# Fermiology and superconductivity of topological surface states in PdTe

###### Abstract

We study the low-energy surface electronic structure of the transition-metal dichalcogenide superconductor PdTe by spin- and angle-resolved photoemission, scanning tunneling microscopy, and density-functional theory-based supercell calculations. Comparing PdTe with its sister compound PtSe, we demonstrate how enhanced inter-layer hopping in the Te-based material drives a band inversion within the anti-bonding -orbital manifold well above the Fermi level. We show how this mediates spin-polarised topological surface states which form rich multi-valley Fermi surfaces with complex spin textures. Scanning tunneling spectroscopy reveals type-II superconductivity at the surface, and moreover shows no evidence for an unconventional component of its superconducting order parameter, despite the presence of topological surface states.

###### pacs:

There has long been interest in systems where spin-polarised electronic states co-exist with superconductivity [1; 2; 3; 4; 5; 6]. This is thought as a promising route to generating a spin-triplet component of the superconducting order parameter, and to realise topological superconductors which host Majorana zero modes at their boundaries or in vortex cores [4; 6]. Such conditions are proposed to be realised if superconductivity can be induced in the spin-momentum locked surface states of materials hosting non-trivial bulk band topology. There have been extensive efforts to achieve proximity-coupled superconductivity in topological surface states by interfacing with conventional superconductors [7; 8; 10; 9; 11] as well as to induce superconductivity by extrinsic doping of bulk topological insulators [12; 13; 14]. The latter approach, however, often suffers from the difficulty of achieving high superconducting volume fractions [12; 15], motivating the search for materials which host topologically non-trivial states and are simultaneously robust superconductors.

In this Letter, we demonstrate that pristine PdTe, an intrinsic bulk superconductor ( K) [16; 17; 18], hosts topological surface states which intersect the Fermi level. We demonstrate, from spin- and angle-resolved photoemission spectroscopy (spin-ARPES) and density-functional theory (DFT), that these form complex multi-pocket Fermi surfaces with intricate spin textures. Nonetheless, tunneling measurements into the surface layer from scanning tunneling microscopy and spectroscopy (STM/STS) indicate the superconductivity at the surface is of a conventional -wave form consistent with Bardeen-Cooper-Schrieffer (BCS) theory.

Single-crystal PdTe and PtSe samples (space group: ) were cleaved in situ at temperatures of K. Spin-resolved ARPES measurements were performed at the APE beamline of Elettra Sincrotrone Trieste using a Scienta DA30 analyser fitted with very low energy electron diffraction (VLEED) based spin polarimeters [19]. Multiple VLEED targets were used through the course of this study, with Sherman functions determined to be between and from comparison measurements on the Rashba-split surface states of Au. Spin-integrated measurements were performed at the I05 beamline of Diamond Light Source using a Scienta R4000 hemispherical analyser [20]. -polarised light and a measurement temperature of 5-15 K was used throughout. STM measurements were performed using a home-built system operating in cryogenic vacuum [21; 22]. Measurements were performed in a magnetic field up to 1T, and the sample temperature was held constant at mK (superconducting state) or K (normal state). STM tips were cut from a platinum/iridium wire. Bias voltages were applied to the sample with the tip at virtual ground. Differential conductance spectra were recorded through a standard lock-in technique with frequency Hz. A superconducting tip was obtained by collecting a small piece of the sample at its apex. Fully-relativistic DFT calculations were performed using the Perdew-Burke-Ernzerhof exchange-correlation functional as implemented in the WIEN2K program [23], using a -mesh. To calculate the surface electronic structure, tight binding supercells containing 100 formula units of PdTe or PtSe stacked along the crystalline -direction were constructed from the bulk DFT calculations using maximally localized Wannier functions with chalcogen -orbitals and transition-metal -orbitals as the projection centres [24; 25; 26].

The electronic structure of PdTe, as well as its sister compound PtSe, are summarised in Fig. 1. While the Pd/Pt states retain a fully-filled configuration, and are thus located well below the Fermi level [27], a conducting state is obtained due to an energetic overlap of predominantly Te/Se-derived bonding and anti-bonding states [29; 28]. In PdTe, this leads to a complex multi-band Fermi surface (Fig. 1(a)). Strong inter-layer interactions for the chalcogen-derived states render this Fermi surface three-dimensional, despite the layered nature of this compound. Many of the observed spectral features are therefore diffuse in our measured Fermi surface from ARPES, reflecting the inherent surface sensitivity, and thus poor resolution, of photoemission. Similarly broad features, including a recently-identified type-II bulk Dirac cone [29; 30] centered at eV, can also be seen in our measured dispersions (Fig. 1(b)) as well as in our supercell calculations where bulk bands at different values are projected onto the surface plane (Fig. 1(c).

A number of much sharper features are also observed. A Dirac cone situated eV below the Fermi level is clearly evident. This has recently been identified as a topological surface state in PdTe and related compounds [31; 34; 29; 32; 33]. This arises from a band inversion that occurs within the Te -orbital manifold, and is induced by a naturally disparate out-of-plane dispersion of and -derived bands along (Fig. 1(d)) [29]. This same mechanism drives the formation of both the type-II bulk Dirac cone and a further topological surface state located eV below , that is visible in our supercell calculations presented here (Fig. 1(c)) and is clearly resolved experimentally in Ref. [29]. These topological states, however, are too far below the Fermi level to play any role in the superconductivity of this system.

On the other hand, our measurements (Fig. 1(b), see inset) and calculations (Fig. 1(c)) reveal an additional pair of sharp spectral features which intersect the Fermi level approximately mid-way along the direction, which we label the and bands. From photon energy-dependent measurements (Supplementary Fig. S1 [35]), we find that these have negligible dispersion in the out-of-plane direction, and we thus assign these as surface states.

The origin of these states is evident from the calculated bulk band structure shown along the out-of-plane direction in Fig. 1(d). In PdTe, the inter-layer hopping between orbitals in neighbouring layers is sufficiently high that the -derived band crosses through both the bonding and anti-bonding -derived bands (lower and upper pair, respectively, of the red coloured bands in Fig. 1(d)) as they disperse along the direction. A protected bulk Dirac point and an inverted band gap with non-trivial topological order are therefore generated above the Fermi level. These are analogous to the crossings which stabilise the type-II Dirac point and topological surface states in the occupied electronic structure discussed in Ref. [29] and introduced briefly above. The inverted band gap above should give rise to topological surface states. While this band inversion is eV above the Fermi level along , the in-plane bandwidths are large, and the relevant band inversion can be traced to the bulk states in the vicinity of the Fermi level at the time-reversal invariant -point. Here, the pair exchange arising from the bulk boundary correspondence of topological surface states [36; 37] enforces that the dispersion of the topological surface states are pulled down towards the Fermi level. Experimentally, we find that both the “upper” (labeled ) and “lower” (labeled ) branch of the topological state cross (Fig. 1(b, inset)), although the occupied bandwidth of the -band along this direction is small ( meV).

To validate the above picture, we compare PdTe with its sister compound PtSe. The occupied electronic structure is similar to PdTe, hosting a type-II bulk Dirac cone and a pair of topological surface states below (Fig. 1(f,h)). These arise from the crossing of the Se -derived band with the bonding Se bands (Fig. 1(e)) [29; 32]. The bandwidth of the Se -derived band is much smaller than that of the Te-derived one, owing to the higher electronegativity of Se than Te. This leads to a stronger metal-chalcogen bond, and therefore weaker interlayer hopping in PtSe. Crucially, although a semi-metallic ground state still occurs (Fig. 1(g) and Supplementary Fig. S1 [35]), the upper and lower branches of Se-derived states remain completely separable in PtSe. The band inversions above that occur for PdTe are therefore absent in this compound (Fig. 1(e-h)), pointing to a powerful opportunity to tune underlying topological phase diagrams in transition-metal dichalcogenides by varying interlayer hopping strengths [29].

Figure 2 shows the Fermi surfaces formed by the topological states which intersect the Fermi level in the Te-based compound. The band forms a small approximately circular electron pocket, located approximately midway along . This electron pocket rapidly shrinks and then vanishes in constant energy slices taken below the Fermi level (Fig. 2(b)), reflecting the narrow occupied bandwidth of this band along . However, this is only a local minimum of the surface state dispersion located within a narrow projected bulk band gap along this direction (see Supplemental Fig. S2 [35]). The band disperses upwards above the Fermi level along the direction perpendicular to before turning over again to form the intense rim of spectral weight that borders the three-dimensional bulk bands that are evident as diffuse filled-in spectral weight spanning out away from the line. With increasing momentum away from this line, these surface states become degenerate with the bulk bands, and ultimately lose all spectral weight. They thus appear to form open-ended arc-like features. We stress that these are not Fermi arcs of the form discussed extensively as spanning surface projections of bulk Dirac and Weyl points [38; 39; 40; 41; 42; 43; 44; 45; 46]. Instead, they reflect the small projected bulk band gaps here (Supplemental Fig. S2 [35]), with the surface states merging into the bulk continuum as they disperse away from the high-symmetry line. As these surface features derive from the same state as the pocket, we label them in Fig. 2.

Similarly, the Fermi crossing of the band forms a small ellipsoidal section of Fermi surface, which again merges into the bulk continuum towards . This opens into apparently disconnected sections in constant energy contours below (see also Supplemental Fig. 4 [35]), again due to the presence of pockets of bulk states; the latter are evident in our measurements as diffuse spectral weight into which the sharp surface-state features merge, and are clearly seen in our measured dispersions shown in Supplemental Fig. S2 [35].

The small projected band gaps of the bulk spectrum leads to only small momentum space regions centered along the directions in which the surface states are well defined. This results in a rich multi-valley Fermi surface (Fig. 1(a)), far removed from the generic isolated near-circular Fermi surfaces of topological surface states in, e.g., BiSe class topological insulators. Nonetheless, we show in Fig. 2 that the Fermi surfaces observed here still maintain a spin texture reminiscent of the simple chirality of a conventional topological surface state.

Our spin-ARPES measurements reveal that the surface states host a strong spin polarisation ( 50% from fits to energy distribution curves (EDCs); Supplemental Fig. S3 [35]). Along , the chiral (perpendicular to ) spin component is dominant (Fig. 2(a)), consistent with the underlying trigonal symmetry of the crystal surface. The and bands have opposite spin polarisation, supporting their assignment as the two branches of a topological surface state. Both Fermi crossings of the band along have the same sign of . The spin, therefore, does not wind around the closed circular -band Fermi surface. Instead, it develops a finite (parallel to ) component of opposite sign on the two sides of the high-symmetry line, canting the spin towards the Brillouin zone boundary in the in-plane direction. Similarly, the two -bands have the same sign of but opposite sign of , as shown extracted from our experimental spin-polarised Fermi surface maps in Fig. 2(d). The overall result is a Fermi surface spin-texture of the -derived Fermi surfaces that has a global chiral winding around the Brillouin zone centre, but with a significant radial component developing away from the line (Fig. 2(c)), similar to what might be expected for a more conventional topological surface state if it develops a hexagonal warping away from circular geometry [47; 48]. While at the Fermi level only the chiral component of the -band is visible, similar radial components, as well as an out-of-plane spin canting, develop for its highly-fragmented surface state contours below (Supplemental Fig. S4 [35]).

These findings raise an exciting prospect to investigate how such topological states, with their complex Fermi surface spin textures, interplay with the bulk superconductivity of PdTe. To this end, we investigate the superconductivity at the surface using low-temperature STM (Fig. 3). Our measured tunneling spectra (Fig. 3(b)) reveal a clearly-resolved U-shaped superconducting gap, indicating nodeless superconductivity. This is well described by a Dynes model fit, indicating a fully-gapped state with a superconducting gap size V. This gives a ratio , comparable to the BCS prediction. A slight underestimation of the Dynes fit to the measured spectra at higher bias voltages may suggest a small anisotropy of the superconducting gap. Crucially, however, there is no evidence of nodes in the superconducting order parameter or of in-gap states. Similar conclusions can be drawn from Josephson tunneling measurements [49] performed using a superconducting tip (Fig. 3(b)), where again a clear lack of any zero-energy bound states is evident.

Measuring the evolution of the superconducting gap size in a magnetic field applied normal to the sample surface, we estimate an upper critical field, mT. This is consistent with the upper critical field of the bulk superconducting state as judged from susceptibility measurements in PdTe [50]. Intriguingly, the bulk superconductivity in PdTe has recently been reported to be of Type I character [50]. We have, however, observed a vortex core in our STM measurements in a magnetic field of 10 mT (Fig. 3(c)), indicating that the superconductivity we probe is of type-II character.

From the measured decay length of the vortex core of nm (Fig. 3(c)), this would suggest an upper critical field of mT for a BCS superconductor, assuming a one-to-one correspondence of the measured decay length to the coherence length. This is in reasonable agreement with our measured value of the surface upper critical field. Finally, we note that scanning tunneling spectroscopy performed as a function of distance away from the centre of the vortex core (Fig. 3(d)) again yields no evidence for the presence of zero-energy bound states. Taken together, these results demonstrate that the surface of PdTe supports a conventional fully-gapped -wave superconducting state, well described by BCS theory. This co-exists with well-defined topologically non-trivial surface states, with complex multi-component and multi-valley Fermi surfaces and rich vorticial spin textures.

The complexity of the surface Fermi surface, as well as the presence of numerous bulk states degenerate in energy but at different in-plane momenta, may ultimately explain why the dominant superconducting pairing remains topologically trivial at the surface. In any case, our results clearly demonstrate that the presence of topologically non-trivial states at the Fermi level is not a sufficient criterion to realise topological superconductivity. Beyond this, our findings highlight the importance of -dependent band inversions within a single orbital manifold for generating topological surface states with rich and complex surface Fermi surfaces. Moreover, they demonstrate how these can be effectively tuned by varying interlayer hopping strengths, paving the way to the design of new topological materials.

###### Acknowledgements.

Acknowledgements: We gratefully acknowledge support from the Leverhulme Trust, the Engineering and Physical Sciences Research Council, UK (Grant Nos. EP/M023427/1 and EP/I031014/1), the Royal Society, CREST, JST (Nos. JPMJCR16F1 and JPMJCR16F2), the Japan Society for Promotion of Science (Grant-in-Aid for Scientific Research (S); No. 24224009 and (B); No. 16H03847), and the International Max-Planck Partnership for Measurement and Observation at the Quantum Limit. OJC, MJN, LB, VS, and JMR acknowledge EPSRC for PhD studentship support through grant Nos. EP/K503162/1, EP/G03673X/1, EP/L505079/1, and EP/L015110/1. IM acknowledges PhD studentship support from the IMPRS for the Chemistry and Physics of Quantum Materials. We thank Diamond Light Source (via Proposal Nos. SI9500, SI12469, SI13438, and SI16262) and Elettra synchrotron for access to the I05 and APE beamlines, respectively, that contributed to the results presented here.## References

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