March 1, 2024

Evidence for superconducting Fermi arcs

ARPES measurements

ARPES measurements were performed on day 1two and 13ARPES end stations26 at the BESSY II synchrotron (Helmholtz-Zentrum Berlin) as well as at the Leibniz-Institut für Festkörper und Werkstoffforschung Dresden (IFW) laboratory using the 5.9 eV laser light source. Samples were cleaved in situ at a pressure less than 1 × 10−10mbar and measured at temperatures of 15 K and 1.5 K in the BESSY II and 3-30 K in the IFW laboratory. The experimental data were obtained using synchrotron light in the photon energy range of 15 to 50 eV with horizontal polarization and laser light with horizontal and circular polarizations. The angular resolution was set to 0.2–0.5° and the energy resolution to 2–20 meV. The findings of the experiments were consistent and reproducible across multiple samples.

The simultaneous presence of non-superconducting and surface superconducting states makes it difficult to detect true coherence peaks with ARPES. Our synchrotron experiments, with energy resolution of the order of 5 meV, turned out to be insufficient to detect even the displacements of the leading edges of the corresponding arc peaks with FWHM of the order of 10 meV and peak-to-background ratio of approximately 5. This occurs because the arc states are always at the top of the bulk continuum. Only by measuring with energy resolution of the order of 1–2 meV were we able to observe sufficiently sharp peaks (Fig. 3c, from Extended Data Fig. 4) and their sensitivity to temperature. The sharpest features need to be found on the surface.

A superconducting gap in the arcs is likely anisotropic. We have included error bars in Fig. 4e to show the influence of a small shift of the beam spot and therefore a slightly different emission angle. Taking into account the very high location in moment space, this could lead to probing a different part of the arc and therefore different kFwhere the superconducting gap is slightly different.

Bulk band structure and Fermi arc position

1, we show ARPES Fermi surface maps obtained using photon energies from 15 eV to 43 eV. A relatively strong variation in the pattern suggests a reasonable kz-sensitivity of our experiment. We found that the ideal value of the internal potential is equal to 10.5 eV. This is in line with the previous study by Jiang et al.17.

In Figure 2 of the extended data, we present further evidence that our assignment of surface and volume features is correct. Extended data Fig. 2a shows EDCs obtained through the Fermi arc for different photon energies (from synchrotron and laser sources), along with the theoretical EDC for the fully integrated system kz. The peak corresponding to the Fermi arc remains clearly visible without any noticeable scatter for different values ​​of kz, while the peaks located further below the Fermi level disperse. Such lack of dispersion is peculiar to surface states.

3, we show an analogue of Fig. 1e-g, but here we compare the experimental data with the results of band structure calculations performed using the linear muffin-tin orbital (LMTO) method in the atomic sphere approximation as implemented in computer code PY LMTO27 . As can be seen from the figure, the agreement is at the same level as before, supporting the previous conclusion regarding the good agreement between the experimental and theoretical 3D band structure.

In Figure 4b of the extended data, we present the clearest EDCs among the various samples and cleavages. Most have FWHM below 3 meV and a peak-to-trough ratio greater than 30.

Band structure calculations

We perform density functional theory calculations using the full potential non-orthogonal local orbital scheme from ref.28 within the general gradient approximation29and extracted a Wannier function model. This allows the determination of bulk projected spectral densities (without surface states) and the spectral densities of semi-infinite slabs via Green’s function techniques.30 . To model semi-infinite plate surface superconductivity, the Wannier model is extended to the BdG formalism with a zero gap function, except for a constant Wannier orbital diagonal singlet gap function matrix in the first three PtBitwo layers. A modification of the Green’s function method is used to accommodate this surface-specific term.

Surface superconductivity calculations

To model a system that has a non-zero gap function only at the surface – in the first 30aB which is 3(PtBitwo) layers – we modified the standard Green’s function technique for semi-infinite slabs. The system is constructed by a semi-infinite chain of identical blocks consisting of 3(PtBitwo) layers, repeating indefinitely far from the surface. Each block has a Hamiltonian Hk for each pseudo-moment k in the plane perpendicular to the surface and a jumping matrix Vk, which couples neighboring blocks. The minimum block size is determined by the condition that H It is V describe all possible jumps. To add superconductivity, the BdG formalism is used by extending the matrices as follows:

$$\begin{array}{rcl}{H}_{k,{\rm{BdG}}} & = & \left(\begin{array}{cc}{H}_{k} & {\varDelta }_{k}\\ {\varDelta }_{k}^{+} & -{H}_{-k}^{* }\end{array}\right),\\ {V}_{k ,{\rm{BdG}}} & = & \left(\begin{array}{cc}{V}_{k} & 0\\ 0 & -{V}_{-k}^{* }\ end{array}\right),\end{array}$$

where we choose \({\varDelta }_{k}={\delta }_{i{i}^{{\prime} }}\left(\begin{array}{cc}0 & {V}_{0}\ \ -{V}_{0} & 0\end{array}\right)\) with I being a spinless Wannier function index and the 2 × 2 matrix to act on the spin subspace of a single Wannier function. This choice also leads \(\varDelta \left[{V}_{k,{\rm{BdG}}}\right]=0\)since V is an off-diagonal part of the full Hamiltonian. To model surface-only superconductivity, we let V0 = 0 for all (infinite) blocks except the first one, which gets a finite block V0 = 2meV.

The standard Green’s function solution to this problem consists of determining the propagator X which encompasses all diagrams that describe paths that start at a given block, propagate anywhere toward the infinite side of that block, and return to that block. X also describes Green’s function G00 of the first block and the self-energy to be added to the Hamiltonian to obtain G00 (a self-consistency condition) \({G}_{00}=X={\left({\omega }^{+}-H-\Sigma \right)}^{-1}\),Σ = VXV+ (In practice, however, self-consistency is achieved by an accelerated algorithm). From this recursion, relations can calculate all other Green’s function blocks. These can be derived by subdividing propagation diagrams into irreducible parts using known components, in particular X.

If the first block is different from all the others (as is the case due to Δk) it is necessary to modify the method as follows. Let the first block have Hamiltonian H and jumps to the second block v (while all other blocks are described by H It is V). Then the irreducible subdivision of the propagation diagrams for G00results in \(g={\left({\omega }^{+}-h\right)}^{-1}\).

$$\begin{array}{l}{G}_{00}=g+gvX{v}^{+}g+(\,gvX{v}^{+})g\\ \,=\,\ frac{1}{{\omega }^{+}-h-vX{v}^{+}}\end{array}$$

which contains the surface Hamiltonian and a modified self-energy depending on the X of the unmodified semi-infinite slab. From this we can derive the Green function of the second block


and everyone else

$${G}_{n+1,n+1}=X+X{V}^{+}{G}_{nn}VX,\quad n > 0$$

which can be used to obtain the spectral density up to a certain penetration depth. Note that in our case BdG \(H={H}_{k,{\rm{BdG}}}\left[{V}_{0}=0\right]\), \(V={V}_{k,{\rm{BdG}}}\left[{V}_{0}=0\right]\) It is \(h={H}_{k,{\rm{BdG}}}\left[{V}_{0}\ne 0\right]\), v= V . The BdG spectral density is particle-hole symmetric and to obtain results that resemble ARPES data, it is necessary to use the particle-particle block Gand is (the upper left room of the Gmatrix) only.

Extended Data Figure 5b shows the spectra resulting from this method along the path indicated in Extended Data in Figure 5a. Note that a gap is opened in the surface band pockets close to the Fermi energy, while the rest of the spectrum remains gapless (if we let V0≠ 0 for all blocks, we obtain a completely empty spectrum). Extended data Fig. 5c shows an enlarged region around the surface state. Note that the bulk bands have no gaps (vertical features in dark blue), while the surface state shows a gap and corresponding backward bending of the band. Particle-hole symmetry becomes apparent, albeit with a greater spectral weight for the occupied part because we use Gand isjust.

Additional discussion

One approach to demonstrate the existence of topologically protected states with a topological insulator is to perform spin-resolved ARPES. In this technique, the spin-locking effect determines the spin structure in the vicinity of the surface’s Dirac node. However, the situation is quite different for Weyl semimetals. Here, there is no specific structure or spin configuration associated with the Weyl nodes, which can occur at generic points in the Brillouin zone. As the inversion is broken and spin-orbital coupling is present, each band in a k-point naturally has a rotation direction, but this rotation texture is smooth. Consequently, spin-resolved ARPES measurements cannot directly reveal Weyl points.

We would like to exclude interpretation of our data based on the order of the density waves, which could, in principle, result in similar features in the spectra. Charge density waves require a redistribution of spectral weight in momentum space, characterized by the particulark-vector (vectors). We always observe almost the same Fermi surface maps and underlying scatterings, regardless of temperature. In line with these observations are the results of STM studies that never detected any type of reconstruction. We never observe any replication of the arcs or the deeper surface states, such as a strong feature at (−0.2, −0.2) in Fig. It is also unclear whichk-vector would be suitable for characterizing the order of the density wave. If the arcs are simply overlapping in terms of momentum, they are all of electron-like topology, then the opening of the hybridization holes seems very unlikely. Finally, the fundamental difference between density wave holes and superconducting holes is that the latter are always fixed at the Fermi level. This is the only energy range where we observe changes in PtBi spectratwowith temperature.

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