# First-principles evidence of type-II Weyl phonons in rock-salt Tin
Chalcogenides (SnS, SnSe & SnTe) materials

Antik Sihi1<EMAIL_ADDRESS>Sudhir K. Pandey2<EMAIL_ADDRESS>1School of Basic Sciences, Indian Institute of Technology Mandi, Kamand -
175075, India
2School of Engineering, Indian Institute of Technology Mandi, Kamand - 175075,
India

###### Abstract

Recent studies on different topological materials in condensed matter physics
have provided the evidence of topological nature for bosonic particle like
phonons by performing various theoretical calculations and experimental
observations. Here, the topological behaviours of phonons of SnS, SnSe and
SnTe materials in rock-salt structure are investigated using $ab$-$initio$
methodology. For all these materials, the tilted linear band touching is
observed along with the presence of band inversion in direction X-W. The
topological point in phonon dispersion curve along X-W direction is found to
be $\sim$2.83, $\sim$2.46 and $\sim$2.51 THz for SnS, SnSe and SnTe,
respectively. The calculated numbers of Weyl points (WPs) is estimated to be
56, 24 and 54 for SnS, SnSe and SnTe, respectively. These WPs have shown
conserved Chiral charges for all corresponding materials. The surface local
density of states is computed for these compounds, where the surface arc is
clearly seen. In case of SnS, the surface arc is found at 2.0 - 2.2 THz energy
region. Also, the isofrequency surface states are investigated to observe the
presence of Fermi arc for SnX (X = S, Se, Te). The present first-principles
calculation predicts that SnS, SnSe and SnTe show type-II Weyl phononic
behaviour along with evidence of topological phononic surface states.

Introduction :- Nowadays, topological materials (TM) bring new era of research
in the field of condensed matter physics because they have shown different
unique properties for example topological insulator, topological semimetal,
quantum hall-effect $etc.$hasan_rev ; bansil_rev ; yan_rev ; viswa_rev  All
these properties give opportunity to the researcher for developing the new
theory of individual field and have provided the possibility of various
applications. In present days, the most emergent applications of different
topological properties are known as valleytronics valley , spintronics based
devices spin , quantized circular photogalvanic effect photogal  $etc$. It is
known that the bulk electronic band structure of any topological insulator
usually show band inversion near the bandgap and the conducting surface states
of this corresponding material are typically protected by time-reversal
symmetry. But, in case of topological semimetals for fermionic system, the
linear band touchings in three-dimensional (3D) crystal may host the
excitation due to Weyl or Dirac fermions in 3D brillouin zone (BZ) viswa_rev ;
vivek . In case of former one, the touching points of two non-degenerate bands
are typically called as Weyl points (WPs). These points generally show non-
zero topological charges based on the Nielsen-Ninomiya theorem nielsen , which
is known as the Chirality ($\mathscr{C}$ = +1 or -1) of the corresponding
point. Each of these opposite chiral points represents the monopole of the
Berry curvature. At a particular energy surface, the opposite two chiral
points are typically connected through the topological surface state, which is
known as the Fermi arc hasan_weyl . Now, depending on the nature of linear
band touching, the Weyl semimetals are sub-divided into type-I and type-II
prl_117 ; nature_527 . In case of type-I, the usual conical band touching is
observed, where point like Fermi surface is generally observed weng_5 . But,
tilted Weyl cones are found for type-II Weyl semimetals. Further, it has been
reported that the topological Lorentz invariance may be possible to be broken
at the energy of the WPs, which may be responsible for these materials to show
different exotic electromagentic responses tamai ; Huang ; chang ; yu ;
udagawa . TaAs is known as the first material in the list of type-I Weyl
semimetal xu ; lv , whereas WTe2 is crowned as the first realized material in
the class of type-II Weyl semimetal nature_527 ; xu_115 . Moreover, the other
examples of type-II Weyl fermionic materials are MoTe2 Huang ; wang , WP2,
MoP2autes , TaIrTe4 koepernik , NbIrTe4liu_17 , Ta3S2chang  $etc.$

Till now, huge amount of theoretical and experimental works have been carried
out to investigate the topological properties of fermionic electrons for many
different materials. Phonon, which is known as the collective excitations of
atoms within the sample, is a bosonic particle. It also shows many exotic
phenomena like thermal conductivity guthrie ; li , mechanical propertiestang ,
phase transitionfleury , superconductivity bohnen ; liu ; zhang  etc.
depending on the corresponding phonon modes of any compound. Recently,
different non-trivial topological properties of these vibrational modes have
been investigated both from theoretical and experimental observations miao ;
tzhang ; ttzhang . In case of this bosonic particle, the topological phenomena
is generally sub-divided into Weyl phonons and Dirac phonons, depending on the
symmetry conservations tzhang ; chen . Different chalcogenides based materials
are already categorized as candidate of topological phonons tzhang ; wang_95 ;
li_97 ; xia_cdte ; liu_103 . But, the list of type-II Weyl phononic sample is
very small, which typically restricts the applicability of this type of
material. Therefore, to find out more realistic type-II Weyl phononic
materials, the first principle calculation is required to be carried out on
different chalcogenides based compounds. In addition to this, if the
electronic band structure of any material shows non-trivial topological nature
along with the non-trivial phononic topological behaviour, then this
particular material will obviously provide more ways for application purpose.
Till our present knowledge, this aspect is not focused in any earlier work. To
perform this work, tin chalcogenides (SnX, X = S, Se, Te) are the best
candidate, which are possible to be experimentally synthesized and the host of
fermionic topological behaviour in rock-salt phase antik_pla ; asihi_sse ;
liu_snte ; liu_snte ; tanaka_snte ; wang_snte ; y_sun ; wang_snse_exp1 ;
jin_snse_exp2 ; mariano_exp_sns ; bilenkii_exp_sns2 ; skelton_sns . Moreover,
the linear band touching is also observed from the phonon band structures of
SnS and SnSe materials, which has motivated us to explore the phononic
topological behaviour of these materials asihi_sse .

In this work, we have investigated the phononic topological behaviour of SnS,
SnSe and SnTe materials in rock-salt structure. The linear band touching along
X-W direction for all these mentioned materials are seen from respective
phonon dispersion curves. The tilted nature of the band touching indicates the
presence of type-II Weyl phonon at frequency value of $\sim$2.83 THz for SnS.
The behaviour is more clearly seen both for SnSe and SnTe, where the touching
points are situated at $\sim$2.46 THz and $\sim$2.51 THz for SnSe and SnTe,
respectively. Moreover, the estimated values of WPs are found to be 56, 24 and
54 numbers for SnS, SnSe and SnTe, respectively, within the first full BZ. The
values of Chirality are estimated to be +1 for half of WPs and -1 for the
other half of WPs for all the materials. Moreover, the presence of surface arc
is suggested from the surface local density of states. The isofrequency
surface states are also computed for getting the information about the Fermi
arc at particular frequency. The Fermi arc is seen at frequency $\sim$2.2THz
and $\sim$2.25 THz for SnS material. The similar behaviour is also observed
for SnSe and SnTe compounds.

(a)

(b)

(c)

(d)

(e)

Figure 1: (Colour online) (a) Phonon dispersion plot of SnS along high-
symmetric direction, phonon dispersion plot along with atomic contribution for
box 1 of (b) SnS and box 2 of (c) SnS, (d) SnSe and (e) SnTe along X-W
direction.

(a)

(b)

Figure 2: (Colour online) (a) The distribution of WPs for the first BZ due to
the touching point 2 as shown in Fig. 1(a) and (b) Normalized Berry curvature
of SnS.

Computational details :- WIEN2k code is used for carrying out the first-
principle calculation using PBEsol exchange-correlation (XC) functional wien2k
; pbesol . The optimized lattice parameters of 5.753 Å, 5.955 Å and 6.291 Å
are utilized for SnS, SnSe and SnTe, respectively antik_jpcm ; antik_jpcm2 .
$F$m$\bar{3}$m space group, which denotes the rock-salt phase of these
compounds, have been chosen for the present calculation along with the Wyckoff
positions at (0.0, 0.0 ,0.0) and (0.5, 0.5, 0.5) for Sn and X (= S, Se, Te),
respectively. The 2$\times$2$\times$2 supercell is taken for performing the
force calculation by setting the force convergence criteria at 0.1 mRy/Bohr.
The non-analytical term correction (NAC) is added for the entire phonon
calculation to lift the degeneracy of longitudinal optical (LO) and transverse
optical (TO) band at q $\rightarrow$ 0 togo_nac . PHONOPY code is used for the
phonon calculation, where the real-space interatomic force constant
($\Phi_{\alpha\beta}(\mathbf{R}_{jl}-\mathbf{R}_{j^{\prime}l^{\prime}})$) is
estimated through finite-displacement method phonopy . In order to investigate
the non-trivial topological properties of phonon modes, the tight-binding
Wannier Hamiltonian for these vibrational modes are formed using
$\Phi_{\alpha\beta}(\mathbf{R}_{jl}-\mathbf{R}_{j^{\prime}l^{\prime}})$, which
are given by liu_2017 ,

$\displaystyle
H=\frac{1}{2}\sideset{}{}{\sum}_{l,m,\alpha}\sideset{}{}{\sum}_{l^{\prime},m^{\prime},\beta}\Phi_{\alpha\beta}(\mathbf{R}_{jl}-\mathbf{R}_{j^{\prime}l^{\prime}})u^{\alpha}_{lm}u^{\beta}_{l^{\prime}m^{\prime}}$
(1)

where, $\alpha,\beta$ represent Cartesian indices, $u^{\alpha}_{lm}$
($u^{\beta}_{l^{\prime}m^{\prime}}$) denotes the displacement component of the
$l$th ($l^{\prime}$th) atom in the $j^{th}$ ($j^{\prime th}$) unit cell along
the direction of $\alpha$ ($\beta$). All the topological properties of the
phonons are explored using WANNIERTOOLS code wanniertools .

Results and discussions :- Phonon dispersion curve of SnS along the high-
symmetric direction is shown in Fig. 1(a), whereas the phonon band structures
of SnSe and SnTe are plotted in Figs. S1(a) and S1(b) of Supplemental Material
supple . For present study, NAC is included to compute all phonon’s related
calculations along with the topological properties. The detailed description
of the phonon band structures for all these materials are already discussed in
different literatures antik_pla ; asihi_sse . Three optical and three acoustic
phonon branches are obtained for these compounds. Some of the optical branches
have shown linear touching along X-W direction as marked by red boxes and have
denoted by numbers 1 and 2, respectively, in Fig. 1(a). Now, in order to
observe the presence of phononic band inversion as compared to electronic
topological materials, these two touching points of SnS are plotted separately
for seeing the clear view of band touchings in Figs. 1(b) and (c), where red
(green) colour denotes the contribution of Sn (X = S, Se and Te) atom to form
this band around the observed energy window. Fig. 1(b) shows the absence of
band inversion at touching point 1. But, in case of point 2, the band
inversion between phonon branches of Sn and S atoms is clearly seen from Fig.
1(c). In such case, the phonon branches are characterized by the corresponding
atoms using projected density of states of individual atom. The linear
touching of two optical branches at the point 2 along X-W directions shows
slightly tilted nature, which provides a convincing reason to classify this
point as type-II Weyl phononic point at $\sim$2.83 THz. For comparison
purpose, similar touching point between the corresponding two bands at point 2
for SnSe and SnTe are shown in Figs. 1(d) and (e), respectively. These
touching points for SnSe and SnTe are found to be $\sim$2.46 THz and
$\sim$2.51 THz, respectively. The presence of phononic band inversion is
evident from these figures for this particular point for all the samples.
Moreover, the nature of tilted band touching is more profoundly seen when one
moves from SnS to SnTe. This behaviour is nicely matched with the previously
predicted ideal type-II Weyl phononic material ZnSe liu_103 .

(a)

(b)

Figure 3: (Colour online) Wannier charge center evolution of SnS around the
(a) positive and (b) negative charges of a pair of Weyl points.

We have found 56 WPs in full Brillouin zone (BZ) for SnS after applying all
symmetry operations, where these points are shown in Fig. 2(a). Here, 28
points have shown $\mathscr{C}$ = +1 and the remaining 28 points have been
found to estimate the value of $\mathscr{C}$ = -1. In case of SnSe and SnTe,
we have estimated the values of 24 and 54 numbers of WPs, respectively, where
the $\mathscr{C}$ of the corresponding sample is found to be conserved. In
order to explain the topological properties of these WPs using the phonon
bands, the Berry curvature needs to be calculated using the following relation
wanniertools ,

$\displaystyle\Omega_{n}^{z}(\mathbf{q})=\nabla_{\mathbf{q}}\times i\langle
u_{n}(\mathbf{q})|\nabla_{\mathbf{q}}|u_{n}(\mathbf{q})\rangle$ (2)

where $u_{n}(\mathbf{q})$ denotes the phonon branches of $n^{th}$ band. It is
known that one pair of Weyl nodes typically contains the positive and negative
Berry curvatures. In momentum space, these positive and negative Berry
curvatures give the information of source and sink of the Berry flux,
respectively. For SnS, Fig. 2(b) illustrates the Berry curvature of crossing
nodes in $\mathbf{q}_{z}$ = 0 plane. In the figure, the source and sink of the
Berry flux are mentioned by the red and blue points, respectively. Further, to
get the evident of positive and negative chiral charges around the WPs, the
Wannier charge centers (WCC) are calculated and plotted in Figs. 3(a) and (b),
respectively, for SnS. Here, WCC is calculated using the Wilson loop
approaches around the corresponding Weyl point. Now, it is noted that the Fig.
3(a) (3(b)) shows the Wannier charge evolution around one particular Weyl
node, which shows positive (negative) chirality. This behaviour is found to be
similar with the previously predicted type-II Weyl phononic materials xia_cdte
; liu_103 . SnSe and SnTe have also shown the similar kind of behaviour as
compared to SnS. However, the comparative study of phononic topological
properties for these compounds needs to be explored further.

Figure 4: (Colour online) The surface local density of states for the (001)
surface of SnS along two dimensional high-symmetric directions. The surface
arc is marked by a black arrow.

(a) Energy = 2.0 THz

(b) Energy = 2.1 THz

(c) Energy = 2.15 THz

(d) Energy = 2.2 THz

(e) Energy = 2.25 THz

(f) Energy = 2.3 THz

Figure 5: (Colour online) The topologically protected non-trivial topological
phononic surface states of SnS of (001) surface at different frequencies for
the top surface of the unit cell.

Now, in order to investigate the exotic topological phononic surface state for
(001) surface, the surface local density of states (LDOS) of SnS is calculated
along the surface high-symmetric direction, which is shown in Fig. 4.
Moreover, the surface LDOS of SnSe and SnTe are shown in Supplemental Material
supple . To calculate the surface LDOS, the second rank tensor of force
constant is estimated from the corresponding phonon calculation. Using this,
we have obtained the tight-binding parameters for the bulk and surface atoms
to study the topological behaviours of phonon. Afterwards, the surface
spectrum is calculated by computing the surface Green’s function (SGF)
iteratively with considering the imaginary part of the SGF surface_green . The
surface BZ of (001) of $F$m$\bar{3}$m spacegroup is a square. It is known that
the (001) surface shows mirror symmetry with respect to (1$\bar{1}$0) mirror
plane tanaka_snte ; y_sun . The high-symmetric points of 3D momentum space are
projected on (001) surface. Therefore, the $\Gamma$, X and W points are now
denoted by $\bar{\Gamma}$, $\bar{X}$ and $\bar{W}$, respectively. We have
chosen this direction because the linear touching is observed along X-W
direction in 3D momentum space. It is clearly seen from the figure that the
presence of surface arc is evident along $\bar{\Gamma}$ \- $\bar{X}$ \-
$\bar{W}$ \- $\bar{\Gamma}$ direction around 2.0 - 2.2 THz energy region,
which is also marked by an black arrow. The similar behaviour is found for
FeSi, which is a famous candidate for a topological phononic material tzhang .
For SnSe and SnTe, we have seen the similar kind of surface arc, which are
shown in Fig. S2(a) and S2(b) of Supplemental Material, respectively supple .
This behaviour of all the materials ($i.e.$ SnX (X = S, Se, Te)) is indicating
the presence of type-II Weyl topological phonon. Moreover, the isofrequency
surface contours provides the information of Fermi arc at particular
frequency. Therefore, we have also computed isofrequency surface states, which
have plotted in Figs. 5(a), (b), (c), (d), (e) and (f) at energy 2.0 THz, 2.1
THz, 2.15 THz, 2.2 THz, 2.25 THz and 2.3 THz, respectively. Here, the
isofrequency surface states are shown for SnS material for the top surface of
the unit cell, whereas the projection of bulk on $k_{1}$-$k_{2}$ plane and the
bottom surface of the unit cell are shown in Figs. S3 and S4 of Supplemental
Material supple  at the different aforementioned frequencies. At this point,
it is important to note that if the bulk projection and any surface ($i.e.$
top/bottom) projection are showing similar behaviour then the Fermi arc due to
topological behaviour of the material is not present at this particular
energy. In present scenario, the evidence of Fermi arc is found at energy 2.2
THz and 2.25 THz because the surface projection and the bulk projection are
showing different spectrum. The similar behaviour is also seen for SnSe and
SnTe materials. Therefore, these three samples ($i.e.$ SnS, SnSe and SnTe) in
rock-salt phase are predicted as the candidates of type-II Weyl topological
phononic materials. The present theoretical study can be possible to verify by
performing the corresponding experiments on these materials. Moreover, this
predicted result enhances the usefulness of SnS, SnSe and SnTe materials for
practical application purposes.

Conclusions :- In this present study, the topological behaviours of phonon are
explored for SnS, SnSe and SnTe materials in rock-salt phase. The tilted
linear band touching is seen from phonon dispersion curves of all the
materials along X-W direction. The band inversion around this point are also
investigated for all the materials, where the frequency value of this
topological point for SnS is found to be $\sim$2.83 THz. In case of SnSe and
SnTe, this topological points are seen at $\sim$2.46 THz and $\sim$2.51 THz,
respectively. The computed values of WPs are 56, 24 and 54 for SnS, SnSe and
SnTe materials, respectively, in the full BZ. The values of Chiral charges,
which are present at these WPs of corresponding materials, are estimated to be
conserved. In case of all the materials, the surface arc is clearly seen from
the surface local density of states. Also, the presence of Fermi arc is
evident from the isofrequency surface state of individual compounds. For SnS,
the Fermi arcs are found at $\sim$2.2THz and $\sim$2.25 THz frequencies. All
these behaviours are suggested to predict SnS, SnSe and SnTe as type-II Weyl
phononic materials. The present study reveals the topological importance in
phononic structure of these famous electronic topological materials, which
will enrich the practical applications of all these materials.

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