range of 0.32–0.44, suggesting that the fitting results are insensitive to the choice of coupling constants. Thus, either interband and intraband coupling strength are comparable or their difference is below the resolution of our experiment. On the other hand, assuming μ₀H_c2(T = 0.35 K) ≈ μ₀H_c2(0), the μ₀H_c2(0) are 17.4(2) and 19.7(4) T for H∥(101) and H⊥(101), respectively. Given the electron–phonon coupling parameter λ_e−ph = 0.5 (typical value for weak-coupling BCS superconductors) [121], the Pauli limiting field μ₀H_p(0) = 1.86T_c(1 + λ_e−ph)^1/2 is 19.8 T [97]. This is nearly the same as the μ₀H_c2,H⊥(101)(T = 0.35 K) and larger than value for H∥(101). It suggests that the spin-paramagnetic effect might also have some influence on the μ₀H_c2(T), but should not be the dominant effect. The situation is rather different in Fe(Te,Se) and Fe(Te,S) where the spin-paramagnetic effect governs μ₀H_c2(T) [110, 117]. The existence of two bands is also confirmed from the Hall measurement [119].

The anisotropy of μ₀H_c2(T), γ(T) (=H_c2,ab(T)/H_c2,c(T)) (insets of figures 12(a)–(c)) decreases with decreasing temperature, reaching ∼1 for all Fe(Te,Se) and Fe(Te,S) compounds. Similar results have been reported in the literature [66, 122]. These results show that Fe(Te,Se) and Fe(Te,S) are high-field isotropic superconductors. When T is close to T_c, the γ of Se-11 is smaller than that of Se-39 but larger than in Fe(Te,S). Although the behavior of γ(T) is similar in Fe(Te,Se)/Fe(Te,S) and FeAs-based superconductors, the value is smaller in the former compounds [103]. For FeSe, the γ(T) (=H_c2,H⊥(101)(T)/H_c2,H∥(101)(T)) is nearly constant and close to 1, but it is difficult to compare this with other FeCh-11 superconductors due to the differences of the orientation.

In AFeCh-122 superconductors, the slopes of μ₀H_c2(T) near T_c0, dμ₀H_c2(T)/dT|_T=Tc(∼6–10 T K⁻¹) are similar to those in FeCh-11 materials for H∥ab but are much smaller (∼1–3 T K⁻¹) than in the latter for H∥c (figure 13) [60, 110, 117, 123–125]. The larger T_c compared to FeCh-11 superconductors leads to the significantly larger H_c2(0) evaluated from equation (11) for H∥ab. The μ₀H_c2(0) values are usually ∼125–275 T and ∼30–60 T for H∥ab and H∥c, respectively [60, 123, 125].

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Recent studies on the μ₀H_c2(T) of K_0.8Fe_1.76Se₂ and Tl_0.58Rb_0.42Fe_1.72Se₂ single crystals up to 60 T show that μ₀H_c2(T) for H∥c presents an almost linear temperature dependence and slight upturn at low temperatures, whereas the curve of μ₀H_c2(T) for H∥ab has a convex curvature with a much larger value and gradually tends to saturate at low temperatures (figure 14) [60, 126]. The μ₀H_c2(T) for H∥c is slightly larger than the value predicted with equation (11) at low temperatures, but is smaller than the theoretical one for H∥ab. For H∥ab, fitting with equation (8) yields α = 5.6 and λ_so = 0.3, indicating that the spin-paramagnetic effect may play an important role in suppressing superconductivity for H∥ab [60]. On the other hand, the enhancement of μ₀H_c2(T) for H∥c at low temperatures may be related to the multiband effect [60]. These behaviors are very similar to those of FeAs-1111 and FeAs-122, but not FeCh-11 compounds. In K_0.8Fe_1.76Se₂, the γ(T) increases with temperature and finally decreases gradually with decreasing temperature. But in Tl_0.58Rb_0.42Fe_1.72Se₂, the γ(T) decreases with temperature in the entire measurement range with larger values when compared to K_0.8Fe_1.76Se₂. Similar to other iron-based superconductors, the decrease of γ(T) with temperature may be related to the multiband effect or to the gradual setting in of pair breaking by the spin-paramagnetic effect, which requires μ₀H_c2,ab = μ₀H_c2,c in the low-temperature and high-field limit [103]. On the other hand, the γ(T) values in K_0.8Fe_1.76Se₂ and Tl_0.58Rb_0.42Fe_1.72Se₂ are much larger than those in FeCh-11 materials. This difference could be related to the decrease of dimensionality when compared to FeCh-11 superconductors.

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For both FeCh-11 and AFeCh-122 superconductors, doping on Fe site is usually detrimental for superconductivity [127, 128]. On the contrary, in FeAs-based superconductors doping on Fe site is beneficial as it induces superconductivity in compounds such as Ba(Fe,Co)₂As₂ [8]. In FeCh-11 materials, doping on Ch site enhances superconductivity [16, 17], whereas it has an opposite trend in AFeCh-122 where S and Te doping decreases T_c [68, 129]. However, the suppression of superconductivity is far less strong when compared to doping on Fe site.

As shown in figure 15(a), the lattice parameters decrease with S doping due to the smaller ionic size of S²⁻ than Se²⁻. The trend of lattice contraction approximately follows the Vegard's law. In addition, doping S into Se site does not lead to random Fe occupation of Fe1 and Fe2 sites. The refinements within single-phase I4/m crystallographic space group show that the Fe1 site has a low occupancy, whereas the Fe2 site is almost fully occupied. Besides K_xFe_2−yS₂, all other crystals show metallic behavior with resistivities below $\rho _{\max }$ at temperatures above T_c (figure 15(b)). It should be noted that the temperature of $\rho _{\max }$ varies non-monotonically with the doping level of S(z), implying that the temperature of crossover may be influenced by both K and Fe deficiencies. Shoemaker et al [130] found that the temperature dependence of resistivity in the normal state can be described well by a two-phase model containing metallic and insulating phases. This result suggests that the metal-semiconductor crossover can be also partially related to the two-phase coexistence in K_xFe_2−ySe₂, i.e. the phase separation in the samples, which has been observed by various measurement technique [32–38]. On the other hand, the crystal with z = 2 is semiconducting even if the Fe deficiency is smaller than in the other samples. Whether there is also phase separation in K_xFe_2−yS₂ needs to be investigated in the future.

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The changes in the ground state phase diagram of K_xFe_2−ySe_2−zS_z are due to the sulfur doping and should not be influenced by the microscopic phase separation. If K_xFe_2−ySe₂ shows superconducting behavior, the T_c is always above 30 K and is never below 30 K [75]. It means that the T_c is not sensitive to the compositional fluctuation of K or Fe once superconductivity appears. So, the gradual decrease of T_c in K_xFe_2−ySe_2−zS_z is difficult to ascribe to compositional fluctuations due to the phase separation. In contrast to $\rho _{\max }$, the T_c is monotonically suppressed to lower temperature with the increase in S and cannot be observed above 2 K for z ⩾1.58. The superconductivity of K_xFe_2−ySe_2−zS_z single crystals with z < 1.58 is confirmed by the magnetization measurement (figure 15(c)). The Curie–Weiss temperature dependence is absent for all samples above 50 K as shown in figure 15(c). Magnetic susceptibilities are weakly temperature dependent with no significant anomalies above 50 K. This suggests the presence of low-dimensional short-range magnetic correlations and/or a long-range AFM order above 300 K. Interestingly, non-superconducting samples show bifurcation between the ZFC and FC curves for both field directions, suggesting spin glass (SG) transition at low temperatures in the 1.58 ⩽ z ⩽ 2 range. The magnetic and superconducting phase diagram of K_xFe_2−ySe_2−zS_z is shown in figure 15(d). Semiconductor-metal crossover can be traced for 0 ⩽z ⩽ 1.58 at high temperatures. In this z region, K_xFe_2−ySe_2−zS_z is a superconducting metal at low temperatures. For z = 1.58, ρ(T) is metallic with no superconducting T_c down to 2 K. For 1.58 ⩽ z ⩽ 2, we observe a drop in χ − T curves that could be related to a SG transition. For z = 2, the K_xFe_2−ySe_2−zS_z becomes a small-gap semiconductor with no metallic crossover and with a SG transition below 32 K.

The gradual changes of T_c are difficult to explain only by the slight variation of Fe and K contents, because usually the superconductivity appears with higher Fe content when K content (x) < 0.85 [131, 75]. As opposed to this trend, the K_xFe_2−ySe_2−zS_z crystals with larger z values have higher Fe contents but lower T_c, indicating that T_c is not only governed by K/Fe stoichiometry or vacancies. The local environment of Fe profoundly changes with S doping, inducing the changes in band structure and physical properties. In the K_xFe_2−ySe_2−zS_z crystal structure refined in I4/m space group, Fe atoms have block-like distribution where four Fe2 sites form a square around Se atom, making a cluster distinct from Fe1 site with low occupancy (inset of figure 15(a)) [31]. Therefore there are Fe1–Fe2 distances as well as intra- and inter-cluster Fe2–Fe2 distances. All cluster distances are unchanged with S doping whereas the Fe1–Fe2 distances decrease significantly (figure 16(a)). A similar magnetization behavior above 50 K and rather different superconducting T_c values as S content varies from 0 to 2 coincide with the nearly unchanged Fe2–Fe2 bond lengths. This result shows that superconductivity is insensitive to the size of Fe2–Fe2 clusters whereas the unchanged high-temperature magnetism could be related to the unchanged Fe2–Fe2 bond lengths. On the other hand, the SG behavior arising at low temperatures for non-superconducting samples can be explained by the non-zero random occupancy of Fe1 (vacancy) site for a higher S content (figure 16(b)), randomly changing the inter-cluster exchange interactions [132].

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The correlation of T_c with the anion height between Fe and Pn (Ch) layers was empirically observed. There is an optimal distance (∼1.38 Å) with a maximum T_c ∼ of 55 K [133]. This seems to be invalid in K_xFe_2−ySe_2−zS_z materials, where there are two Fe and two Ch sites and four Fe–Ch heights. There is no monotonic decrease as T_c is tuned to 0, whereas both Se and S end members have rather similar anion heights (figure 16(c)). The bond angle α between Pn(Ch)–Fe–Pn(Ch) is more instructive since T_c in iron pnictides is optimized when Fe–Pn (Ch) tetrahedron is regular (α = 109.47°) [134]. In K_xFe_2−ySe_2−zS_z, the Ch2–Fe1–Ch2 angle changes toward the optimal value with increasing S content (figure 16(d)), but the local environment of Fe2 site exhibits inverse trend. Among six angles in the Fe2–Ch1(2) tetrahedron, three (Ch1–Fe2–Ch2) are nearly unchanged with S doping (figure 16(e)). The other three (Ch1–Fe2–Ch1) change significantly (maximum 6°) and deviate from optimal value from Se-rich to S-rich side (figure 16(f)). Hence, the increasing distortion of Fe2–Ch tetrahedron with S doping is closely correlated to the suppression of T_c. This distortion may lead to carrier localization, decreasing the density of states at the Fermi energy, which is confirmed by the measurement of thermal properties as discussed below. Therefore, the regularity of Fe2–Ch1(2) tetrahedron is an important structural factor governing the formation of the metallic states in K_xFe_2−ySe_2−zS_z, and consequently the T_c.

Thermal properties indicate that the density of states at the Fermi energy decreases with S doping, resulting in the suppression of correlation strength [135, 136]. As shown in figure 17(a), the suppression of T_c with the increase of sulfur concentration is confirmed by the shift of transition temperature where the Seebeck coefficient S becomes 0 because Cooper pairs carry no entropy in the superconducting state. The diffusive Seebeck response of a Fermi liquid dominates and is expected to be linear versus temperature in the zero-temperature limit, with a magnitude proportional to the strength of electron correlations in the simple Boltzmann picture [137]. This is similar to the temperature linearity of electronic specific heat, C_e/T = γ. In a one-band system both can be described by

$$\begin{equation} S/T=\pm \frac{\pi^{2}}{2}\frac{k_{\mathrm{B}}}{e}\frac{1}{T_{\mathrm{F}}}=\pm \frac{\pi^{2}}{3} \frac{k_{{\rm B}}^{2}}{e}\frac{N(E_{\mathrm{F}})}{n}, \end{equation} \tag{ 29 }$$

$$\begin{equation} \gamma _{n}=\frac{\pi^{2}}{2}k_{\mathrm{B}}\frac{n}{T_{\mathrm{F}}}=\frac{\pi^{2}}{3} k_{\mathrm{B}}^{2}N(E_{\mathrm{F}}), \end{equation} \tag{ 30 }$$

where e is the electron charge and T_F is the Fermi temperature, which is related to E_F and N(E_F) (=3n/2E_F = 3n/k_BT_F) [137]. For superconducting crystals, the Seebeck coefficient in the normal state is independent of magnetic field and exhibits a linear relationship with temperature which can be fitted using equation (29) very well in the low-temperature range (figure 17(a)). With S doping, S/T is suppressed from −0.48 μV K⁻² to a very small value of ∼0.03 μV K⁻² for crystals without superconducting transition. Similar to S/T, the electronic Sommerfeld coefficient in the normal state γ_n is also gradually suppressed with the increase in sulfur content (inset of figure 17(b)). According to equations (29) and (30), S/T and γ_n are related to n and N(E_F). Since sulfur has an identical electronic configuration to selenium, there should be no simple change in the carrier concentration with sulfur doping because the elemental analysis is consistent with full occupancy of Se (S) sites [68]. The absolute value of the dimensionless ratio of the Seebeck coefficient to specific heat, q = N_AveS/Tγ_n, where N_Av is the Avogadro number, gives the carrier density n. The values of q do not exhibit significant change [136]. Therefore, the suppressions of S/T and γ_n reflects a suppression of N(E_F).

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The ratio of T_c to T_F characterizes the correlation strength in superconductors. In unconventional superconductors, such as CeCoIn₅ and YBa₂Cu₃O_6.67, this ratio is about 0.1, but it is only ∼0.02 in BCS superconductors, such as LuNi₂B₂C [135]. In Fe_1+yTe_1−xSe_x, T_c/T_F it is also near 0.1, pointing to the importance of electronic correlations [138]. The ratio T_c/T_F ∼ 0.04 for K_0.64Fe_1.44Se₂ implies a weakly correlated superconductor [135] when compared to Fe_1+y Te_1−xSe_x. A theoretical study pointed out that the ordered Fe vacancies could induce band narrowing and consequently decrease the correlation strength needed for the Mott transition [139]. With increasing sulfur content, the value of T_c/T_F decreases further [136]. This implies a suppression of electron correlation strength as the system is tuned toward semiconducting states. The effective mass m, derived from k_BT_F = ℏ²k²_F/2m, is also suppressed with the increase in S content, consistent with the decrease of correlation strength with S doping [136].

S doping affects not only T_c and properties in the normal state, but also μ₀H_c2(T). As shown in figure 18(a), with S doping, for K_0.70Fe_1.55Se_1.01S_0.99 (S-99), dμ₀H_c2(T)/dT|_T=Tc is 2.74(7) and 0.649(7) T K⁻¹ for H∥ab and H∥c, respectively, and for K_0.76Fe_1.61Se_0.96S_1.04 (S-104), dμ₀H_c2(T)/dT|_T=Tc is 3.15(4) and 0.499(4) T K for H∥ab and H∥c, respectively [140]. Both are much smaller than in K_0.64Fe_1.44Se₂ (S-0). Therefore, the μ₀H_c2(T) values obtained with equation (10) are significantly smaller than those in S-0 for both field directions. Using λ_e−ph = 0.5, the Pauli limiting field μ₀H_p(0) = 1.86T_c(1 + λ_e−ph)^1/2 is 53.5 and 41.0 T for S-99 and S-104, respectively. Both values are larger than extrapolated μ₀H_c2,c(0) using equation (11) for S-99 and S-104. It should be noted that for the μ₀H_c2,c(T) of S-99 and S-104, there is no upturn at low temperatures when compared to undoped AFeSe-122 materials [60, 126]. Moreover, the fits using simplified WHH theory are rather satisfactory for temperatures far below T_c for H∥c for both S doped samples. All this implies that for S-99 and S-104 the spin-paramagnetic and multiband effects are negligible when the magnetic field is applied along the c-axis. This is different from undoped AFeSe-122 materials [60, 126]. The changes of the μ₀H_c2,c(T) with S doping could be due to the changes of band structure. Experiments in higher field and lower temperature are needed to shed more light on the behaviors of μ₀H_c2,ab(T).

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To characterize the evolution of the anisotropy γ(T) with S doping near T_c, the angular-dependent resistivity ρ_ab(θ,μ₀H,T) is measured at various magnetic fields and temperatures, where θ is the angle between the direction of external filed and the c-axis of the samples. According to the anisotropic Ginzburg–Landau (GL) model, the effective upper critical field μ₀H^GL_c2(θ) can be represented as [141]

$$\begin{equation} \mu _{0}H_{\mathrm{c}2}^{\mathrm{GL}}(\theta )=\mu _{0}H_{\mathrm{c}2,ab}/(\sin ^{2}\theta +\gamma ^{2}\cos ^{2}\theta )^{1/2}, \end{equation} \tag{ 31 }$$

where γ = H_c2,ab/H_c2,c = (m_c/m_ab)^1/2 = ξ_ab/ξ_c. Since the resistivity in the mixed state depends on the effective field H/H^GL_c2(θ), the resistivity can be scaled with H/H^GL_c2(θ) and data should collapse onto one curve in different magnetic fields at a certain temperature when a proper γ(T) value is chosen [141]. Figures 18(b)–(d) show the relation between resistivity and scaling field μ₀H_s = μ₀H(sin²θ + Γ²cos²θ)^1/2 for S-0, S-99 and S-104, respectively. Good scaling is obtained by adjusting γ(T). The temperature dependence and value of γ(T) for S-0 (the inset of figure 18(b)) are similar to reported results [126]. For S-99 and S-104, the γ(T) exhibits the same trend as that for S-0, i.e. increasing with temperature. However, the value increases gradually with increasing S content (figures 18(c) and (d)). It changes from ∼3 for S-0 to ∼6 for S-104. The larger anisotropy with increasing S content may suggest that the two-dimensional Fermi surface is becoming less warped with S doping [123].

According to equation (14), the Maki parameter α can be calculated as α = [(H*_c2(0)/H_c2(0))² − 1]^1/2. It can be also obtained from the fitting using equation (8) or (27). We summarize the α of iron-based superconductors reported in the literature for H∥ab and H∥c (figure 19). There are three general trends for α: (i) α_H∥ab are larger than α_H∥c for all of superconductors, indicating that the spin-paramagnetic effect is weaker for H∥c than H∥ab. (ii) FeCh-11 shows the largest α for both field directions with α > 1, reflecting that the spin-paramagnetic effect is the dominant pair-breaking effect for both field directions. (iii) For FeAs-based superconductors where the spin-paramagnetic effect is considered, the α_H∥ab are usually larger than 1 but the α_H∥c are usually smaller than 1, implying that the spin paramagnetic effect is dominant for H∥ab. The larger α_H∥ab when compared to α_H∥c can be understood from equations (15) and (16). When assuming other parameters are the same for H∥ab and H∥c, if v_ab is larger than v_c it will lead to the larger α_H∥ab than α_H∥c. Theoretical calculations confirms this assumption [42, 142]. On the other hand, assuming that the relaxation time of the conduction electrons τ is isotropic and independent of Fermi velocities, the anisotropy of resistivity in the normal state γ_ρ(0) is proportional to <v²_ab > _FS/ < v²_c > _FS, where <··· > _FS denotes the average on the FS [143]. The experimental results of γ_ρ(0) also confirmed that v_ab is larger than v_c [143]. Briefly, because of v_i = ∂_k/∂k_i, the cylinder-like Fermi surface in iron-based superconductors will usually result in v_ab ≫ v_c, therefore α_H∥ab larger than α_H∥c. Moreover, in the case of a cylinder-like Fermi surface, the open electronic orbits along the c-axis at low temperatures will also make the orbitally limiting upper critical field unlikely [60]. On the other hand, the evolution of spin-paramagnetic effect in iron-based superconductors (FeCh-11, AFeCh-122 > FeAs-111 > FeAs-122 > FeAs-1111) can be partially related to the evolution of the enhancement of effective mass in these materials according to equations (15) and (16) (figure 20) [41]. Moreover, the low carrier density n in iron-based superconductors [42, 43], resulting in the smaller E_F according to the free electron model, will also lead to the larger α.

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We now comment on the recently discovered phase separation in K_xFe_2−ySe₂ crystals [32–38]. Since the wide-gap semiconducting phase is locked at high temperatures, (super)conductivity originates from the volume fraction of the sample that is conducting below the high-temperature structural transition. This should not have a major effect on μ₀H_c2(T) phase diagrams, fits and estimate of mechanism of pair breaking in high fields. The values of J_c could be affected to some extent because the Bean model was applied to inhomogeneous samples. Yet, due to the rather small scale of phase separation and Josephson coupling that bridges insulating nanoislands [35], values of J_c represent a lower estimate of J_c and should be close to the intrinsic values for the superconducting phase. A similar argument can be applied for diffusion thermopower S and conclusions based on the correlation strength in the normal state since S does not depend on the sample geometry. In contrast, resistivity values represent the upper limit of ρ for the superconducting phase, similar to polycrystalline samples with grain boundary contributions.