Large enhancement of superconducting transition temperature of SrBi3 induced by Na substitution for Sr

The Matthias rule, which is an empirical correlation between the superconducting transition temperature (Tc) and the average number of valence electrons per atom (n) in alloys and intermetallic compounds, has been used in the past as a guiding principle to search for new superconductors with higher Tc. The intermetallic compound SrBi3 (AuCu3 structure) exhibits a Tc of 5.6 K. An ab-initio electronic band structure calculation for SrBi3 predicted that Tc increases on decreasing the Fermi energy, i.e., on decreasing n, because of a steep increase in the density of states. In this study, we demonstrated that high-pressure (~ 3 GPa) and low-temperature ( < 350 °C) synthesis conditions enables the substitution of Na for about 40 at.% of Sr. With a consequent decrease in n, the Tc of (Sr,Na)Bi3 increases to 9.0 K. A new high-Tc peak is observed in the oscillatory dependence of Tc on n in compounds with the AuCu3 structure. We have shown that the oscillatory dependence of Tc is in good agreement with the band structure calculation. Our experiments reaffirm the importance of controlling the number of electrons in intermetallic compounds.

The most straightforward way to decrease n in SrBi 3 is to replace divalent Sr ions with monovalent alkali metal ions, such as K or Na. However, to our knowledge, such substitution has not successfully been performed in usual intermetallic compounds, because alkali metals are far more reactive compared to alkaline earth metals. The synthesis conditions of the substituted samples are thus entirely different from those of the pristine ones; in particular, the substituted samples require low-temperature and tightly sealed conditions in order to prevent the evaporation of volatile alkali metals. We realized such conditions by using a cubic-anvil-type high-pressure (HP) apparatus and succeeded in synthesizing (Sr,Na) Bi 3 . As expected, the T c of (Sr,Na)Bi 3 increases with Na concentration, reaching up to 9.0 K. Based on the present results, we demonstrated that there is a new and higher peak in the oscillatory relationship between T c and n for materials with an AuCu 3 structure and that the relationship results from the n-dependence of N(E), which is characteristic of this crystal structure.

Results
Introduction of Na and enhancement in T c through the high-pressure, low-temperature synthesis. Figure 2 shows the χ(T) of the samples with nominal compositions of (Sr 1-x Na 1.5x )Bi 3 (x = 0.5) synthesized under various temperatures (T syn ) ranging from 300-450 °C for 6 h. For the case of T syn = 350 and 300 °C, samples were slowly cooled (20 °C/h) to 225 and 200 °C, respectively. All the samples exhibit higher T c compared to pristine SrBi 3 (T c = 5.6 K). Moreover, T c increases on lowering T syn , up to 9.0 K (indicated by an arrow) for T syn = 300 °C. The T c of 9.0 K is the second highest among the superconductors possessing the AuCu 3 structure, the highest being T c = 9.54 K for InLa 3 8 . The lattice parameter of the samples, a, decreases with decreasing T syn ; for example, a = 5.013, 4.992, and 4.989 Å for T syn = 450, 400 and 300 °C, respectively. (a = 5.04 Å for SrBi 3 ). The decrease in the lattice parameter is due to the increase in Na substitution at the Sr sites, as will be elaborated in a later section.
It should be noted that HP, low-temperature synthesis conditions promote the Na substitution for Sr. Indeed, the sample synthesized under ambient conditions (x = 0.5, annealed at 300 °C in an evacuated quartz tube) exhibits a lower T c of 6.5 K and larger a of 5.033 Å.
Change in the structure upon Na substitution: powder X-ray diffraction patterns. As the next step, we synthesized series of samples with various values of x -0, 0.2, 0.4, 0.5, and 0.6 -by reacting the starting materials at 350 °C for 6 h, following which annealing was performed at (or with slow cooling down to) 275-225 °C for 6 h at a pressure of 3.4 GPa.
The powder XRD patterns of the reacted samples are shown in Fig. 3. Major peaks can be indexed on the basis of the cubic unit cell expected for the AuCu 3 structure. The a value of SrBi 3 (x = 0) calculated using a least-squares fitting is 5.043 Å, which is in good agreement with the previously reported value (5.035-5.04 Å) 9 . The overall X-ray patterns do not change with increase in x, while the peak width broadens, which is due to the disorder associated with the Na substitution at the Sr sites. Diffraction peaks corresponding to Bi are observed for all samples, with their intensity increasing with x. For the samples with x = 0.5 and 0.6, peaks corresponding to NaBi are also observed, indicating that the introduced Na is not completely incorporated into the samples and that the solubility limit is around x = 0.4-0.5.
The x-dependence of lattice parameter a and T c . Figure 4(a) shows χ(T) for the samples with various values of x. SrBi 3 (x = 0) shows a sharp superconducting transition at 5.6 K, which is in good agreement with the reported value. As x increases, T c monotonously shifts to higher temperatures. Note  that the transition width does not change with changing x, suggesting that Na is uniformly incorporated into the samples. The shielding volume fraction calculated from the ZFC susceptibility value at 5 K exceeded 100% for all samples, indicating bulk superconductivity.
In Fig. 4(b), the lattice parameter a and T c are plotted as functions of x. The lattice parameter a decreases linearly with x up to x = 0.3 and then saturates above x = 0.4. The decrease of a results from the substitution of smaller Na + (with ionic radius (XII coordination) of 1.39 Å 10 ) into the Sr 2+ (1.44 Å) sites. The substitution of larger K + (ionic radius of 1.64 Å) into the Sr 2+ sites was not successful. As seen in Fig.  4(b), T c and a exhibit similar x-dependences; they change linearly with x up to x = 0.3 and then saturate in the vicinity of x = 0.4. These behaviours again indicate that the Na solubility limit is around x = 0.4.

Discussion
In the present study, we demonstrated that the T c of SrBi 3 increases with decreasing n. The relationship between T c and n for compounds with the AuCu 3 structure was first discussed by Havinga et al. 9 . They showed that T c exhibits an oscillatory dependence on n with peaks at n = 3.75 and 4.00. For example, LaSn 3 (T c = 6.02 K) and ThPb 3 (T c = 5.55 K) have n = 3.75 and 4.00, respectively. Most known superconductors with the AuCu 3 structure have n values in the range 2.75 ≤ n ≤ 4.00. SrBi 3 (n = 4.25) has an exceptionally large n, which enabled us to investigate T c for 4.00 ≤ n ≤ 4.25. Figure 5 shows the general relationship between n and T c for superconductors with the AuCu 3 structure that contain Pb, Bi, or Tl at the crystallographic Cu site. Their band structures near E F are similar; the band structures are dominated by Bi (or Pb, Tl) 6p orbitals, as shown in Fig. 1. In Fig. 5, one can recognize a clear oscillation of T c with respect to n. In particular, a new T c peak corresponding to (Sr,Na) Bi 3 is observed at n ~ 4.15. This peak is higher than those located around n = 3.25 and 3.70. These T c peaks reflect the shape of N(E) shown in Fig. 1. In Fig. 1, the N(E) peak immediately below E F corresponds to n = 4.0, and the next two N(E) peaks at energies of ~ 0.41 Ry and ~ 0.38 Ry correspond to n ~ 3.5 and n ~ 3.25, respectively. These peaks are mainly attributed to the Bi-6pπ-bonding bands, which have narrow band widths. Considering the crudeness of the rigid band model, the agreement between the experiments and theory is reasonably good.
In summary, we have demonstrated that the T c of SrBi 3 increases by tuning the number of valence electrons on the basis of the prediction from the band structure calculation. High-pressure and low-temperature synthesis conditions enables to substitute a large amount of Na for Sr in SrBi 3 . Consequently, the T c of SrBi 3 increases from 5.6 K to as high as 9.0 K by decreasing the n. We have shown that a new high oscillation peak appears on the n dependence of T c in compounds with the AuCu 3 structure and the T c oscillation is in good agreement with the band structure.

Methods
Electronic band structure calculation. Our calculation is based on the local-density approximation (LDA) and implemented using the computer code KANSAI-94 and TSPACE 11 . Spin-orbit interaction is included in a second-variational procedure. We used the experimental value of 5.043 Å for the lattice parameter a. The muffin-tin radii were set as 0.252a for Sr and 0.231a for Bi.
Preparation of (Sr,Na)Bi 3 samples. Series of polycrystalline samples of (Sr,Na)Bi 3 were synthesized through a solid-state reaction by using a cubic-anvil-type HP apparatus. The starting materials were Bi, Na 3 Bi, and Sr 5 Bi 3 powders. Na 3 Bi powder was prepared by heating an appropriate amount of Na and Bi chunks at 900 °C in an alumina crucible sealed in a stainless-steel vessel 12 . Sr 5 Bi 3 was prepared by heating a mixture of Bi and Sr powders to 950 °C by using the HP synthesis method or a method similar to that used for preparing Na 3 Bi powder.
An appropriate amount of the starting materials were ground with agate mortar in a nitrogen-filled glove box and pressed into a pellet. Excess Na (50%) was added to the starting compositions to compensate for possible Na loss during the heat treatment. The pellet with a nominal composition of (Sr 1-x Na 1.5x ) Bi 3 was placed in a BN crucible and assembled into an HP cell 13 . The sample was heated under a pressure of 3.4 GPa. As we have described, the T c of samples strongly depends on the sample synthesis temperature (T syn ); T c monotonously increases with decreasing T syn , even when starting from the same nominal values of x. The resulting samples were handled in a nitrogen-or argon-filled glove box because of their reactivity in air.
Material characterization. Powder X-ray diffraction (XRD) patterns were measured at room temperature using Cu K α radiation. Because the samples are easily degraded by reactions with oxygen and/or moisture in air, a polyimide adhesive tape was placed on the sample, and the XRD pattern was collected for 8 min using a diffractometer equipped with a high-speed detector system (Rigaku, D/teX Ultra). Temperature (T)-dependent magnetic susceptibility (χ(T)) measurement was performed using a magnetic property measurement system (MPMS) (Quantum Design, MPMS-XL7) under a magnetic field of 0.001 T. The data were collected during warming after zero-field cooling (ZFC) and then during field cooling (FC). T-dependent electrical resistivity (ρ(T)) was measured using the four-probe method under magnetic fields of up to 2.4 T. Because the sample is unstable in air, the electrodes were coated with a silver paste in an argon-filled glove box and covered with APIEZON grease before exposure to air.