Superconducting order parameter of the nodal-line semimetal NaAlSi

Nodal-line semimetals are topologically non-trivial states of matter featuring band crossings along a closed curve, i.e. nodal-line, in momentum space. Through a detailed analysis of the electronic structure, we show for the first time that the normal state of the superconductor NaAlSi, with a critical temperature of $T_{\rm c} \approx$ 7 K, is a nodal-line semimetal, where the complex nodal-line structure is protected by non-symmorphic mirror crystal symmetries. We further report on muon spin rotation experiments revealing that the superconductivity in NaAlSi is truly of bulk nature, featuring a fully gapped Fermi-surface. The temperature-dependent magnetic penetration depth can be well described by a two-gap model consisting of two $s$-wave symmetric gaps with $\Delta_1 =$ 0.6(2) meV and $\Delta_2 =$ 1.39(1) meV. The zero-field muon experiment indicates that time-reversal symmetry is preserved in the superconducting state. Our observations suggest that notwithstanding its topologically non-trivial band structure, NaAlSi may be suitably interpreted as a conventional London superconductor, while more exotic superconducting gap symmetries cannot be excluded. The intertwining of topological electronic states and superconductivity renders NaAlSi a prototypical platform to search for unprecedented topological quantum phases.


I. INTRODUCTION
In the course of the advent of topological insulators, several classes of non-trivial topological semimetals have been proposed and experimentally sought after: Weyl semimetals, Dirac semimetals, and nodal-line semimetals [1][2][3][4][5] . In their essence, all these types of topological matter arise from band inversion, often along with non-symmorphic symmetries. As opposed to insulators, the topological nature of a semimetal is given by a more intricate version of topological bulk invariants and bulk boundary correspondence. For the most elementary instance, the surface Fermi arcs of a Weyl semimetal are localized away from the Weyl cone projection points at the surface, and derive their localization length scale from the direct gap between the underlying bulk and valence bands at the given surface momentum. For nodal-line semimetals, the surface states feature intriguing structures referred to as drumhead states. Additional complexity can arise if the closed nodal-line takes on more complicated forms in momentum space, which yields structures called nodal knot semimetals. The higher the semimetallic topological complexity the harder it appears to find quantum matter realizations of such states, so classical metamaterial platforms often seem to be the only viable alternative 6,7 .
In many respects, however, only quantum material realizations of topological semimetals promise a complete unfolding of their rich phenomenology 8,9 . The interplay of emergent quantum effects such as magnetism, charge-ordering, and superconductivity, together with nontrivial band topology has been identified as a promising platform for the realization of exotic quasi particles, such as e.g. Majorana fermions [10][11][12][13]  The ternary compound NaAlSi investigated here crystallizes in the centrosymmetric space group P 4/nmm of PbFCl structure-type, as shown in Fig 1(a). This compound is isostructural to the "111" Fe-based superconductors LiFeAs and LiFeP 26 . NaAlSi has been reported to be a type-II superconductor with a critical temperature of T c ≈ 7 K at ambient pressure 27 , while the isostructural and isoelectronic NaAlGe is not superconducting. NaAlSi and NaAlGe have almost exactly the same band structure except for one missing piece of small Fermi surface. This small Fermi surface is rather unusual, and further obscured by the small but seemingly important interlayer coupling along the crystalline c-axis 28 . Even though one expects the phonon spectra for these two compounds to be fairly similar, and as such the onset of superconductivity if phonons were responsible for the pairing, the isoelectric cousin NaAlGe does not exhibit any kind of superconductivity down to low temperatures.

H. B. Rhee et al. have shown by first-principles electronic structure calculations that
NaAlSi is what they refer to as "a naturally self-doped semimetal", with charge-transfer between the covalent bands within the substructure and two-dimensional free-electron-like bands within the Al-Si layers 28 . This electronic structure results in an unusually small Fermi surface and a very low density of states at the Fermi level. Both characteristics, together with the reasonably high critical temperature of NaAlSi, are contradictory to conventional BCS theory predictions, where the critical temperature T c depends expotentially on the density of states at the Fermi-level. Given these special electronic features, it was proposed that the superconductivity in this material may not be of phononic origin. Hence, the microscopic origin of superconductivity still remains to be unambiguously identified. It calls for a detailed study of the electronic structure of this nodal-line semimetal, and in particular of how small electronic deviations between NaAlGe and NaAlSi might affect unconventional pairing tendencies from a weak coupling perspective. An unconventional pressure dependence of the superconductivity in NaAlSi has been reported: The critical temperature was found to slightly increase up to a transition temperature of T c ≈ 9 K under an external pressure, and to disappear abruptly at a pressure of p = 4.8 GPa in the absence of a structural transition 29 .
Here, we report on detailed band structure calculations, showing that NaAlSi is a nodalline semimetal for the first time, and on its superconducting order parameter analysis. Our muon spin rotation measurements reveal that the superconductivity in NaAlSi is truly of bulk nature and that the Fermi-surface is fully gapped with an s + s-wave symmetrical gap

A. DFT Calculations
The DFT calculations were performed using the VASP package 30 using the standard pseudopotentials for Na, Si and Al. The experimental geometries were taken from Ref. 29 .
The reducible Brillouin zone was sampled by a 9 × 9 × 9 k-mesh for the self-consistent calculations. A Wannier interpolation using 18 bands was performed by projecting onto an atomic-orbital basis centered at the atomic positions, consisting of Na-3s,Al-3s and 3p as well as Si-3s and 3p orbitals. The nodal-lines were calculated via the package wanniertools 31 based on this Wannier interpolation.

B. Sample Preparation
Blue-metallic, highly crystalline samples of NaAlSi were prepared by solid-state synthesis.
In a first step, the elements Na (purity 99.99 %), Al (purity 99.999 %) and Si (purity 99.9999 %) were mixed in a ratio of Na:Al:Si = 3:1:1. The excess of Na is necessary for obtaining phase pure products, it partly acts as a sodium flux. This mixture (2g) was sealed in an argon-filled tantalum tube in order to minimize Na loss during the reaction. The tantalum tube was sealed in a quartz ampule in order to prevent the oxidation of the tantalum reaction vessel. The reaction was carried out at 700 • C for three days (heating rate 50 • C/h). To increase the crystallinity of the product, the sample was slowly cooled (5 • C/h) to 600 • C, held for three days at this temperature, and finally cooled to room temperature (5 • C/h).
In a second step, the excess sodium was removed by distillation of the product under a dynamic vacuum at 200 • C for 100 h. The phase purity and crystal structure of the sample was verified by powder and single crystal x-ray diffraction using a D8 Focus diffractometer with Cu K α1 radiation (Bruker AXS GmbH, Karlsruhe, Germany).

C. µSR Measurements
Spin-polarized muons (µ + ) are extremely sensitive local magnetic probes that were here used to investigate the field distribution of the vortex state in the type-II superconductor NaAlSi. Transverse field (TF) and zero field (ZF) µSR experiments were carried down to T = 250 mK, well below the superconducting transition temperature of NaAlSi. Pressed pellets of NaAlSi were transferred under inert atmosphere with a portable glovebox into the cryostat. The µSR spectra have been analyzed using the MUSRFIT software package 32 .

III. RESULTS AND DISCUSSION
A. Electronic Structure of NaAlSi Charge balanced materials with 8 valence electrons of the form XAlSi (X =Li, Na) are expected to be semiconductors due to a completely filled Si p−shell that is separated by a gap from the empty s-shells of Li + and Al 3+ . While LiAlSi, which crystallizes in the cubic Half-Heusler structure, is indeed a semiconductor with a large direct band gap, NaAlSi shows metallic transport behaviour and becomes superconducting below 7 K. This can be attributed to the fact that NaAlSi does not crystallize in a Half-Heusler structure, but in the tetragonal spacegroup P 4/nmm, which consists of edge-sharing AlSi 4 tetrahedra with short Al-Al distances of about 2.9Å.
The short distance leads to considerable bonding between the Al-atoms in the x − y plane, which is indicated by a dispersive Al-s-band close to the Fermi energy (E F ) with a bandwidth of over 4 eV as shown in Fig 1(b). As expected from the ionic picture discussed  Magnetism, if present in the samples, may also enhance the muon depolarization rate. The temperature dependence of the muon spin depolarization rate σ sc , which is proportional to the second moment of the field distribution (see Method section), of NaAlSi in the superconducting state is shown in figure 3d. Below T c the relaxation rate σ sc starts to increase from zero with decreasing temperature due to the formation of the FLL. Interestingly, the form of the temperature dependence of σ sc , which reflects the topology of the superconducting gap, shows the saturation upon lowering the temperature below approximately 2 K.
We show in the following how these behaviours indicate the presence of the two isotropic s-wave gaps on the Fermi surface of NaAlSi.
N a A l S i G a u s s i a n R a t e L o r e n t z i a n R a t e Z e r o F i e l d R e l a x a t i o n R a t e s (

C. Probing the nonuniform field distribution in the vortex state
In order to investigate the symmetry of the superconducting gap, we note that λ(T ) is related to the relaxation rate σ sc (T ) by the equation 40 where γ µ is the gyromagnetic ratio of the muon, and Φ 0 is the magnetic-flux quantum. Thus, the flat T -dependence of σ sc observed for low temperatures (see figure 3(d)) Superconducting order parameter of the nodal-line semimetal NaAlSi is consistent with a nodeless, fully-gapped superconductor, in which λ −2 (T ) reaches its zero-temperature value exponentially.
To proceed with a quantitative analysis, we consider the local (London) approximation (λ ξ, where ξ is the coherence length) and employ the empirical α-model. The model, widely used in previous investigations of the penetration depth of multi-band superconductors [41][42][43][44][45] assumes that the gaps occurring in different bands, besides a common T c , are independent of each other. The superfluid density is calculated for each component separately 46 and added together with a weighting factor. For our purposes, a two-band model suffices, yielding where λ(0) is the penetration depth at zero temperature, ∆ 0,i is the value of the i-th superconducting gap (i = 1, 2) at T = 0 K, and ω i is the weighting factor which measures their relative contributions to λ −2 (i.e. ω 1 + ω 2 = 1).
The results of this analysis are presented in figure 4, where the temperature dependence of λ −2 for NaAlSi is plotted. We consider two different possibilities for the gap functions: are therefore reduced to very excotic superconducting symmetries, such as e.g. helical superconductors, or (slightly more realistically) a relative sign change between the two s-wave sheets. The experiments at hand cannot alone distinguish between sign changing s +− , hence topological, and s ++ trivial pairing states. Generally, s ++ can be considered as more likely. However, the high sensitivity of the superconducting state in NaAlSi to disorder, might suggest that a s +− might be realized in this material. Further surface probes such as, e.g., the Kerr effect or phase sensitive measurements, however, would be needed to reach conclusive statements on such questions.

IV. CONCLUSION
In summary, we have reported on a detailed analysis of the electronic structure of the compound NaAlSi. We here describe for the first time that the superconductor NaAlSi has a topological non-trivial nodal-line band structure. Its complex nodal-line structure is thereby protected by the symmetry of its crystal structure, in particular by the non-symmorphic mirror symmetries of space group P 4/nmm. We have characterized the microscopic superconducting properties of NaAlSi by a series of µSR experiments. The TF µSR experiments reveal for the first time unambiguously that the superconductivity in NaAlSi is of bulk nature. The measured temperature-dependant magnetic penetration depth λ corresponds to a fully-gapped Fermi-surface. In our analysis an s + s-wave symmetrical gap with ∆ 1 = 0.6(2) meV and ∆ 2 = 1.39(1) meV was sufficient to explain the observed behavior. The ZF µSR experiments above and below the critical temperature indicated the preservation of time reversal symmetry.
Our results indicate that superconductivity in this topologically non-trivial material may be explained as a conventional London superconductor. These results, however, also do not exclude some more exotic superconducting gap symmetries, such as e.g. a s +− symmetric gap. It may be speculated, whether the observed disappearance of superconductivity under pressure or the absence of superconductivity in isoelectronic and isostructural NaAlGe could be tied to the change in topological class in this material. Further theoretical as well as experimental (especially measurements of the bandstructure and thermodynamic measurements of the superconducting properties) work are crucial for understanding the interplay of superconductivity and topology in this material. We expect the results at hand to generally motivate significant additional studies into materials that couple topology to emergent quantum effects.

V. ACKNOWLEDGEMENT
We