Full-gap superconductivity in spin-polarised surface states of topological semimetal β-PdBi2

A bulk superconductor possessing a topological surface state at the Fermi level is a promising system to realise long-sought topological superconductivity. Although several candidate materials have been proposed, experimental demonstrations concurrently exploring spin textures and superconductivity at the surface have remained elusive. Here we perform spectroscopic-imaging scanning tunnelling microscopy on the centrosymmetric superconductor β-PdBi2 that hosts a topological surface state. By combining first-principles electronic-structure calculations and quasiparticle interference experiments, we determine the spin textures at the surface, and show not only the topological surface state but also all other surface bands exhibit spin polarisations parallel to the surface. We find that the superconducting gap fully opens in all the spin-polarised surface states. This behaviour is consistent with a possible spin-triplet order parameter expected for such in-plane spin textures, but the observed superconducting gap amplitude is comparable to that of the bulk, suggesting that the spin-singlet component is predominant in β-PdBi2.

gap opening in the TSS. Quasiparticle interference (QPI) imaging combined with numerical simulations has been utilized to explore the spin textures at the surface. We find that the observed QPI patterns are qualitatively consistent with the spin textures predicted by the first-principles calculations, which reveal that not only the TSS but also all other states at the surface are spin polarized. This suggests that the surface of β-PdBi 2 can potentially harbour the mixed spin-singlet and triplet superconductivity. A fully-opened single SC gap ∆ ∼ 0.8 meV has been identified in the tunnelling spectrum without any detectable residual spectral weight near E F , being consistent with the result of air-cleaved sample [21]. These results indicate that the electronic states detected by SI-STM including the TSS are all spin-polarized and exhibit fully-gapped superconductivity. We argue that the spin-triplet order parameter expected from the spin textures is consistent with the observed isotropic gap. The observed gap amplitude, however, is comparable to that of the bulk [23], suggesting that the spin-singlet component is dominant. Therefore, the topological nature of the observed superconductivity, if any, may manifest itself at lower temperatures.

Calculating band structures and spin textures at the surface
We first examine the spin textures at the surface of β-PdBi 2 by first-principles calculations within the framework of the density functional theory. The surface exposed after cleaving a single crystal along (001) plane is expected to be composed of Bi atoms since the weakest bond in the crystal structure is the van der Waals bond between adjacent Bi layers ( Fig. 1a inset). Therefore, we mainly focus on low-energy electronic states composed of Bi 6p orbitals at the topmost layer predominating the tunnelling current. (Details of the calculation is described in Methods section.) The calculated band dispersions and the constant energy contours (CECs) at energy E = +30 meV are shown in Figs. 1a and 1b, respectively. Each of the surface and bulk states is labelled in the same manner as the previous report [20]. The surface states are found in the close vicinity of the bulk bands. The trivial surface state S1 is located near the bulk β band, whereas the TSS S2 is located at the bottom of the bulk γ and δ bands.
For comparison with the experimental results, we performed a Wannier transformation as written in the Methods section. Figure 1c shows the spectral function at +30 meV with the Wannier transformation. The spectral weights of bulk γ and δ bands are very small. For the TSS S2 surrounding M points, the weight is negligibly small except for the section parallel to the M-X direction.
Figures 1d, 1e and 1f show x, y and z components of the spin polarizations weighted by the spectral function. It is clear that the spin polarizations are predominantly aligned in-plane, as expected from the symmetry of the crystal structure. It should be noted that not only the TSS S2 but also S1, and even the bulk bands are all spin polarized. The spin orientations of S1 and S2 are consistent with the results of spin-resolved ARPES [20].
The spin polarizations of the bulk bands are likely to be induced by local broken inversion symmetry at the Bi site in the bulk as well as at the surface [24] (Fig. 1a inset). The spin textures of these bulk bands have never been studied before and play an important role in QPI as we shall show below.

Spin textures revealed by QPI imaging
In order to confirm the calculated spin textures experimentally, we utilize QPI imaging.
The QPI effect is nothing but the formation of electronic standing waves. In the presence of elastic scatterers, two quantum states on the same CEC with a scattering wave vector q may interfere and generate a standing wave with a wavelength of 2π/|q|. This yields E-dependent spatial oscillations in the local density-of-states distribution that manifest themselves in the tunnelling conductance dI/dV | V =E/e images, where I is the tunnelling current, V is the bias voltage and e is the elementary charge. Fourier analysis allows us to identify various QPI signals appearing at different q vectors. The prerequisite to have QPI signals at q is that q must connect two states with high enough spectral weights. In addition, the spin polarizations of these two states should not be antiparallel because spin-antiparallel states are orthogonal and thereby cannot interfere with each other. In this way, the QPI imaging can provide information on the spin textures in the k space [25, 26].
Here we outline expected QPI patterns from the calculated spin textures at the surface (Figs. 1d-1f). In the case of non-magnetic scatterers, any intra-band back-scattering of spin-polarized bands is forbidden because the spin polarizations of the two states relevant for scattering are always antiparallel due to time-reversal symmetry. The inter-band back-scatterings from S1 to Γ-centred S2 and from α to Γ-centred S2 are also forbidden, because the spin orientations of S1, S2 and α are the same in the Γ-M direction. The inter-band forward-scatterings from S1 to Γ-centred S2 and from S1 to α are allowed but the expected q vectors are small; it may be difficult to distinguish these QPI signals from extrinsic small q modulations associated with inhomogeneously distributed unavoidable defects in a real material. As a result, three scattering channels may govern the QPI patterns; inter-band forward-scattering from α to Γ-centred S2 (q 1 ), inter-band back-scatterings from S1 to β band (q 2 ) and from Γ-centred S2 to β band (q 3 ) (Fig. 1c).  Fig. 1a, respectively. Namely, the observed QPI signals may correspond to q 1 , q 2 and q 3 .
We confirm this conjecture by comparing experimental data with the numerically simu- The result of the full simulation at E = +30 meV is shown in Fig. 3a. Three parallel line-like QPI signals normal to the Γ-M direction are identified. These features correspond to q 1 , q 2 and q 3 indicated in Fig. 1c and their energy evolutions reasonably agree with the experimental observation ( Fig. 3b), indicating that q 1 , q 2 and q 3 scatterings indeed dominate the QPI patterns.
The key role of the spin textures becomes evident if we compare the result of full simulation with that of spinless simulation (Fig. 3c), and the experimental data. Figure 3d compares line profiles along the Γ-M direction from experimental and simulated QPI patterns. In the spinless simulation, there are three sharp peaks but their |q| values do not agree with the experimental observations. These sharp peaks correspond to the intra-surface-state back-scatterings in S1 (q S1−S1 ) and in Γ-centred S2 (q S2−S2 ), and inter-band back-scattering from S1 to Γ-centred S2 (q S1−S2 ). Because the spin orientations are anti-parallel between initial and final states of these scatterings, they are suppressed in the full simulation that well captures the experimental observations.
The above comprehensive approach combining experiments and calculations provide the following important implications. Firstly, our SI-STM data primarily reflect the electronic state of the top-most Bi layer as we assumed. Secondly, the spin degrees of freedom is crucial for QPI. Finally, the calculated spin textures shown in Figs. 1d-1f adequately capture the real spin textures at the surface. Notably, the four-fold QPI pattern is similarly identified at E F in the normal state (T = 1.5 K, B = 12 T) as shown in Supplementary Fig. 2. This indicates that the spin textures discussed here indeed exist at E F in the normal state.

Superconductivity at the surface
Given the surface sensitivity of our measurement evidenced in the QPI patterns, we are able to argue the nature of superconductivity at the surface. Figure 4a shows the temperature T evolution of the tunnelling spectrum in the SC state. At T = 0.4 K, the SC gap fully opens and there is no residual spectral weight inside the gap, indicating that all of the states at the surface are gapped. Each spectrum can be fitted well with the Dynes function for the single isotropic gap, meaning that the SC gap is k independent. We also investigate QPI patterns near the SC gap energy and find no clear superconductivityinduced QPI signals except at q = 0 ( Supplementary Fig. 2), which means that the SC gap is spatially uniform. The SC gap amplitude ∆(T ) is estimated to be 0.8 meV at T = 0.4 K, being reasonably consistent with the result of air-cleaved sample [21]. We note that any noticeable spectroscopic features are not observed near E F at step edges ( Supplementary   Fig. 3). The temperature dependence of the SC gap well follows the BCS behaviour (Fig. 4b).
We next examine the electronic state of vortices that may host Majorana fermions [6,27,28]. Isotropic vortices are clearly imaged in the dI/dV map at E = 0 meV (Fig. 4c inset), as being similar to the previous result obtained at the surface of the air-cleaved bulk crystal [21]. The line profile of the vortex core is also reasonably consistent with the previous result [29] (Supplementary Fig. 4). To estimate the in-plane coherence length and compare it with that obtained from H c2 measurements, detailed magnetic field dependence of the vortex core size is required as proposed recently [29]. Considering the good agreement with our result and those in ref. 29, we believe that a similar coherence length as discussed in ref. 21, 29 would be expected in our samples. The spectrum taken in the vortex core is almost flat (Fig. 4c). This result indicates that the core is in the dirty limit (mean free path l < ξ) where vortex bound states as well as the Majorana zero mode are not well defined. Although the zero-energy local density-of-states peak in the vortex core cannot be an evidence of the  28], for example. We anticipate that β-PdBi 2 will be a good touchstone for these theories in future because it is a stoichiometric material that can be made cleaner in principle.

Discussion
Our concurrent investigations of the surface electronic states and superconductivity reveal that the SC gap fully opens in all of the spin-polarized surface states. Since the mixing of spin-singlet and triplet order parameters is generally expected in the presence of nondegenerate spin-polarized surface states [1,2] and the triplet component can possess nodes, it is intriguing to argue the possible nodal structure of the SC states of β-PdBi 2 . It has been shown that the spin-triplet order parameter d(k) favours to be aligned along the spin direction when space inversion symmetry is broken [30]. If this is the case at the surface of β-PdBi 2 where the point group symmetry is lowered from D 4h to C 4v , the triplet pairing is of p-wave type with nodes along the out-of-plane direction [30]. Since the surface states are two dimensional in nature, the out-of-plane nodes would not be active and the SC gap should look like isotropic as observed.
Next we discuss the amplitude of the SC gap. The SC gap amplitude in the presence of the parity mixing is given by |Ψ s (k)| ± |d(k)| where Ψ s denotes s-wave order parameter that represents the bulk gap [2,[30][31][32]. Thus the difference between the SC gap amplitudes of bulk and surface gives us an estimate of the amplitude of the p-wave component. The bulk SC gap has been estimated to be about 0.9 meV by the specific heat measurement [23], which is close to the value of 0.8 meV that we observed at the surface. According to the theory [32], the difference between the surface SC gap and the bulk counterpart depends on how far the surface state is separated from the bulk band at E F in k space. We estimate the 32, to be about 0.05 (k D F and k F denote the Fermi momenta of the TSS and the bulk band, respectively). This is reasonably small to explain the similar SC gap amplitudes between the surface and the bulk. Namely, even though it is allowed, the p-wave component at the surface of β-PdBi 2 is small. The absence of edge states shown in Supplementary Fig. 3 might be due to the small p-wave component.
In order to detect this small p-wave component, if any, we need to avoid comparing the results of two different experiments, STM and specific heat, as well as to improve the energy resolution. As the bulk bands appear at the surface, too, it would be possible to detect the surface and bulk SC gaps simultaneously by STM alone, provided higher energy resolution could be achieved. To this end, the use of a SC tip at lower temperatures is important.

Sample preparation and STM measurements.
The β-PdBi 2 single crystals were grown by a melt growth method [20], and characterized by x-ray diffraction (XRD) and transport measurements. Pd (3N5) and Bi (5N) at a molar ratio of 1 : 2 were encapsulated in an evacuated quartz tube, pre-reacted at temperature above 1000 • C until it completely melted and mixed. Then, it was kept at 900 • C for 20 h, cooled down at a rate of 3 • C/h down to 500 • C, and finally rapidly quenched into cold water. PdBi 2 has two different crystallographic and superconducting phases: α-phase with the space group C2/m (T c ∼ 1.7 K [33]) and β-phase with I4/mmm (T c ∼ 5.4 K). The last quenching procedure is important to selectively obtain the β phase. Any trace amount of α-phase was not detected by XRD. For STM measurements, we used single crystals with residual resistivity ratio of 15, larger than those of the previous reports [19, 21, 23]. The crystals were cleaved along the (001) plane at room temperature in ultra high vacuum conditions to obtain clean and flat surfaces needed for STM. A commercial 3 He-based STM system (UNISOKU USM-1300) modified by ourselves [34] was employed in this study. We used electrochemically etched tungsten tips, which were cleaned and sharpened by field ion microscopy. The tips were subsequently treated and calibrated on clean Au(100) surfaces before used for β-PdBi 2 . We applied bias voltages to the sample whereas the tip was virtually grounded at the current-voltage converter (Femto LCA-1K-5G). Tunnelling spectra were measured by the software-based lock-in detector included in the commercial STM control system (Nanonis).

Calculation of the electronic state.
To calculate the surface electronic structure and its corresponding QPI pattern, we first performed a DFT calculation using the Perdew-Burke-Ernzerhof exchange-correlation functional [35] as implemented in the WIEN2K program [36]. Relativistic effects including spinorbit coupling were fully taken into account. For all atoms, the muffin-tin radius R MT was chosen such that its product with the maximum modulus of reciprocal vectors K max become R MT K max = 7.0. Considering the tetragonal phase of β-PdBi 2 , the corresponding Brillouin zone was sampled using a 20 × 20 × 5 k-mesh. For the surface calculations, a 100 unit tight binding supercell was constructed using maximally localized Wannier functions. [37][38][39].
The 6p-orbitals of Bi atoms were chosen as the projection centres.

QPI simulation.
We simulated QPI patterns by incorporating the eigenvalues and eigenvectors obtained from our tight-binding supercell calculations into the standard T -matrix formalism [40].
The topmost Bi 6p orbitals were considered in the calculations. For all the simulations, the broadening factor was chosen to be 5 meV, and a localized, spin-and orbital-preserving scatterer of strength 0.1 eV were employed. For comparison with the experimental results, we performed a basis transformation from the lattice model to the continuum model with the Wannier function [41] constructed by projecting the Bi 6p orbitals. We calculated the local density of state at 0.5 nm above the topmost Bi atoms. The basis transformation is also applied to the spectral function shown in Fig. 1c to present the electronic structure used for the QPI calculation. For the spinless simulation, the sum of the Green's functions constructed from the spin-up and spin-down subspaces were used for the T -matrix calculations.
superconductor. Nat. Phys. 10, 943-950 (2014).     states (S1 and S2) and the bulk states (α, β, γ and δ). The spin orientations of S1 (red), S2 (blue) and β (black) are schematically shown by arrows. (c) Spectral function at E = +30 meV with the Wannier transformation written in the Methods. The inter-band forward-scattering from α to Γ-centred S2, the inter-band back-scattering from S1 to β, and from Γ-centred S2 to β along the Γ-M direction is indicated as q 1 , q 2 and q 3 , respectively. A dashed square denotes the first  q S1-S2 q S1-S1 q S1-S1 q S1 The intra-band back-scatterings in S1 (q S1−S1 ) and in Γ-centred S2 (q S2−S2 ), the inter-band backscattering from S1 to Γ-centred S2 (q S1−S2 ) appear.  where r, z, V , κ and ρ s is the lateral location on the sample surface, the tip-sample distance, the bias voltage applied to the sample, the decay length inverse and the local density of state of the sample, respectively. By differentiating this tunnelling current with respect to V , we obtain the differential conductance: In the constant current mode, the tunnelling current is regulated to be constant by the feedback loop, which is opened during the spectroscopic measurement, so that Supplemen- where V set , I set is the set-point bias voltage and the set-point current, respectively. The denominator of Supplementary Eq.(S3) represents the set-point effect [42].
To examine the set-point effect on Fourier-transformed QPI images, let us consider to integrate the QPI signal at a wavevector q with respect to the bias-voltage from 0 to V set as follows: Vset 0 ∂I ∂V exp(−iq · r)dr dV = exp(−iq · r)dr Vset 0 ∂I ∂V dV = I set δ(q) This means that the integration from 0 to V set will be always zero at q = 0. To satisfy this requirement, each Fourier component except at q = 0 will change its sign as a function of V and should become zero at least once between 0 and V set . The suppression of QPI signals at around +100 mV in Fig. 2d in the main text is due to this zero-crossing.