# Dirac surface states, multiorbital dimerization and superconductivity
in Nb- and Ta-based A15 compounds

Raghottam M. Sattigeri ID<EMAIL_ADDRESS>International
Research Centre Magtop, Institute of Physics, Polish Academy of Sciences,
Aleja Lotników 32/46, PL-02668 Warsaw, Poland  Giuseppe Cuono ID
<EMAIL_ADDRESS>International Research Centre Magtop, Institute of
Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw,
Poland  Ghulam Hussain ID International Research Centre Magtop, Institute of
Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw,
Poland  Xing Ming  College of Science, Guilin University of Technology, Guilin
541004, People’s Republic of China.  Angelo Di Bernardo ID Dipartimento di
Fisica ’E.R. Caianiello’, Universitá degli Studi di Salerno, via Giovanni
Paolo II 132, I-84084 Fisciano (SA), Italy  Carmine Attanasio ID Dipartimento
di Fisica ’E.R. Caianiello’, Universitá degli Studi di Salerno, via Giovanni
Paolo II 132, I-84084 Fisciano (SA), Italy  Mario Cuoco ID
<EMAIL_ADDRESS>Consiglio Nazionale delle Ricerche CNR-SPIN, UOS
Salerno, I-84084 Fisciano (Salerno), Italy  Dipartimento di Fisica ’E.R.
Caianiello’, Universitá degli Studi di Salerno, via Giovanni Paolo II 132,
I-84084 Fisciano (SA), Italy  Carmine Autieri ID<EMAIL_ADDRESS>International Research Centre Magtop, Institute of Physics, Polish Academy of
Sciences, Aleja Lotników 32/46, PL-02668 Warsaw, Poland

###### Abstract

Using first-principle calculations, we investigate the electronic, topological
and superconducting properties of Nb3X (X = Ge, Sn, Sb) and Ta3Y (Y = As, Sb,
Bi) A15 compounds. We demonstrate that these compounds host Dirac surface
states which are related to a nontrivial $\mathbb{Z}_{2}$ topological value.
The spin-orbit coupling (SOC) splits the eightfold degenerate R point close to
the Fermi level enhancing the amplitude of the spin Hall conductance. Indeed,
despite the moderate spin-orbit of the Nb-compounds, a large spin Hall effect
is also obtained in Nb3Ge and Nb3Sn compounds. We show that the Coulomb
interaction opens the gap at the R point thus making more evident the
occurrence of Dirac surface states. We then investigate the superconducting
properties by determining the strength of the electron-phonon BCS coupling.
The evolution of the critical temperature is tracked down to the 2D limit
indicating a reduction of the transition temperature which mainly arises from
the suppression of the density of states at the Fermi level. Finally, we
propose a minimal tight-binding model based on three coupled Su-Schrieffer-
Heeger chains with t2g Ta- and Nb-orbitals reproducing the spin-orbit
splittings at the R point among the $\pi$-bond bands in this class of
compounds. We separate the kinetic parameters in $\pi$ and $\delta$-bonds, in
intradimer and interdimer hoppings and discuss their relevance for the
topological electronic structure. We point out that Nb3Ge might represent a
$\mathbb{Z}_{2}$ topological metal with the highest superconducting
temperature ever recorded.

###### pacs:

71.15.-m, 71.15.Mb, 75.50.Cc, 74.40.Kb, 74.62.Fj

##  I Introduction

Topological superconductivity is a captivating phase of condensed matter
physics characterized by unique electronic properties such as the Majorana
fermionsBeenakker (2013). These particles, distinct for their non-Abelian
statistics, hold promise for fault-tolerant quantum computation. This
enigmatic phase of matter has ignited a surge of research, with potential
applications spanning quantum computing to quantum information storage.
Understanding and harnessing topological superconductivity holds the key to
obtaining new quantum technologies. Therefore, a growing interest in
topological superconductivity has been rising in the last decade. Yonezawa
_et al._ (2017); Sharma  _et al._ (2022) Topological superconductivity can
arise from the coexistence of Bardeen–Cooper–Schrieffer (BCS)
superconductivity and Dirac states which can lead to mixed pairing order
parameters or topological superconductivity, therefore, the Dirac surface
states of superconductors are platforms for investigating the interplay
between superconductivity and topologically nontrivial Fermi surfacesYang  _et
al._ (2014); Gao  _et al._ (2022).

The most common k-space topological phase is characterized by a non-zero
$\mathbb{Z}_{2}$ topological invariant. The $\mathbb{Z}_{2}$ topological
insulators have gapped bulk band structure and gapless surface states. These
surface states are protected by time-reversal symmetry. $\mathbb{Z}_{2}$
topological metals are conducting materials with gapless bulk band structures
and gapless surface statesCheng  _et al._ (2023); Zhao and Wang (2016); Cerjan
and Loring (2022). In the last years, different families of materials have
been proposed to be $\mathbb{Z}_{2}$ topological metals with a superconductive
ground state. We can mention some of the most representative members of these
families as the kagomeOrtiz  _et al._ (2020) CsV3Sb5, the non-symmorphic
ZrOSSiGhosh1  _et al._ (2023) and KHgAs compoundsDai  _et al._ (2023) and the
van der Waals Ta2Pd3Te5 materialHigashihara  _et al._ (2021); Yang  _et al._
(2023). Dirac surface states were also found in several undoped iron-based
systems such as BaFe2As2 and LiFeAsZhang  _et al._ (2018); Ma  _et al._
(2022).

The Nb-based A15 compounds were widely studied in the past due to their
superconductivity with a high critical temperature ($T_{c}$). The
superconductivity in Nb-based A15 compounds was found to be BCS-like Stewart
(2015), namely the pairing of the superconducting electrons is via
electron–phonon coupling. The Fermi level of these compounds is close to a
peak in the density of states deriving from dimerized one-dimensional Nb
chains. In silicides and germanides of transition metals, the highest $T_{c}$
was found in V3Si among all the known binary compounds Hardy and Hulm (1954).
The A15 claimed the title of the highest $T_{c}$ superconductors in 1954 when
$T_{c}$ = 18 K was first observed in Nb3SnMatthias  _et al._ (1954). Other Nb-
based superconductors were then found as for example Nb3Al with $T_{c}$=18.8
KWillens  _et al._ (1969), Nb3Ga with $T_{c}$=20.3 K inWebb  _et al._ (1971)
and Nb3Ge with $T_{c}$=22.3 KGavaler (2003). Recently, several Nb-based
compounds were investigated for their exotic superconducting properties such
as the van der Waals NbX2(X=S, Se)Wang  _et al._ (2022, 2020); Cossu  _et al._
(2020) and the non-centrosymmetric NbReCaputo  _et al._ (2017); Kakhaki  _et
al._ (2023) Both theoretical and experimental investigations have extensively
explored the properties of A15 compounds Sundareswari  _et al._ (2010); Muller
(1980); Wu  _et al._ (2020); Stewart (2015); Ding  _et al._ (2014); Marzi  _et
al._ (2013) based on niobium and tantalum, due to their high critical
temperature and high critical magnetic fields. Due to the discovery of the
unconventional high $T_{c}$ superconductivity in heavy fermions and cuprates,
the superconductive phase of the A15 compounds has received less scientific
attention in recent years. Recently, the A15 compounds regained attention due
to the large degeneracy at the R point, these A15 compounds are multifold
fermion metalsBradlyn  _et al._ (2016) with a notable spin-Hall effectDerunova
_et al._ (2019); Hou  _et al._ (2021) and non-trivial band structure
topologyKim  _et al._ (2019); Wieder  _et al._ (2016). Dirac points are
emergent along the R–M path due to the C4 rotational symmetry.Hou  _et al._
(2021) The large spin-Hall conductivity in these compounds is due to the fact
that they have bands close to the Fermi level that present crossings
unprotected under the action of the spin-orbit coupling interaction (SOC)
Derunova  _et al._ (2019). The Ta-based A15 compounds have the same filling as
the Nb-based A15 compounds, with the Ta having a larger SOC. The Ta3Sb
compound with A15 crystal structure was proposed to be a topological
superconductorChapai  _et al._ (2022, 2023).

Regarding the realization of devices, Nb3Sn superconductors have significant
applications in constructing high-field magnetsXu (2017). Nb3Sn can be used as
a coating for producing superconducting surfacesSchäfer  _et al._ (2020) and
for particle acceleratorsSundahl  _et al._ (2021). Nb3Sn thin films are
promising candidates for future applications in superconducting radio
frequency cavitiesSchäfer  _et al._ (2020).

Figure 1: (a) Crystal structure of the A3B compound. The $A$ sites are shown
in yellow and $B$ sites are shown in brown. The A sites belong to the dimers
along the $a$, $b$ and $c$-axis of the unit cell. We indicate with a red bond
the dimer along the a-axis, with a green bond the dimer along the b-axis and
with a blue bond the dimer along the c-axis. (b) The irreducible bulk
Brillouin zone with (001) surface projections is used for computing surface
states. (c) Slab of A3B compounds periodic along (001) crystal direction with
the top surface layer composed of A2 and bottom surface layer composed of AB
elements. The composition of the bottom and top surface layers is preserved
along the paper.

In this paper, we study the electronic, topological and superconductive
properties of Nb3X (X = Ge, Sn, Sb) and Ta3Y (Y = As, Sb, Bi) A15 compounds
and we demonstrate that all these compounds are $\mathbb{Z}_{2}$ topological
metals hosting Dirac surface states. We study the interplay between the
electronic and topological properties with the spin-Hall and BCS
superconductivity. These topological properties can be explained by a tight-
binding model with three coupled Su–Schrieffer–Heeger (SSH) chains. Nb3Sb and
Ta3Y (Y = As, Sb, Bi) have half-filling p- and d-orbitals, while Nb3Ge and
Nb3Sn have one electron less. The paper is organized as follows. In the next
Section, the results of our ab initio calculations are reported. In more
detail, this Section is divided into many Subsections: in Subsection A the
structural and electronic properties of Nb3X and Ta3Y are investigated, in
Subsection B we study the Spin Hall conductivity, while in Subsections C and D
we discuss the topological properties for the Ta-based and Nb-based compounds,
respectively. Subsection E is devoted to the superconductivity, while
Subsection G is dedicated to the thickness-dependent density of states. In
Section III, we report our tight-binding model composed of the three coupled
chains of the SSH model with t2g orbital basis. Finally, Section IV is devoted
to the discussion, conclusions and outlook.

##  II Results

###  II.1 Structural and Electronic properties of Nb3X and Ta3Y

Figure 2: Electronic structures of (a) Ta3As, (b) Ta3Sb and (c) Ta3Bi with
spin-orbit coupling. Surface states of (d) Ta3As, (e) Ta3Sb and (f) Ta3Bi
indicating spin-momentum locked Dirac dispersions at $\Gamma$ point. Slab band
structures of (g) Ta3As, (h) Ta3Sb and (i) Ta3Bi with the red bands indicating
the contribution from the top surface layer and the blue bands indicating the
contribution from the bottom surface layer of the slab presented in Fig 1. The
legends on the right in panels (d-i) indicate the spectral weights of the top
and bottom layers. The Fermi level is set to zero in all panels.

A15 compounds are governed by the Pm$\overline{3}$n (No. 223) space group
which exhibits an intermetallic nature arising from a chemical composition of
A3B, where site A is occupied by a transition metal/d-block element and site B
is occupied by the p-block element. The crystal structure presented in Fig.
1(a) is a typical unit cell of an A15 compound with inversion symmetry
containing eight atoms with site A forming one-dimensional chains along the
edges which are orthogonal to neighboring faces and the B site forming a body-
centered cubic lattice. The presence of spatial inversion symmetry is due to
the non-symmorphic space group governing the system which involves a screw
axis in the [001] crystal direction. The A sites in A3B composition occupy 6c
Wyckoff positions (0.25,0.00,0.50) and the B sites occupy 2a Wyckoff positions
(0.00,0.00,0.00). The crystal structure presents three dimers of the A atoms,
along the $a$, $b$ and $c$ axes, as shown in Fig. 1(a). A typical slab of A15
compound periodic in [001] crystal direction is presented in Fig. 1(c) with
surface Brillouin zone highlighted in Fig. 1(b). This exposes two unique
surfaces, the top surface originates from the A2 atomic arrangement and the
bottom surface originates from the AB atomic arrangement. The optimized
lattice constants (a) after structural relaxation for Nb3Ge, Nb3Sn and Nb3Sb
are 5.177 Å, 5.324 Å and 5.303 Å, respectively which are in agreement with
literature.Devantay  _et al._ (1981); Tarutani and Kudo (1977); Furuseth and
Kjekshus (1964) While the optimized lattice constants (a) for Ta3As, Ta3Sb and
Ta3Sn are, 5.203 Å, 5.329 Å and 5.394 Å, respectively.

Figure 3: Electronic structures of (a) Nb3Ge, (b) Nb3Sn and (c) Nb3Sb with
spin-orbit coupling. Surface states of (d) Nb3Ge, (e) Nb3Sn and (f) Nb3Sb
indicating spin-momentum locked Dirac dispersions at $\Gamma$ point. Slab band
structures of (g) Nb3Ge, (h) Nb3Sn and (i) Nb3Sb with the red bands indicating
the contribution from the top surface layer and the blue bands indicating the
contribution from the bottom surface layer of the slab presented in Fig 1. The
legends on the right in panels (d-i) indicate the spectral weights of the
bottom and top layers. The Fermi level is set to zero in all panels.

The computational framework is described in Appendix B. In Figs. 2 and 3 the
electronic structures of Ta3Y and Nb3X compounds are shown, respectively. Both
band structures host fourfold rotational symmetries. This is due to the
presence of non-symmorphic symmetry operations involving fourfold rotations
and fractional translations with respect to the [001] crystal directions.
Apart from the symmetries due to time-reversal and spatial inversion symmetry,
we observe additional degeneracies in the momentum. Due to the non-symmorphic
symmetries, we have 4-fold and 8-fold degeneracies at the R point. As we
demonstrate in Appendix A, in the low-energy sector at the R-point two groups
of bands are present: the bands related to the $\pi$ and $\delta$ intradimer
bonds. In Figs. 2(a,b,c) and 3(a,b,c) the band structures in the range between
-1 eV and 1 eV are shown, at the R point we can see in this range the
$\pi$-bond bands. The $\delta$-bond bands are around 1.5 eV below the Fermi
level. The parabolic band appearing at the $\Gamma$ point for the Ta-based
compounds is the 6s band of Ta. The 6s band crosses the p-d bands for Ta3As
and Ta3Sb and increases the density of states.

However, at the R point in the momentum space, we have mild variations in Ta3Y
and Nb3X compounds with eigenvalues that are four and eight times degenerate
as shown in Figs. 2 and 3 which is in agreement with the literature. While in
Ta-based compounds, the strong SOC at the R-point opens the gap between
$\pi$-orbital bands, as we can see in Fig. 2 (a, b, c), in the Nb-based there
is a smaller splitting at R but the bands with camelback shape around R are
still crossing keeping conduction and valence sectors entangled, as it is
shown in Figs. 3 (a, b,c).

The crystal symmetries are responsible for orbital hybridizations in the
compounds, once we apply the SOC the anticrossings strongly contribute to the
Berry curvature and spin Berry curvature. The multiple crossings observed in
the electronic structures here give rise to large spin Berry curvatures which
effectively translates to a large spin Hall effect in such compounds since the
Fermi level lies within gaped crossings.

At the Fermi level, we observe a high density of states with large
contributions (due to multiple crossings) from transition metals Ta or Nb and
minor contributions from other constituent elements of the A3B composition, as
shown in Appendix C. In the momentum space, the conduction band minima at
$\Gamma$ point originates from the s-orbitals of group III elements or
pnictogens in the A3B composition while the valence band maxima at $\Gamma$
point originate from the d-orbitals of transition metals. Since these
dispersions vary between the Ta3Y and Nb3X compositions, their relative
positions define the density of states at the Fermi level and in turn the
superconducting critical temperature ($T_{c}$) of the system.

In Ta3As and Ta3Bi, the conduction band minima with s-orbital contributions
are below the valence band maxima with d-orbital contributions at $\Gamma$
point as evident from Fig. 2 (a,c). However, in the case of Ta3Sb and Nb3X,
the conduction bands and valence bands are well separated throughout the
Brillouin zone as evident from Fig. 2(b) and Fig. 3, respectively. In this
last case, the s-orbital contributions are farther away from the Fermi level
as compared to the Ta3Y family. The well-resolved band manifolds in Ta3Sb and
Nb3X are accompanied by band inversions across the Fermi level which could
produce topological properties. We will see that the presence of the
s-orbitals band produces additional anticrossings and additional bands that
increase the spin Hall conductivity (SHC) and $T_{c}$ respectively.

###  II.2 Spin Hall conductivity

Figure 4: Spin Hall conductivity as a function of the energy for Ta3As and
Ta3Bi compounds. The black solid line indicates Fermi level.

It is evident from the electronic structures of these compounds that they host
multiple crossings and anticrossings, therefore we have a large change in the
spin Berry curvature which indicates that the spin Hall effect should be
large. It is known that the SHC is inversely proportional to the spin-orbit
induced gap. Accordingly, in the case of Ta3As, Ta3Sb and Ta3Bi (spin-orbit
induced gap in electronic structure in increasing order) we find that the SHC
at the Fermi level is $-$ 1492.8 ($\hbar/e$) Scm-1, $-$ 1423.86 ($\hbar/e$)
Scm-1 and $-$ 1320.2 ($\hbar/e$) Scm-1 respectively (which has decreasing
trend as compared to the increasing order of the spin-orbit induced gap). The
SHC of Ta3As and Ta3Bi close to the Fermi level are shown in Fig. 4. The SHC
exhibits a wide peak that encompasses the energy range where the gapped
crossings are located. In both cases of Ta3As and Ta3Bi the peak is very close
to the Fermi level since the gapped crossings are located near the Fermi
level, as shown in Fig. 2(a,c). A similar trend is observed in the case of
Nb3X (with group III elements) i.e., for Nb3Ge and Nb3Sn (with the spin-orbit
induced gap in electronic structure in increasing order) we have SHC at the
Fermi level of $-$ 1691.4 ($\hbar/e$) Scm-1 and $-$ 983.1 ($\hbar/e$) Scm-1
respectively. In the composition of Nb3X, Nb3Sb is an outlier with
significantly low and positive SHC of 155.3 ($\hbar/e$) Scm-1 since it is a
pnictogen substitution as compared to the other two which are group III
elements.

Large values of the SHC are usually associated with strong orbital texture.
Indeed, the A15 systems host a strong orbital texture.Kim  _et al._ (2019)
Even if the orbital moment is zero, since the compounds present inversion
symmetry and highly symmetric crystal structure, the breaking of the inversion
symmetry at the surface will generate an orbital magnetic moment. This makes
the study of the surface states of A15 compounds interesting.

Figure 5: (a) Electronic structure of Nb3Ge for U = 0, 2, 3 and 4 eV with
spin-orbit coupling. (b) Surface states of Nb3Ge for U = 0, 2, 3 and 4 eV
indicating the presence of spin-momentum locked Dirac dispersions shifting
farther from the Fermi level with the increase in U. The legends on the right
in panel (b) indicate the spectral weights. The Fermi level is set to zero in
all panels. Figure 6: (a) Electronic structure of Nb3Sn for U = 0, 2, 3 and 4
eV with spin-orbit coupling. (b) Surface states of Nb3Sn for U = 0, 2, 3 and 4
eV indicating the presence of spin-momentum locked Dirac dispersions shifting
farther from the Fermi level with the increase in U. The legends on the right
in panel (b) indicate the spectral weights. The Fermi level is set to zero in
all panels. Figure 7: (a) Electronic structure of Nb3Sb for U = 0, 2, 3 and 4
eV with spin-orbit coupling. (b) Surface states of Nb3Sb for U = 0, 2, 3 and 4
eV indicating the presence of spin-momentum locked Dirac dispersions shifting
farther from the Fermi level with the increase in U. The legends on the right
in panel (b) indicates the spectral weights. The Fermi level is set to zero in
all panels.

###  II.3 Topological properties of Ta3Y (Y = As, Sb, Bi)

In this subsection, we discuss first the topological properties of the Ta3Sb
that is the ideal case, later, we discuss the topological properties of Ta3As
and Ta3Bi with the presence of the s-band at the Fermi level. Since the Ta3Sb
compound shows well-resolved band manifolds, we compute the surface states
projected on [001] crystal direction as presented in Fig. 2(e). Clearly, this
compound hosts spin-momentum locked surface states with Dirac dispersions at
$\Gamma$ point. We also represent the corresponding slab band structure in
Fig. 2(h), which shows that the Dirac dispersion at $\Gamma$ originates from
the top surface layers. Although these are slightly away from the Fermi level,
one can realize them at the Fermi level in experimental conditions by varying
the carrier concentrations. Albeit, as the band manifolds are well resolved in
the case of Ta3Sb, we compute the $\mathbb{Z}_{2}$ topological invariants
using the Wilson loop method around the Wannier charge centers. Therefore, for
Ta3Sb, the four $\mathbb{Z}_{2}$ 3D topological invariants are
($\nu_{0}$,$\nu_{1}\nu_{2}\nu_{3}$)=(1;000) indicating a strong topological
insulator character.

Since the conduction bands and valence bands are degenerate at the $\Gamma$
point due to the presence of the 6s band in the case of Ta3As and Ta3Bi, we do
not calculate the $\mathbb{Z}_{2}$ invariants for these compounds. Although
$\mathbb{Z}_{2}$ is not well-defined, from the band structure we can see the
Dirac surface states for Ta3As and Ta3Bi as shown in Figs. 2(d,g) and 2(f,i),
respectively, where we show both the surface states and the slab band
structures. However, when we include the 6s band in the tight-binding model
with the Wannier basis, the Dirac surface states are blurred by the
hybridization with the 6s band (see Appendix E for more details). The Dirac
surface states would be difficult to detect in Ta3As and Ta3Bi, while they
should be observable in all other compounds investigated in this paper.

Figure 8: Phonon dispersion curves of (a) Nb3Ge and (b) Nb3Sn alongside the
corresponding anisotropic Migdal-Eliashberg spectral functions
($\alpha^{2}$F($\omega$)) and phonon density of states (F($\omega$)).

###  II.4 Topological properties of Nb3X (X = Ge, Sn, Sb): effects of Coulomb
repulsion

Although $\mathbb{Z}_{2}$ invariants are not well defined for Nb3X compounds
in the absence of Coulomb repulsion (U), from the band structures, we can see
the Dirac surface states as shown in Fig. 3(d,g) for Nb3Ge, 3(e,h) for Nb3Sn
and 3(f,i) for Nb3Sb. To investigate further, we perform DFT + U calculations
for Nb3Ge, Nb3Sn, and Nb3Sb. The band structures within DFT + U are shown in
Figs. 5, 6 and 7. The Coulomb repulsion opens a global gap in the momentum
space which is evident from the evolution of bulk band structures for
different values of U as shown in Figs. 5(a) for Nb3Ge, 6(a) for Nb3Sn and
7(a) for Nb3Sb. The corresponding surface states are instead shown in Figs.
5(b) for Nb3Ge, 6(b) for Nb3Sn and 7(b) for Nb3Sb. As we can see from the
surface states, for all three compounds, the Dirac point at $\Gamma$ is buried
in the bulk at U = 0 eV, while already at U=2 eV it is clearly visible and at
U = 4 eV the band gap is opened making the calculation of four
$\mathbb{Z}_{2}$ invariants possible.

After the opening of the gap, the calculation of $\mathbb{Z}_{2}$ is well-
defined and we obtain the four $\mathbb{Z}_{2}$ topological invariants
($\nu_{0}$,$\nu_{1}\nu_{2}\nu_{3}$)=(1;000), showing that all these Nb-based
compounds are $\mathbb{Z}_{2}$ topological metals hosting Dirac surface states
similarly to Ta3Sb compound. This is a clear signature of non-trivial
topological states appearing not only in heavy Ta-based compounds but also in
Nb-based compounds.

Figure 9: Total density of states (DOS) for bulk Nb3Sn (a) and for various
thicknesses (b,c,d,e). The Fermi level is set to zero in all panels. The trend
of increasing DOS as a function of thickness is shown in panel (f). The red
line represents the DOS value of the bulk.

###  II.5 Bulk superconductivity in Nb3X and Ta3Y

Generally, superconductivity requires metallic states at the Fermi level
whereas topological insulators are gaped due to spin-orbit interactions, and
they present conducting surface states. Hence, finding a stoichiometric
composition where bulk superconductivity and topological surface states
coexist is a tough task i.e., the surface states should lie at the Fermi level
while the bulk remains fully gaped superconductor in the critical temperature
regime. Typically in such systems the Dirac points exist farther from the
Fermi level in the conduction bands making it challenging to be observed in
experiments like Angle-resolved photoemission spectroscopy (ARPES).

Several compounds have been investigated to this effect with A15 compounds not
being an exception due to their metallic character.Derunova  _et al._ (2019);
Kim  _et al._ (2019) Studies have been dedicated to Ta3Sb as a potential
candidate for topological superconductivity due to the presence of well-
resolved spin-orbit induced band manifolds and a superconducting critical
temperature of 0.7 K.Kim  _et al._ (2019) We revisited this compound and find
that in agreement with the previous studies, the Dirac dispersions in surface
states are around 500 meV away from the Fermi level in the conduction bands as
evident in Fig 2(e) with a superconducting critical temperature of 0.81 K.
However, these Dirac dispersions on the top surface merge with the s-bands of
the pnictogens and on the bottom surface merge with the d-bands of Ta at the
$\Gamma$ point in the momentum space. Hence it is highly unlikely that Ta3Sb
will become a topological superconductor as has been observed in some
noncentrosymmetric binary compound BiPd where the Dirac dispersions lie away
from the Fermi level in the superconducting regime. This explanation holds
true for Ta3As and Ta3Bi as well which have similar characteristics on the
surface states (presented in Fig. 2 (d,f)) with superconducting critical
temperatures of 3.0 K and 1.16 K respectively. The s-bands in the conduction
band minimum (CBM) of Ta3Bi and Ta3As are responsible for higher $T_{c}$ as
compared to Ta3Sb since they increase the density of states near the Fermi
level.

On the other hand, the $T_{c}$ for Nb3Ge and Nb3Sn compounds is higher as
compared to Ta3Y and Nb3Sb. Owing to the electronic structure near the Fermi
level, Nb3Sb has a $T_{c}$ of 2.21 K which is comparable to that of Ta3As and
Ta3Bi. However, Nb3Sb is an outlier when compared to Nb3Ge and Nb3Sn which
have a $T_{c}$ of 15.25 K and 15.66 K respectively. This distinction
originates from the lattice dynamics which is evident from the phonon
dispersion curves. As compared to Nb3Sn which exhibits anomalous vibrational
properties such as soft modes in the $\Gamma\rightarrow$ X $\rightarrow$ M
directions in the momentum space (as evident from Fig. 8(b)), the phonon
softening is not observed in Nb3Sn presented in Fig. 14(d) (see Appendix D).
The anomaly of longitudinal acoustic modes of Nb3Sn softening at lower
temperature scales is accompanied by large neutron scattering linewidths which
is a function of the electron-phonon coupling coefficient $\lambda$($\omega$),
hence resulting in higher $T_{c}$.Stewart (2015)

We present the phonon dispersion curves alongside the Migdal-Eliashberg
spectral functions $\alpha^{2}$F($\omega$) and phonon density of states
F($\omega$) for Nb3Ge and Nb3Sn are presented in Fig. 8 and for Ta3Y, Nb3Sb
are presented in Fig. 14 (see Appendix D). The electron-phonon coupling
coefficients $\lambda$($\omega$) for Nb3Ge and Nb3Sn are 1.41 and 2.03,
respectively. The bulk superconductivity in Nb3X compounds is quite sensitive
to the method of crystal growth and various experimental conditions vary the
experimental $T_{c}$. However, Nb3Ge has been found to exhibit a maximum
$T_{c}$ of 23.2 K which along with Nb3Sn gives further scope to explore the
interplay between the topology and superconductivity owing to large SHC at the
Fermi level.

###  II.6 Thickness dependence of the electronic and superconducting
properties in Nb3Sn thin films

We calculate the DOS as a function of the thickness to asses the
superconducting properties of the Nb3Sn thin film. The electronic properties
reported in this subsection were calculated with the computational framework
described in Appendix C. In Fig. 9(a-e), we show the DOS calculated for the
bulk and for the stoichiometric slabs with different thicknesses. The DOS as a
function of the thickness is reported in Fig. 9(f), we observe a decreasing
value of the DOS when the thickness got reduced, this will reflect in a
reduction of the critical temperature for the thin films of Nb3Sn. For ultra-
thin films, the DOS is reduced approximately by half with respect to the bulk
value. These calculations show that Nb3Sn has a different behavior due to its
complex band structure properties compared to other BCS superconductors like
V, Ta or Nb where one normally finds that, as soon as the thin film thickness
is equal or greater than the coherence length, $T_{c}$ is very close to the
bulk $T_{c}$. The experimental superconducting critical temperature of the
thin films strongly depends on the sample quality, of the substrate and
usually tends to reduce with the thickness reduction.Il’in  _et al._ (2010);
Asano  _et al._ (1988)

##  III Tight-binding model with three SSH chains for t2g orbitals

Here, we report a model that includes only the t2g orbitals of the Nb/Ta
atoms. This tight-binding model allows us to understand the character of the
orbitals and the Fermi level and which hopping parameters tune the opening of
the topological gap. If we want to produce a minimal model for the d-orbitals
of the Nb/Ta atoms, we must include all the 6 Nb/Ta atoms per unit cell. If we
could decouple p- and d-electrons, eg are below in energy with respect to t2g
given the crystal field due to the position of other atoms, or anyway, we can
assume this in first approximation. Since the charge transfer is zero, the
Nb/Ta atoms are in d5 electronic configuration. We developed a tight-binding
model for the t2g subset for the 6 Nb/Ta-atoms. Within the t2g tight-binding
model we can easily include the spin-orbit coupling. The crystal structure
presents three dimers of Nb/Ta atoms, along the $a$, $b$ and $c$ axes, as
shown in Fig. 1a). We consider in our model the intradimer hybridizations and
the interdimer hoppings by including the first and the second nearest
neighbours.

Figure 10: Top view of the crystal structure of the A3B compound. The $A$
sites are shown in yellow and $B$ sites are shown in brown. We indicate with a
green bond the dimer along the b-axis, we have deleted the other dimers for a
better visualization. The hopping parameters t1α and t1β have the same
geometric structure as the hopping in the SSH chain.

The spinful tight-binding model is reported in more detail in Appendix A and
it is composed of three coupled SSH chains with t2g orbitals. The SSH chain
along the b-axis is shown in Fig. 10. Very few recent examples of a three-
dimensional SSH model have been reported in the literatureKim  _et al._
(2023); Kłosiński  _et al._ (2023), with significant differences from the case
proposed in this paper. The parameters of the model are described in detail in
Appendix A, we have two onsite energies E1 and E2 for the $\pi$ and
$\delta$-bonds, respectively. The hopping parameters t1α, t1β, t2α, t2β for
the intradimer elements and t3 and t4 for the interdimer elements. The band
structure reproduced of the model without including SOC is reported in Figs.
11. A magnification of the band structure at the R point is shown in 12a) and
12b), without and with SOC respectively. We include the SOC within the t2g as
the SOC of L=$-$1\. We have a gap between the conduction and the valence band
when the gap at R is open. The opening of the gap at R allows us to explicitly
calculate the topological invariants. The opening of the topological gap at R
is controlled by both the difference t1α-t1β and the spin-orbit coupling, also
the Coulomb repulsion controls the gap as it was proved within DFT+U. The SSH
model is a spinless and chiral 1D model, while the model that we have proposed
is spinful and not chiral. Despite these differences regarding the material
class, the relevant quantity for the topological properties is the difference
between the hopping parameters due to the dimerization in both cases.

Figure 11: Band structure reproduced with the minimal model without including
the SOC interaction. The parameters used are: E1=6.359 eV, E2=3.205 eV,
t1α=1.691 eV, t1β=1.594 eV, t2α=$-$ 0.516 eV, t2β=$-$ 0.518 eV, t3=$-$0.065 eV
and t4=$-$0.495 eV. The value of the onsite energy E1 controls the position of
the bands that are at around 1 eV at the R point, while the value of E2
controls the position of the bands manifold which is at around -3 eV at the R
point. The Fermi level is set at zero energy.  Figure 12: Zoom of the band
structure reproduced with the minimal model along the lines $\Gamma$-R-X a)
without SOC and b) with the SOC interaction included. The parameters used are:
E1=6.359 eV, E2=3.205 eV, t1α=1.691 eV, t1β=1.594 eV, t2α=$-$ 0.516 eV,
t2β=$-$ 0.518 eV, t3=$-$0.065 eV and t4=$-$0.495 eV. The SOC parameter is
$\lambda$=0.3 eV. The energy difference between the bands at the R point
without SOC is roughly proportional to the module of the difference between
the hoppings t1α and t1β. The Fermi level is set at zero energy.

##  IV Discussion, conclusions and outlook

Ta3Sb shows strong orbital textures in the topological surface statesKim  _et
al._ (2019). Since the material class is the same and topological surface
states are qualitatively the same, we expect the orbital texture even in Nb3Ge
and Nb3Sn. The proximity-induced superconductivity in the Dirac surface states
can generate Majorana zero mode localized at the vortexKim  _et al._ (2019).
Further studies using the tight-binding model for the bulk and the surface of
A15 could shed light on the interplay between topological and superconductive
properties.

In conclusion, using a first-principle approach, we provide an extensive
description of the electronic, topological and superconductive properties of
the Nb- and Ta-based A15 compounds by calculating band structure, SHC,
superconducting $T_{c}$, and topological surface states. All compounds Nb3X (X
= Ge, Sn, Sb) and Ta3Y (Y = As, Sb, Bi) have metallic band structures. Nb3Ge
and Nb3Sn have one electron less respect to the other compounds, they have a
larger density of states and high superconducting $T_{c}$. Ta3As and Ta3Bi
host the Ta-6s band at the Fermi level producing larger $T_{c}$ and larger SHC
compared to Ta3Sb due to the presence of additional DOS and additional
anticrossings close to the Fermi level. The spin-Hall conductivity is
relatively large also for the lighter elements due to the presence of several
anticrossings in the Brillouin zone. All the superconducting $T_{c}$ are
sizeable. One of the most interesting results among our outcomes is the
presence of Dirac surface states at the $\Gamma$ point for all compounds at
the same filling. The ideal case is the Ta3Sb where we have $\mathbb{Z}_{2}$
metallic compounds with net separation between conduction and valence bands.
In all the other compounds, the conduction and the valence bands cross due to
the Ta-6s for the Ta-compounds and due to the weaker SOC in the Nb-compounds.
Despite the crossing between conduction and valence bands, the band inversion
at the high-symmetry points persists in Nb-based compounds producing always
the Dirac surface states at the $\Gamma$ point. Unfortunately, the Dirac
surface states are blurred when we include the hybridization with the s-bands
in the Ta-based compounds. The surface Dirac points can be tuned by the
Coulomb repulsion, in the case of transition metal termination the Dirac point
is around the Fermi level. Even if the $\mathbb{Z}_{2}$ topological invariant
cannot be calculated in all cases, the presence of the Dirac surface states is
persistent. Therefore, we can assume that the Nb3X (X = Ge, Sn, Sb) and Ta3Y
(Y = As, Sb, Bi) compounds are all $\mathbb{Z}_{2}$ topological metals. These
Dirac surface states could explain the low resistivity of the Nb3Sn surface
observed experimentallySchäfer  _et al._ (2020). Additionally, we provide a
minimal tight-binding model composed of three coupled SSH chains and based on
the t2g orbitals. This tight-binding reproduces the relevant electronic and
topological features for these compounds at the Fermi level. With a $T_{c}$ of
23.2 K, Nb3Ge could be the $\mathbb{Z}_{2}$ topological metal with the highest
$T_{c}$ ever reported amongst the A15 compounds.

The surfaces of the A15 will host an interplay between $\mathbb{Z}_{2}$
topology, strong orbital texture, breaking of the inversion symmetry and BCS
superconductivity with relatively large $T_{c}$. Therefore, the surfaces of
A15 are a platform to search for exotic superconductivity. Once it will be
grown the thin film of A15, it will be possible to construct superlattices,
junctions, or heterostructures of superconductors and topological compounds in
order to study the topological superconductivity via the proximity effect. The
non-trivial surface properties of A15 thin films can also represent an
interesting platform for the realization of gate-controllable superconducting
devices, where recent studies have suggested that surface properties are key
for the observation of the suppression of a critical current under an applied
gate voltage Ruf  _et al._ (2023).

## Acknowledgments

The work is supported by the Foundation for Polish Science through the
International Research Agendas program co-financed by the European Union
within the Smart Growth Operational Programme (Grant No. MAB/2017/1). C. A.
acknowledges Erasmus+ for a training scholarship. We acknowledge the access to
the computing facilities of the Interdisciplinary Center of Modeling at the
University of Warsaw, Grant g91-1418, g91-1419 and g91-1426 for the
availability of high-performance computing resources and support. We
acknowledge the CINECA award under the ISCRA initiative IsC99 ”SILENTS”,
IsC105 ”SILENTSG”, IsB26 ”SHINY” and IsB27 ”SLAM” grants for the availability
of high-performance computing resources and support. We acknowledge the access
to the computing facilities of the Poznan Supercomputing and Networking Center
Grant No. 609.

##  Appendix A Three coupled SSH chains with a t2g orbital basis

Our model includes just the t2g electrons for the 6 Nb-atoms. The crystal
structure presents three dimers of Nb atoms, along the $a$, $b$ and $c$ axes.
We consider in our model the intradimer hybridizations and the interdimer
hoppings by considering the first and the second nearest neighbours.

The lattice constant is d=5.139 Å. The basis in the Hilbert space is given by
the vector

$\phi_{i}^{\dagger}=(\phi_{1a}^{\dagger},\phi_{2a}^{\dagger},\phi_{3c}^{\dagger},\phi_{4c}^{\dagger},\phi_{5b}^{\dagger},\phi_{6b}^{\dagger}),$
(1)

where

$\phi_{1a}^{\dagger}=(\phi_{1a,xz}^{\dagger},\phi_{1a,yz}^{\dagger},\phi_{1a,xy}^{\dagger}),$
(2)

the indices 1,.., 6 indicate the Nb atoms from the first to the sixth, the
letters $a$, $c$, $b$ indicate the fact that these sites belong to the dimers
along the cell directions $a$, $c$ and $b$, while xz, yz and xy indicate the
t2g orbitals dxz, dyz and dxy.

Our Hamiltonian is the following:

$H=\begin{bmatrix}H_{aa}&H_{ac}&H_{ab}\\\ H_{ca}&H_{cc}&H_{cb}\\\
H_{ba}&H_{bc}&H_{bb}\end{bmatrix},$

where the subscript $a$ indicates the dimer along the $a$ direction, composed
of Nb1 and Nb2 atoms, $c$ the dimer along the $c$ direction, composed of Nb3
and Nb4 atoms and $b$ indicates the dimer along the $b$ direction, namely the
dimer of Nb5 and Nb6 atoms. The $aa$, $bb$ and $cc$ blocks of the Hamiltonian
contain the intradimer hybridizations, while the off-diagonal blocks contain
the interdimer hoppings.

The intradimer submatrices are of this type:

$H_{aa}=\begin{bmatrix}H_{1a1a}&H_{1a2a}\\\ H_{2a1a}&H_{2a2a}\end{bmatrix},$

where

$H_{1a1a}=\begin{bmatrix}E_{1a_{xz}}&0&0\\\ 0&E_{1a_{yz}}&0\\\
0&0&E_{1a_{xy}}\\\ \end{bmatrix},$

and

$H_{1a2a}=\begin{bmatrix}H_{1a_{xz}2a_{xz}}&0&0\\\ 0&H_{1a_{yz}2a_{yz}}&0\\\
0&0&H_{1a_{xy}2a_{xy}}\\\ \end{bmatrix},$

The on-site energies belong to two groups. The energy $E_{1}$ belongs to the
orbitals that form intradimer $\pi$-bond, while the energy $E_{2}$ belongs to
the orbitals that form intradimer $\delta$-bond :

$\begin{split}E_{1a_{xz}}=E_{1}\qquad E_{1a_{yz}}=E_{2}\qquad
E_{1a_{xy}}=E_{1}\\\ E_{2a_{xz}}=E_{1}\qquad E_{2a_{yz}}=E_{2}\qquad
E_{2a_{xy}}=E_{1}\\\ E_{3c_{xz}}=E_{1}\qquad E_{3c_{yz}}=E_{1}\qquad
E_{3c_{xy}}=E_{2}\\\ E_{4c_{xz}}=E_{1}\qquad E_{4c_{yz}}=E_{1}\qquad
E_{4c_{xy}}=E_{2}\\\ E_{5b_{xz}}=E_{2}\qquad E_{5b_{yz}}=E_{1}\qquad
E_{5b_{xy}}=E_{1}\\\ E_{6b_{xz}}=E_{2}\qquad E_{6b_{yz}}=E_{1}\qquad
E_{6b_{xy}}=E_{1}\end{split}$

The intradimer elements have a hopping form similar to the SSH modelSu  _et
al._ (1979) as we can see in Fig. 10, therefore, the tight-binding model that
describes the A15 is composed of three coupled SSH chains. The intradimer
Hamiltonian elements have the following form:

$\displaystyle H_{1a_{xz}2a_{xz}}$ $\displaystyle=$ $\displaystyle
t_{1\alpha}e^{-ik_{x}d/2}+t_{1\beta}e^{ik_{x}d/2}$ $\displaystyle
H_{1a_{yz}2a_{yz}}$ $\displaystyle=$ $\displaystyle
t_{2\alpha}e^{-ik_{x}d/2}+t_{2\beta}e^{ik_{x}d/2}$ $\displaystyle
H_{1a_{xy}2a_{xy}}$ $\displaystyle=$ $\displaystyle
t_{1\beta}e^{-ik_{x}d/2}+t_{1\alpha}e^{ik_{x}d/2}$ $\displaystyle
H_{3c_{xz}4c_{xz}}$ $\displaystyle=$ $\displaystyle
t_{1\beta}e^{-ik_{x}d/2}+t_{1\alpha}e^{ik_{x}d/2}$ $\displaystyle
H_{3c_{yz}4c_{yz}}$ $\displaystyle=$ $\displaystyle
t_{1\alpha}e^{-ik_{x}d/2}+t_{1\beta}e^{ik_{x}d/2}$ $\displaystyle
H_{3c_{xy}4c_{xy}}$ $\displaystyle=$ $\displaystyle
t_{2\alpha}e^{-ik_{x}d/2}+t_{2\beta}e^{ik_{x}d/2}$ $\displaystyle
H_{5b_{xz}6b_{xz}}$ $\displaystyle=$ $\displaystyle
t_{2\alpha}e^{-ik_{x}d/2}+t_{2\beta}e^{ik_{x}d/2}$ $\displaystyle
H_{5b_{yz}6b_{yz}}$ $\displaystyle=$ $\displaystyle
t_{1\alpha}e^{-ik_{x}d/2}+t_{1\beta}e^{ik_{x}d/2}$ $\displaystyle
H_{5b_{xy}6b_{xy}}$ $\displaystyle=$ $\displaystyle
t_{1\beta}e^{-ik_{x}d/2}+t_{1\alpha}e^{ik_{x}d/2}$

where t1 are $\pi$-bonds hoppings and t2 are $\delta$-bonds hoppings. As
expected, we have t1 $>$ t2 if we extract the parameters from the
wannierization of the DFT band structure. The configuration of $\alpha$ and
$\beta$ are the left and right hopping, their configuration is related to the
symmetries of the system. The topological gap at the R-point is controlled by
the difference between t1α \- t1β and enhanced by the spin-orbit coupling.

The interdimer submatrices are of the type:

$H_{ac}=\begin{bmatrix}H_{1a3c}&H_{1a4c}\\\ H_{2a3c}&H_{2a4c}\end{bmatrix},$

where we have:

$H_{1a3c}=\begin{bmatrix}H_{1a_{xz}3c_{xz}}&H_{1a_{xz}3c_{yz}}&H_{1a_{xz}3c_{xy}}\\\
H_{1a_{yz}3c_{xz}}&H_{1a_{yz}3c_{yz}}&H_{1a_{yz}3c_{xy}}\\\
H_{1a_{xy}3c_{xz}}&H_{1a_{xy}3c_{yz}}&H_{1a_{xy}3c_{xy}}\\\ \end{bmatrix},$

Regarding the interdimer hybridizations, the intraorbital elements that are
different from zero in our model are the following:

$\displaystyle H_{1a_{xz}3c_{xz}}$ $\displaystyle=$ $\displaystyle
2t_{3}e^{i(k_{x}+k_{z})d/4}cos(k_{y}d/2)$ $\displaystyle H_{1a_{xz}4c_{xz}}$
$\displaystyle=$ $\displaystyle 2t_{3}e^{i(k_{x}-k_{z})d/4}cos(k_{y}d/2)$
$\displaystyle H_{1a_{xy}5b_{xy}}$ $\displaystyle=$ $\displaystyle
2t_{3}e^{i(-k_{x}+k_{y})d/4}cos(k_{z}d/2)$ $\displaystyle H_{1a_{xy}6b_{xy}}$
$\displaystyle=$ $\displaystyle 2t_{3}e^{i(-k_{x}-k_{y})d/4}cos(k_{z}d/2)$
$\displaystyle H_{2a_{xz}3c_{xz}}$ $\displaystyle=$ $\displaystyle
2t_{3}e^{i(-k_{x}+k_{z})d/4}cos(k_{y}d/2)$ $\displaystyle H_{2a_{xz}4c_{xz}}$
$\displaystyle=$ $\displaystyle 2t_{3}e^{i(-k_{x}-k_{z})d/4}cos(k_{y}d/2)$
$\displaystyle H_{2a_{xy}5b_{xy}}$ $\displaystyle=$ $\displaystyle
2t_{3}e^{i(k_{x}+k_{y})d/4}cos(k_{z}d/2)$ $\displaystyle H_{2a_{xy}6b_{xy}}$
$\displaystyle=$ $\displaystyle 2t_{3}e^{i(k_{x}-k_{y})d/4}cos(k_{z}d/2)$
$\displaystyle H_{3c_{yz}5b_{yz}}$ $\displaystyle=$ $\displaystyle
2t_{3}e^{i(-k_{y}+k_{z})d/4}cos(k_{x}d/2)$ $\displaystyle H_{3c_{yz}6b_{yz}}$
$\displaystyle=$ $\displaystyle 2t_{3}e^{i(k_{y}+k_{z})d/4}cos(k_{x}d/2)$
$\displaystyle H_{4c_{yz}5b_{yz}}$ $\displaystyle=$ $\displaystyle
2t_{3}e^{i(-k_{y}-k_{z})d/4}cos(k_{x}d/2)$ $\displaystyle H_{4c_{yz}6b_{yz}}$
$\displaystyle=$ $\displaystyle 2t_{3}e^{i(k_{y}-k_{z})d/4}cos(k_{x}d/2)$

and the interorbital Hamiltonian elements are the following:

$\displaystyle H_{1a_{yz}3c_{xy}}$ $\displaystyle=$ $\displaystyle
2t_{4}e^{i(k_{x}+k_{z})d/4}cos(k_{y}d/2)$ $\displaystyle H_{1a_{yz}4c_{xy}}$
$\displaystyle=$ $\displaystyle-2t_{4}e^{i(k_{x}-k_{z})d/4}cos(k_{y}d/2)$
$\displaystyle H_{1a_{yz}5b_{xz}}$ $\displaystyle=$
$\displaystyle-2t_{4}e^{i(-k_{x}+k_{y})d/4}cos(k_{z}d/2)$ $\displaystyle
H_{1a_{yz}6b_{xz}}$ $\displaystyle=$ $\displaystyle
2t_{4}e^{i(-k_{x}-k_{y})d/4}cos(k_{z}d/2)$ $\displaystyle H_{2a_{yz}3c_{xy}}$
$\displaystyle=$ $\displaystyle-2t_{4}e^{i(-k_{x}+k_{z})d/4}cos(k_{y}d/2)$
$\displaystyle H_{2a_{yz}4c_{xy}}$ $\displaystyle=$ $\displaystyle
2t_{4}e^{i(-k_{x}-k_{z})d/4}cos(k_{y}d/2)$ $\displaystyle H_{2a_{yz}5b_{xz}}$
$\displaystyle=$ $\displaystyle 2t_{4}e^{i(k_{x}+k_{y})d/4}cos(k_{z}d/2)$
$\displaystyle H_{2a_{yz}6b_{xz}}$ $\displaystyle=$
$\displaystyle-2t_{4}e^{i(k_{x}-k_{y})d/4}cos(k_{z}d/2)$ $\displaystyle
H_{3c_{xy}5b_{xz}}$ $\displaystyle=$
$\displaystyle-2t_{4}e^{i(-k_{y}+k_{z})d/4}cos(k_{x}d/2)$ $\displaystyle
H_{3c_{xy}6b_{xz}}$ $\displaystyle=$ $\displaystyle
2t_{4}e^{i(k_{y}+k_{z})d/4}cos(k_{x}d/2)$ $\displaystyle H_{4c_{xy}5b_{xz}}$
$\displaystyle=$ $\displaystyle 2t_{4}e^{i(-k_{y}-k_{z})d/4}cos(k_{x}d/2)$
$\displaystyle H_{4c_{xy}6b_{xz}}$ $\displaystyle=$
$\displaystyle-2t_{4}e^{i(k_{y}-k_{z})d/4}cos(k_{x}d/2)$

Figure 13: Local density of states of (a) Nb3Ge, (b) Nb3Sn and (c) Nb3Sb. The
Fermi level is set to zero in all panels.

##  Appendix B Computational details for the Quantum Espresso calculations

To compute material properties, we employed density functional theory based
first-principles calculations as implemented within the Quantum ESPRESSO
code.Giannozzi  _et al._ (2009) We use generalized-gradient approximations
with Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional Perdew  _et
al._ (1996) implemented in norm-conserving pseudopotential with optimized
kinetic energy cut-off of 110 Ry for Nb3X (X = Ge, Sb, Sn) and 70 Ry Ta3Y (Y =
Bi, As, Sb). A uniform Monkhorst-Pack grid of 20 $\times$ 20 $\times$ 20 was
used in all the calculations which corresponds to 1342 special k-points in the
irreducible Brillouin zone. In phonons of Nb3Sn, we observed a numerical
artifact at high symmetry point $\Gamma$ with PBE pseudopotential, hence made
additional calculations using local-density approximation with Perdew-Wang 91
gradient-corrected functional only for this system.Perdew and Wang (1992) The
optimized kinetic energy cut-off of 70 Ry was used for this purpose.

The dynamical calculations pertaining to vibrational properties have been
performed within the density functional perturbation theory Baroni  _et al._
(2001) wherein the dynamical matrices were sampled within the irreducible
Brillouin zone using a q-mesh of at least 3 $\times$ 3 $\times$ 3\. The
Migdal-Eliashberg spectral functions were calculated using Eq. 3 presented
below.Baroni  _et al._ (2001)

$\alpha^{2}F(\omega)=\frac{1}{2\pi
N(E_{F})}\sum_{qv}\frac{\gamma_{qv}}{\omega_{qv}}\delta(\omega-\omega_{qv})$
(3)

where, $\gamma_{qv}$ is phonon linewidth and $\omega_{qv}$ is the phonon eigen
frequency. Integrating over these spectral functions as presented in Eq. 4, we
obtain the electron-phonon coupling coefficient $\lambda$($\omega$).

$\lambda(\omega)=2\int_{0}^{\infty}\frac{\alpha^{2}F(\omega)}{\omega}d\omega$
(4)

Following this, the superconducting critical temperature ($T_{c}$) was
calculated using Allen-Dynes modification of the McMillan formula Allen and
Dynes (1975) presented in Eq. 5 where $\mu^{*}$ is effective Coulomb repulsion
parameter $\omega_{ln}$ is weighted logarithmic average phonon frequencies.

$T_{c}=\frac{\omega_{log}}{1.2}exp\left(-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\right)$
(5)

To compute the topological surface states, the $\mathbb{Z}_{2}$ topological
invariants and spin Hall conductivity, we used the Wannier90 Mostofi  _et al._
(2014) and WannierTools codes Wu  _et al._ (2018). The basis of the tight-
binding model was composed of the Wannier functions Marzari and Vanderbilt
(1997) obtained from the d-orbitals of Nb or Ta and the p-orbitals of the
other atoms. The atomic orbital configuration is s2d3 for Nb and Ta, but it
becomes d5 in crystals, therefore Ta and Nb have half-filled d-orbitals. The
tight-binding model is composed of 60 d-bands and 12 p-bands for a total of 72
bands. The topological gap is observed at half-filling. Of these 36 occupied
bands, 6 are mainly p-bands and 30 are mainly d-bands. The momentum mesh used
for calculating spin Hall conductivity
${\sigma}_{\mathit{\text{xy}}}^{\text{spin}z}(\omega)$ was 200 $\times$ 200
$\times$ 200\. This was done using the Kubo-Greenwood formula as implemented
in Wannier90 Guo  _et al._ (2008); Qiao  _et al._ (2018).

Figure 14: Phonon dispersion curves of (a) Ta3As, (b) Ta3Sb, (c) Ta3Bi and (d)
Nb3Sb alongside the corresponding anisotropic Migdal-Eliashberg spectral
functions $\alpha^{2}$F($\omega$) and phonon density of states F($\omega$).
Figure 15: Surface states of (a) Ta3As, (b) Ta3Sb and (c) Ta3Bi indicating
spin-momentum locked Dirac dispersions at $\Gamma$ point submerged below the
s-bands. Slab band structures of (d) Ta3As, (e) Ta3Sb and (f) Ta3Bi with the
red bands indicating the contribution from the top surface layer and the blue
bands indicating the contribution from the bottom surface layer of the slab
presented in Fig 1. The legends on the right in all the panels indicate the
spectral weights. The Fermi level is set to zero in all panels.

##  Appendix C Slab calculation and density of states of the Nb-based
compounds using VASP

For the slab calculations, additional first-principles calculations are
performed via the Vienna ab initio simulation package (VASP) Blöchl (1994);
Kresse and Furthmüller (1996) using density functional theory. The generalized
gradient approximation with PBE form Perdew  _et al._ (1996) and PBEsol Perdew
_et al._ (2008) is adopted to calculate the lattice parameters and the density
of states (DOS). An energy cutoff of 350 eV and a mesh of 16 × 16 × 1 k-points
were chosen for the different thicknesses of Nb-based compounds. The
calculations were converged with the convergence criteria of 0.01 eV/Å, and
energy $10^{-5}$ eV. We have constructed stoichiometric slabs of Nb3Sn with
Nb2 and NbSn terminations as shown in Fig. 1(c) and the internal degrees of
freedom were relaxed.

The density of states (DOS) is a relevant quantity to examine the
superconducting properties. We report the DOS Nb3Ge, Nb3Sn and Nb3Sb in Fig.
13(a-c). From the DOS, we can see that we have the d-orbitals roughly between
-3 eV and +5 eV sandwiched between p-orbitals spectral weight. In these cases,
it is extremely challenging to decouple the d-orbitals from the p-orbitals due
to their strong hybridization as shown for other materials with such p-d
spectral weightCuono  _et al._ (2019); Wadge  _et al._ (2022). To perform the
wannierization, we must include the d-orbitals of Nb and the p-orbitals of Ge,
Sn and Sb. The DOS results were obtained within the VASP code and matched with
the results obtained with Quantum Espresso.

We report the calculation of the equilibrium lattice constant within PBEsol
and PBE. The system is cubic and does not have internal degrees of freedom,
therefore, the only parameter to optimize is the lattice constant $a$. The
lattice constant results were obtained within the VASP code and matched with
the results obtained with Quantum espresso. The PBE exchange-correlation
functional overestimates the experimental lattice constant by 1%, indeed the
lattice constant for Nb3Ge, Nb3Sn and Nb3Sb are 5.188 Å, 5.339 Å and 5.314 Å,
respectively. The PBEsol gives slightly better results underestimating the
lattice constant by 0.1% (i.e., lattice constant for Nb3Ge, Nb3Sn and Nb3Sb
are 5.135 Å, 5.280 Å and 5.259 Å, respectively.

##  Appendix D Phonon dispersions and Migdal-Eliashberg spectral functions for
Ta3Y (Y = Ge,Sn,Sb) and Nb3Sb compounds

In this Section, we report the phonon dispersion curves, phonon density of
states, Migdal-Eliashberg spectral functions and the electron-phonon coupling
values for all the compounds except that for Nb3Ge and Nb3Sn which are
described in the main text. The results are reported in Fig. 14. The electron-
phonon coupling coefficient $\lambda$($\omega$) for Ta3As, Ta3Sb, Ta3Bi and
Nb3Sb are 0.56, 0.41, 0.45 and 0.47, respectively. Correspondingly the $T_{c}$
is slightly higher in Ta3As, Ta3Bi and Nb3Sb as compared to Ta3Sb (see the
main text for the numerical values), this is essentially due to the density of
states at the Fermi level in the electronic structure. Although, Ta3Sb shows
softening of the optical phonon modes which implies that the $T_{c}$ should be
as high as Ta3Bi and Nb3Sb, however, the density of states at the Fermi level
in the electronic structure dominates the superconducting behavior leading to
lower $T_{c}$.

##  Appendix E Hybridization of the Dirac surface states with the s-bands for
Ta-based systems

In this Section, we want to include the s-band in our tight-binding model and
see the effect on the Dirac surface states. While the results of the main text
are obtained with high numerical accuracy, the process of including the s-band
in the tight-binding Hamiltonian is obtained within strong approximations in
the wannierization process (just including the s-band in the frozen window).
The presence of the s-band in the Ta-based system affects the Dirac surface
states by submerging the Dirac points below them. We report in Fig. 15 the
surface states for the Ta3Y which include the s-bands. In our calculations, we
can observe the Dirac surface states are pushed below the s-bands and the
Dirac surface states are blurred by the hybridization with the s-bands.

The parity of the s-band and the parity of the other highest valence bands are
both positive, therefore, we expect that this s-band would produce an
adiabatic transition without affecting the topologyFu and Kane (2007); Dzero
_et al._ (2016). Indeed, in the literature, it was shown that the Dirac
surface states of Ta3Sb can coexist with the s-bandsDerunova  _et al._ (2019).

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