Two-gap superconductivity in R₂Fe₃Si₅ (R=Lu, Sc) and Sc₅Ir₄Si₁₀

Abstract

R₂Fe₃Si₅ (R= Sc, Y, Lu) contains nonmagnetic iron and has a relatively high superconducting transition temperature T_c among iron-containing superconductors. An anomalous temperature dependence of specific heat C(T) has been reported for polycrystalline samples down to 1 K. We have grown R₂Fe₃Si₅ single crystals, confirmed the anomalous C(T) dependence, and found a second drop in specific heat below 1 K. In Lu₂Fe₃Si₅, we can reproduce C(T) below T_c, assuming two distinct energy gaps 2Δ ₁/k_BT_c = 4.4 and 2Δ ₂/k_BT_c = 1.1, with nearly equal weights, indicating that Lu₂Fe₃Si₅ is a two-gap superconductor similar to MgB₂. Hall coefficient measurements and band structure calculation also support the multiband contributions to the normal-state properties. The specific heat in the Sc₂Fe₃Si₅ single crystals also shows the two-gap feature. R₅Ir₄Si₁₀ (R = Sc, rare earth) is also a superconductor where competition between superconductivity and the charge-density wave is known for rare earths but not for Sc. We have performed detailed specific heat measurements on Sc₅Ir₄Si₁₀ single crystals and found that C(T) deviates slightly from the behavior expected for weak-coupling superconductors. C(T) for these superconductors can also be reproduced well by assuming two superconducting gaps.

Introduction

Several exotic silicide superconductors were discovered around 1980. R₂Fe₃Si₅ and R₅Ir₄Si₁₀ (R= Sc, Y, rare earth elements) are two typical examples of such superconductors [1, 2]. Another example is the first heavy fermion superconductor CeCu₂Si₂ [3]. R₂Fe₃Si₅ crystallizes in a tetragonal Sc₂Fe₃Si₅ structure and contains nonmagnetic iron with square planar and linear arrangements as shown in figure 1(a) [1]. R₅Ir₄Si₁₀ crystallizes in a tetragonal Sc₅Co₄Si₁₀ structure, and R atoms form chains in the pentagonal and hexagonal Ir–Si network along the c-axis, as shown in figure 1(b) [2]. Peculiar properties have been reported in both R₂Fe₃Si₅ and R₅Ir₄Si₁₀ in the superconducting and normal states.

Figure 1.

Crystal structures of (a) R₂Fe₃Si₅ (R = Sc, Lu) and (b) Sc₅Ir₄Si₁₀. Large green circles are Lu or Sc atoms, medium-sized brown circles are Fe or Ir atoms, and small blue circles are Si atoms.

R₂Fe₃Si₅ has one of the highest superconducting transition temperatures T_c among iron-containing superconductors (Lu₂Fe₃Si₅: T_c = 6.0 K) [1, 4], except for the recently found iron-based oxypnictides (LaFeAsO_{1 - x}F_x: T_c = 26 K, SmFeAsO_{1 - x}F_x: T_c = 55 K) [5, 6]. One of the most marked properties of R₂Fe₃Si₅ is an anomalous temperature dependence of specific heat C(T) found in polycrystalline samples down to 1 K [7]. The jump of specific heat at T_c, ΔC/γ _nT_c (γ _n is the electronic specific heat coefficient) is reduced from the BCS (Bardeen–Cooper–Schrieffer) value of 1.43, and there are apparent residual values of specific heat coefficients in the low-temperature limit. These anomalies of C(T) have been confirmed in high-quality polycrystalline samples of Lu₂Fe₃Si₅ [8]. However, more detailed discussion on the anomalous nature, including anisotropy, requires high-quality single crystals. In the present study, we have grown single crystals of Lu₂Fe₃Si₅ and Sc₂Fe₃Si₅ and measured the temperature dependence of specific heat down to 0.35 K. All anomalous specific heat features mentioned above are reproduced also in single crystals. In addition, we find a second drop in the specific heat coefficient below 1 K. We can reproduce C(T) reasonably well by assuming two energy gaps. These findings strongly suggest that Lu₂Fe₃Si₅ and Sc₂Fe₃Si₅ are two-gap superconductors similar to MgB₂ [9]. Band structure calculation and Hall coefficient measurements also support the multiband contributions to the normal state transport properties. We also discuss the anisotropic properties of R₂Fe₃Si₅ and the significance of two-gap superconductivity in this system.

R₅Ir₄Si₁₀ is also a superconductor where the competition between superconductivity and the charge-density wave is known for all R except for Sc. We have grown high-quality single crystals of Sc₅Ir₄Si₁₀ and performed detailed specific heat measurements. C(T) in Sc₅Ir₄Si₁₀ is slightly different from that in a weak-coupling superconductor and it is again explained well by assuming two superconducting gaps. However, comparison with R₂Fe₃Si₅ suggests that Sc₅Ir₄Si₁₀ is a superconductor with an anisotropic energy gap.

Experiments

Single crystals of Lu₂Fe₃Si₅, Sc₂Fe₃Si₅, and Sc₅Ir₄Si₁₀ have been grown by the floating-zone technique using an image furnace. The starting rods for the single-crystal growth are prepared by melting a stoichiometric ratio of constituent elements in an arc furnace. We have used lumps of Lu (3Nup), Sc (3Nup), Fe (4N), Ir powder (4N), and Si chips (6N) as starting materials. In the case of R₂Fe₃Si₅, as-grown crystals often show lower T_c and broader transitions. In order to improve these characteristics, we have annealed single crystals at ∼1250 °C for an extended period. Lattice parameters for all three single crystals show excellent agreement with published data. Chemical analyses using energy dispersive x-ray spectroscopy reveal that chemical compositions for all three compounds are nearly stoichiometric. Resistivity measurement is performed by the conventional four-probe technique under magnetic fields up to 5 T, and from the results, we estimate the anisotropic upper critical field. The Hall coefficient is measured in the six-probe configuration. Magnetization measurements are performed with a SQUID magnetometer (MPMS-XL5, Quantum Design). Specific heat is measured by the relaxation method with home-built electronics down to 0.35 K using a ³He cryostat.

Results and discussion

Lu₂Fe₃Si₅

Figure 2 shows an example of the evolution of resistivity in Lu₂Fe₃Si₅ with the duration of annealing at ∼1250 °C. Resistivity decreases and the residual resistivity ratio increases with annealing time. The paramagnetic background magnetization measured at H= 50 kOe rapidly decreases at the annealing time of about one week, and it levels off beyond about two weeks, as shown in the inset of figure 2. We speculate that this paramagnetic signal originates either from small impurity phases dispersed in the crystal or the iron defects in the crystal.

Figure 2.

Effect of annealing at 1250 °C on temperature dependence of resistivity in Lu₂Fe₃Si₅. Inset shows the magnetization at H = 50 kOe for different annealing times.

Figures 3(a) and (b) show temperature dependences of Hall coefficients for two configurations, I∥c, H∥ab and I∥abH∥c, respectively [10]. Hall coefficients are negative at room temperature, and their magnitudes are about ten times smaller than those in iron-oxypnictides [11]. However, they show strong temperature dependences, including a sign change for I∥ab, H∥c, suggesting contributions from multiple bands, as in iron-oxypnictides. Band structure calculation is carried out by a full potential linearized augmented plane wave (FLAPW) method with the local density approximation for the exchange correlation potential. Resulting Fermi surfaces in Lu₂Fe₃Si₅ reveal three bands, as shown in figure 4: two hole bands and one electron band. These results strongly suggest that multiple kinds of carriers are responsible for the normal-state properties in this system.

Figure 3.

Temperature dependence of Hall coefficient for (a) I∥c, H∥ab and (b) I∥ab, H∥c in Lu₂Fe₃Si₅ [10].

Figure 4.

Fermi surfaces of Lu₂Fe₃Si₅ calculated by the FLAPW method [10]. Hole-like Fermi surfaces from the (a) 155th and (b) 156th bands, and (c) electron-like Fermi surface from the 157th band are shown.

The temperature dependence of the upper critical field for Lu₂Fe₃Si₅ is determined by the midpoint of resistive transition at a constant field. It turns out that H_c2^c (T) is larger than H_c2^ab (T) with an anisotropy parameter of γ∼2.0, indicating that Lu₂Fe₃Si₅ is a weakly anisotropic superconductor with one-dimensional anisotropy (not shown) [10, 12]. This value of γ is consistent with the ratio of ρ _c and ρ _ab [10, 12], since ρ _ab /ρ_c=(H_c2^c(T)/H_c2^ab (T))². It should be noted that both H_c2(T) dependences show extended linear regions in the measured temperature range. Recent measurements of H_c2(T) down to 0.4 K confirm this tendency [13].

Figure 5 shows the temperature dependence of specific heat in Lu₂Fe₃Si₅. The electronic specific heat jump at T_c, ΔC_e, is strongly reduced from the BCS value of ΔC_e /γT_c = 1.43. At 0.2T_c, where a BCS superconductor has a small electronic contribution, an appreciable C_e /γ _nT_c is observed, which is followed by a second drop below 0.2T_c. All these anomalies of C_e /γ _nT_c are similar to those of MgB₂ with two distinct superconducting gaps originating from the two-dimensional σ-band and three-dimensional π-band [9]. In a two-gap superconductor, electronic specific heat is the sum of two contributions with different gap values, C_e (T) =x₁C₁ (T) +x₂C₂ (T), where C₁ (T) and C₂ (T) correspond to electronic specific heat from bands 1 and 2, and x₁ and x₂ are the fractional densities of states for bands 1 and 2, respectively. The solid line in figure 5 shows the best fit based on the two-gap model. The two gap values obtained for Lu₂Fe₃Si₅ are 2Δ ₁ /k_BT_C = 4.4 and 2Δ ₂ /k_BT_C = 1.1, where x₁ /x₂ = 47/53 and k_B is the Boltzmann constant. The Arrhenius plot of C_e (T) in the inset of figure 5 shows the slope corresponding to the smaller gap. It should be noted that a similar two-gap analysis has successfully explained the temperature dependence of the penetration depth in Lu₂Fe₃Si₅ [14]. However, which of the three bands are responsible for the large or the small gaps is not yet clear.

Figure 5.

Temperature dependence of specific heat for Lu₂Fe₃Si₅ [10]. C_e(T) based on the BCS model is shown by dashed line. The solid line shows the best fit using the two-gap model with D₁/k_BT_B=44 and D₂/k_BT_B=1.1. Inset shows an Arrhenius plot of C_e(T) and suggests the presence of the smaller gap.

Sc₂Fe₃Si₅

Figure 6 shows the temperature dependence of resistivity in Sc₂Fe₃Si₅. The inset shows a magnification near the transition temperature of ∼4.8 K. A small residual resistivity and sharp superconducting transition guarantee the high quality of the measured Sc₂Fe₃Si₅ crystal.

Figure 6.

Temperature dependence of resistivity along ab-axis for Sc₂Fe₃Si₅. Inset shows a magnification close to the superconducting transition temperature.

Figure 7 shows the temperature dependence of the upper critical field determined by the midpoint of resistive transition for Sc₂Fe₃Si₅. Similar to the case of Lu₂Fe₃Si₅, H_c2^c (T) is larger than H_c2^ab (T), indicating that Sc₂Fe₃Si₅ is also a weakly one-dimensional superconductor. The anisotropy parameter γ is about 2.0, and it is only weakly temperature dependent.

Figure 7.

Temperature dependence of upper critical fields, H_c2^c and H_c2^ab, estimated from the magnetization-temperature curves for Sc₂Fe₃Si₅.

Figure 8 shows the temperature dependence of specific heat in Sc₂Fe₃Si₅. The jump in specific heat is again reduced from the BCS value. The solid line in figure 8 shows the best fit based on the two-gap model similar to the case of Lu₂Fe₃Si₅. The two gap values obtained for Sc₂Fe₃Si₅ are 2Δ ₁ /k_BT_C = 3.53 and 2Δ ₂/k_BT_C = 1.7 with x₁/x₂ = 36/64. It is natural to expect a similar two-gap behavior in Sc₂Fe₃Si₅ since Sc₂Fe₃Si₅ has the same crystal structure as Lu₂Fe₃Si₅ with only a slight difference in the lattice parameters.

Figure 8.

Temperature dependence of specific heat for Sc₂Fe₃Si₅. C_e(T) based on the BCS model is shown by dashed line. The solid line shows the best fit using the two-gap model with D₁/k_BT_B=3.53 (36%) and D₂/k_BT_B=1.70 (64%).

Here, we stress the uniqueness and importance of two-gap superconductivity in R₂Fe₃Si₅. In the case of MgB₂, a well-established two-gap superconductor, the small size of single crystals is an obstacle to many measurements. In R₂Fe₃Si₅, however, crystals of millimeter size are readily available. The impurity effect is an important diagnosis to study the gap structure of unconventional superconductors, as in the case of high-temperature superconductors. In R₂Fe₃Si₅, we can prepare a complete solid solution of Lu₂Fe₃Si₅ and Sc₂Fe₃Si₅, where the 3d band of iron, responsible for the superconductivity, is intact. In the case of MgB₂, both carbon doping and aluminum doping inevitably introduce charge carriers into the system, in addition to the random potential. This masks the effect of interband scattering. In addition, the orthogonality of σ- and π-bands makes the inter-band scattering probability extremely small, reducing the impurity effects. In R₂Fe₃Si₅, no such special situation is anticipated. Hence, this system gives us a unique opportunity to study the effect of interband scattering in two-gap superconductors, which is expected to suppress the larger gap and enhance the smaller gap until they merge. Such a study is now in progress and will be reported elsewhere.

Sc₅Ir₄Si₁₀

Figure 9 shows the temperature dependence of the upper critical field in Sc₅Ir₄Si₁₀ with T_c = 8.4 K. We define H_c2 on the basis of the onset of diamagnetism in the magnetization–temperature (M–T) curve or the break in the M–H curve. In Sc₅Ir₄Si₁₀, H_c2 for fields along the c-axis, H_c2^c, is larger than for fields in the ab-plane H_c2^ab [15]. The anisotropy parameter γ is again about 2.0, and it is only weakly temperature dependent. This result indicates that Sc₅Ir₄Si₁₀ is an anisotropic superconductor with weak one-dimensional anisotropy [15].

Figure 9.

Temperature dependence of the upper critical fields, H_c2^c and H_c2^ab, estimated from the M–T (circles) and M–H (squares) curves for Sc₅Ir₄Si₁₀.

Figure 10 shows the temperature dependence of specific heat in Sc₅Ir₄Si₁₀. The jump of specific heat at T_c is larger than the BCS value, suggesting that Sc₅Ir₄Si₁₀ is a strong-coupling superconductor. The obtained value of the electronic specific heat coefficient γ is 30.5 mJ (mol K²)^{- 1}. It should be noted, however, that this value of γ is considerably smaller than the previously reported value of 9.93 mJ (mol K²)^-1 for Sc₅Ir₄Si₁₀ and is closer to the value of 30.9 mJ (mol K²)^{- 1} for Sc₅Co₄Si₁₀ [16]. Actually, recent nuclear magnetic resonance studies on Sc₅Co₄Si₁₀, Sc₅Ir₄Si₁₀, and Sc₅Rh₄Si₁₀ show rather similar values of the density of states at the Fermi level in these three compounds [17]. Therefore, we believe that our value of γ is more intrinsic. Coming back to figure 10, the broken line shows the BCS temperature dependence of the electronic specific heat coefficient with 2Δ/k_BT_c = 3.53. A systematic deviation is obvious. The BCS formula with different values of 2Δ/k_BT_c also does not fit the data. Such a discrepancy between the experimental result and the theory can be amended by assuming two superconducting gaps. The solid line in figure 10 shows the best fit based on the two-gap model similar to the case of Lu₂Fe₃Si₅. The two gap values obtained for Sc₅Ir₄Si₁₀ are 2Δ ₁/k_BT_c = 4.1 and 2Δ ₂/k_BT_c = 1.9 with x₁/x₂ = 78/22. The Arrhenius plot of C_e (T) in the inset of figure 10 shows the slope corresponding to the smaller gap.

Figure 10.

Temperature dependence of specific heat for Sc₅Ir₄Si₁₀. C_e(T) based on the BCS model is shown by dashed line. The solid line shows the best fit using the two-gap model. Inset shows Arrhenius plot of C_e(T). The solid line is a guide for the eye suggesting the presence of smaller gap.

Here, we discuss the plausibility of the two-gap scenario in Sc₅Ir₄Si₁₀. Strictly speaking, C_e (T) in most superconductors deviates from the simple weak-coupling BCS formula. Recently, the two-gap feature was also reported in Nb₃Sn [18]. Huang et al analyzed C_e (T) in YNi₂B₂C with different gap symmetries including the two-gap model [19]. They concluded that both the point-node model and the two-gap model fit the experimental data equally well. In the case of Lu₂Fe₃Si₅, and possibly also Sc₂Fe₃Si₅, two-gap analyses have a rather firm ground since we do observe a clear second drop in C_e (T) at lower temperatures. In other words, the smaller gap has a markedly different gap value from the larger one with significant fraction. In the case of Sc₅Ir₄Si₁₀, the deviation of C_e (T) from the BCS model is too small, and correct gap structure can be identified only by combining other experimental results such as impurity effects. In this sense, T_c being almost independent of the Lu content in (Lu_{1 - x}Sc_x)₅Ir₄Si₁₀ [20] might suggest that Sc₅Ir₄Si₁₀ is a superconductor with a weak gap anisotropy rather than a two-gap superconductor.

Summary

The temperature dependence of specific heat in the silicide superconductors Lu₂Fe₃Si₅, Sc₂Fe₃Si₅, and Sc₅Ir₄Si₁₀ were investigated down to 0.35 K. The phenomenological two-gap model reproduced C_e (T) reasonably well in all compounds. Although the two-gap scenario for superconductivity has a firm ground for Lu₂Fe₃Si₅ and Sc₂Fe₃Si₅, a superconductor with a weak gap anisotropy might be more appropriate for Sc₅Ir₄Si₁₀. It was also found that all these compounds are anisotropic superconductors with weak one-dimensional anisotropy.