# Spin-orbit insulating phase in SnTe cubic nanowires:
consequences on the topological surface states

Ghulam Hussain ID<EMAIL_ADDRESS>International Research Centre
MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotników
32/46, PL-02668 Warsaw, Poland  Kinga Warda  International Research Centre
MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotników
32/46, PL-02668 Warsaw, Poland  Faculty of Applied Physics and Mathematics,
Gdansk University of Technology, Gdańsk 80-233, 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  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

We investigate the electronic, structural and topological properties of the
SnTe and PbTe cubic nanowires using ab initio calculations. Using standard and
linear-scale density functional theory, we go from the ultrathin limit up to
the nanowires thicknesses observed experimentally. Finite-size effects in the
ultra-thin limit produce an electric quadrupole and associated structural
distortions, these distortions increase the band gap but they get reduced with
the size of the nanowires and become less and less relevant. Ultrathin SnTe
cubic nanowires are trivial band gap insulators, we demonstrate that by
increasing the thickness there is an electronic transition to a spin-orbit
insulating phase due to trivial surface states in the regime of thin
nanowires. These trivial surface states with a spin-orbit gap of a few meV
appear at the same $k$-point of the topological surface states. Going to the
limit of thick nanowires, we should observe the transition to the topological
crystalline insulating phase with the presence of two massive surface Dirac
fermions hybridized with the persisting trivial surface states. Therefore, we
have the co-presence of massive Dirac surface states and trivial surface
states close to the Fermi level in the same region of the $k$-space. According
to our estimation, the cubic SnTe nanowires are trivial insulators below the
critical thickness tc1=10 nm, and they become spin-orbit insulators between
tc1=10 nm and tc2=17 nm, while they transit to the topological phase above the
critical thickness of tc2=17 nm. These critical thickness values are in the
range of the typical experimental thicknesses, making the thickness a relevant
parameter for the synthesis of topological cubic nanowires. Pb1-xSnxTe
nanowires would have both these critical thicknesses tc1 and tc2 at larger
values depending on the doping concentration.

##  I Introduction

Majorana fermions are intriguing particles that have garnered significant
attention in the field of condensed matter physics. Majorana fermions are non-
Abelian anyons, meaning their quantum states exhibit intriguing noncommutative
properties. They obey non-Abelian statistics and their topological protection
could be crucial for quantum computing and topological quantum information
processing. Their potential application in topological quantum computation
promises quantum algorithms that are robust against decoherence and errors.
Majorana-based qubits are expected to be more robust than other qubit
technologies due to their topological protection and are envisioned to play a
vital role in quantum error correction protocolsSau  _et al._ (2020);
Beenakker (2013).

One of the platforms proposed to host Majorana fermions is a topological
nanowire with superconductivity in an applied magnetic field. However, the
realization of Majorana’s fermions is challenging as it requires the growth of
a nanowire crystal to cultivate a column of atoms measuring approximately 100
nm in diameter. Subsequently, this system needs to be integrated with a
circuit that exhibits the necessary sensitivity to monitor individual
electrons as they traverse through it. Moreover, all of this intricate work
must be conducted at temperatures merely one-hundredth of a degree above
absolute zero and within a magnetic field that is 10,000 times stronger than
Earth’s Frolov (2021). Nanowires made from various materials like
semiconductor-superconductor hybrids or topological insulators with proximity-
induced superconductivity have been studied to realize Majorana
fermionsStanescu and Tewari (2013).

IV-VI semiconductors have exhibited interesting properties such as
thermoelectricity Wang  _et al._ (2011); Wood (1988), ferroelectricityLebedev
and Sluchinskaya (1994); Liu  _et al._ (2021) and superconductivityMatsushita
_et al._ (2006); Mazur  _et al._ (2019). In particular, the discovery of
topological crystalline insulating phase (TCI) in SnTeFu (2011); Hsieh  _et
al._ (2012); Lau and Ortix (2019); Tanaka  _et al._ (2012); Okada  _et al._
(2013) and some of their substitutional alloys such as Pb1-xSnxTeXu  _et al._
(2012) and Pb1-xSnxSeDziawa  _et al._ (2012); Kazakov  _et al._ (2021) have
aroused tremendous research interest to further study this class of materials.
The TCI phase is different from the conventional topological insulating phase,
indeed the linearly dispersing Dirac states on the high-symmetry surfaces of
the TCIs are protected by crystal symmetries Fu  _et al._ (2007) and not by
time-reversal symmetry. SnTe and its substitutional alloys are TCIs with
mirror symmetry, and the presence of the surface states is indicated by a non-
zero integer topological invariant named mirror Chern number Teo  _et al._
(2008); Safaei  _et al._ (2015); Hsieh  _et al._ (2012); Cuono  _et al._
(2022); Rechciński  _et al._ (2021). Furthermore, it has been shown that SnTe
is a helical higher-order topological insulator Schindler  _et al._ (2018);
Kooi  _et al._ (2020); van Miert and Ortix (2018). It has been observed that
the characteristic properties change by changing the size or dimensions (2D or
3D phase) of materials, namely the properties will change if we move from bulk
Plekhanov  _et al._ (2014); Wang  _et al._ (2020); Barone  _et al._ (2013) to
thin films Sławińska  _et al._ (2020); Volobuev  _et al._ (2017); Liu  _et
al._ (2018). For instance, SnTe is a trivial insulator at low thickness but
becomes topological above some critical thicknessLiu  _et al._ (2014, 2015)
quite robust against impurity dopingArroyo-Gascón  _et al._ (2022). Recently,
the twinning in thin films as a function of the mirror Chern number was
studiedSamadi  _et al._ (2023). By using scanning tunneling microscopy for
Pb1-xSnxSe, it was shown that atomically flat terraces in the Se sublattice
separated by step edges of various heights are present. Enhancements of the
local DOS have been found at odd step edges, and by using tight binding models
spin-polarized flat bands connecting Dirac points have been shown Sessi  _et
al._ (2016). The properties of the low-energy states at the surface atomic
steps and the behavior of such edge channels under doping were investigated
Sessi  _et al._ (2016); Brzezicki  _et al._ (2019); Mazur  _et al._ (2019);
Wagner  _et al._ (2012).

Going to the one-dimensional (1D) case, the topological crystalline insulator
SnTe nanowires (NWs) can be even more interesting than the bulk due to a
larger contribution from the topological surface states and to the 1D
confinement effectShen and Cha (2014). Methods to produce SnTe NWs by using
grapheneSadowski  _et al._ (2018), high yield and alloy nanoparticles as
growth catalystsLiu  _et al._ (2021) or to obtain the smallest possible SnTe
NWs by using single-walled carbon nanotubesVasylenko  _et al._ (2018) were
found. TCI NWs are a versatile platform for the confinement and manipulation
of Dirac fermions Skiff  _et al._ (2023a). The Pb1-xSnxTe cubic NWs were also
synthesized Safdar  _et al._ (2015); Dad  _et al._ (2022); Saghir  _et al._
(2015); Xu  _et al._ (2016). Recently, pentagonal nanowires with a perfect C5
symmetry have been synthesized in Pb1-xSnxTe. This non-cubic crystal structure
(impossible in a pure ionic system) rises from the mixture of covalent and
ionic bonds. Additionally, the pentagonal phase could be a platform for high-
order topology. The results of the investigation on pentagonal nanowires will
be presented elsewhereHussain  _et al._ (2023).

Theoretically, the SnTe NWs were investigated and it was shown that different
topological states can be realized under the application of various magnetic
fields Nguyen  _et al._ (2022), bulk Majorana modes are present when there is
inversion symmetry, while under symmetry-breaking fields at the ends of the
NWs, Majorana zero modes appear while the Majorana bulk modes are gapped
Nguyen  _et al._ (2022). Ultrathin SnS and SnSe NWs were also studied by means
of ab initio calculations Shukla  _et al._ (2022), while the NWs of trivial
PbTe were shown to be also interesting for the possible presence of Majorana
zero modesSong  _et al._ (2023); Jung  _et al._ (2022).

The 3D and 2D cases of these rock-salt chalcogenides are widely investigated
from first principles, while the one-dimensional structures like NWs are
poorly investigated so far due to the huge number of atoms to be considered
when we break the boundary conditions along 2 dimensions. We find that the
thin cubic NWs show a robust insulating behavior. We investigate the evolution
of the band gap as a function of the thickness for the cubic NWs and we find
the critical thickness at which the system becomes topological. The paper is
organized as follows: in the next Section the computational details are
described, the third section is devoted to the theoretical investigation of
the structural and the electronic properties of the cubic NWs, while in the
fourth section, the topology of the NWs is investigated. In the last Section,
we draw our conclusions.

Figure 1: a) Side view and b) top view of the crystal structure of the cubic
SnTe nanowires with an even number of atoms along the x- and y-direction. We
indicate ’$t$’ as the thickness of the nanowire. The unit cell is composed of
2N$\times$2N$\times$2 atoms with periodic boundary conditions along the
z-axis.

##  II Computational details

The results were obtained within the framework of the first-principles density
functional theory (DFT), the calculations are fully relativistic by
considering spin-orbit coupling (SOC) if not mentioned otherwise. The system
presents inversion symmetry since we do not assume ferroelectricity, as a
consequence, the systems host the Kramer’s degeneracy due to the combination
of time-reversal and inversion symmetry. Therefore, all bands reported in this
paper are double degenerate and we plot the band structure only along the
$k$-path $\Gamma$-Z. The system has a mirror along the (110) and
(1$\overline{1}$0) planes and a non-symmorphic symmetry respect to the (100)
and (010) planes.

We investigated the cubic SnTe and PbTe NWs with [001] orientation. The
lattice constant $c$ is equal to 6.2964 and 6.4277 Å for SnTe and PbTe,
respectively, which are the experimental values. These lattice constants are
equivalent to twice the anion-cation distance. We limit the study to an even
number of atoms along the x- and y-direction to avoid odd-even effects and to
keep the crystal structure stoichiometric. Indeed, depending on the odd or
even number of atoms the system hosts the mirror or non-symmorphic
symmetryBrzezicki  _et al._ (2019). For the VASP calculations, we investigate
unit cells containing 2N$\times$2N$\times$2 nanowires, where N=1, 2, 3… is the
number of atoms as we can see in Fig. 1a). When we use ONETEP, we double the
cell along the z-axis studying 2N$\times$2N$\times$4 nanowires with the
lattice constant of the superlattice $c_{sup}$=2$c$. We define $t$ as the in-
plane thicknesses of the nanowires as shown in Fig. 1b).

Structural and electronic structure calculations of the thin nanowires were
performed with a plane wave basis set and projector augmented wave method
using the VASPKresse and Furthmüller (1996) package. A plane-wave energy cut-
off of 250 eV has been used. As an exchange-correlation functional, the
generalized gradient approximation (GGA) of Perdrew, Burke, and Ernzerhof
(PBE) has been adoptedPerdew  _et al._ (1996). The description of the IV-VI
semiconductors and more in general narrow band semiconductors suffers of the
band gap problem that overestimates the topological region of the phase
diagram giving a wrong band ordering. To ease this problem and obtain a
description closer to the experiments, we need to go beyond the standard GGA
approximation Cuono  _et al._ (2022, 2023); Islam  _et al._ (2019); Hussain
_et al._ (2022). In the case of nanowires, a more realistic band order and
band gap can be obtained with the meta-GGA approach named SCAN, namely, the
strongly constrained and appropriately normed functional Sun  _et al._ (2015),
which is one of the few exchange-correlation functional working well in the
presence of SnTe and PbTe surfaces. Both results with GGA and meta-GGA are
qualitatively similar but the meta-GGA gives a larger gap that is much closer
to the experimental results. We have performed the VASP self-consistent
calculations using 1$\times$1$\times$17 for GGA and 1$\times$1$\times$12
$k$-points centered in $\Gamma$ for SCAN, while for the band structures, we
have used 60 $k$-points from $\Gamma$ to Z.

The number of atoms in the system is NATOMS=8N2, standard DFT codes scale as
O(N${}_{ATOMS}^{3}$), therefore, they are scaling as the O(N6). This poses
serious limitations in the study of thick NWs. In order to increase the
thickness of cubic SnTe NWs, we need to study a large number of atoms with
SOC. This was accomplished through the use of ONETEP Skylaris  _et al._
(2005); Prentice  _et al._ (2020), a linear-scaling density functional theory
(LS-DFT) approach. In the ONETEP code orbitals that are nonorthogonal
generalized Wannier functions (NGWFs) are expressed within the basis of
periodic sinc (psinc) functions, providing accuracy comparable to the
traditional plane-wave DFT software Skylaris and Haynes (2007).
Simultaneously, the developed approach offers parallel computation, which
allows the simulations of thousands of atoms Skylaris  _et al._ (2008). The
ONETEP calculations were performed with the use of PBE functional and PAW
pseudopotentials from the JTH library Jollet  _et al._ (2014). The
investigation of the bandstructure was carried out with the use of spectral
function unfolding methodology Constantinescu and Hine (2015). The sampling of
the reciprocal space consisted of 110 $k$-points, spaced equally along the
z-reciprocal lattice vector. With each of the Sn and Te atoms, there were
assigned 9 NGWFs. Considering the study of large systems, for all NGWFs the
radius was determined as $R_{\phi}\textnormal{ = 5 }a_{0}$, allowing the
convergence and optimal accuracy to be maintained. To ensure the requirements
of the bandstructure unfolding formalism, the diameter of the NGWFs could not
exceed half the length of the simulation cell. Therefore, the size of the
nanowire in the axial direction had to be doubled compared to the structure
studied within VASP. The basis set consisted of the psinc functions distant
from each other by $d_{\textnormal{psinc}}$ = 0.36 Å, resulting in the density
kernel cutoff $E_{\textnormal{c}}$ equal 447 eV. The length of the simulation
cell in the z-direction was equal to $c_{sup}$ = 12.5928 Å, with the
imposition of periodic boundary conditions (PBCs). The lengths along the x-
and y-directions were selected as multiplies of the $d_{\textnormal{psinc}}$
depending on the thickness of the structure, with the vacuum spacing applied.
The system investigated consisted of up to 784 atoms for the SOC calculations
and 2704 atoms for non-relativistic calculations. The density kernel
truncation was not applied. Considering that ONETEP is based on localized
Wannier functions and it is developed for molecules and surfaces, the surface
states calculated within ONETEP would be quite reliable.

Figure 2: Phonon dispersions for undistorted (a) and distorted (b) cubic NWs.
This indicates that the distortions in the cubic systems are important for the
dynamical stability. (c) Quadrupolar distortion in the case of the 2$\times$2
SnTe nanowire. The arrows represent the displacements with respect to the
ideal NaCl crystal structure. As in Fig. 1, the green balls represent the Te
atoms, while the dark grey balls represent the Sn atoms. Figure 3: Bond
lengths for [001] oriented cubic nanowires as a function thickness for (a)
SnTe and (b) PbTe cubic nanowires, respectively. The anion-cation distance is
the first neighbor distance, while the anion-anion and cation-cation distances
are the second neighbor distances. The second neighbor distances are larger by
a factor $\sqrt{2}$ with respect to the first neighbor distances. The line is
a guide to the data points.

##  III Structural and Electronic properties of the SnTe and PbTe cubic
nanowires

In this Section, we investigate the structural and electronic properties of
the NWs grown along the [001] direction focusing on the ultrathin nanowires
using the VASP code. The (001) surfaces in cubic SnTe are not polar, the
charge present on the inner atoms is almost the same of the surface atoms,
therefore, we do not need to introduce dangling bonds.

Figure 4: Thickness-dependent electronic band structures of SnTe (left panel)
and PbTe (right panel) cubic NWs grown along [001] without SOC. All the cubic
nanowires reveal trivial insulating behavior. The Fermi level is set to zero.
The conduction bands become closer and closer to the Fermi level as the
thickness increases, which decreases the band gap. The thickness is shown in
Å, and the conversion to the size is reported in Table 1. Calculations were
performed using the VASP code.

Ultrathin nanowires show distortions from the NaCl phase. The phonon
dispersion of the perfectly cubic ultrathin nanowires shows in Fig. 2a)
negative frequencies and imaginary phonon modes. Once we include these
distortions, all the phonon frequencies of the system become positive as we
can see in Fig. 2b). The finite-size effects produce an electric quadrupole
with the Sn atoms attracting and the Te repelling each other as in Fig. 2c).
We have the clear formation of an electric quadruple with positive and
negative charges that are slightly different due to the different atomic
numbers, this produces non-zero forces on the atoms and displacements along
the diagonals. This is very evident in the simple case of the 2x2 nanowires as
shown in Fig. 2c). The quadrupolar distortions slightly affect the electronic
and structural properties at low thicknesses. When we increase the thickness,
these quadrupolar distortions are still present on the surfaces of the
nanowires while being reduced in the inner parts of the nanowire. The
quadrupolar distortions do not break the inversion, mirror, and non-symmorphic
symmetries.

In Figs. 3a)-b), we show the bond lengths as a function of the thickness for
both the SnTe and PbTe cubic nanowires oriented along the [001] direction. We
can see that the bond lengths become closer to the theoretical values without
relaxation when we go to larger thicknesses. Therefore, these quadrupolar
distortions are less and less relevant for thicker nanowires and they will not
be considered in the study of the topology. Also experimentally, it was
reported that polar distortions in thin films are more relevant at low
thicknessesOkamura  _et al._ (2022).

Size | $t$ | Gap SnTe | Gap PbTe
---|---|---|---
4x4 | 1.33 | 0.762 | 1.215
6x6 | 2.24 | 0.560 | 1.028
8x8 | 3.14 | 0.453 | 0.929
10x10 | 4.05 | 0.388 | 0.865

Table 1: We report the size of the NW, the thicknesses $t$ (nm) equal for both
SnTe and PbTe NWs, and the total band gaps (eV) from SOC calculations for SnTe
and PbTe cubic nanowires after structural relaxation using the VASP code.

Moving to the electronic properties, as all the structures are one-
dimensional, we do not need to provide the density of states (DOS) since the
band structure gives information about the entire Brillouin zone. We first
present the electronic band structures without SOC of the SnTe and PbTe cubic
NWs oriented along [001] direction in Fig. 4. All these NWs with size
10$\times$10 or smaller exhibit the trivial nature of insulators thereby
revealing explicit band gaps. Going to the SOC calculations, Table 1 displays
the quantitative values of band gaps for SnTe and PbTe cubic NWs, with the
band gap decreasing as a function of the thicknesses. As for the bulk systems,
we show that also for nanowires the trivial band gap of PbTe is larger than
the gap of SnTe. The alloying of Pb1-xSnxTe would likely produce an
intermediate gap between PbTe and SnTe. Moreover, the nature of the band gap
is different for the two compounds i.e. SnTe cubic NWs show indirect band
gaps, whereas direct band gaps appear for PbTe cubic NWs.

Figure 5: Band gap of SnTe at Z point as a function of NWs thickness for GGA
(blue) and SCAN (red). Undistorted (UD) nanowires band gaps are reported for
GGA and SCAN in green and purple, respectively. The solid lines are a guide
for the eyes. The in-plane number of atoms is 2N$\times$2N. Calculations were
performed using the VASP code. Figure 6: a) Schematic band structure of one-
dimensional TCI with the surface Dirac point close to the Z point with
Z=$\frac{\pi}{c}$. The vertical dashed line represents half of the Brillouin
zone. b) Schematic band structure of one-dimensional TCI with the folding of
the $k$-path (due to the doubling of the real space unit cell). The Z point is
folded in $\Gamma$, the surface Dirac point moves close to $\Gamma$. The new
Brillouin zone ranges from $\Gamma$ to the Z of the supercell
(Zsup=$\frac{\pi}{2c}$) that is half of the initial Z. The horizontal dashed
line is the Fermi level set to zero. Figure 7: Band structure of the SnTe
nanowires with different sizes. a) Band gap insulating phase for the
10$\times$10$\times$4 NW, b) spin-orbit insulator for the
14$\times$14$\times$4 NW and c) magnification of the 14$\times$14$\times$4 NW.
The calculations have been performed within ONETEP, the Brillouin zone ranges
from $\Gamma$ to Zsup. The Fermi level ($E_{\textnormal{F}}$) is set to zero.
Figure 8: Band gap of SnTe at $\Gamma$ point as a function of NWs thickness
without and with SOC. The in-plane number of atoms is 2N$\times$2N.
Calculations were performed using the ONETEP code with four atoms in the unit
cell along the c-axis. The gap at the $\Gamma$ point with four atoms along the
c-axis is equivalent to the gap at the Z-point reported in Fig. 5. Blue points
represent the gap without SOC, while yellow points represent the gap with SOC.
The square points represent the band gap calculated with VASP without
distortions. The solid lines are a guide for the eyes. We extrapolate the band
gap going to zero from a parabolic fitting of the ONETEP data with SOC. The
extrapolation of the SCAN data was performed with a linear fitting for gaps
above 0.4 eV and a parabolic trend for the gap below 0.4 eV as from the band
gap without SOC.

##  IV Phase diagram of the cubic SnTe NWs as a function of the thickness

The topological crystalline phase and the associated mirror Chern number are
protected by mirror symmetry with respect to the (110) plane. The bulk SnTe
has 6 surface Dirac points in the Brillouin zone, and the (001) surface of 2D
topological SnTe hosts 4 surface Dirac points at non-time-reversal-invariant
momentaTanaka  _et al._ (2013). When the 2D Brillouin zone is projected on the
1D Brillouin zone, two Dirac points are projected in $\Gamma$ and become
gapped due to their hybridization. The other 2 surface Dirac points of the
(001) topological NWs rise close to +Z and -Z. Due to the large number of
electronic bands, we cannot perform the wannierization and calculate the Chern
number as it was done for the quaternary alloy bulkCuono  _et al._ (2022).
However, we can study the band gap closure to search for the surface Dirac
cones in the band structure of the NWs.

Here, we investigate the band gap of the SnTe cubic NWs as a function of the
thickness using GGA and SCAN metaGGA exchange functional correlations and with
the inclusion of the SOC within VASP (see Fig. 5). We study the band gap at
the high-symmetry points that are the relevant points for the topological
properties. As the thickness of the NWs increases, the band gap decreases for
both the functionals considered. In the metaGGA approximation, the gap is
larger and the system is more trivial. For all the values of the thicknesses
investigated within VASP, the system is trivial in both approximations. To
check the effects of the structural relaxation on the band gap, we also
calculated the band gap for the undistorted system, namely without performing
structural relaxation. Although the gap without structural distortions is
lower in energy than the gap of the distorted structure, in all cases analyzed
the system is trivial.

In order to increase the thickness of the SnTe nanowires, we need to study
larger supercells and we approach this using the linear scaling ONETEP
codeSkylaris  _et al._ (2005). Due to technical reasons, within the ONETEP
code, we are forced to double the structure along the z-axis. This doubling in
the real space produced a halving of the first Brillouin zone in the
$k$-space. The high-symmetry point Z [see Fig. 6a)] is folded at the $\Gamma$
point, therefore, the band closing present close to Z moves close to $\Gamma$.
The Brillouin zone of the supercell extends up to Z of the supercell (Zsup).
This scheme is shown in Fig. 6b). Therefore, the band gap at the point
$\Gamma$ within the supercell calculations (ONETEP) must be compared with the
band gap at Z of the unit cell (VASP).

We plot in Fig. 7 the whole band structure for the two representative cases,
namely the 10$\times$10$\times$4, the 14$\times$14$\times$4 NWs and its
magnification. Fig. 7a) shows the band insulator phase that appears in the
ultrathin limit for the 10$\times$10$\times$4 NW or thinner. Increasing the
size, the cubic SnTe NWs show some trivial surface states that fill the gap as
shown in 7b). The spin-orbit opens a few meV gap between the valence and
conduction bands creating an insulator as shown in Fig. 7c), since the gap was
not created at a high-symmetry point this is not a topological insulating
phase but a trivial spin-orbit insulating phase. Considering the folding, the
position of the trivial surface state is around the $k$-point
0.90$\frac{\pi}{c}$ in the unfolded Brillouin zone. When the band gap at the
high-symmetry point reaches around 0.25 eV, we have the first transition from
band gap insulator to spin-orbit insulator. These trivial surface states
appear on a wider range of thicknesses with respect to the topological surface
states, therefore, they are quite robust even if not topologically protected.
These trivial surface states lie at the same energy and $k$-point where the
topological surface states should form the Dirac point. Since these trivial
surface states would persist in the topological phase, we would have a co-
presence of trivial and topological surface states at around
0.90$\frac{\pi}{c}$. The presence of these trivial bands would increase the
number of carriers in SnTe and in general in 1D Pb1-xSnxTe systems. These
electronic and topological properties are strongly dependent on the nanowire’s
thickness as experimentally confirmedXu  _et al._ (2015).

The ab initio calculations will be closer to the experimental materials with
respect to the model Hamiltonian where the same SOC term and the same hopping
parameters are used for Sn and Te atoms. Within ab initio calculations, we
find these additional trivial surface states that are present also in the
model Hamiltonian calculationsNguyen  _et al._ (2022); Skiff  _et al._
(2023b), however, the spin-orbit gap is much smaller with DFT. Therefore, the
presence of trivial surface bands at the Fermi level beyond the surface Dirac
bands is not an artifact of the application of the 3D model Hamiltonian to the
1D system but it is a real property of the IV-VI NWs. These trivial surface
states hybridize with the surface Dirac points since they are at the same
energy and the same $k$-point; this hybridization produces massive Dirac
pointsRechciński  _et al._ (2021). To have a closed Dirac point, we need to go
to very high thicknesses Schindler  _et al._ (2018) to avoid hybridization
between the surfaces. Some elements of high-order topology were reported in
model Hamiltonian papers even without breaking the mirror symmetry, indeed,
the hinge states in cubic NWs are present for large thicknesses at the point
ZNguyen  _et al._ (2022) or very close to ZSkiff  _et al._ (2023b).

In Fig. 8, we report the band gap of SnTe at $\Gamma$ point as a function of
the film thickness without and with SOC by using the supercell with 4 atoms
along the c-axis and the ONETEP code. We compare it with the band gap
calculated with VASP within GGA and SCAN. In all cases, the gaps go to zero
increasing the number of atoms and therefore the number of hoppings and the
total bandwidth. Examining the ONETEP calculations with and without SOC, we
observe that the band gap becomes smaller with SOC due to the band splitting
in the valence band and conduction band. Increasing N, the reduction of the
gap becomes lower and lower, this is due to the increasing hybridization at
the $\Gamma$ point between the topmost valence band and the lowest conduction
band. The quantitative difference between VASP and ONETEP is attributed to
different implementations and approximations of the two DFT codes. The SCAN
results are expected to be the ones closer to the experimental results, we
extrapolate the SCAN results in order to obtain realistic values for the
topological transition thicknesses.

Figure 9: Phase diagram as a function of the thickness $t$ for cubic
Pb1-xSnxTe NWs. The different phases are separated by the critical thickness
values tc1 and tc2. The blue bands are trivial, while the surface massive
Dirac fermions are plotted in red. We have a co-presence of trivial and
topological surface states at around 0.90$\frac{\pi}{c}$. The horizontal
dashed lines represent the Fermi level. Extrapolating the data from SCAN
calculations, realistic critical values for pure cubic SnTe NWs are tc1=10 nm
and tc2=17 nm. We expect the same phase diagram for Pb1-xSnxTe NWs but with
larger values of tc1 and tc2 depending on the doping concentration.

Within the ONETEP calculations, the first transition from band insulator to
SOC-insulator happens at N=6 when the gap at $\Gamma$ goes below 0.3 eV, while
the topological transition happens at N=14 when the band gap of the system
with SOC goes to zero according with the extrapolation of Fig. 8. The critical
thickness of the nanowires is given by $t=(2N-1)d_{SnTe}$, where $d_{SnTe}$ is
the distance between Sn and Te which is half of the lattice constant. From
this formula, we calculate the critical values and we obtain
$t_{c1}^{GGA}$=3.5 nm and $t_{c2}^{GGA}$= 8.5 nm for the GGA ONETEP
calculations.

However, we know that a more realistic value for the topological transition
would be within the metaGGCA SCAN. We extrapolate from the SCAN data the
behavior up to 0.4 eV, below 0.4 we assume that the reduction of the gap is
slowed down by the hybridization between conduction and valence bands as we
can see in the ONETEP results without SOC. Therefore, we assume that behavior
for the gap. From our assumptions, we get the first transition at N=16 and the
second transition at N=27. Since SCAN is good also for surfacesSi  _et al._
(2020), we start from the SCAN results and extrapolate the data adding the
behavior of the NOSOC that we were able to send to zero. We obtain that the
system reaches the gap of circa 0.25 eV for N= 16 which corresponds to tc1=10
nm, while the critical value of the topological transition is N=27 which is
equivalent to tc2=17 nm. The trivial surface states appear in a wider range
than the topological surface states; SnTe NWS with thicknesses 17 nm or below
were synthesizedHoskam (2018); Sadowski  _et al._ (2018), therefore, we
conclude that these NWs are not topological but they are in the SOC insulating
phase.

##  V Discussion and Conclusions

In order to investigate Majorana fermions, the SnTe nanowires should host
superconductivity beyond the topological properties demonstrated in the
previous Section. The superconductivity can be realized using proximity
effectKlett  _et al._ (2018), by doping Mitobe  _et al._ (2021) or interfacing
SnTe with PbTe Fogel  _et al._ (2001); Tang and Fu (2014).

The possibility of ferroelectricity of the cubic phase is addressed in a
recent experimental paperLiu  _et al._ (2021), further studies are necessary
on ferroelectricity and its interplay with the topology. In SnTe, the
ferroelectricity produces the Rashba effect because the inversion symmetry is
broken; in PbTe we have strong SOC, then in Pb1-xSnxTe, due to the combination
of lack of inversion symmetry and strong SOC, we can have ferroelectric Rashba
topological systemWang  _et al._ (2023); Shimada  _et al._ (2020).

In conclusion, we have shown using the DFT that the thin cubic SnTe and PbTe
nanowires are trivial insulators in the ultrathin limit. A quadrupolar
distortion relevant on the surface of the NWs slightly increases the band gap.
At the same thicknesses, the PbTe NWs have a larger trivial gap than SnTe NWs.
The cubic SnTe NWs undergo an electronic transition from the band gap
insulator to a spin-orbit insulator at the thickness tc1=10 nm. Increasing
again the thickness, the cubic SnTe NWs become topological crystalline
insulators for a thickness larger than tc1=17 nm. In the topological phase,
the SnTe NWs host two massive Dirac surface states at non time-reversal
invariant momenta close to the Z point hybridizing with trivial surface
states. The SnTe cubic nanowires experimentally reported by different authors
with thicknesses ranging between 10 and 200 nmSafdar  _et al._ (2013);
Sadowski  _et al._ (2018); Liu  _et al._ (2021); Hoskam (2018) are mainly
topological systems with the exception of the thin NWs with a thickness
smaller than tc2=17 nm. The IV-VI NWs are topologically more trivial than the
IV-VI bulk systems, therefore the Sn-concentration necessary to activate the
topological transition in Pb1-xSnxTe will be higher than the bulk values and
dependent on the thickness of the NWs. Moving to Pb1-xSnxTe NWs, the values of
tc1 and tc2 could sensibly increase especially close to the level of doping
that induces the topological transition in the bulk. Since experimental SnTe
cubic NWs are topological and could host superconductivity as well, we
conclude that these systems are a suitable platform for the investigation of
Majorana fermions.

###### Acknowledgements.

G. H. and K. W. contributed equally to this work. The authors thank Tomasz
Dietl for encouraging us to work on topological nanowires. We acknowledge R.
Buczko, S. Samadi, A. Kazakov, W. Brzezicki, C. M. Canali and S. Sattar for
constructive discussions. K. W. acknowledges S. Winczewski and J. Dziedzic for
useful suggestions about the use of the ONETEP code. 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). 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” and IsB26
”SHINY” 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 Grants No. 609.

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