Electronic structure and superconductivity of the non-centrosymmetric Sn$_4As$_3$

In a superconductor that lacks inversion symmetry, the spatial part of the Cooper pair wave function has a reduced symmetry, allowing for the mixing of spin-singlet and spin-triplet Cooper pairing channels and thus providing a pathway to a non-trivial superconducting state. Materials with a non-centrosymmetric crystal structure and with strong spin-orbit coupling are a platform to realize these possibilities. Here, we report the synthesis and characterisation of high quality crystals of Sn$_4$As$_3$, with non-centrosymmetric unit cell ($R3m$). We have characterised the normal and superconducting state using a range of methods. Angle-resolved photoemission spectroscopy shows a multiband Fermi surface and the presence of two surface states, confirmed by Density-functional theory calculations. Specific heat measurements reveal a superconducting critical temperature of $T_c\sim 1.14$ K and an upper critical magnetic field of $H_c\gtrsim 7$ mT, which are both confirmed by ultra-low temperature scanning tunneling microscopy and spectroscopy. Scanning tunneling spectroscopy shows a fully formed superconducting gap, consistent with conventional $s$-wave superconductivity.


INTRODUCTION
Identification of a spin-triplet superconductor would provide us with a potential solid state platform for topological quantum computations, a variant that is particularly robust against decoherence -one of the main impediments to realization of larger scale quantum calculations. There are different routes to realizing spin-triplet superconductivity (SC): either through engineered heterostructures, or in materials where triplet pairing is allowed or even promoted. Here, we focus on the latter path. One class of materials where a triplet component becomes allowed are non-centrosymmetric SC, where mixing of spin-singlet and spin-triplet Cooper pairing channels is possible and Rashba spin-orbit coupling (SOC) can lead to a lifting of Kramers degeneracy for electronic states in the bulk of the material[1].
A possible triplet component is expected to manifest in a number of observables: the upper critical magnetic field will be much higher than for singlet SC and the SC gap will exhibit a more complex structure than the hard gap predicted by the Bardeen-Cooper-Schrieffer (BCS) theory for a singlet SC. One would also expect topologically protected bound states near defects and boundaries that could be detected in local measurements. Experimentally, evidence for this mixing has been found in the non-centrosymmetric heavy fermion CePt 3 Si, where the strong SOC gives rise to a SC gap with line nodes [2]. However, the degree of mixing is not only determined by the strength of the SOC but also by the dominant pairing interaction [3]. If spin-singlet pairing interactions are dominant, the SC in the noncentrosymmetric material will follow the predictions by the BCS theory, as has been found in the case of BiPd [4], and it is independent of the SOC strength. Nevertheless, in BiPd the breaking of inversion symmetry together with strong SOC leads to Dirac-cone surface states [5] with an intricate spin texture [6].
In Sn-based compounds, the development of unconventional SC has been suggested, particularly in the case of the topological crystalline insulator SnTe with Indium doping, where strong SOC has been shown to play an important role [7,8]. A non-centrosymmetric crystal structure in Sn-based materials could then, in principle, favour the appearance of a triplet component. Here we focus on the non-centrosymmetric material Sn 4 As 3 . Transport measurements reveal a metallic nature and SC critical temperature, T c , in the range 1. 16-1.19 K [9], although other measurements showed a partly compensated semimetal [10]. It was only recently that the crystal structure of Sn 4 As 3 was identified as belonging to the non-centrosymmetric space group R3m (no. 160), with a hexagonal unit cell [11]. Additionally, band structure calculations confirmed that it is metallic, although a depletion of the density of states (DOS) is found at ∼ 0.4 eV above the Fermi energy. However, a direct measure of its electronic structure has been missing. Moreover, there have been no recent reports on its SC, apart from other SnAs-based superconductors such as SnAs [12,13] and NaSn 2 As 2 [14][15][16] which exhibit centrosymmetric crystal structures, revealing SC consistent with spin-singlet pairing.
In this work, we report a detailed study of the properties of single crystal samples of Sn 4 As 3 , through thermodynamic and spectroscopic characterization of both normal and SC states using angle-resolved photoemission spectroscopy (ARPES) and ultra-low temperature scanning tunneling microscopy and spectroscopy (STM/STS). The experimental results from ARPES and STM are directly compared with bulk and slab Density Functional Theory (DFT) calculations.

Sample
Growth. Sn 4 As 3 single crystals were synthesized from high purity elemental Sn (99.99%) and As (99.9999%), weighted in stoichiometric molar ratio (4:3). Synthesis was performed inside a quartz ampoule with Ar atmosphere at a residual pressure of 0.2 bar. The ampoule was put into a furnace and heated up to 600 • C for 24 hours. The temperature was then increased to 650 • C, followed by slow cooling down to room temperature at a rate of 2 • C/h. The samples were characterized by energy-dispersive x-ray spectrometry (EDS) and x-ray diffraction (XRD), revealing a chemical composition of Sn 3.8 As 3 and lattice constants of a = 4.0891Å and c = 36.0524Å, in agreement with Refs. [10,11].
Scanning tunneling microscopy and spectroscopy. STM measurements were performed with a home-built ultra-low temperature STM, mounted in a dilution refrigerator [17] with a base temperature of 10 mK and in a superconducting magnet with maximum field of 14 T.
The Pt-Ir tip was cut from a wire and conditioned by field-emission on a Au single crystal prior to measuring. The Sn 4 As 3 sample was cleaved in-situ at low temperatures (T ≈ 20 K).
The bias voltage was applied to the sample. Differential conductance spectra were recorded using a lock-in amplifier (f = 437 Hz), with amplitudes of modulations set at 15 mV and 25 µV for measurements at 11 K and 50-900 mK, respectively.
Angle-resolved photoemission spectroscopy. ARPES measurements were performed at the I05 beamline of Diamond Light Source, UK [18]. Single-crystal samples were cleaved in-situ in a vacuum better than 2·10 −10 mbar and measured at temperatures of 20 K. Measurements were performed using linear horizontal (LH) and linear vertical (LV) polarized synchrotron light with variable photon energy, using a Scienta R4000 hemispherical electron energy analyzer with an angular resolution of 0.2 • and an energy resolution of 20 meV.
Density functional theory calculations. Bulk electronic band structure calculations were performed for the experimental crystal structure of Sn 4 As 3 from Kovnir et al. [11] in the generalized gradient approximation (GGA) using WIEN2k [19], taking into account SOC.
These were used to produce a three dimensional (3D) Fermi surface, as well as 2D cuts at different k z planes. Additionally, bulk and slab calculations were carried out with the Quantum Espresso package [20] using GGA within the framework of Perdew-Burke-Ernzerhof [21], employing optimized norm-conserving Vanderbilt pseudopotentials [22,23], for the same crystal structure as before. We chose a plane wave (PW) cutoff of 80 Ry, Gaussian smearings of 0.02 Ry, and a 24 × 24 × 6 Monkhorst-Pack k-grid to sample the Brillouin zone (BZ). We checked that we reproduce the results for the bulk bands calculated with WIEN2k.
To simulate STM images of the pristine surface and of the surface with a Sn vacancy, we considered a 3 × 3 × 1 supercell, while to simulate STM images with Sn or As vacancies at different subsurface layers, we also considered larger 4 × 4 × 1 supercells. The BZs were sampled using Monkhorst-Pack k-grids (5 × 5 × 1 for the 3 × 3 × 1 supercell, 4 × 4 × 1 for the 4 × 4 × 1 supercell). We chose a vacuum region of 10Å in the case of the slab calculations, SOC was neglected for all STM simulations and all DOS calculations have been performed with a denser 36 × 36 × 8 (36 × 36 × 1 for the slab) BZ grid and a Gaussian smearing of 0.01

Ry.
Specific heat measurements. Specific heat of Sn 4 As 3 crystals was measured by thermal relaxation technique, using a PPMS-9 (Quantum Design) with a 3 He calorimeter. The mass of the sample was m = 11 mg. Measurements were performed at temperatures 0.37-2 K and magnetic fields of 0-20 mT.

Crystal structure and Surface topography
The crystal structure of Sn 4 As 3 belongs to the non-centrosymmetric group R3m, whose hexagonal unit cell is shown in figure 1 (a). The unit cell is composed of three seven-layer blocks of alternating Sn and As layers stacked along the c-axis. Inside each block, pairs of atoms that would otherwise be symmetrically equivalent (Sn1 and Sn2; Sn3 and Sn4; As1 and As3) are inequivalent: the Sn atoms in different layers form distorted octahedra with the surrounding atoms, which are responsible for the lack of inversion centre [11]. The weakest bond in the unit cell occurs between Sn3-Sn4 atoms from two different blocks (indicated by the dashed line in figure 1(a)), with the larger bond distance of 3.24Å, comparable to that observed in other layered SnAs-based materials [14]. Thus, the crystal is expected to cleave well in the (0001) plane, between the Sn3-Sn4 layers. The exposed surface can be either Sn4 or Sn3, which are structurally identical. A Sn4-terminated surface is illustrated in figure 1 figure 4(a). The measurements are consistent with Sn 4 As 3 being metallic, with several bands crossing the Fermi level, in agreement with Kovnir et al. [11]. DFT calculations for the bulk band structure neglecting SOC are shown in figure 4(b). There is good agreement between experiment and calculations over an energy range of several electronvolts, indicating a weakly correlated nature of the Sn 4 As 3 electronic structure. Small discrepancies between experimental data and calculations arise from the strong 3D character of the electronic dispersion and finite k z -averaging in the photoemission experiment. In addition to the bulk bands predicted by the DFT calculations, the measurements show additional bands at energies close to −1 eV (indicated by a white arrow), which are split by ∼ 100 meV. These split bands appear clearly in the slab DFT calculations (red lines in figure 4(c)), similar to those for SnAs [13]. The consistency with ARPES data confirms that they are surface states (SS), which have a splitting of ∼ 108 meV. Including SOC effects in the calculations did not improve the agreement with the experimental data. The DOS of the slab DFT calculation shows two additional peaks, mainly due to contributions from the SS. The STM differential conductance (dI /dV ) spectrum can be taken as proportional to the local DOS. In the dI /dV measurements ( figure 4(d)), two peaks can be identified in the energy range corresponding to the SS, with a splitting of 110 meV (indicated by black arrows), consistent with the splitting observed in ARPES and obtained from the slab calculations. This can be directly compared to the local DOS calculations, which shows the increase in intensity at the SS energies. The STM spectrum shows a depletion of the LDOS around the Fermi level, evidenced by a low differential conductance, which is also consistent with the calculations of the local DOS.

Thermodynamic measurements
Specific heat measurements of Sn 4 As 3 are shown in figure 5. In zero applied magnetic field ( figure 5(a)), a clear jump in C/T is observed at temperatures close to 1.1 K, typical of a superconducting transition. From the local entropy conservation, the critical temperature was found to be T c = 1.14 ± 0.01K, close to the reported values of 1.16 − 1.19 K [9]. The small width of the superconducting transition is indicative of the high quality of the sample.
At low temperatures (well below the Debye temperature) the specific heat of a metal can be written as C/T = γ n + βT 2 , where γ n and β are the electronic and the phonon contributions, respectively. The C/T curve for the normal state at 20 mT field (which fully suppresses the superconducting transition) is shown in figure 5(a). It has a parabolic shape, consistent with a metallic behaviour. A second order polynomial fit (red solid line) yields γ n = 6.66 ± 0.20 mJ mol −1 K −2 and β = 0.933 ± 0.143 mJ mol −1 K −4 5 .
The electronic specific heat, C el /T , can be obtained by subtracting the phonon contribution. Figure 5(b) shows C el /γ n T as a function of T , at zero magnetic field. The jump in specific heat at this temperature is ∆C el = 9.74 mJ mol −1 K −2 . The relative magnitude of the jump, ∆C el /γ n T c = 1.30, is close to the BCS prediction of ∆C el /γ n T c = 1.43. The red line in figure 5(b) shows the fit of the electronic specific heat in the superconducting state derived from the BCS theory: with the temperature dependence of the gap described by where ∆ 0 = rk B T c is the superconducting gap at T = 0 K. f (ε) is the Fermi function and a = 1.138 obtained from fitting the BCS mean field behaviour [4] with equation 2.
Fitting r = ∆ 0 /k B T c yields r = 1.76 ± 0.02 6 in excellent agreement with BCS theory.
Using the measured T c = 1.14 K and the BCS approximation, the superconducting gap is 5 Errors from 95% confidence bounds of the parabolic fit. 6 Error from 95% confidence bounds from the fit.
8 ∆ 0 = 1.76k B T c = 0.177 ± 0.003 meV. Use of more complex models (introducing anisotropy or using two gaps) does not give significant improvement of the fits. The magnetic field dependence of C/T is shown in figure 5(c). SC is found to be completely suppressed already in magnetic fields of H c ∼ 7 mT.

Superconducting gap
In order to obtain further evidence of the superconducting gap structure, we have performed STM/STS measurements at temperatures below 1 K. A well resolved superconducting gap is observed in high resolution tunneling spectra, dI /dV, taken in an energy range of ±1 mV at 50 mK, shown in figure 6(a). The coherence peaks can be clearly identified, while the DOS is completely suppressed around the Fermi energy, as expected from BCS theory with s-wave symmetry. A Dynes equation [24] for a single isotropic gap was fitted to the data, taking into account both thermal and lock-in broadening [17]. The fitting parameters were the superconducting gap ∆ and the electronic temperature, T elec . Here, the additional broadening in the Dynes equation, Γ, was fixed to be very small (Γ ∼ 10 −4 meV). The fit yielded ∆ = 0.182 ± 0.018 meV 7 and T elec ≈ 223 mK. The electronic temperature is dominated by the lock-in modulation (V L =25 µV RMS) used in the experiment, whose contribution to broadening is larger than the thermal broadening. Using the BCS relation ∆/k B T c = 1.76, the gap size yields a critical temperature of T c = 1.20 ± 0.14 K, which is consistent with the reported values [9] and in excellent agreement with the specific heat measurements. broadening. It can be seen that the SC is suppressed already at a temperature of 900 mK, lower than the expected temperature from the thermodynamic measurements. The apparent lower critical temperature can be due to both thermal and lock-in modulation broadening.
The magnetic field dependence of the superconducting gap can be seen in figure 6 (c).
The measurements show suppression of SC above magnetic fields of 8 mT, again consistent with the specific heat measurements. 7 Error from 95% confidence bounds from the fit.

STM topographies show a surface consistent with a cleave between adjacent Sn layers,
where the bond between atoms is expected to be weakest. The surface shows atomic defects that we identify as Sn vacancies in the topmost surface layer from comparison with DFT slab calculations and identification of the defect site. The occurrence of these defects is consistent with the chemical composition of Sn 3.8 As 3 determined from post-growth compositional analysis by EDS.
In addition to these defects in the top surface layer, we find a random distribution of bright triangular patterns. The lack of periodicity indicates that they are not due to the presence of a charge density wave. Additionally, bias dependent imaging and dI /dV spectroscopy maps (not shown) reveal that these are static in energy, suggesting that they are not generated from electron scattering off defects. Simulated STM images from slab DFT calculations show that missing Sn and As atoms from deeper layers produce bright triangular shapes at the topmost layer, which resemble the observed patterns. Thus, we attribute the origin of these patterns as coming from randomly distributed defects throughout different layers of the sample. The less abundant 'triforce' defect does not seem to be captured by these calculations, and is possibly due to an adatom at the surface.
The ARPES measurements confirm the metallic nature of the material with several bands crossing the Fermi level, consistent with tunneling spectra, specific heat and DFT calcula- tions. The Fermi surface shows significant dispersion along all directions, including z direction, which is evidence of a 3D character of the electronic structure despite the layered crystal structure. The ARPES measurements do reveal a pair of surface states that look at first like Rashba-spin split states, but are fully captured in calculations without SOC.
Comparison with bulk and slab DFT calculations reveal that this splitting is due only to the symmetry breaking at the surface.
Despite the non-centrosymmetric crystal structure of the material, the SC properties are found to be fully consistent, within the experimental errors, with what would be expected from BCS theory. The STM tunneling spectra shows a fully formed gap, which is spatially uniform and has the shape characteristic of an isotropic s-wave SC gap. These results follow the trend of other SnAs-based superconductors, where conventional BCS SC with s-wave symmetry has been found [12,13,15,16].
The low upper critical field on its own already provides strong indication that any triplet component of the order parameter is negligible in this system. Taken together with the observations of rather conventional SC in other non-centrosymmetric materials [4,25], this does confirm that to observe a sizeable triplet component of the superconducting order parameter requires a material system where pairing is mediated by a mechanism other than electron-phonon coupling [26,27].

CONCLUSIONS
We successfully synthesized Sn 4 As 3 in the non-centrosymmetric crystal structure (R3m).
Comprehensive characterisation of the normal state electronic structure shows metallic be-   For better comparison, bulk and slab DFT DOS and ARPES data were normalised by the intensity at -3.9 eV and at -1.8 V for the STM spectrum.