Weakly-correlated nodeless superconductivity in single crystals of Ca3Ir4Sn13 and Sr3Ir4Sn13 revealed by critical fields, Hall effect, and magnetoresistance measurements

We report a study of single crystal Ca3Ir4Sn13 (CIS) and Sr3Ir4Sn13 (SIS) by measuring the longitudinal and Hall resistivities, upper and lower critical fields and magnetoresistance, as well as the magnetization. The sign change in the Hall coefficient observed on both the CIS and SIS provides direct evidence for the Fermi surface reconstructing during the superlattice phase transition. Both materials are of current interest due to indications of superconductivity associated with charge-density-wave (CDW) ordering. Observations of the diamagnetic feature and the lower critical field Hc1(T) in both CIS and SIS can be realized by means of the nodeless single-gap BCS theory. In addition, a weak electronic correlation in both systems has been revealed by the small values of the spin exchange energy, upper critical field and Δ(0)/kBTc ratio, derived respectively from the normal-state Hall effect, resistive transition and temperature-dependent Hc1. It is noticeable that the magnetoresistance of SIS shows a rapid increase below T′ ∼ 40 K, following Kohler’s scaling rule. The results of the magnetic susceptibility and Hall coefficient also exhibit anomalous features near T′. With respect to these observations, this suggests that the existence of an additional phonon mode with energy of about 4.0 meV in SIS is responsible for the presence of lattice instability toward a phase transition.


Introduction
The unique properties that develop in quantum critical matter have become a major focus of research over the past two decades. Within the quantum-mechanical phase transition, the singular quantum critical point (QCP) at absolute zero produces a wide region of unusual behavior at a finite temperature. Here, a QCP can be tuned by a nonthermal-control parameter, such as the magnetic field, pressure or chemical composition, and crucially influences the physical properties in correlated electronic systems. For example, it is believed that hightemperature superconducting (HTS) cuprates and a whole host of other superconductors with quasi-linearly temperature-dependent resistivity in the normal state are driven by quantum criticality. Recent investigations into (Sr, Ca) 3 Ir 4 Sn 13 have demonstrated the pressure-and composition-induced structure quantum phase transition in these cubic superconductors [1,2]. Unlike the case of high-temperature superconductors, superconducting iron-based pnictides or some heavy-fermion superconductors in which superconductivity emerges as the spin-density-wave (SDW) order is suppressed [3], the superconductivity in the (Sr, Ca) 3 Ir 4 Sn 13 system is driven by the suppression of charge-density-wave (CDW) ordering induced in their superlattice structure. This new kind of quantum phase transition provides a new opportunity to understand the interplay between superconductivity and structural instabilities. Sr 3 Ir 4 Sn 13 (SIS) has been found to be a superconductor of T c = 5 K and has distinct anomalies in the electrical and thermal transport properties, magnetic susceptibility and nuclear magnetic resonance (NMR) at T* ∼ 147 K, as characterized in previous research [4]. Applying the chemical pressure by alloying calcium in (Sr, for CIS and SIS, respectively. Here, the demagnetization effect arisen from the geometric factor is considered, and the corrected effective magnetic susceptibility is calculated using χ eff (T) = χ/ (1−χN), where χ = M/H and N is the demagnetization factor, which can be obtained from the magnetization in the Meissner state [12]. The demagnetization factors of N = 0.974 and 0.972 are thus obtained for CIS and SIS, respectively. As can be seen, both CIS and SIS reveal a diamagnetic feature in the entire temperature range investigated, and the behaviors of χ eff (T) in cases of out-of-plane (H// [110]) and in-plane (H⊥[110]) configurations show similar results, indicating weak anisotropy in this system. The anomalies of χ eff (T) appearing around T* = 35 and 147 K due to CDW transition for CIS and SIS, respectively, are consistent with previous observations [1,10]. The insets of figures 1(a) and (b) show the zero-field-cooling magnetization in H = 10 Oe around superconducting transition temperatures for CIS and SIS, respectively. The observation of T c ∼ 5 and 7 K for CIS and SIS, respectively, are also consistent with previous reports [5,8]. The right inset of figure 1(a) shows the field-dependent magnetization for CIS at 50 K within magnetic fields of up to 2 T and confirms the diamagnetic feature in it. Such an observation is in accordance with that reported by Wang and Petrovic [10] but is contrary to that reported by Yang et al [8]. The observation also confirms that both SIS and CIS have a weak diamagnetic property, being consistent with the results of electronic structure calculations [2] as well as with recent muon spin rotation (μSR) measurements [9,13]. In addition, the low-temperature χ eff (T) for SIS shows an anomalous increase at temperature T′ of ∼40 K, as shown in the right inset of figure 1(b). This temperature corresponds to some anomalies appearing in the Hall coefficient and magnetoresistance and will be discussed later. Figure 2(a) shows the temperature-dependent resistivities for CIS and SIS samples. Obviously, two pronounced anomalies of resistivity appearing around T* = 35 and 147 K due to CDW transitions for CIS and SIS, respectively, are observed, which are also consistent with previous observations on magnetization results. The inset of figure 2(a) shows resistive transition near the superconducting critical temperatures for the CIS and SIS samples. Sharp superconducting transitions with superconducting critical temperatures of 7.1 and 5.1 K for CIS and SIS samples, respectively, have been observed. Figure 2(b) illustrates the temperature dependence of the Hall coefficient R H for both CIS and SIS measured at H = 3 T. As shown, the R H values at temperatures above CDW transition temperature T* are negative and signify that electron-type carriers dominate the electrical transport in both CIS and SIS single crystals. Below T*, a sharp reduction in |R H | value accompanying a sign change at around T* is observed on both the CIS and SIS. Such a sign change in R H can be attributed to the sudden change in band structure accompanying the Fermi surface reconstructing during the superlattice phase transition. Recent electronic structure calculations for SIS have shown the characteristics of nesting Fermi surface sheets and illustrate a difference in the total Fermi energy between the superlattice phase and its parent phase, which causes the Fermi surface sheets to reconstruct [1,2]. This reconstructing should lead into the change in type of the major electric carriers. The inset of figure 2(b) shows the magnetic-field dependence of the transverse Hall resistivity ρ xy for the CIS sample at different temperatures. Similar to that observed on SIS crystal [4], the straight dashed lines correspond to a linear dependence, revealing an ordinary Hall effect. As seen, the slope of the ρ xy −H curve changes signs from negative to positive upon cooling from 30 to 20 K, carefully confirming the observation shown in figure 2(b). One may notice that this sign change in R H for CIS was not observed by Wang and Petrovic [10], and their Hall measurement showed the Hall coefficients of CIS only in the low-temperature region. The sign change in R H observed on both CIS and SIS provides direct evidence for the Fermi surface reconstructing during the superlattice phase transition. Also, note that a downturn in R H of the SIS sample below 30 K is observed. The origin of this feature is not clear at this moment and deserves further investigation. It will be further explored by magnetoresistance in the following discussion.

Results and discussion
The basic superconducting transport properties presented in figures 3(a) and (b) show resistivity as a function of temperature in magnetic fields parallel to the crystalline [110] orientation for CIS and SIS samples, respectively. As seen, the resistivity under a magnetic field shows a broadening behavior due to thermally activated flux motion, which has been proposed by Anderson and Kim [14,15] and can be described by where U is the activation energy and is normally both fieldand temperature-dependent. In addition, the upper critical field H c2 (T) can be determined by taking the 50% transition of resistivity in various fields. The inset of figure 3(a) shows the field-dependent activation energies extracted from Anderson and Kim's theory via the Arrhenius plots for CIS and SIS crystals. In the Anderson-Kim model, the activation energy U is an indication of the magnitude of the effective pinning energy. As seen in the inset, the activation energies for CIS are slightly higher than those for SIS, and the values of U for CIS and SIS are approximately two orders of magnitude smaller than those of several 10 4 K for YBa 2 Cu 3 O y (YBCO) [16], indicating a relatively weak vortex pinning in CIS and SIS systems. Also seen in the inset of figure 3(a) is that U can be fitted with an approximate field dependence of U ∝ H −α with α ≈ 0.8 and 1.0 for CIS and SIS, respectively. The present observation of thermally activated behavior for SIS in a mixed state does scale with the predicted H −1 dependence of U [17]. The inset of figure 3(b) shows the temperature dependence of upper critical field H c2 for CIS and SIS single crystals, where a linear-like behavior for temperatures near T c can be observed. According to these results, the upper critical field H c2 (0) can be estimated using the formula derived by Werthamer et al where |dH c2 /dT| Tc can be derived from the linear fitting in the inset of figure 3(b). By using equation (1), the obtained H c2 (0) values are 4.9 and 1.8 T for CIS and SIS single crystals, respectively, which are slightly smaller than 5.5 T for CIS and 3.5 T for SIS, as reported in other studies [5,7,8]. Meanwhile, the larger values of H c2 Additionally, it is known that the upper critical field H c2 (0) is limited by μ 0 H c2 (0) (Tesla) = 1.86 T c (kelvin) for weak electron-phonon coupling [19], being one of the indications of electronic correlation in superconductivity. Taking into account T c = 7.1 and 5.1 K for CIS and SIS, respectively, one can see that the obtained H c2 (0) values are significantly lower than the calculated weak-coupling-limit values of 13.2 and 9.5 T for CIS and SIS single crystals, respectively. These results indicate that superconductivities of both CIS and SIS are weak-coupling and resemble that observed on the Y 3 Pt 4 Ge 13 compound in which an s-wave symmetry single-energy gap for its superconducting phase was inferred [20]. The lower critical field H c1 (T), corresponding to the field at which the presence of vortices into the superconductor and related to the magnetic penetration depth, can provide key information regarding the thermodynamic properties and superconducting energy gap. The temperature dependence of the magnetic penetration depth λ(T) can be determined from H c1 using the formula, μ 0 H c1 = (Φ 0 /4πλ 2 )lnκ, where κ = λ/ξ is the Ginzburg-Landau parameter. Meanwhile, according to the nodeless BCS superconducting band gap theory [21,22], the term λ −2 can be expressed by (T c /T−1)] 0.51 }, the Boltzmann constant k B and the residual penetration depth λ(0). Thus, the formula for the temperature dependence of H c1 is   (2) with adopted κ = 48 and 32 for CIS and SIS, respectively [5,7,8]. The obtained zero-temperature superconducting gaps for CIS and SIS are 9.52 (0.82 meV) and 7.43 K (0.64 meV), respectively. Meanwhile, the obtained residual penetration depths λ(0) for CIS and SIS are 294 and 381 nm, respectively, which are close to those values previously reported [5,8]. In view of the superconducting pair coupling, the obtained superconducting gaps for CIS and SIS are then considered. One can see that the ratios of Δ(0)/k B T c ≈ 1.36 and 1.49 for CIS and SIS, respectively, are slightly smaller than that of 1.76, predicted by the traditional BCS theory.
Furthermore, we try to fit the temperature dependence of H c1 for CIS and SIS using the d-wave model of the gap symmetry [24] in which the H c1 (T) can be denoted by The best fitting curves (dashed lines) of equation (3) with Δ(0) = 10.5 and 8.5 K, respectively, for CIS and SIS are also shown in the inset of figure 4(a). However, we find that the curves obtained within the d-wave model of gap symmetry cannot be reconciled well with our experimental data at low temperatures and exclude the possibility of d-wave gap symmetry in CIS and SIS. These results again demonstrate a possibly weak-couple pairing in CIS and SIS superconductors. Additionally, the fitting of equation (2) also confirms that the gap symmetry in SIS and CIS can be described by the nodeless single-gap BCS theory and excludes the possibility of ferromagnetic spin fluctuation in these CDW-related superconductors, which are consistent with the observed results of diamagnetic magnetic susceptibility, as shown in figures 1(a) and (b). In addition, it may be worth pointing out that Biswas et al [13] recently obtained the lower and upper critical fields for SIS via magnetization and μSR measurements. Their value of H c2 (0) = 1.44 T is slightly smaller than that of 1.8 T extracted by the transport  [13] also inferred the existence of a nodeless s-wave energy gap in the superconducting state of SIS according to their temperature-dependent penetration depth λ, being consistent with the argument shown here.
A normal-state Hall measurement can provide further insight into the spin exchange energies in superconductors. Figures 5(a) and (b) plot cotθ H as defined by cotθ H = ρ xx /ρ xy versus T 2 for CIS and SIS measured in the fields of 3 and 6 T, respectively. As can be seen, the data almost fall on a straight line in the temperature range below T* and can be fitted to the equation of cotθ H = ΛT 2 + C with the parameters Λ of 1.063 (0.023) K −2 and C of 182.6 (5.7) for CIS (SIS) in an applied field of 6 T, where C corresponds to the impurity contribution. Also shown in the inset of figure 5(b) is the T 2 behavior of cotθ H for CIS and SIS at temperatures above T* with the parameters Λ and C of 32.34 (34.20) K −2 and 475.2 (946.6) for CIS (SIS) in an applied field of 3 T. The Λ value for CIS is almost two orders of magnitude larger than those observed in HTS cuprates, MgB 2 or superconducting pnictides (10 −3 ∼ 10 −2 K −2 ) [25][26][27][28]. It has been pointed out that the temperature dependence of cotθ H for many superconductors with a wide range of doping retains its simple ΛT 2 + C form at temperatures above T c even though the temperature dependence of ρ xx changes considerably [29]. The T 2 behavior of cotθ H for CIS and SIS seemingly conforms to this contention because of the nonlinear-T resistivity observed in SIS at temperatures below T*, as seen in figure 2(a), unlike HTS cuprates in which a linear-T resistivity is commonly observed. So far, the T 2 dependence of cotθ H has been observed in HTS cuprates, MgB 2 , superconducting pnictides and CDW-related superconductors, as shown here, regardless of their temperature dependence of ρ xx . In fact, R H or ρ xy itself is complex, and many different explanations have been proposed for the anomalous normal-state Hall effect. A widely known approach proposed by Anderson [30] argues that the system has non-Fermi liquid (Luttinger liquid) properties, and the temperature dependence of R H (T) could be better understood in terms of two apparently decoupled scattering rates with the spin-charge separation scenario. Following Anderson's theory, Chien et al analyzed the normal-state Hall angle in single-crystal YBa 2 Cu 3−x Zn x O y [31] and derived a correlation between parameter Λ and the spin exchange energy as where J is the bandwidth of spin excitations (or the spin exchange energy in Anderson's theory), and n is the planar carrier density. The spin exchange energy J for YBa 2 Cu 3−x Zn x O y was determined by the experimental value of Λ to be ∼830 K. A similar estimation for Tl 2 Ba 2 CuO y has recently shown J ≈ 800 K [32]. By substituting the experimental values of Λ and the carrier concentration derived from the Hall coefficient into equation (4), the analog exchange energy J for the superconducting CDW system of CIS and SIS can be determined to be 88 and 464 K, respectively. The obtained spin exchange energy for CIS is much smaller than those of HTS cuprates and is in accordance with the extremely small T c /T F value of ∼0.001 derived from the Seebeck coefficient measurement [10]. Surprisingly, both the Hall and thermo-electric measurements agree with the aspect of a weak electronic correlation in CIS. On the other hand, the obtained spin exchange energy for SIS is larger than that of CIS but is still much smaller than those of HTS cuprates, again implying that the mediate couplings for forming superconducting pairs in SIS are relatively weak in comparison with HTS cuprates. Recently, the T 2 behavior of cotθ H has also been observed in many heavy-fermion superconductors. For example, compounds of CeIrIn 5 and CeCoIn 5 exhibit cotθ H = ΛT 2 + C with the parameters Λ of ∼1.1 and 0.254 K −2 [33,34], respectively, which are comparable to that of 1.063 for CIS but are much larger than those for HTS cuprates or superconducting pnictides. Both CeIrIn 5 and CeCoIn 5 are suggested to be d-wave-symmetry superconductors in which the microscopic coexistence of superconductivity and antiferromagnetic fluctuation (AF) near a QCP has been reported [33,34]. Interestingly, the larger Λ values imply that the spin exchange energies in these AFrelated heavy-fermion superconductors would not be large.
In addition to the anomalous normal-state Hall effect for the general understanding of superconductivity in CDW-related superconductors, it is of crucial importance to clarify the normal-state charge transport in fields. In particular, the classical orbital magnetoresistance (MR) offers the key to an understanding of the electron transport on the Fermi surface since it involves the same scattering processes as the Hall current. Harris et al [35] have shown that the classical orbital MR can measure the variance of a local Hall angle around the Fermi surface and is closely related to the temperature dependence of the measured Hall angle. The relationship between MR and the Hall angle can be expressed as  (5), the first term 〈θ(s) 2 〉 corresponds to the variance in in-plane conductivity, −Δσ xx /σ xx , whereas the second term 〈θ(s)〉 2 corresponds to the observed Hall angle (θ H ) 2 [35]. Equation (5) also involves the elementary case of zero orbital MR for a circular Fermi surface of an isotropic metal [35]. figure 6(a) shows the temperature dependence of MR orb measured with the resistivity changes between H = 0 and 3 T for CIS and SIS, respectively. As can be seen, the MR orb values of SIS are much larger than those of CIS at temperatures below 30 K and increase quickly at temperatures below 40 K, while there is no significant change in the MR orb of CIS. This MR behavior corresponds seemingly to the R H behavior, as shown in figure 2(b), in which the SIS sample shows a downturn in R H around 30 K, while the R H of CIS shows a monotonous increase when the temperature is decreased to be below 35 K. Also seen in figure 6(a) is a fitting curve which shows a roughly T −2 -dependent MR orb for SIS at temperatures below 40 K. The inset of figure 6(a) shows the temperature dependences of MR ⊥ and MR // for SIS. Both MR ⊥ and MR // also show rapid increases at temperatures below 40 K, implying that the MR is affected by a strong spin-flip scattering in the low-temperature region. It is worthy to note that the temperature of ∼40 K corresponds to the temperature T′, where an anomalous increase in effective magnetic susceptibility χ eff (T) is observed, as previously shown in figure 1(b). Figure 6(b) plots MR orb vs. H for SIS in the low-temperature normal-state regime. Also shown in the inset of figure 6(b) is the plot of MR orb vs. H 2 corresponding to the data in figure 6(b). Clearly, a nearly H 2 dependence of MR orb is observed for the SIS single crystal at temperatures below 50 K. As a result, SIS indeed exhibits the typical H 2 -dependent MR character at the low-temperature region. In conventional metals, the magnetoresistance MR orb due to an orbital motion of carriers can be scaled as a function of the term H/ρ xx (0), which is regarded as following the Kohler's scaling rule [36]. To examine this relation, figure 7(a) plots the MR orb as a function of [H/ρ xx (0)] 2 for SIS at temperatures ranging from 8 to 50 K, a temperature region which is far below the Debye temperature (∼184 K) [5] but well above T c , where the resistances due to the electron-phonon-dominated interaction, superconducting fluctuation and flux motion can be neglected. As seen, the MR orb can be fitted to be proportional to the term [H/ρ xx (0)] 2 with a merged temperature-independent slope; that is a scaling of Kohler's rule. Another pronounced MR characteristic is the modified Kohler's scaling, which relates the magnetoresistance with the resistivity and Hall angle [35,36]. shows the corresponding data of MR orb against tan 2 θ H for the SIS sample. As can be seen, the data at different temperatures show distinctly different curves, indicating that the MR orb of SIS does not obey the modified Kohler's rule. Contrary to this result of SIS, it has been demonstrated that the scaling of Kohler's rule breaks down in cases of optimum-doped YBCO, La 2−x Sr x CuO 4 , NaFe 1−x Co x As and BaFe 2 (As 1−x P x ) 2 [28,35,37] in which the MR violates the Kohler's scaling and can be scaled by the square of the Hall angle (modified Kohler's rule) in these optimally doped systems. The scenario of SDW QCP has recently been proposed to describe the normal-state transport properties [38,39]. It has even been proposed that the SDW QCP is a central organizing principle of organic, iron-pnictide, heavy-fermion and HTS cuprates [36,40]. Recently, heavy-fermion superconductors, such as CeIrIn 5 and CeCoIr 5 compounds, continue to be a central focus of investigation, as mentioned previously. Most of them belong to the class of field-tuned or pressure-tuned heavy-fermion systems near a QCP in which the electrical transport properties have been suggested to be governed by the AF spin fluctuation. It has been reported that the MR in CeIrIn 5 and CeCoIr 5 does not obey Kohler's rule but can be scaled by tan 2 θ H (modified Kohler's rule) [34,41,42]. Apparently, the normal-state MR effect in the SIS superconductor, which has superconductivity that is driven by the suppression of CDW ordering accompanying a pressure-induced structure quantum phase transition, reveals a significantly different property from those of SDW-suppressed or AF-related heavy-fermion superconductors. In addition, an interesting compound, MgCNi 3 , having a perovskite structure without any oxygen, exhibits a T c of ∼8 K even though it contains a high proportion of Ni [43,44]. The MR behavior in MgCNi 3 shows that the classical Kohler's rule is valid [43]; meanwhile, the estimated H c2 (0) satisfies the value expected within the same weak-coupling BCS theory [44]. This situation in MgCNi 3 is analogous to that observed here in SIS and implies that the normal-state MR behaviors in weakly-correlated superconductors tend to follow the scaling of the classical Kohler's rule.
It is noteworthy that MgB 2 reveals the T 2 dependence of cotθ H with a small Λ value of 0.005 K −2 , as mentioned previously [27], but shows a large normal-state MR in which Kohler's rule is not obeyed [45]. Meanwhile, the breakdown of Kohler's rule in MgB 2 has been attributed to the multiband effect [45]. Being different from CIS and SIS, MgB 2 reveals the paramagnetic feature with small magnetic moments in the normal state [46]. According to equation (4), a small Λ value should correspond to a larger exchange energy J in MgB 2 . Compared with the cases of CIS and SIS in which a diamagnetic feature accompanying a larger Λ value has been observed, as presented previously, the spin exchange in MgB 2 seemingly can be enhanced in a paramagnetic environment even though the spins in MgB 2 are not active [27]. It is well known that the diamagnetic properties of CIS and SIS, as demonstrated by our magnetization measurement or the μSR studies [9,13], should originate from the so-called Larmor diamagnetism, which is probably sufficient to exceed the paramagnetic Pauli spin susceptibility of the conduction carriers in CIS and SIS [6]. This is consistent with the smaller spin exchange energy obtained for CIS or SIS. Kohler's scaling of MR behavior for SIS at low temperatures also implies that the conduction carriers in SIS should be dominated by a single electronic band. So far, we have seen that these QCP systems indeed behave in unconventional normal-state transport properties (the unusual Hall effect or MR following the modified Kohler's rule) as indicated by Naira et al [36], regardless of the CDW-or SDW-related superconductors they belong to. Furthermore, in this study, we also find that the MR behavior, which obeys the conventional Kohler's rule or the modified Kohler's rule, is influenced by the magnitude of spin exchange.
Looking deeper into the MR behavior in SIS via equation (5), one can see that the MR orb varies as ∼T −2 (figure 6(a)), while the Hall measurement shows a roughly T −4 dependence of (θ H ) 2 due to the T 2 -dependent cotθ H (here, θ H ≈ tanθ H ∝ T −2 ). This observation indicates that the MR orb in SIS is dominated by the first term, 〈θ(s) 2 〉 (corresponding to −Δσ xx /σ xx ), not the second term, 〈θ(s)〉 2 (corresponding to (θ H ) 2 ). As a result, the MR orb of SIS certainly does not obey the modified Kohler's rule, as seen in figure 7(b). Moreover, it has been pointed out that the numerical value of the ratio MR orb /(θ H ) 2 can provide further information about the sign changing of the local Hall angle, i.e. the reconstruction of the Fermi surface [35]. This idea can be understood by looking closely at equation (5). The second term 〈θ(s)〉 2 in equation (5) corresponding to the observed (θ H ) 2 will be of an extremely small value in comparison with the first term 〈θ(s) 2 〉, as the local Hall angle is changing in sign over some segments of the Fermi surface due to band reconstruction, resulting in a larger ratio MR orb /(θ H ) 2 . The inset of figure 7 plots the temperature dependence of MR orb /tan 2 θ H (≈MR orb /(θ H ) 2 ) for CIS and SIS samples. As can be seen, the value of MR orb /tan 2 θ H for CIS is much larger than that for SIS due to an extremely small Hall angle in CIS even though a small MR orb value is observed in CIS, as shown in figure 6(a). The MR orb /tan 2 θ H for CIS increases gradually with the rising temperature and shows a rapid increase at temperatures approaching the CDW transition temperature T* of ∼35 K. The MR orb /tan 2 θ H for SIS shows another upturn with lowering the temperature below T′ of ∼40 K. The MR orb /tan 2 θ H for SIS also exhibits a rapid increase at the temperature T* of ∼147 K (not shown) because of the sign change in R H and θ H ≈ 0 at T*. Following the argument previously stated, these observed rapid increases in MR orb /tan 2 θ H can denote the occurrence of band structure reconstructions, which lead to some anomalies in the magnetic susceptibility, Hall coefficient and MR observed. An interesting subject is the origin of anomalies at T′ for SIS. A recent electronic structure calculation has shown an additional soft phonon mode (relative to the absence in CIS) of energy of 4.0 meV at the R point for SIS [2]. This energy corresponds to ∼46 K and indicates the presence of lattice instability toward a phase transition at that temperature. This is seemingly consistent with the observed temperature T′ of 40 K in SIS. Certainly, more theoretical or experimental studies on the anomalous properties of SIS at low temperatures are necessary.

Conclusions and outlook
In summary, the longitudinal and Hall resistivities, upper and lower critical fields, magnetoresistance and magnetizations of superconducting CIS and SIS were investigated to clarify both magnetic and transport properties of these CDW-related superconductors. Both CIS and SIS show basic features with the CDW transition temperature T* of 35 and 147 K, respectively, and the superconducting critical temperature T c of 7.1 and 5.1 K, respectively. The Hall coefficient R H values for CIS and SIS at temperatures above T* are negative and change sign at around T*, implying a sudden change in band structure accompanying the Fermi surface reconstructing during the superlattice phase transition. The resistivities under magnetic fields show thermally activated behavior with a power-law magnetic-field dependence of activation energy U ∝ H -α observed. The values of U for CIS and SIS are approximately two orders of magnitude smaller than those of several 10 4 K for HTS cuprates, indicating a relatively weak vortex pinning in CIS and SIS systems. Meanwhile, the upper critical field H c2 (0) values obtained are significantly lower than the calculated weak-coupling-limit values for both CIS and SIS, indicating that the superconductivities of CIS and SIS are weak-coupling. As to magnetic properties, both CIS and SIS reveal a diamagnetic feature in the entire investigated temperature range. The lower critical H c1 (T) extracted by the deviation from the Meissner-state linear M(H) curve can be described by equation (2), which is deduced from the nodeless BCS superconducting band gap theory. These results confirm that the gap symmetry in SIS and CIS can be described by the nodeless single-gap BCS theory and excludes the possibility of ferromagnetic spin fluctuation in CDW-related superconductors. The obtained ratios of Δ(0)/k B T c ≈ 1.36 and 1.49 for CIS and SIS, respectively, are slightly smaller than that of 1.76, predicted by the traditional BCS theory, and again demonstrate a possibly weak-couple pairing in CIS and SIS superconductors. Moreover, the normalstate Hall angle is observed to follow cotθ H = ΛT 2 + C in both CIS and SIS crystals. According to equation (4), the spin exchange energy J for CIS and SIS can be determined to be 88 and 464 K, respectively. Smaller values of spin exchange energy, the upper critical field and the Δ(0)/k B T c ratio agree with the thought of a relatively weak electronic correlation in CIS and SIS.
In MR measurements, the MR orb values of SIS are much larger than those of CIS at temperatures below 30 K and increase quickly at temperatures below 40 K. The MR orb of SIS is found to follow a scaling of Kohler's rule but does not obey the modified Kohler's rule, revealing a significantly different property from those of SDWsuppressed superconductors. It is found that the MR behavior, which obeys the conventional Kohler's rule or the modified Kohler's rule, is influenced by the magnitude of spin exchange. The MR effect in SIS shows a rapid increase with decreasing temperature below the temperature T′ of ∼40 K, which is consistent with the temperature of anomalies observed in magnetic susceptibility and the Hall coefficient, implying the reconstructions of band structure occurring around T′. The anomalies at T′ for SIS originate seemingly from an additional soft phonon mode (relative to the absence in CIS) of energy of 4.0 meV, which corresponds to ∼46 K, leading to the presence of lattice instability toward a phase transition at this temperature.