Superconductivity in the nonsymmorphic line-nodal compound CaSb$_2$

We found superconductivity in CaSb$_2$ with the transition temperature of 1.7 K by means of electrical-resistivity, magnetic-susceptibility, and specific-heat measurements. This material crystallizes in a nonsymmorphic structure and is predicted to have multiple Dirac nodal lines in the bulk electronic band structure protected by symmetry even in the presence of spin-orbit coupling. We discuss a possible topological superconductivity for the quasi-2-dimensional band originating mainly from one of the antimony sites.


I. INTRODUCTION
Topological quantum phenomena have been one of the hottest topics in recent condensed-matter physics [1][2][3]. Many fascinating topological phenomena such as the surface Dirac cone and spin current have been actively investigated in topological insulators. Moreover, superconductors are also classified by topology [4][5][6]. Topological superconductors are intimately related to topological insulators, and indeed topological superconductivity was found in a doped topological insulator [7][8][9][10].
Nowadays, Dirac and Weyl semimetals have been attracting much attention as new kinds of topological materials [11,12]. These materials are characterized by the topologically or symmetrically protected gapless point in the bulk electronic states in the reciprocal space. Such gapless points lead to peculiar properties such as ultrahigh mobility, chiral anomaly, and surface Fermi arcs. It is also predicted that Dirac and Weyl semimetals can become topological superconductors [13][14][15]. Thus, superconductivity experimentally found in pressurised and doped Dirac materials [16,17] can be leading candidate for realization of topological superconductors.
More recently, the concept of the Dirac and Weyl semimetals has been extended to the line-nodal materials, in which the bulk energy gap closes along a line in the Brillouin zone [18][19][20]. These materials are predicted to exhibit several notable phenomena including a longranged Coulomb interaction [21], a large surface polarization charge [22], and quasitopological electromagnetic response [23]. Furthermore, if superconductivity is induced in the materials with torus-shaped Fermi surfaces derived from nodal loops, topological crystalline and second-order topological superconductivity is expected to be realized [24].
Although many materials can have nodal lines originating from band crossings in the absence of spin-orbit coupling, such nodal lines generally become gapped in the presence of spin-orbit coupling and the material turns * a.ikeda@scphys.kyoto-u.ac.jp into an insulator or a point-nodal semimetal [19,25]. An additional symmetry such as nonsymmorphic (screw or glide) symmetry is necessary to retain the gapless nodal lines [18]. Thus, so far, only a limited number of materials are proposed as line-nodal materials in the presence of spin-orbit coupling [26][27][28][29], and superconductivity in materials with such nodal lines robust against spin-orbit interaction is even more limited [30].
In this paper, we report discovery of superconductivity with the transition temperature T c = 1.7 K in the nonsymmorphic metal CaSb 2 . A band structure calculation shows that CaSb 2 has several nodal lines protected by the screw and mirror symmetries of the space group P 2 1 /m even in the presence of spin-orbit coupling and that some nodes cross the Fermi energy [31]. We present evidence of bulk superconductivity obtained by means of magnetic susceptibility, resistivity and specific heat. The zerotemperature upper critical field is found to be around 0.2 T, corresponding to the coherence length of 40 nm. We also discuss the possibility of topological superconductivity. This superconductivity in a nonsymmorphic nodal-line material can trigger exploration of possible novel superconducting phenomena originating from the robust nodal lines.

II. EXPERIMENT
Stoichiometric mixture of Ca (Sigma-Aldrich, 99.99%) and Sb (Rare Metallic, 99.99%) was placed in an alumina crucible (IRIE Corporation, SSA-S), and the crucible was sealed in a quartz tube under 0.2 atm of argon at room temperature. The tube was heated up to 1,000 • C in 3 h, kept at that temperature for 24 h using a box furnace (Denken, KDF 80S), and quenched in water in order to prevent the phase splitting into Ca 11 Sb 10 and Sb [34].  [31,32]. The bars at the bottom indicate the reported peak positions of CaSb2 (blue bars; PDF 01-071-0135) and Al2O3 (black bars; PDF 01-089-7716). For our sample, very small peaks of the latter are attributable to a contamination from crucibles. The inset shows the crystalline structure of CaSb2 with the nonsymmorphic space group P 21/m (No. 11, C 2 2h ) with the screw axis along b. The structural image is produced using the program VESTA [33].
nealing temperature was chosen to be below the CaSb 2 -Sb eutectic temperature of 587 • C [34]. We also tried a method similar to the previously reported one, starting from a molar ratio of Ca:Sb = 1:3.9 [31] and slow cooling from 700 • C to 570 • C, but it resulted in a sample containing a lot more additional Sb side product.
Powder x-ray diffraction (PXRD) was measured with a commercial diffractometer equipped with an array of 192 detectors (Bruker AXS, D8 Advance) using the Cu-Kα radiation. We did not find any trace of decomposition of CaSb 2 even after the sample was kept in air at room temperature for more than two weeks.
Direct-current (DC) magnetization was measured with a commercial magnetometer using a superconducting quantum interference device (Quantum Design, MPMS-XL) equipped with a 3 He refrigerator (IQUANTUM, iHelium3). A cut pellet of 50.6 mg (approximately 3.3 × 3.4 × 1.0 mm 3 ) was used. Alternating-current (AC) magnetic susceptibility was measured with a lockin amplifier (Stanford Research Systems, SR830) using a miniature susceptometer [35] compatible with an adiabatic demagnetization refrigerator (ADR) of a commercial apparatus (Quantum Design, PPMS).
The electrical resistivity was measured by a standard four-probe method using the ADR option of PPMS. Gold wires with a diameter of 25 µm were attached to the pellet using room-temperature-cure silver paste (DuPont Electronics Material, 4929N) with n-butyl acetate as solvent. We comment that zero resistance was not observed when we used silver epoxy (Epoxy Technology, H20E), possibly because the sample reacted with air or with the silver epoxy during the curing process at 120 • C.
Heat capacity from 0.42 to 3.4 K was measured using a calorimeter for a 3 He option of PPMS, by using the relaxation method. A cut pellet of 15.7 mg (approximately 2 × 2 × 1 mm 3 ) was fixed on the sample stage with a thin layer of grease (Apiezon, N-grease) to obtain good thermal connection. For each measurement, a heat pulse raised the sample temperature by 3% of the bath temperature and the temperature relaxation curve was fitted using a double exponential function to evaluate the heat capacity.

III. RESULTS
First, we present the PXRD pattern in Fig. 1. The pattern revealed that our CaSb 2 sample is single phased, except for a slight Al 2 O 3 impurity attributable to contamination from the crucible. The lattice parameters of the CaSb 2 phase were extracted to be a = 0.4741 nm, b = 0.4182 nm, c = 0.9073 nm, and β = 106.3 • with the space group P 2 1 /m (No. 11, C 2 2h ) by the Rietveld analysis with a weighted pattern R value R wp = 12.7% and goodness of fit S = 1.21. The fitting errors were about 0.002%. These lattice parameters are consistent with the previous reports within 0.06% [31,32].
Next, we show the DC magnetic susceptibility in Fig. 2. Strong diamagnetism indicates superconductivity below 1.8 K. The susceptibility reaches −1.54 at 0.5 K under 0.5 mT. Assuming a rectangular sample shape, the superconducting shielding fraction after demagnetization correction is estimated to be 79% [36]. Later in this paper, the volume fraction is evaluated from the specific heat to be 65%. The susceptibility is suppressed to −0.33 under 5 mT, suggesting a lower critical field H c1 smaller than 5 mT. Figure 3 shows the AC susceptibility of CaSb 2 . Diamagnetism in the real part of the susceptibility, as well as a small peak in the imaginary part, was observed with an onset of 1.7 K, with an AC field of 3.3 or 8.2 µT. In addition to the transition at 1.7 K, we observed an- other broad feature at around 1 K signaled by an additional increase in the diamagnetic signal and by a large and broad peak in the imaginary part. For the following three reasons, we consider that this second feature reflects the percolation of the superconducting current among the polycrystalline grains. Firstly, the shielding fraction changes with the amplitude of the AC field only below the second anomaly as shown in Fig. 3. Secondly, the second feature is absent in the DC magnetic susceptibility as shown in Fig. 2. Thirdly, the peak of the imaginary part of the AC susceptibility is about 40 times larger at 1 K than at 1.6 K under 3.3 µT, even though the diamagnetism seen in the real part is comparable. These behaviors cannot be explained by the bulk effect and are consistent with the Josephson coupling across the grain boundary. The shielding fraction decreases with increasing AC field because the induced supercurrent exceeds the critical current of the grain boundary. In addition, there is a large energy dissipation due to rapid movement of vortices along the boundary. Similar grain-boundary effect has been reported in various polycrystalline or inhomogeneous materials [37,38]. The temperature dependence of the resistivity is presented in Fig. 4(a). Under cooling at zero field, the resistivity starts to decrease at 1.8 K and reaches zero at 1.2 K. The large resistivity value of 9 mΩ cm in the normal state is probably dominated by scatterings at the grain boundaries. The superconductivity is suppressed as we apply the magnetic field, and above 120 mT the zero resistivity was not observed down to 0.16 K. Figure 4(b) shows the superconducting phase diagram deduced from the field dependence of the resistivity. The transition temperature at each field was defined as the temperature at which the resistivity becomes 5, 50, and 95% of the normal-state value. As we can see especially in T c determined by the 95% criteria, the slope of the phase boundary becomes steeper above 40 mT. This behavior might be related to the second feature observed in the AC susceptibility (Fig. 3). Small applied magnetic field may break the superconductivity at the grain boundary. As a result, the resistance increases and consequently the transition temperature rapidly decreases with small field. At relatively low temperature, or in the high-field region in the phase diagram, superconductivity at the grain boundary seems less sensitive to magnetic field. Due to this enhancement, we adopted a linear extrapolation of the upper critical field at the low temperatures instead of the ordinary Werthamer-Helfand-Hohenberg relation.
Then the upper critical field at 0 K is estimated to be 0.17-0.26 T depending on the definition of T c . These values correspond to the Ginzburg-Landau coherence length of ξ GL = Φ 0 /(2πµ 0 H c2 ) = 36-44 nm. Using the information of µ 0 H c1 < 5 mT, the penetration depth λ and the Ginzburg-Landau parameter κ = λ/ξ GL are evaluated to be λ > 240 nm and κ > 5.4 by solving the equa- tion µ 0 H c1 = (ln κ)Φ 0 /(4πλ 2 ) [39]. Finally, we show the temperature dependence of the electronic specific heat C e (T ) in Fig. 5(a). We observed a broad but pronounced anomaly starting at 1.5 K. This broadening is most likely due to inhomogeneous sample property as reflected by the AC susceptibility in Fig. 3.
We fitted the data under zero field assuming that T c has a spacial distribution described by the lognormal (LN) distribution f LN (x): where C P (T ) is the specific heat including the phononic contribution βT 3 + αT 5 and the Schottky specific heat A/T 2 , and C BCS (T ; T c ) is the theoretical specific heat of a fully gapped superconductor derived from the Bardeen-Cooper-Schrieffer (BCS) theory with a critical temperature of T c [39]. The Schottky specific heat likely originates from the quadrupole splitting of the Sb nucleus or from the Zeeman splitting of the electronic spin of possible impurities. For C P under 20 mT, we also used C BCS (T ; T c ) valid for H = 0. The Sommerfeld coefficient and the phononic specific heat coefficients were determined to be γ = 2.044(3) mJ mol −1 K −2 , β = 0.537(2) mJ mol −1 K −4 , and α = 0.0037(2) mJ mol −1 K −6 from the data under µ 0 H = 80 mT using Eq. (1) with C e = γT . From γ and β, we evaluated the electronic density of states per spin at the Fermi energy to be D(E F ) = 0.4336(7) eV −1 per formula unit and the Debye temperature to be Θ D = 221.5(2) K. From the entropy balance, the thermodynamic T c was determined to be T c = 1.2 K under 0 mT and T c = 1.1 K under 20 mT. First, we extrapolated the almost linear part of the experimental data to higher temperature by using a linear fitting. The fitting range was determined so that the root mean square of the residuals becomes the smallest. Then, we defined the thermodynamic T c as the temperature where the entropies of the extrapolation line and experimental data balance. As we can see in Fig. 4(b), the thermodynamic T c is almost equal to the temperature at which the resistivity drops to 5% of that in the normal state. Figure 5(b) presents the temperature dependence of the resultant volume-fraction density f LN (T o − T ). From the fitting, we obtained the onset temperature T o = 1.542(8) K under zero field. This probability distribution, restricted to 0 ≤ T ≤ T o , has a mode of T mode = 1.42 K and a median of T med = 1.07 K. We can see that T med coincides with the temperature at which C e /T is maximum. Moreover, T o and T mode roughly correspond to T c 's determined from resistivity using the 95% and 50% criteria, respectively. The superconducting volume fraction To T f LN (T o − T c )dT c is estimated to be 55% at T = 0.5 K and 65% at T = 0 K. Here, we assume that the entire part of the residual C e /T of 35% originates from the normal state, as a result of the superconducting volume fraction smaller than 100%. This evidences the bulk superconductivity in CaSb 2 .

IV. DISCUSSION
The first-principles calculation shows that CaSb 2 hosts two sets of Fermi surfaces near the zone center (attributable mainly to Sb2) and the zone boundary (Sb1) with almost the same carrier density [31]. Interestingly, the Fermi surface near the zone boundary consists of a pair of concentric cylinders. These cylinders along the AZ line (parallel to the c axis) emerges from a symmetryprotected nodal line after it is slightly self-doped with electrons. Since the nonsymmorphic space group P 2 1 /m forces an additional orbital degeneracy in addition to the Kramers spin degeneracy, this nodal line near the bottom of the band is four-fold degenerate [31].
Due to such quartet degeneracy, the Fermi surface consists of a pair of cylinders and exhibits a strong mixing among orbitals and spins. Thus, in a similar manner to doped topological insulators [7] and doped Dirac semimetals [13,15], topological superconductivity is naturally expected if the inter-orbital pairing interaction is dominant. In fact,the zig-zag chains consisting of two Sb1 sites in the unit cell may host such inter-orbital Cooper pairing on each cylinder. If the inter-orbital pairing involving inter-site Cooper pairs is realized, an odd-parity state with nodal lines protected by the nonsymmorphic symmetry at the Brillouin-zone boundary is expected [40]. Even for the intra-orbital pairing, an oddparity state is possible besides even-parity states, but they would be fully gapped.
Another interesting possibility is a time-reversalbreaking chiral p-wave superconductivity [24]. Whereas the present system has a cylindrical Fermi surface not a torus one discussed in Ref. [24], and the relevant space group symmetry is different, it has the same feature that the Fermi surface originates from a nodal line. Therefore, the mechanism for chiral p-wave superconductivity in Ref. [24] might work as well.
Experimental evidence for the nodal structure of the superconducting state will be given by the temperaturedependence of the specific heat much below T c . However, the present data with a broad transition is insufficient for determination of the gap structure. When analyzing the experimental data, we used for simplicity a theoretical curve for a fully gapped superconductivity, which includes some of the p-wave superconducting states such as the Balian-Werthamer (BW) state as well as the s-wave states. Determination of the superconducting gap structure using samples with a sharp transition is a challenge of future work.

V. CONCLUSION
We report the discovery of superconductivity in CaSb 2 , which has nodal lines in the bulk electronic band structure crossing the Fermi energy, protected by a nonsymmorphic crystalline symmetry. Both DC magnetization and specific heat indicate the bulk superconductivity. The transition temperature is determined to be 1.7 K from magnetic susceptibility and resistivity. The lower critical field is smaller than 5 mT, and the upper critical field is estimated to be 0.17-0.26 T. From these critical fields, the Ginzburg-Landau parameter is evaluated to be κ > 5.4. We hope this work stimulates further investigations of the superconductivity in nonsymmorphic Dirac-line materials.