S-wave pairing symmetry in non-centrosymmetric superconductor Re W 3 8 0 0 2 n ∗ a Jing Yan, Lei Shan, Qiang Luo, Weihua Wang, Hai-Hu Wen J 2 1 National Laboratory for Superconductivity, Institute of Physics and Beijing ] n o National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, c - r P. O. Box 603, Beijing 100080, P. R. China p u s . t a m Abstract - d n o The alloys of non-centrosymmetric superconductor, Re3W, which were reported c [ to have an α-Mn structure [P. Greenfield and P. A. Beck, J. Metals, N. Y. 8, 1 v 265 (1959)] with Tc = 9 K were prepared by arc melting. The ac susceptibility 9 7 8 and low-temperature specific heat were measured on these alloys. It is found that 1 1. there are two superconducting phases coexisting in the samples with Tc1 ∼ 9 K 0 8 and Tc2 ∼ 7 K, both of which have a non-centrosymmetric structure as reported 0 : previously. By analyzing the specific heat data measured in various magnetic fields, v i X we found that the absence of the inversion symmetry does not lead to the deviation r a from a s-wave pairing symmetry in Re3W. Key words: Re3W, Non-centrosymmetric, Superconductivity, Specific heat ∗ Corresponding author. Email address: [email protected] (Hai-Hu Wen). Preprint submitted to Elsevier 2 February 2008 1 Introduction Very recently the scientific community has paid a lot of attention in under- standing the supercondictivity of the non-centrosymmetric superconductors, since the superconducting properties of such materials are expected to be un- conventional [1,2,3,4,5,6,7]. In a lattice with inversion symmetry, the orbital wave function of the cooper pair has a certain symmetry and the spin paring will be simply in either the singlet or triplet state. The noncentrosymmetry in the lattice may bring a complexity to the symmetry of orbital wave func- tion. This effect with the antisymmetric spin-orbital coupling gives rise to the broken of the spin degeneracy, thus the existence of the mixture of spin singlet and triplet may become possible[2,5]. So there might be something unconventional, such as spin triplet pairing component, existing in the non- centrosymmetric superconductors. Recently, a spin-triplet pairing component was demonstrated in Li2Pt3B both by penetration depth measurement[4] and nuclear magnetic resonance (NMR)[5], as was ascribed to the large atomic number of Pt which enhances the spin-orbit coupling. Re3W is one of the rhenium and tungsten alloys’ family. Up to now, two superconducting phases of Re3W were reported with Tc ∼ 9 K[8] and Tc ∼ 7K[9]. Both phases belong to the α-Mn phase (A12, space group I¯43m)[10], which has a non-centrosymmetric structure. Moreover, atomic numbers of Re and W are 75 and 74, respectively, being close to that of Pt. Therefore, similar spin-triplet pairing component as that inLi2Pt3Bare expected in Re3W. Most recently, it was found that the superconducting phase of Re3W with Tc ∼ 7 K is a weak-coupling s-wave BCS superconductor by both penetration depth [9] and Andreev reflection measurements [11]. 2 In this paper, we report the measurements of the ac susceptibility and low- temperature specific heat of Re3W alloys. Both the measurements imply that our samples have two superconducting phases with critical temperatures near 9 K and 7 K, respectively, and the high temperature phase near 9 K accounts for nearly 78%-87% in total volume. The specific heat data can be fitted very well by the simple two-component model, which is based on the isotropic s- wave BSC theory. Furthermore, a linear relationship is found between the zero-temperature electronic specific heat coefficient and the applied magnetic field. These results suggest that the absence of the inversion symmetry does not result in novel pairing symmetry in Re3W. 2 Experiment The Re3W alloys are prepared by arc melting the Re and W powders (purity of 99.9% for both) with nominal component 3 : 1 in a Ti-gettered argon atmo- sphere. Normally,theobtainedalloyisahemisphere inshapewithadimension × of 5 mm (radius) 5 mm (height). Some pieces of the alloy had been cut from the original bulk (e.g. sample ♯1 and sample ♯2). The ac susceptibility of these samples has been measured at zero dc magnetic field to identify their super- conducting phases, whereas, all of them have two superconducting transitions at about 9 K and 7 K, as shown in Fig. 1. The specific heat was measured by a Physical Property Measurement System (PPMS, Quantum Design). The data ata magnetic fieldwere obtainedwith increasing temperature afterbeing cooled in field from a temperature well above Tc, namely, field cooling process. 3 0.6 #1 (a) 0.4 " g) 0H=0, 0.05, 0.1, 0.15, m 0.2 0.2, 0.25, 0.3 T 2 / m A 8 - 0 0.0 1 ( ’ " d 1 " (b) n a 0 ’ -1 -2 ’ -3 -4 -5 0H=0, 0.25, 0.3, 0.4, 0.6, 1, 1.5, 2, 3, 4, 5, 7 T -6 2 3 4 5 6 7 8 9 10 T (K) Fig. 1. (Color online) Temperature dependence of ac susceptibility on sample ♯1 underdifferentdcmagnetic fields,with acfield h= 1 Oeandfrequencyf = 333 Hz. 3 Results and discussion ′ ′′ The temperature dependence of ac susceptibility (χ = χ +iχ ) at different dc magneticfieldsfrom0 Tto7 TisshowninFig.1.Onecanseethattwodistinct superconducting transitions occur at Tc1 ∼ 9 and Tc2 ∼ 7 K in χ′(T) curve at ′′ H = 0 [Fig. 1(b)], and double peaks in χ (T) show up at the corresponding temperatures. These two phases are consistent with the previous reports in ′′ which they are proofed to be non-centrosymmetric[8,9]. The peaks of χ shift to lower temperatures as the magnetic field increases, showing the continuous suppression of superconductivity by the magnetic field. The low-T peak shifts to lower temperatures more slowly than the high-T one, indicating distinct behaviors of the upper critical fields in these two superconducting phases. As 4 80 #2 ) 2 K60 ol m J/ m40 ( T C/ 0H=0.0T 0H=0.5T 20 0H=1.0T 0H=2.0T 0H=3.0T 0H=4.0T 0H=5.0T 0H=7.0T 0 0 40 80 120 160 2 2 T (K ) 2 Fig. 2. (Color online) Specific heat data of sample ♯2 plotted as C/T versus T at various fields. H increases to ∼ 7 T, the χ(T) curves are completely flat, showing no sign of superconducting transition. Similar results were obtained on sample ♯2 and other samples. We thus measured the specific heat of sample ♯2 and in Fig. 2 we present 2 the data of C/T versus T at various magnetic fields. On each curve, there are two jumps related to the superconducting transitions consistent with the measurements of ac susceptibility. From the zero field data in low tempera- ture region, one can see that the residual specific heat coefficient γ0 is close to zero, implying the absence of non-superconducting phase. The superconduct- ing anomaly is suppressed gradually with increasing magnetic field, and from the curve at 7 T there is no sign of superconductivity above 1.8 K, consistent with the observation in χ(T) curve. The low temperature part of the normal state specific heat at H = 7 T in Fig. 2 is not a straight line, implying that 3 the specific heat of phonon does not satisfy the Debye’s T law. We may need 7 a T term to fit the normal state specific heat well: 2 4 6 Cn/T = γn +β3T +β5T +β7T . (1) 5 40 Ce=0.78 Ce1(Tc1=9K)+0.22 Ce2(Tc2=6.75K). 2 ) 30 K ol m J/ 20 m ( /Te10 0H=0.0T 0H=0.5T C 0H=1.0T 0H=2.0T 0 0H=3.0T 0H=4.0T 0H=5.0T 0H=7.0T 0 2 4 6 8 10 12 T (K) Fig. 3. (Color online) Specific heat of electrons plotted as Ce/T versus T. The solid lines are the calculating results which separate the electronic specific heat into two components with different Tc by using specific heat formula based on the BCS theory. The first termis theelectronic specific heat inthe normalstate,and theothers are the contributions of the phonons. Fitting the data of 7 T to Eq. (1), the coefficients γn = 17 ± 0.1 mJ/mol K2, β3 = 0.185± 0.001 mJ/mol K4, β5 = (1.63±0.01)×10−3 mJ/mol K6, and β7 = (−2.087±0.005)×10−6 mJ/mol K8 are determined. From the relation: 4 12π NAkBZ β3 = 3 , (2) 5 Θ D where NA = 6.02 × 1023 is the Avogadro constant, and Z = 4 the number of atoms in one unit cell, we obtained the Debye temperature of our alloys ΘD = 347.9 K. These coefficients and Debye temperature are all very close to the results of other works on Re-W alloys[12,13,14,15,16,17,18]. By subtracting the phonon contribution, the electronic specific heat Ce is obtained, which is shown in Fig. 3 as Ce/T versus T. Before a quantitative analysis, the low temperature specific heat at low fields has presented a strong 6 0.2 (a) 0.0 0H=0T " -0.2 d an -0.4 ’ -0.6 -0.8 0.85 -1.0 #2 ) (b) #2 2 K 30 ol m J/ 20 m ( T 10 / e C 0 Ce=0.78 Ce1(Tc1=9K)+0.22 Ce2(Tc2=6.75K). 0 2 4 6 8 10 12 14 T (K) Fig. 4. (a) shows the ac susceptibility of sample ♯2 on which the specific heat have been measured. (b) shows the zero field specific heat data, and the black line is the calculating result based on the BCS theory. evidence that Re3W has a nodeless gap function. For a nodal superconductor (expected by the strong mixing of spin-singlet and spin-triplet pairing com- ponents in a heavily non-centrosymmetric superconductor such as Li2Pt3B), the low temperature C/T vs. T relation should be a power law like. However, as denoted by the dashed lines in Fig. 3, if a linear relationship is assumed, the specific heat at zero field would be negative when the temperature ap- proaches to zero. In the following section, by using a quantitative analysis, we will demonstrate that both phases of Re3W have an isotropic gap function, which is in goodagreement with the expectation of an s-wave superconductor. Figure 4 shows the ac susceptibility and specific heat data at zero dc field measured on the same sample(♯2). The ac susceptibility data have been nor- 7 ± malized. The high temperature phase occupies nearly 85 1% in the whole superconducting volume. In order to fit the zero field electronic specific heat, we attempt to use the formula derived from thermodynamic relations based on the BCS theory[19] ∞ 4N(0) eζ/kBT T d∆2(T) Ces = kBT2 Z (1+eζ/kBT)2(ε2 +∆2(T)− 2 dT )dε, (3) 0 where ζ = ε2 +∆2(T), and ∆(T) is an isotropic s-wave gap which depends q on temperature in the same way as expected by BCS theory. Since there are two coexistent phases in our samples, we use two separate terms of C and H CL to take into account the contributions of the high Tc and low Tc phases, respectively. Thus the total specific heat can be expressed as follows: Ce = ωHCH +ωLCL,(ωH +ωL = 1), (4) inwhichω andω indicatetheweight ofthecontributionsforthetwophases. H L According to Eq. (3) and Eq. (4) we can nicely simulate the experimental data very well as presented in Fig. 4(b) by a solid line. The parameters for the best fit are ∆0H = 1.4 meV, ωH = 0.78 for TcH = 9 K and ∆0L = 1.1 meV, ωL = 0.22 for TcL = 6.75 K and ∆0 is the gap value at zero temperature. Interestingly, ω = 0.78 found here is very close to the relative weight 85% H of the high temperature phase which was obtained from the ac susceptibility data in Fig.4(a).Furthermore, ∆0L ∼ 1.1 meV is ingoodagreement with that from the penetration depth and Andreev reflection experiments[9,11]. These results give a strong evidence that there is no novel pairing symmetry in our alloys. To get further evidence for this argument, we did similar calculations for the 8 20 15 ) 2 K ol m 10 J/ m ( 5 0 0 1 2 3 4 5 0H (T) Fig. 5. The electronic specific coefficient γ(H) at zero temperature obtained from the calculation based on BCS theory. specific heat in the mixed state using the same weights of the two phases obtained from the zero field calculation. In the mixed state, there are two dif- ferent regions, namely the core region and the outside core region. Therefore we adopted a simple two-component model[20,21] which separates the elec- tronic specific heat into two components. The electronic specific heat is thus written as H H Ce = α γnT +(1−α )Ces. (5) Hc2(0) Hc2(0) Here α is an adjustable parameter. The first part on the right hand side is the quasi-particle density of states (DOS) coming from the normal vortex core regions, and the second part comes from the superconducting regions outside the cores. The results of the quantitative calculations are plotted as solid lines in Fig. 3, and they are in good agreement with the experimental data for all magnetic fields. In the superconducting state, Ce = γT, where γ is the electronic specific heat coefficient that is dependent on temperature and field. According to 9 Eq. (5), the zero temperature electronic specific heat coefficient γ(H) is equal to αH/Hc2(0)γn, which is shown in Fig. 5 as solid squares, and the solid line is a linear fit to the data. The obvious linear relationship of γ vs. H presents further evidence that Re3W is a conventional superconductor in which γ(H) is proportional to the number of vortex cores and hence to the applied field. For a nodal superconductor with novel pairing symmetry, on the other hand, a nonlinear γ(H) relation should be expected, which is obviously not the case in our present samples[22]. 4 Conclusion In summary, we have synthesized Re3W alloys by arc melting. From the mea- surements of ac susceptibility and specific heat on the alloys two distinct superconducting phases were found. Both the qualitative and quantitative analysis were done on the specific heat data in zero field and the mixed state. We found that the simple two-component model based on the BCS theory with an isotropic s-wave gap can fit our experimental data very well, and we obtained a linear γ(H) relationship. All these results indicate that the absence of the inversion symmetry does not result in any novel pairing symmetry in Re3W for both Tc ∼ 7 K and Tc ∼ 9 K phases. Acknowledgments This work was supported by the National Science Foundation of China, the Ministry ofScience andTechnologyofChina(973Project:No.2006CB601000, No. 2006CB921802, No. 2006CB921300), the Knowledge Innovation Project 10