Abstract
The topological aspects of electrons in solids can emerge in real materials, as represented by topological insulators. In theory, they show a variety of new magnetoelectric phenomena, and especially the ones hosting superconductivity are strongly desired as candidates for topological superconductors. While efforts have been made to develop possible topological superconductors by introducing carriers into topological insulators, those exhibiting indisputable superconductivity free from inhomogeneity are very few. Here we report on the observation of topologically protected surface states in a centrosymmetric layered superconductor, βPdBi_{2}, by utilizing spin and angleresolved photoemission spectroscopy. Besides the bulk bands, several surface bands are clearly observed with symmetrically allowed inplane spin polarizations, some of which crossing the Fermi level. These surface states are precisely evaluated to be topological, based on the Z_{2} invariant analysis in analogy to threedimensional strong topological insulators. βPdBi_{2} may offer a solid stage to investigate the topological aspect in the superconducting condensate.
Introduction
Topological insulators are characterized by the nontrivial Z_{2} topological invariant acquired when the conduction and valence bands are inverted by spin–orbit interaction (SOI), and the gapless surface state appears^{1,2,3}. This topologically nontrivial surface state possesses the helical spin polarization locked to momentum, and is expected to host various kinds of new magnetoelectric phenomena. Especially, the ones realized with superconductivity are theoretically investigated as the candidates for topological superconductor^{2,3,4}, whose excitation is described as Majorana Fermions, that is, the hypothetical particles originating from the field of particle physics^{5,6,7,8}. Experimentally, several superconductors developed by utilizing topological insulators are reported thus far, such as Cuintercalated Bi_{2}Se_{3} (refs 9, 10, 11, 12), Indoped SnTe^{13} and M_{2}Te_{3} (M=Bi, Sb) under pressure^{14,15}. While the previous studies of pointcontact spectroscopy on Cu_{x}Bi_{2}Se_{3} (refs 9, 10) and InSnTe^{13} suggest the existence of Andreev bound states thus raising the possibility of topological superconductivity, the scanning tunnelling microscope/spectroscopy reports the simple swavelike full superconducting gap^{16}. Theoretically, this contradiction has been discussed in terms of the possible peculiar bulk oddparity pairing^{17}, which awaits experimental verifications by various probes^{18,19}. However, partly due to the inhomogeneity effect accompanied by doping or pressurizing, the unambiguous clarification of superconducting states in doped topological insulators has been hindered until now. The halfHeusler superconductor RPtBi (R: rare earth) is another class of material recently reported as a candidate for topological superconductors^{20,21}. Practically, however, its low critical temperature (T_{c}) of T_{c}<2 K and the noncentrosymmetric crystal structure without a unique cleavage plane may pose some difficulties for its further investigation.
In this work, we introduce a superconductor βPdBi_{2} with a centrosymmetric tetragonal crystal structure of space group I4/mmm^{22,23,24} as shown in Fig. 1a. It has a much simpler structure compared with the related noncentrosymmetric superconductor αPdBi, recently being discussed as a possible topological superconductor^{25,26}. Pd atoms, each of them located at the centre of the square prism of eight Bi atoms, form the layered bodycentred unit cell. PdBi_{2} layers are stacked in van der Waals nature, making it a feasible compound for cleaving. We investigate the electronic structure of βPdBi_{2} using (spin) angularresolved photoemission spectroscopy, (S)ARPES. With the large single crystals of good quality, exhibiting the high residual resistivity ratio (∼14) and a clear superconducting transition at T_{c}=5.3 K, several spinpolarized surface states are clearly observed in addition to the bulk bands. On the basis of the relativistic firstprinciples calculation on bulk and the slab calculation on surface, we find that the observed surface states can be unambiguously interpreted to be topologically nontrivial.
Results
Bulk and surface band structures
Here we present the ARPES result obtained using the singlecrystalline βPdBi_{2}. The resistivity and magnetic susceptibility of the sample as shown in Fig. 1b,c clearly indicate the sharp superconducting transitions. The band structure of βPdBi_{2} observed by ARPES is shown in Fig. 2b,c. For simply describing the (S)ARPES results hereafter, we use the projected twodimensional (2D) surface Brillouin zone depicted in Fig. 2a by a green square. The projected highsymmetry points are , and , and we define k_{x} as the momentum along –. The ARPES image in Fig. 2c is recorded along – and –, respectively. Bands crossing the Fermi level (E_{F}) are predominantly derived from Bi 6p components with large dispersions from the binding energy (E_{B}) of E_{B}∼6 eV to above E_{F}. On the other hand, bands mainly consisting of Pd 4d orbitals are located around E_{B}=2.5∼5 eV with rather small dispersions. Near E_{F}, two hole bands (α, β) and one electron band (γ) are observed along –, whereas for –, the large ARPES intensity from another electron band (δ) is additionally observed. As we can see in Fig. 2b, the experimental Fermi surface mapping mostly well agrees with the 2D projection of the calculated bulk Fermi surfaces (Fig. 2a).
To compare with ARPES, the calculation of bulk band dispersions projected into 2D Brillouin zone is shown in Fig. 2d. Considering that the ARPES intensity includes the integration of finite k_{z}dispersions due to the surface sensitivity, the overall electronic structure is in a good agreement with the calculation; nevertheless, several differences can be noticed. The most prominent one appears in the orange rectangles in Fig. 2c,d. A sharp Diracconelike dispersion is experimentally observed where the calculated bulk bands show a gap of ∼0.55 eV around the point. To confirm its origin, we performed a slab calculation for 11 PdBi_{2} layers (Fig. 2e). Apparently, a Diracconetype dispersion appears in the gapped bulk states, showing a striking similarity to ARPES (Fig. 2c). It clearly presents the surface origin of this Diraccone band.
Now we focus on the observed surface Diraccone band. The closeup of the surface Dirac cone is demonstrated in Fig. 3a, indicating its crossing point at E_{B}=E_{D}=2.41 eV (E_{D}: the energy of Dirac point where the bands cross each other). Such a clear Diracconeshaped band strongly reminds us of the helical edge states in threedimensional (3D) strong topological insulators. We can see the very isotropic character of surface Dirac cone in its constantenergy cuts (Fig. 3b), appearing as the perfectly circularshaped contour even at E_{B}=E_{D}–0.8 eV with a large momentum radius of 0.3 Å^{−1}. It is in contrast to the warping effect often appearing in trigonal strong topological insualtors^{27,28}. The spin polarization of surface Dirac cone is also directly confirmed by SARPES experiments as depicted in Fig. 3c (ref. 29). Figure 3e,f shows the results for the ycomponent spin, measured along k_{x} (–). Because of C_{4v} symmetry, x and zcomponents are forbidden (Supplementary Note 1; Supplementary Fig. 1). The red (blue) curves in Fig. 3f, indicating the energy distribution curves of spinup (down) components, clearly show the spinpolarized band dispersions. As easily seen in the SARPES image (Fig. 3e), the spin polarization with spinup (spindown) for negative (positive) dispersion of surface Dirac cone is confirmed. The observed spinpolarized surface Dirac cone thus presents a strong resemblance to the helical surface state in strong topological insulators.
Analysis of the topological invariant
To evaluate whether the observed surface state is topologically nontrivial, we derive the Z_{2} invariant ν_{0} for βPdBi_{2}, in analogy to 3D strong topological insulators^{30}. For 3D band insulators with inversion symmetry, ν_{0} obtained from the parity eigenvalues of filled valence bands at eight timereversal invariant momenta (TRIM) classifies whether it is a strong topological insulator (ν_{0}=1) or not (ν_{0}=0). The bulk βPdBi_{2} is apparently a metal; nevertheless, here we define a gap in which there is no crossing of the bulk band dispersions through the entire Brillouin zone. By considering this gap, we discuss its topological aspect by calculating ν_{0}. The calculated bulk bands without and with SOI are shown in Fig. 4a,b, respectively. The valence bands are identified by numbers (from 1st to 10th) as indicated on the right side of respective graphs. The bands are numbered by the energy (E) at the Z point. Note that all bands are doubly spindegenerate. By comparing Fig. 4a,b, we notice that many anticrossings are introduced by SOI, including the ∼0.55 eV gap opening in the green rectangle region where the surface Dirac cone appears. Here we focus on the gap between the 7th and 6th bulk bands, namely gap 7−6, shaded by pink in Fig. 4b. The distribution of the direct gap between the 7th and 6th bands can be evaluated by the joint density of states as a function of the gap energy E_{g}, defined as . Here, E_{6}(k) and E_{7}(k) represent the respective eigenenergies of the 6th and 7th bands at momentum k with k=(k_{x}, k_{y}, k_{z}). The result for gap 7−6 is shown in Fig. 4e, which guarantees the minimum value of 0.105 eV gap opening between the 7th and 6th bands through the entire Brillouin zone.
By considering the obtained gap, we discuss its topological aspect by calculating ν_{0} in analogy to 3D strong topological insulators. As shown in Fig. 4g, the eight TRIM in the Brillouin zone of βPdBi_{2} with I4/mmm symmetry are Γ, Z, two X and four N points. Considering these TRIM, Z_{2} invariant for the gap between the (N+1)th and Nth bulk bands, ν_{0}(N), can be calculated by , where represents the parity eigenvalue (±1) of the mth band at ith TRIM. Note that since there are even numbers of X and N points, only Γ_{i}=Γ and Z contribute to the calculation of ν_{0}(N), that is, . Thus, ν_{0} can be calculated by considering solely Γ and Z points, whose symmetries of wavefunctions are listed in Fig. 4d for respective bands. Those indicated by red (black) is of odd (even) parity. We find that gap 7−6 is characterized by ν_{0}(6)=1, indicating its analogy to 3D strong topological insulators. This requires an odd number of surface states connecting the 7th and 6th bands, to topologically link the bulk βPdBi_{2} and a vacuum. The observation of spinhelical surface Dirac cone in gap 7−6 clearly represents the characters of such topologically protected surface states.
Topological surface state crossing E_{F}
By further looking at the list of ν_{0} in Fig. 4d, we notice ν_{0}(8)=1 for gap 9−8 shaded by blue in Fig. 4b, which has a minimum gap of 0.127 eV as confirmed by the calculation (Fig. 4f). It suggests that the topological surface states connecting the 9th and 8th bands must exist, where we may observe the effect of superconductivity if located close enough to E_{F}. To clarify this possibility, the closeup of ARPES image near E_{F} is shown with the calculation in Fig. 5b,c. The green curves in Fig. 5c indicate the calculated surface states crossing E_{F} separately from the 2D projected bulk bands shaded by grey. They appear at the smallerk_{x} side of β (8th) and γ (9th) bands. Experimentally, the sharp peaks indicative of 2D surface states are observed in momentum distribution curve at E_{F}, as denoted by S1 and S2 in Fig. 5a. As can be seen in the list of ν_{0} in Fig. 4d, S2 should be the topological surface state connecting the 9th and 8th bands, whereas S1 appearing in gap 8−7 must be trivial.
The spin polarization of the topological surface state S2 as well as the trivial surface state S1 is also confirmed experimentally. As shown in Fig. 5d, the yoriented spin polarizations of S1 (#2–5) and S2 (#7–10) along k_{x} (–) are clearly observed in the spinresolved spectra. Here, the peak positions for S1 and S2 (bulk β) bands are depicted by green circles (black squares). We can see that S1 and S2 are both spinpolarized with spinup for k_{x}>0, whereas they get inverted for k_{x}<0 (Fig. 5e,f) as required by the timereversal symmetry. These clearly indicate that both topological and trivial surface states crossing E_{F} possess the inplane spin polarizations.
Discussion
The Z_{2} analysis shows that odd number of gapless surface states in gap 9−8, connecting the 9th and 8th bands, must exist between and . To confirm whether the experimentally observed S2 indeed corresponds to this topological surface state, we need to carefully look at the slab calculation since S2 crosses E_{F} and extends to the unoccupied state. By tracking the calculated data from towards (Fig. 6a), we first notice that S2 is derived from the local minimum of the 9th (γ) band. S2 then crosses E_{F} and reaches up to E–E_{F}=2 eV without merging into the bulk states. At , although it gets overlapped with 2D projected bulk bands, we can distinguish S2 forming a Rashbalike crossing point at E–E_{F}=2.4 eV. After the crossing, S2 band eventually gets merged into the 8th (β) band. It thus shows that S2 indeed connects the 9th and 8th bands. The crossing of S2 surface band at is more clearly seen, by comparing the 2D projected bulk (Fig. 6b) and the slab (Fig. 6c) calculations magnified near the crossing point. The crossing of the S2 surface band at is distinguished in Fig. 6c, by following the eigenenergies highlighted with the red markers. Note that no such crossing exists for the calculation of bulk in Fig. 6b. S2 thus possesses a similarity to the Dirac cone that connects the gap with the crossing at , and is indeed a topologically protected surface state.
Here we note that the spinpolarized topological S2 and the surface Dirac cone are both derived as a consequence of SOI, but in different processes. For the case of S2 in gap 9−8, we see that ν_{0} changes to 1 by including the SOI. It thus indicates the band inversion associated with the 8th, 9th and 10th bands occurring at Γ (see Fig. 4c,d) induced by the SOI. This situation is fairly similar to the topological phase transition being discussed in 3D strong topological insulators^{31}. For the surface Dirac cone in gap 7−6, on the other hand, ν_{0}=1 is realized already in the nonrelativistic case (Fig. 4c), due to the inversion of A_{1g} and A_{2u} bands introduced by Bi6p–Pd4d mixing. This nonrelativistic situation should be rather similar to the 3D Dirac semimetals^{32,33}, as represented by the bulk Dirac points appearing along Z–M and Z–X (Fig. 4a), which may accompany the spindegenerate surface states (Fermi arcs). The role of the SOI in this case is the gap opening at these bulk Dirac points, giving rise to the spinpolarized surface Dirac cone connecting the gap edges.
The next future step for βPdBi_{2} should be the direct elucidation of the superconducting state. Lowtemperature ultrahighresolution ARPES will surely be a strong candidate for such investigation^{34,35}. There may be a chance to observe nontrivial superconducting excitations, by selectively focusing on the surface and bulk band dispersions as experimentally presented in Bi_{2}Se_{3}/NbSe_{2} thin film^{34}. Scanning tunnelling microscope/spectroscopy, on the other hand, can locally probe the superconducting state around the vortex cores. As theoretically suggested, it may capture the direct evidence of Majorana mode^{4,11,36,37}. We should note that βPdBi_{2} will also provide a solid platform for bulk measurements such as thermal conductivity and nuclear magnetic resonance, which are expected to give some information on the oddparity superconductivity^{18,19}. It may thus contribute to making the realm of superconducting topological materials, and pave the way to various new findings such as the direct observation of Majorana fermions dispersion and/or surface Andreev bound states^{36,37}, clarification of its relation to the possible oddparity superconductivity^{11,17} and bulksurface mixing effect^{36,38}.
Methods
Crystal growth
Single crystals of βPdBi_{2} were grown by a melt growth method. Pd and Bi at a molar ratio of 1:2 were sealed in an evacuated quartz tube, prereacted at high temperature until it completely melted and mixed. Then, it was again heated up to 900 °C, kept for 20 h, cooled down at a rate of 3 °C h^{−1} down to 500 °C and rapidly quenched into cold water. The obtained single crystals had good cleavage, producing flat surfaces as large as ∼1 × 1 cm^{2}. The resistivity shown in Fig. 1b and the magnetic susceptibility shown in Fig. 1c exhibit the clear superconducting transition at T_{c}=5.3 K.
Angularresolved photoemission spectroscopy (ARPES)
ARPES measurement with the HeIα light source (21.2 eV) were made at the Department of Applied Physics, The University of Tokyo, using a VUV5000 Hedischarge lamp and an R4000 hemispherical electron analyzer (VGScienta). The total energy resolution was set to 10 meV. Samples were cleaved in situ at around room temperature and measured at 20 K.
Spin and angularresolved photoemission spectroscopy (SARPES)
SARPES with the HeIα light source (21.2 eV) was performed at the Efficient SPin REsolved SpectroScOpy (ESPRESSO) end station attached to the APPLEIItype variable polarization undulator beamline (BL9B) at the Hiroshima Synchrotron Radiation Center (HSRC)^{29}. The analyzer of this system consists of two sets of verylowenergy electron diffraction spin detectors, thus enabling the detection of the electron spin orientation in three dimension^{39}. The angular resolution was set to ±1.5° and the total energy resolution was set to 35 meV. Samples were cleaved in situ at around room temperature and measured at 20 K.
Band calculations
Firstprinciples electronic structure calculations within the framework of the density functional theory were performed using the fullpotential linearized augmented planewave method as implemented in the WIEN2k code^{40}, with the generalized gradient approximation of Perdew, Burke and Ernzerhof exchangecorrelation function^{41}. SOI was included as a second variational step with a basis of scalarrelativistic eigenfunctions.
The experimental crystal data (a=3.362 Å, c=12.983 Å, z(Bi)=0.363) were used for the bulk calculations. The (001) surface was simulated by a slab model; a stacking of 11 PdBi_{2}triple layers along the c axis with a 15 Å of vacuum layer, forming a tetragonal crystal structure of space group P4/mmm with the lattice constants of a=3.362 Å and c=83.423 Å.
The planewave cutoff energy was set to R_{MT}K_{max}=9, where the muffin tin radii are R_{MT}=2.5 a.u. for both Bi and Pd. The Brillouin zone was sampled with the MonkhorstPack scheme^{42} with momentum grids finer than Δk=0.02 Å^{−1} (for example, a Γcentred 38 × 38 × 38 kpoint mesh was used for the Fermi surface visualization, corresponding to Δk=0.009 Å^{−1}).
Additional information
How to cite this article: Sakano, M. et al. Topologically protected surface states in a centrosymmetric superconductor βPdBi_{2}. Nat. Commun. 6:8595 doi: 10.1038/ncomms9595 (2015).
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Acknowledgements
We thank R. Arita for fruitful discussion, A. Kimura, H. Namatame and M. Taniguchi for sharing SARPES infrastructure. M.S. is supported by Advanced Leading Graduate Course for Photon Science (ALPS). M.S., K.O. and M.K. are supported by a research fellowship for young scientists from JSPS. This research was partly supported by Precursory Research for Embryonic Science and Technology (PRESTO), Japan Science and Technology Agency; the Photon Frontier Network Program of the MEXT; Research Hub for Advanced Nano Characterization, The University of Tokyo, supported by MEXT, Japan; and GrantinAid for Scientific Research from JSPS, Japan (KAKENHI KibanB 24340078 and KibanA 23244066). The experiments were approved by the Proposal Assessing Committee of the Hiroshima Synchrotron Radiation Center (Proposal No. 14B7).
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M.S. and H.S. carried out ARPES. M.S., T.O. and K.I. carried out SARPES. K.O., M.K. and T.S. synthesized and characterized the single crystals. T.S. carried out the calculations. M.S. and K.I. analysed (S)ARPES data and wrote the manuscript with input from K.O., M.K., T.O. and T.S. K.I. conceived the experiment. All authors contributed to the scientific discussions.
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Sakano, M., Okawa, K., Kanou, M. et al. Topologically protected surface states in a centrosymmetric superconductor βPdBi_{2}. Nat Commun 6, 8595 (2015). https://doi.org/10.1038/ncomms9595
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DOI: https://doi.org/10.1038/ncomms9595
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