2023

[1]Sergey Borisenko

1]Leibniz Institute for Solid State and Materials Research, Helmholzstraße 20,
Dresden, 01069, Germany

2]Kyiv Academic University, 36 Vernadsky blvd, Kyiv, 03142, Ukraine

3]Helmholtz-Zentrum Berlin für Materialien und Energie, BESSY II, Berlin,
12489, Germany

4]Centro Atómico Bariloche, Instituto de Nanociencia y Nanotecnología (CNEA-
CONICET) and Instituto Balseiro, Av. Bustillo, 9500, Argentina

5]Max Planck Institute for Solid State Research, Heisenbergstraße 1,
Stuttgart, 70569, Germany

# Superconducting Arcs

Andrii Kuibarov  Oleksandr Suvorov  Riccardo Vocaturo  Alexander Fedorov  Rui
Lou  Luise Merkwitz  Vladimir Voroshnin  Jorge I. Facio  Klaus Koepernik
Alexander Yaresko  Grigoriy Shipunov  Saicharan Aswartham  Jeroen van den
Brink  Bernd Büchner<EMAIL_ADDRESS>[  [  [  [  [

###### Abstract

An essential ingredient for the production of Majorana fermions that can be
used for quantum computing is the presence of topological superconductivity
Sato_2017 ; Sharma_2022 . As bulk topological superconductors remain elusive,
the most promising approaches exploit proximity-induced superconductivity
Proximity_Fu  making systems fragile and difficult to realize NanoWires_Mourik
; Frolov ; YSRIBS_Wang ; BorisenkoNPJQM . Weyl semimetals due to their
intrinsic topology Vishwanath  belong to potential candidates too Sato_2017 ;
Sharma_2022 , but search for Majorana fermions has always been connected with
the superconductivity in the bulk, leaving the possibility of intrinsic
superconductivity of the Fermi surface arcs themselves practically without
attention, even from the theory side.

Here, by means of angle-resolved photoemission spectroscopy and ab-initio
calculations, we unambiguously identify topological Fermi arcs on two opposing
surfaces of the non-centrosymmetric Weyl material PtBi2 Veyrat_Nanoletters .
We show that these states become superconducting at different temperatures
around 10K. Remarkably, the corresponding coherence peaks appear as the
strongest and sharpest excitations ever detected by photoemission from solids,
suggesting significant technological relevance. Our findings indicate that
topological superconductivity in PtBi2 occurs exclusively at the surface,
which not only makes it an ideal platform to host Majorana fermions, but may
also lead to a unique quantum phase - an intrinsic topological SNS Josephson
junction.

###### keywords:

ARPES, topological superconductivity, Weyl semimetal, surface Fermi arcs

##  1 Introduction

Figure 1: 3D Band Structure of PtBi2. a Crystal structure of PtBi2. b Fragment
of the band structure. One Weyl point (WP) is included. c Fermi surface, Weyl
and high symmetry points. Color scale corresponds to Fermi velocity. d
$\Gamma$MK plane of the BZ zone with projections of the Weyl points. Magenta
(blue) colors stand for positive (negative) chirality. e Fermi surface maps
taken using different photon energies and corresponding results of the band
structure calculations. We note, that fixed photon energy probes a sphere of
the large radius in the k-space, matching theoretical data formally only at
one point in the center. Theoretical Fermi maps were averaged of over a range
of 1/10 of the BZ size in the $k_{z}$ direction to account for experimental
uncertainties. The intensities of the theoretical Fermi maps were normalized
to the DOS for different $k_{z}$ points. f, g left: Energy-momentum intensity
distributions at 21 eV and 19 eV respectively along the cuts indicated by blue
dashed arrows in panel e, right: corresponding energy-momentum spectra taken
from the band structure calculation.

The realisation of topological superconductivity, which leads to robust
Majorana fermions in new materials has so far been hindered by numerous
experimental challenges. Among them are the sophisticated growth of nanowire
single crystals and heterostructures as well as fine-tuning of the composition
of non-stoichiometric compounds. Additionally, the rarity of spin-triplet
superconductors and extremely small inverted gaps in iron-based
superconductors BorisenkoNPJQM , proposed as intrinsic heterostructures
Zhang2018 , are responsibe for the lack of success in already existing
materials.

Weyl semimetals bear non-degenerate spin states both in the bulk and at the
surface and the doped version of either time-reversal-breaking or
noncentrosymmtric ones can become superconducting Meng_WeylSupc . The search
for topological superconductivity in such systems has been focused on finding
bulk superconductivity which would lead to Majorana fermion surface states.
The possibility of intrinsic superconductivity of the arcs themselves, related
to topology of the bandstructure with Weyl nodes, has virtually not been
considered. Although the arcs cannot support superconductivity in time-
reversal-breaking Weyl semimetals Sato_2017 , the noncentrosymmetric ones
remain an option.

Trigonal PtBi2 has emerged recently as a type-I Weyl semimetal, which together
with reported superconductivity Shipunov_PRM ; Veyrat_Nanoletters  makes it an
attractive candidate for topological superconductivity. Scanning tunneling
spectroscopy (STM) experiments confirmed the presence of surface
superconductivity by observing typical spectra of superconducting gaps
Schimmel_gap ; Shipunov_PRM . Here we found that the topological Fermi surface
arcs bear the superconductivity in PtBi2. This occurs on both of the two
nonequivalent surfaces of PtBi2.

##  2 Three-dimensional band structure

The electronic structure of trigonal PtBi2 has been studied both
experimentally and theoretically earlier YaoPRB2016 ; Setti ; Jiang_JAP2020 ;
Feng2019 ; Veyrat_Nanoletters ; Gao2018 . The material crystallizes in $P31m$
space group Shipunov_PRM  and exposes two different surfaces upon cleaving,
which we refer to as A and B below (Fig. 1a). The band structure (Fig. 1b)
arises mostly due to hybridization of Bi 6p, Pt 5d and Pt 6s states. Two sets
of Weyl points are located in momentum space as shown in Fig. 1c, d having the
energy of 47 meV above the Fermi level. In order to set a baseline from which
the 3D band structure can be resolved we have recorded 16 ARPES datasets
covering at least the first 3D Brillouin zone and approximately 1 eV in energy
using the photon energies from 15 to 43 eV (see also Fig. S1 in SI). This
allowed us to identify high-symmetry points along the $k_{z}$ direction and
find the value of the inner potential ( $V_{0}$ = 10.5 eV ). In Fig. 1e we
show the Fermi surface maps taken using the photon energies corresponding to
high symmetry points along the $k_{z}$ direction and approximately half-way in
between them. The latter two (left column) are easy to recognize due to
pronounced $C_{3}$ symmetry of the pattern, rotated by 60∘ with respect to
each other. The map taken with 19 eV photons has a higher degree of hexagonal
shape, which corresponds to the A-point of the BZ. The map taken with 29 eV
photons at the level of the $\Gamma$-point bears a certain degree of
resemblance with the almost featureless calculated intensity as suggested by
Fig. 1c. The experimental pattern in this case is connected with the finite kz
resolution of ARPES. In Fig. 1f, g we also show the comparison of the
dispersions along the lines indicated in Fig. 1e. Although there is no one-to-
one correspondence, the features look very similar being shifted in energy or
momentum without any signatures of strong renormalization etc (see also Fig.
S2 in SI). There are several reasons why the intensity distributions do not
match exactly. First is that DFT calculations never perfectly reproduce the
electronic structure, second is the influence of matrix elements on the
photoemission intensity, third is the moderate $k_{z}$-resolution of ARPES
and, finally, the band structure calculations in Fig. 1 do not consider the
presence of the surface. Nevertheless,the data presented in Fig. 1e-g suggest
a reasonable general agreement between experiment and theory, which is in
accord with all previous ARPES studies YaoPRB2016 ; Feng2019 ; Setti ;
Jiang_JAP2020 . The experimental confirmation of all the main features of the
band structure near the Fermi level thus implies that PtBi2 is indeed a Weyl
semimetal, which we will fully confirm below.

##  3 Surface states on two terminations

Figure 2: Fermi surface arcs. a High-resolution Fermi surface maps ($h\nu$ =
17 eV, T = 1.5 K) from both terminations. Arcs in the first BZ are indicated
by the arrows. Note their presence in the equivalent positions in the first
and repeated BZ. b Fermi surface maps at different photon energies, all
showing the presence of the arcs measured at 15K. The sketch in the middle is
the guide to the eye. c Arcs as seen in the calculations. d,e Energy-momentum
intensity plots for terminations A and B along the cuts through the arcs
highlighted by blue dashed arrows in panels a.

In Fig. 2a we present high-resolution Fermi surface maps from both
terminations. Although seemingly different, closer inspection suggests that
they share mostly the same pattern provided intensity variations are taken
into account. The number of localized features can be clearly distinguished in
the map from termination A, at approximately 3/4 of $\Gamma$M distance and
equivalent locations. These features have been overlooked in earlier ARPES
studies YaoPRB2016 ; Feng2019 ; Setti ; Jiang_JAP2020 . Since the calculated
bulk continuum displayed in Fig. 1b does not contain any similar electronic
states in this region, we consider those as originating from the surface. The
termination B map also shows similarly located features, but they are more
clearly seen in the second BZ. The underlying bulk-related intensity is
higher, masking the presence of the surface states. In order to establish
their presence unambiguously, we show eight Fermi surface maps taken using
different photon energies in Fig. 2b. All maps exhibit all of the above-
mentioned features at the same location - approximately 3/4 of $\Gamma$M
distance, as in the case with termination A. Since it is unlikely that a
particular bulk feature would be present in all of the recorded FS maps for a
material with a highly three-dimensional electronic structure, we conclude
that also these features represent the surface. The schematic plot (in the
middle of Fig. 2b) summarizes our observations regarding the locations of the
arcs made from considering the Fermi surface maps.

Detected spots of intensity, which we identified above as surface states,
remarkably coincide with the results of calculations which take into account
the presence of the surface (Fig. 2c). Since PtBi2 is a Weyl semimetal, one
does expect the presence of the topological Fermi arcs, different for
terminations A and B. At the same time, in an idea type-I Weyl semimetal, the
location of the starting and end points of the arcs should be identical as
those are the projections of the Weyl points (see Fig. 1d). The Weyl points,
non-degenerate crossings of the bands in three-dimensional k-space, are almost
impossible to detect by ARPES directly because of the finite resolution, but
the corresponding Fermi arcs have been repeatedly seen experimentally in
various materialsarcs1_Yang2015 ; arcs2_Lv ; arcs3_Deng ; arcs4_Eric ;
arcs5_Borisenko2019 .

Fig. 2d,e demonstrates a comparison of the intensity distribution along the
paths marked in Fig. 2a, which run through the arcs. The arcs are situated
very close to the Fermi level and are well-distinguishable from the smeared-
out by $k_{z}$-resolution bulk dispersions.

Taking into account the discrepancies in the experimental and theoretical 3D
band structure (Fig. 1), we do not expect exact correspondence between the
calculated Fermi arcs and the ARPES data, but the observed agreement proves
not only that the experimental features are indeed the topological Fermi arcs,
but also that PtBi2 is a Weyl semimetal.

##  4 Robust Fermi arcs from laser-ARPES

Figure 3: Laser ARPES. a Fermi surface map taken using $h\nu$ = 5.9 eV at 3K.
Arcs are seen together with other bulk-originated features. b Underlying
dispersion along the momentum cuts indicated by arrows in a. c Typical energy
distribution curves (EDC) from b. Bulk EDC is taken close to zeroth momentum,
while surface EDC corresponds to the arc. d One of the narrowest and strongest
EDCs detected in the present study. e Arcs seen along the different cuts
through the BZ in different experimental geometries. f Intensity distribution
taken using horizontally polarised light along the path crossing two arcs. g
The same momentum and energy range as in f from the calculations. Note, that
also the surface states at around 200 meV binding energies are reproduced. h
Circular dichroism from the same region of the $k$-space.

In order to study the detected at the Fermi surface maps spots of intensity in
more details, we carried out ARPES experiments using a laser setup. Because of
the low kinetic energy of photoelectrons ( $\sim$ 1.7 eV), the part of the BZ
accessible during these experiments is very limited. We have concentrated on
detecting at least one of the arcs in the portion of the $k$-space marked in
the sketch of Fig. 3a. Several representative cuts through the key features
seen in the map are shown in Fig. 3b. Now the arc is better resolved, but
still very localized in terms of both momentum and energy. We estimate the
momentum extension to be of the order of 0.04 Å-1, which is in excellent
agreement with theory (Fig. 2c).

The most striking characteristic of the arc states is their energy
distribution. In Fig. 3c we compare the EDCs corresponding to the bulk and
surface states. Since the data are taken with very high resolution and at
extremely low temperature, the $k_{F}$-EDC representing bulk states has a well
defined maximum (FWHM 30 meV) near the Fermi level and leading edge width of
approximately 5 meV. However, the sharpness and peak-to-background ratio of
the EDC representing the surface arc is unprecedented. We have routinely
observed the peaks having FWHM below 3 meV and $\sim$50 peak-to-background
ratio in numerous cleaves of many samples (see Fig. S3 in SI). One of such
curves is shown in Fig. 3d. To our best knowledge, such a sharp lineshape has
never been observed in any photoemission experiment from solids earlier.

In Fig. 3e we show further appearances of the arcs in the momentum-energy
plots from different cleaves and different terminations. The sharpness and
flatness remain the robust characteristics of the feature in all our
experiments at the lowest temperatures. We noticed that for A and B surfaces
the arc states are supported by the strongly and weakly dispersing bulk states
respectively, exactly as expected from theory.

Direct comparison with the calculations taking into account the presence of
the topological surface states is presented in Fig. 3f,g. The agreement with
the experiment is remarkable: bulk and surface related dispersions are
captured not only qualitatively but also quantitatively. The difference
between the spectra taken with the right- and left-circularly polarised light
(Fig. 3h) allows us to identify additional features in the intensity
distribution, making the agreement with the theory even better.

In spite of the clear correspondence between laser ARPES data and DFT
calculations there is one detail which remains unexplained — the striking
flatness of the surface band without any signature of the Fermi level
crossings.

##  5 Superconductivity at the surface

Figure 4: Superconducting arcs. a Temperature dependence of the arcs
dispersion from the terminations A and B. b Zoomed-in datasets showing
underlying dispersion of the arcs. c EDCs corresponding to the coloured arrows
in b. d Leading edge and peak positions from b. e Averaged values of the peak
positions closest to the Fermi level as a function of temperature for
different samples and terminations. Left panels correspond to termination A,
right panels - to termination B. f Shift of the EDCs with temperature. g
Difference plots showing the changes of the intensity as a function of
temperature. h Results of the calculated spectral weight taking into account
the superconductivity at the surface. i Schematics of the electronic structure
of PtBi2. Green contours represent the Majorana states suggested by the
topogical superconductivity at the surfaces.

Record-high sharpness of the arc EDCs strongly resembles coherence peaks in
ARPES data from superconductors (e.g. Kushnirenko_LiFeAs ). In order to find
out whether the electronic states in question bear any other characteristic
features of superconductivity, we have carried out temperature dependent
measurements. In Fig. 4a we show the datasets recorded at 3 and 30K for both
terminations. The comparison of the spectra taken at different temperatures
underlines their flatness at lowest temperature. The arcs clearly lose
spectral weight and gain dispersion - just as is to be expected when the
system enters the normal state. The apparent asymmetry of the arcs dispersion
stems from the openness of the Fermi contour made by an arc - conventional
electron-like Fermi surface pockets would be supported by the symmetric with
respect to the bottom dispersions crossing the Fermi level.

We have also reproducibly observed another peculiar for the superconducting
state behavior. In Fig. 4b, where the arcs are measured with the highest
resolution, the typical back-bending of the dispersion from the side where the
states most closely approach the Fermi level is clearly seen. This is
illustrated in panels c and d of Fig. 4, where we plot EDCs at several
momentum values as well as their peaks and leading edge positions. We further
track the behaviour of the peak positions as a function of temperature in Fig.
4e. Typical for superconductivity shifts are observed when going through the
critical temperatures $T_{c}$. Such shifts measured at the $k_{F}$ give rather
precise estimate of the superconducting gap $\Delta$. Our measurements yield
$T_{cA}=14\pm 2$K and $T_{cB}=8\pm 2$ K whereas the corresponding
superconducting energy gaps are equal to $1.4\pm 0.2$ meV and $2\pm 0.2$ meV
respectively. The transition for the termination A seems to be broader, and
$T_{c}$ higher compared to the surface B, which suggests that slightly
differing superconducting states set in on the opposing surfaces. Taking into
account the different electronic structure of the two surfaces with Fermi arcs
it is not surprising that the superconducting orders are not fully equivalent.
Signatures of Berezinskii-Kosterlitz-Thoules transition seen by transport
Veyrat_Nanoletters  may explain the unusual $\Delta/T_{c}$ ratios.

In panels f and g of Fig. 4 we present additional evidence for essentially
surface-related superconductivity observed in two different samples. While the
EDC’s corresponding to the surface states are clearly shifted upon varying the
temperature (Fig. 4f), the bulk-related spectral weight remains virtually
intact, showing only weak changes caused by slightly different width of the
Fermi function. This is illustrated with the aid of 2D difference plots (Fig.
4g), where the clearly stronger variations of spectral function occur in the
region where the arcs are located.

We can reproduce the experimental spectral function of PtBi2 including both
the flatness and back-bending of the surface states by switching on super-
conductivity only at the surface via a solution of the Bogoliubov-de Gennes
Hamiltonian for a semi-infinite solid with a gap function of $V_{0}=2$ meV in
the first 3 PtBi2 layers. The result is shown in Fig. 4h for the momentum and
energy intervals corresponding to Fig. 4g (see also Fig. S4 in SI). Note, that
only the electron-electron part of the BdG spectral density is plotted to
model the ARPES signal and that only the states around the Fermi arcs acquire
a gap at the Fermi level.

The superconductivity of arcs in PtBi2 follows not only from the emergence of
unusually strong and sharp coherence peaks at low temperatures, flatness and
back-bending of the dispersion as well as characteristic shifts of the EDCs.
It follows also from the striking agreement with recent STM data Schimmel_gap
— results of another surface-sensitive experiment on the crystals from the
same batch. There (see Fig. 3 in Schimmel_gap ) the authors observed typical
tunneling conductance of superconductors characterized by the superconducting
gaps varying in space. Remarkably, the average value of the gap closely
corresponds to the gap values determined by ARPES. Moreover, rather unusual
considerable zero-biased conductance observed by Schimmel et alSchimmel_gap
has now a very natural explanation in terms of a bulk contribution which
remains ungapped. As can be seen from the ARPES data, the integrated
contribution from the states associated with the bulk can easily reach a
noticeable fraction of the signal from the surface, despite the dominant
intensity of the arcs. The spot size of the laser beam in our study is of the
order of 0.1 mm. This explains the agreement between the determined gap values
with averaged STM data and does not exclude the possible existence of higher
$T_{c}$ regions, the detection of which by ARPES would require an application
of micro- or nano- variations of the technique.

PtBi2 emerges as a stoichiometric Weyl semimetal with superconducting surfaces
(Fig.4i) and opens up a plethora of possibilities to manipulate topological
and superconducting phases in a single material. For instance, by varying the
thickness of the single crystal, one can obtain a tunable intrinsic Josephson
junction. Topological superconductivity at the surface should generate
Majorana states at the edges. We tentatively assign the momentum-independent
spectral weight at the Fermi level (apparently seen e.g. in Fig. 2d), enhanced
when probing the regions of the surface with a number of terraces, to their
presence. Further studies are needed to unambiguously identify and control
both the high-$T_{c}$ superconductivity and Majorana fermions at the surfaces
and edges of PtBi2 single crystals and nano-structures.

##  6 Methods

PtBi2 single crystals with size up to 1cm were grown via self-flux method. For
the ARPES study best crystals with dimensions 5 $\times$ 5 mm were selected.
An important and convenient for ARPES experiments property of the single
crystals is apparent absence of the stacking faults resulting in essentially
the same and only termination upon multiple cleaving of the given sample.

ARPES measurements were carried out on the $1^{2}$\- and $1^{3}$-ARPES end-
stations at BESSY (HZB), as well as in the IFW laboratory using 5.9 eV laser
light source. Samples were cleaved in situ at a pressure lower than $1\times
10^{-10}$ mbar and measured at the temperatures of 15K and 1.5K at BESSY, and
3-30 K in the laboratory. The experimental data were obtained using the
synchrotron light in the photon energy range from 15 to 50 eV with horizontal
polarization and laser light with horizontal and circular polarizations .
Angular resolution was set to 0.2∘ \- 0.5∘ and energy resolution - to 2 - 20
meV. The findings from the experiments were consistent and reproducible across
multiple samples.

The simultaneous presence of bulk non-superconducting and surface
superconducting states makes the detection of true coherence peaks in ARPES
challenging. Our experiments at the synchrotron with the energy resolution of
the order of 5 meV turned out to be not sufficient to detect even the shifts
of the leading edges of the corresponding arc peaks having FWHM of the order
of 10meV and peak-to-background ratio of approximately 5. This is because the
arc states are always located on top of the bulk continuum. Only measuring
with energy resolution of the order of 1-2 meV did we manage to observe
sufficiently sharp peaks (Fig. 3 c,d and Fig. S3) and their sensitivity to the
temperature. The sharpest features need to be found on the surface.

Superconducting gap on the arcs is most likely anisotropic. We included to
error bars in Fig. 4e the possible influence of a small shift of the beam spot
and thus slightly different emission angle. Taking into account the very high
localization in momentum space, this could potentially lead to probing a
different part of the arc and thus different $k_{F}$, where the
superconducting gap is slightly different.

We performed DFT calculations using FPLOKLAUS_FPLO  within the general
gradient approxiamtion (GGA)per96  and 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 techniquesSancho1985 . To model surface superconductivity of the
semi-infinite slab the Wannier model is extended into the Bogoliubov-de Genne
formualism with a zero gap function in general except for a constant Wannier
orbital diagonal singlet gap function matrix at the first 3 PtBi2 layers. A
modification of the Green’s function method is used to accomodate this surface
specific term. Further details are found in the supplementary material.

##  7 Acknowledgements

We thank the Helmholtz-Zentrum Berlin für Materialien und Energie for for the
allocation of synchrotron radiation beamtime. This work was supported withing
the Collaborative Research Center SFB 1143 “Correlated Magnetism—From
Frustration to Topology” and by the Dresden-Würzburg Cluster of Excellence
(EXC 2147) “ct.qmat—Complexity and Topology in Quantum Matter”. SA
acknowledges DFG via AS 523/4-1. S.A. and B.B. also acknowledge the support of
DFG through Project No. 405940956. JIF acknowledges the support of the
Alexander von Humboldt Foundation via the Georg Forster Return Fellowship and
ANPCyT grants PICT 2018/01509 and PICT 2019/00371. We thank Ulrike Nitzsche
for technical assistance.

##  8 Supplementary information

Here we present additional experimental and theoretical datasets which support
our conclusions drawn in the main text.

In Fig. S1 we show ARPES Fermi surface maps obtained using the photon energies
from 15 eV to 43 eV. Relatively strong variation of the pattern suggests a
reasonable $k_{z}$-sensitivity of our experiment. We found the optimal value
of inner potential to be equal to 10.5 V. This is in agreement with the
previous study of Jiang et al. Jiang_JAP2020

In Fig. S2 we show analog of Fig. 1 e-g, but here we compare experimental data
with the results of band structure calculations carried out using the linear
muffin-tin orbital (LMTO) method in the atomic sphere approximation (ASA) as
implemented in PY LMTO computer code. As is seen from the figure, the
agreement is at the same level as earlier, underpinning the previous
conclusion as regards the good agreement between experimental and theoretical
3D band structure.

To model a system which has a non-zero gap-function only at the surface - in
the first 30aB which is 3(PtBi2) layers - we modified the standard Green’s
function technique for semi infinite slabs. The system is build by a semi-
infinite chain of identical blocks consisting of 3(PtBi2) layers, repeating
indefinitely away from the surface. Each block has a Hamiltonian $H_{k}$ for
each pseudo momentum $k$ in the plane perpendicular to the surface and a
hopping matrix $V_{k}$, which couples neighboring blocks. The blocks’ minimum
size is determined by the condition that $H$ and $V$ describe all possible
hoppings. In order to add superconductivity the Bogoliubov de Genne (BdG)
formalism is used by extending the matrices in the following way

$\displaystyle H_{k,\mathrm{BdG}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}H_{k}&\Delta_{k}\\\
\Delta_{k}^{+}&-H_{-k}^{*}\end{array}\right),$ $\displaystyle
V_{k,\mathrm{BdG}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}V_{k}&0\\\
0&-V_{-k}^{*}\end{array}\right),$

where we choose
$\Delta_{k}=\delta_{ii^{\prime}}\left(\begin{array}[]{cc}0&V_{0}\\\
-V_{0}&0\end{array}\right)$ with $i$ being a spinless Wannier function (WF)
index and the $2\times 2$-matrix to act in a single WFs spin subspace. This
choice also leads to $\Delta\left[V_{k,\mathrm{BdG}}\right]=0$, since $V$ is
an off-diagonal part of the full Hamiltonian. To model surface-only
superconductivity we let $V_{0}=0$ for all (infinite) blocks, except the first
one, which gets a finite $V_{0}=2\mathrm{meV}$.

The standard Green’s function solution for this problem consists of
determining the propagator $X$ which encompasses all diagrams, which describe
paths that start at a certain block, propagate anywhere towards the infinite
side of that block and returning to that block. $X$ also describes the Green’s
function $G_{00}$ of the first block as well as the self energy to be added to
the Hamiltonian to obtain $G_{00}$ (a self consistency condition)
$G_{00}=X=\left(\omega^{+}-H-\Sigma\right)^{-1}$, $\Sigma=VXV^{T}$ (in
practice self consistency is obtained by an accelerated algorithm, though).
From this recursion relations can be used to calculate all other Greens
function blocks. These can be derived by subdividing propagation diagrams into
irreducible parts using known components, in particular $X$.

If the first block differs from all the others (as is the case due to
$\Delta_{k}$) one needs to modify the method in the following way. Let the
first block have Hamiltonian $h$ and hoppings to the second block $v$ (while
all other blocks are described by $H$ and $V$). Then the irreducible
subdivision of the propagation diagrams for $G_{00}$ results in
($g=\left(\omega^{+}-h\right)^{-1}$)

$\displaystyle G_{00}$ $\displaystyle=$ $\displaystyle
g+gvXv^{+}g+\left(gvXv^{+}\right)g$ $\displaystyle=$
$\displaystyle\frac{1}{\omega^{+}-h-vXv^{+}}$

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

$G_{11}=X+Xv^{+}G_{00}vX$

and all others

$G_{n+1,n+1}=X+XV^{+}G_{nn}VX,\quad n>0$

which can be used to obtain the spectral density up to a certain penetration
depth. Note, that in our BdG case $H=H_{k,\mathrm{BdG}}\left[V_{0}=0\right]$,
$V=V_{k,\mathrm{BdG}}\left[V_{0}=0\right]$ and
$h=H_{k,\mathrm{BdG}}\left[V_{0}\neq 0\right]$, $v=V$. The BdG spectral
density is particle-hole symmetric and in order to obtain results, which
resemble ARPES data, one needs to use the particle-particle block
$G^{\mathrm{ee}}$ (the upper left quarter of the $G$ matrix) only.

Fig. S4b shows the EDC of this method along the path denoted in Fig. S4a. Note
that a gap is opened at the surface band pockets close to the Fermi energy,
while the rest of the spectrum stays gap-less (if we let $V_{0}\neq 0$ for all
blocks we get a completely gapped spectrum). Fig. S4c shows a zoomed-in region
around the surface state. Note, that the bulk bands are gap-less (dark blue
vertical features) while the surface state shows a gap and corresponding band-
back-bending. The particle-hole symmetry becomes apparent, although with a
larger spectral weight for the occupied part due to the fact that we use
$G^{\mathrm{ee}}$ only.

Figure S1: Fermi surface maps from the 16 datasets taken using different
photon energies.  Figure S2: a, Fermi surface maps taken using different
photon energies and corresponding results of the band structure calculations.
We note, that fixed photon energy probes a sphere of the large radius in the
k-space, matching theoretical data formally only at one point in the center.
b, c Energy-momentum intensity distributions at 21 eV and 19 eV respectively
along the cuts indicated by blue dashed arrows in panel e.

In Fig. S3 we present the sharpest EDCs from different samples and cleaves.
Most of them have FWHM below 3 meV and peak-to-background ration of more than
30.

Figure S3: Energy distribution curves taken close to $k_{F}$ of the arcs for
different $k$, samples (S) and cleaves (C)  Figure S4: a The path in the BZ. b
The surface-only superconducting spectral function of the (00-1)-surface for
$\Delta$ = 2meV and penetration depth 30aB ($G_{ee}$ only). Note that the gap
is only open around the surface state pocket. c the blow-up of this pocket.
Note, that the gap is open for the surface state but closed for the bulk bands

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