#  Electronic structures and stability investigation of the new class of large
band gap topological insulators $M$Tl4Te3 ($M$ = Cd, Hg)

Ying Li  Wuhan National High Magnetic Field Center $\&$ School of Physics,
Huazhong University of Science and Technology, Wuhan 430074, China  Gang Xu
<EMAIL_ADDRESS>Wuhan National High Magnetic Field Center $\&$ School of
Physics, Huazhong University of Science and Technology, Wuhan 430074, China

###### Abstract

By means of ternary chemical potential phase diagram and phonon spectrum
calculations, we propose that $M$Tl4Te3 ($M$ = Cd, Hg), the derivatives of
Tl5Te3, are thermodynamically and dynamically stable in the body centered
tetragonal crystal structure with $I$4/$mcm$ symmetry. Our electronic
structures calculations confirm that a robust $s$-$p$ band inversion occurs
near the Fermi level in $M$Tl4Te3, and a topological band gap about 0.13 eV in
CdTl4Te3 is induced by the spin-orbit coupling. These results suggest that
$M$Tl4Te3 are a new class of large band gap 3D strong topological insulators
that are stable and synthesizable in experiment, which could be used to design
efficient spin torque equipment and spin device.

##  I Introduction

In the past two decades, topological electronic materials, including
topological insulators (TIs) PhysRevB.74.085308 ; science.1133734 ;
PhysRevB.78.195125 ; RevModPhys.82.3045 ; RevModPhys.83.1057 , topological
semimetals PhysRevB.83.205101 ; PhysRevLett.107.186806 ;
PhysRevLett.107.127205 ; PhysRevB.85.165110 ; PhysRevLett.108.140405 ;
PhysRevB.85.195320 ; PhysRevB.88.125427 ; hsieh2012topological ;
xu2012observation , and topological superconductors(SC) RevModPhys.83.1057 ;
PhysRevLett.100.096407 ; PhysRevLett.102.187001 ; PhysRevLett.104.057001 ;
PhysRevLett.105.097001 ; PhysRevLett.117.047001 ; PhysRevB.81.134508 , have
attracted great interest and reshaped perception of the materials. In
particular, large band gap three-dimensional (3D) TIs have many intriguing
properties in both fundamental physics PhysRevB.78.195424 ; science.1167733 ;
li2010dynamical  and device applications brumfiel2010topological ;
pesin2012spintronics ; APLMaterials.5.035504 ; PhysRevLett.121.116801 ;
wu2021magnetic . TIs with the spin momentum locked surface states are robust
against perturbations, and thus have high performance on the spin torque
Mellnik2014 ; RN37 , topotronic PhysRevB.92.205310 ; RN35  and spintronic
devices zhang2009topological ; hsieh2009tunable . Furthermore, many exotic
topological states are realized by modulating the TIs. For example, the
quantum anomalous Hall (QAH) effect has been achieved by the magnetic doping
in Bi2Te3 films PhysRevLett.61.2015 ; PhysRevLett.90.206601 ;
PhysRevLett.101.146802 ; AdvPhys.64.227 ; science.1187485 ; science.1234414 ;
xia2009observation ; ge2020high . Besides, topological superconductivity is
reported in the Cu-intercalated Bi2Se3 (CuxBi2Se3) PhysRevLett.104.057001 ;
PhysRevLett.106.127004 ; PhysRevLett.105.097001 ; wray2010observation , Bi2Te3
under high pressure Zhang24  and TI/SC heterostructures science.1216466 .
However, easily synthesized TIs with relatively large band gap and clear two-
dimensional (2D) Dirac-cone surface states as the Bi2Se3 family
zhang2009topological ; xia2009observation ; hsieh2009tunable ; science.1173034
; PhysRevLett.103.146401 ; arakane2012tunable  remain rare until now.
Therefore, it is desirable to search for large band gap 3D TIs for potential
utilizations.

First-principles calculations have played remarkable roles in the development
of topological physics and topological materials. Many topological materials
are predicted by first-principles calculations firstly, and then confirmed by
experiments, including HgTe quantum well science.1133734 ; science.1148047 ,
bismuth antimony alloy Bi1-xSbxPhysRevB.76.045302 ; hsieh2008topological ;
science.1167733 , Bi2Se3-family of TIs zhang2009topological ;
xia2009observation ; hsieh2009tunable ; science.1173034 ;
PhysRevLett.103.146401 ; arakane2012tunable , topological crystalline
insulator (TCI) SnTe hsieh2012topological ; xu2012observation , topological
semimetals PhysRevB.85.195320 ; PhysRevB.88.125427 ; science.1245085 ;
science.1256742 ; yi2014evidence ; jeon2014landau ; PhysRevLett.113.246402 ;
PhysRevB.83.205101 ; PhysRevLett.107.186806 ; PhysRevX.5.011029 ;
PhysRevX.5.031023  and so on. Recently, Tl5Te3 schewe1989crystal  has been
found to be a topological material that host Dirac surface states at 0.5 eV
above the Fermi level (EF). After then, by using 4$c$ site substitution, many
derivatives $M$Tl4Te3 ($M$ = Cu, Sn, Mo, Pb, Bi, Sb, La, Nd, Sm, Gd, Tb, Dy,
Er, Tm) bradtmoller1994crystal ; imamalieva2008new ;
PhysicsandChemistryofSolidState.21.492 ; RN34 ; bradtmoller1993darstellung
have been experimentally synthesized and reported. Among them, SnTl4Te3 with
8-electron configuration is expected to be a TI. Unfortunately, the band
inversion in SnTl4Te3 disappears so that it becomes a trivial insulator
PhysRevLett.112.017002 ; APLMaterials.3.041507 .

Inspired by the above understanding, we propose that CdTl4Te3 and HgTl4Te3 are
a new class of large band gap 3D strong topological insulators that are stable
and synthesizable experimentally, through the 4$c$ site substitution of Cd or
Hg. For this purpose, ternary chemical potential phase diagram with precursors
and convex hull diagram are constructed. Both of them demonstrate that
CdTl4Te3 is thermodynamically stable and easily synthesized in Cd-rich,
Tl2Te3-rich and Tl-poor condition. The phonon spectrum reveal that CdTl4Te3
adopts the body centered tetragonal structure with $I$4/$mcm$ symmetry.
Further electronic structures calculations identify that a robust band
inversion between Cd-5$s$ and Te-5$p$ orbitals exists at $\Gamma$ point even
without spin-orbit coupling (SOC). When SOC is considered, a topological band
gap about 0.13 eV is induced in CdTl4Te3, which is larger than the energy
scale of room temperature in theory. As a result, one single Dirac cone formed
by the topological surface states is discovered at $\bar{\Gamma}$ point of the
surface. The corresponding left-hand momentum locking texture is also studied,
which can be applied on the efficient spin torque equipment and spin device
design. Finally, the topological electronic structures and stability of the
other derivative HgTl4Te3 are discussed, which is expect to possess the same
crystal structure and strong TI nature as CdTl4Te3.

##  II Crystals structures and methodology

In our work, the same crystal structure of SnTl4Te3, i.e., the body centered
tetragonal phase with $I4/mcm$ space group(No. 140, D${}^{18}_{4h}$) as shown
in Fig. 1(a), is used for the calculation of $M$Tl4Te3 ($M$ = Cd, Hg). The
corresponding first Brillouin zone (BZ) SETYAWAN2010299  and its projection on
the (100) surface of the primitive cell are displayed in Fig. 1(c). In Fig.
1(a), $M$ atom locates at Wyckoff position 4$c$ ($0.5,~{}0.5,~{}0.0$) which is
the center of the corner-sharing CdTe6 octahedron; Tl is located at 16$l$
($x_{1},~{}0.5+x_{1},~{}z_{1}$); two types of Te locate at 4$a$
($0.5,~{}0.5,~{}0.25$) and 8$h$ ($-x_{2},~{}-x_{2}+0.5,~{}0.0$). The lattice
parameters, atomic coordinates $x_{1}$, $x_{2}$ and $z_{1}$ are fully relaxed.
The detailed structure information of MTl4Te3 is summarized in Table. 1.
Besides, all the compounds used in our ternary phase diagram calculation are
based on their ground phase in experiment, and their crystal parameters are
fully relaxed as tabulated in Tabel. 1 to make the energies comparable.

Our first-principles calculations are performed by the Vienna ab initio
simulation package KRESSE199615 ; PhysRevB.54.11169  with the projected
augmented wave method PhysRevB.50.17953 . The energy cutoff is set as 400 eV,
and $9\times 9\times 7$ k-meshes are adopted. Local-density approximation
(LDA) type of the exchange-correlation potential PhysRevB.23.5048  is used in
all calculations. All the different compositions CdlTlmTen are fully relaxed
until the Hellmann-Feynman forces on each atom are less than 0.01 eV/Å and the
total energy converge up to 10-6 eV. The ternary phase diagram is constructed
by calculating the total energy of CdlTlmTen without SOC. The phonon spectrum
calculations are carried out by the PHONOPY code TOGO20151  with a $2\times
2\times 2$ supercell through the density-functional perturbation theory
approach PhysRevB.55.10355 . The band inversion is further confirmed by the
modified Becke-Johnson (MBJ) potential PhysRevLett.102.226401  with the MBJ
parameter CMBJ setting as 1.35. We note that CMBJ = 1.1 $\sim$ 1.7 is usually
used for semiconductors including IIB-VIA compounds as proposed by Tran and
Blaha PhysRevLett.102.226401 ; PhysRevB.85.155109 . Its reliability and
accuracy have been identified to be at the same level as the hybrid functional
heyd2003hybrid  and GW methods PhysRev.139.A796  for a wide variety of
semiconductors PhysRevB.83.195134 ; PhysRevB.82.155145 ; PhysRevB.82.205212 ;
PhysRevB.82.205102 ; PhysRevB.93.045304 ; PhysRevB.93.115104 ;
PhysRevB.101.245163 ; PhysRevB.85.155109 . The maximally localized Wannier
functions are constructed by the Wannier90 package MOSTOFI20142309  based on
the MBJ+SOC calculations. The surface states are calculated by iterative
Green’s function method as implemented in the WannierTools package WU2018405 .

Figure 1: (a) The unit cell of CdTl4Te3. (b) Top view of the unit cell. (c)
Bulk BZ and its projection to the (100) surface of its primitive cell. High
symmetry $k$ path is indicated.

##  III The stability of CdTl4Te3

In this section, we would like to take CdTl4Te3 as example to study its
thermodynamical and dynamical stability. By choosing three stable compounds
Cd,Tl and Tl2Te3 as the precursor materials, the target compound CdTl4Te3 can
be synthesized by the following reaction:

$Tl_{2}Te_{3}+Cd+2Tl~{}\rightarrow~{}CdTl_{4}Te_{3}.$ (1)

A phase diagram as function of the precursors’ chemical potentials could
determine the ranges of the experimental synthesis condition within which the
target compound can be stabilized and within which the undesired competing
phases are formed by varying the composition of precursors RN36 ;
IntegratedFerroelectrics.175.186 ; ma13194303 ;
JournaloftheAmericanCeramicSociety.72.2104 ; RN21 ; RN22 . In general, the
synthesis process can be understood as the exchange of elemental components
between precursors and the forming phases. Therefore, the formation energies
of all forming compounds CdlTlmTen could be expressed as:

$\displaystyle E^{F}(Cd_{l}Tl_{m}Te_{n})~{}={}$ $\displaystyle
l\Delta\mu(Cd)+\frac{n}{3}\Delta\mu(Tl_{2}Te_{3})$ (2)
$\displaystyle+(m-\frac{2n}{3})\Delta\mu(Tl).$

where $\Delta\mu$ (i) = $\mu$(i) - ET(i) with i = Cd, Tl and Tl2Te3 are the
chemical potentials of the precursors referenced to the total energy of their
ground states. Therefore, Eq. 2 makes a connection between $\Delta\mu(i)$ and
experimental condition, which means a rich condition of the corresponding
precursor if $\Delta\mu(i)$ is close to zero, and a poor condition if
$\Delta\mu(i)$ has a large negative value. Based on our calculation, the
formation energy of CdTl4Te3 is -1.032 eV/formula with respect to the
precursors, which leads to two requirements for the chemical potentials. One
is that each $\Delta\mu(i)$ could only vary between 0 and -1.032 eV. The other
is that there are only two independent variable chemical potentials.
Therefore, the phase diagram can be visualized by a 2D graph with variables
$\Delta\mu$(Tl2Te3) and $\Delta\mu$(Cd) as shown in Fig. 2(a).

The competing phases such as CdTlTe2, TlTe, Tl5Te3, CdTe and Te are
considered. Their formation energies EF(CdlTlmTen) with respect to the
precursors are calculated and listed in Table 2. Then the phase diagram of
CdTl4Te3 as function of the chemical potentials $\Delta\mu$(Cd),
$\Delta\mu$(Tl2Te3) and $\Delta\mu$(Tl) is constructed in Fig. 2(a) by using a
general scheme IntegratedFerroelectrics.175.186 ; ma13194303 ; RN22 . More
details are described in supplementary material (SM) SM . Yielding to the
constraint $\Delta\mu$(i) = 0 $\sim$ -1.032 eV, the whole allowed chemical
potential region is restricted in the triangle surrounded by $\Delta\mu$(Cd)=0
(blue solid line), $\Delta\mu$(Tl)=0 (green solid line) and
$\Delta\mu$(Tl2Te3)=0 (black solid line). Our results reveal that CdTl4Te3 is
most stable against to other competing compounds in the orange region
manifested by the phase separation lines $\Delta\mu($Tl5Te3) = 0,
$\Delta\mu$(CdTlTe2) = 0, $\Delta\mu$(TlTe) = 0 and $\Delta\mu$(Te) = 0. Here,
$\Delta\mu$(CdlTlmTen) = $\mu$(CdlTlmTen) - EF(CdlTlmTen) is the chemical
potential of competing phase that is the function of precursor’s chemical
potentials. The competing phase CdlTlmTen will precipitate out at
$\Delta\mu$(CdlTlmTen) = 0, and become unstable with a negative value. These
results clearly demonstrate that CdTl4Te3 is easy to synthesize at Cd-rich,
Tl2Te3-rich and Tl-poor condition. We would like to recall that the abundance
of precursors is relative yielding to EF(CdTl4Te3) = -1.032 eV and Eq. 2. By
increasing $\Delta\mu$(Tl) along the black arrow in Fig. 2(a), it means that
Tl grows more and more rich while Cd and Tl2Te3 become poor. When the arrow
cross $\Delta\mu$(Tl5Te3)=0 and $\Delta\mu$(CdTlTe2)=0 lines, CdTl4Te3 becomes
unstable accompanied with the precipitation of Tl5Te3 and CdTlTe2, and the
following decomposition will take place,

$\begin{split}CdTl_{4}Te_{3}~{}&\rightarrow~{}Tl_{5}Te_{3}+3TlTe+2Cd\\\
CdTl_{4}Te_{3}~{}&\rightarrow~{}CdTlTe_{2}+TlTe+2Tl\\\ \end{split}$ (3)

According to our calculations, the energy of the right products is 0.234 eV
and 0.564 eV higher than that of left, respectively.

The convex hull analysis is another useful method to investigate the
thermodynamical stability JChemPhys.154.234706 ; PhysRevLett.123.097001 . In
this way, the formation energy of all possible atomic configurations with
respect to constituent elements needs to be calculated, which is defined as
$\widetilde{E}^{F}=E^{T}(Cd_{l}Tl_{m}Te_{n})-lE^{T}(Cd)-mE^{T}(Tl)-nE^{T}(Te)$.
Using this definition, our calculations demonstrate that CdTl4Te3 is
thermodynamically stable against to elements with formation energy of -0.213
eV/atom. With all possible $\widetilde{E}^{F}$ as listed in Table. 2, the
convex hull diagram is constructed in Fig. S1 SM , which shows that only the
binary compounds are on the convex hull, and all ternary compounds such as
CdTl4Te3 and CdTlTe2 are within a viable energy window for potentially
metastable phases. Even so, we estimate that CdTl4Te3 is potentially
synthesizable based on the following facts. One is that CdTl4Te3 is just a
little above the convex hull with small energy and even 0.020 eV/atom lower
than CdTlTe2. Since CdTlTe2 has already been synthesized in 1969
GUSEINOV1969807 , it is expected that CdTl4Te3 is highly feasible in proper
condition, especially at Cd-rich, Tl2Te3-rich and Tl-poor condition in the
reaction Eq. 2.

Table 1: Detailed crystallographic information of corresponding compounds used in Fig. 2. Compounds | Space group | $a,b,c$ |  | Atomic coordinates |
---|---|---|---|---|---
|  | Å |  | (fractional) |
Cd | $P6_{3}/mmc$ |  $a$=2.9179 | Cd |  $2$c $(\frac{1}{3}$ $\frac{2}{3}$ 0.25) |
|  |  $c$=5.3842 |  |  |
Tl | $P6_{3}/mmc$ |  $a$=3.4108 | Tl |  2$c$ ($\frac{1}{3}$ $\frac{2}{3}$ $0.25$) |
|  |  $c$=5.4269 |  |  |
Te | $P3_{1}21$ |  $a$=4.2786 | Te |  3$a$ ($0.2879$ $0$ $\frac{1}{3}$) |
|  |  $c$=5.9249 |  |  |
CdTe | $F-43m$ |  $a$=6.4082 | Cd |  4$a$ ($0.0$ $0.0$ $0$) |
|  |  | Te |  4$c$ (0.25 0.25 0.25) |
TlTe | $I4/mcm$ |  $a$=12.6445 | Tl | 16k (0.0783 0.2300 0.0) |
|  | $c=6.0775$ | Te1 |  8$h$ (0.1663 0.6663 0.25) |
|  |  | Te2 |  4$d$ (0.0 0.5 0.0) |
|  |  | Te3 |  4$a$ (0.0 0.0 0.25) |
Tl2Te3 | $C2/c$ |  $a$=12.8449 | Tl1 |  8$f$ (0.8947 0.6452 0.5521) |
| $\alpha=\gamma=90^{\circ}$ |  $b$=6.3989 | Te1 |  8$f$ (0.6817 0.6379 0.0860) |
| $\beta=144.9264^{\circ}$ |  $c$=13.0071 | Te2 |  4$e$ (0.0 0.8703 0.25) |
CdTlTe2 | $P-3m1$ |  $a$=4.2425 | Cd |  1$a$ (0.0 0.0 0.0) |
|  |  $c$=7.3925 | Tl |  1$b$ (0.0 0.0 0.5) |
|  |  | Te |  2$d$ ($\frac{1}{3}$ $\frac{2}{3}$ 0.2236) |
Tl5Te3 | $I4/mcm$ |  $a$=8.6689 | Tl1 |  4$c$ (0.5 0.5 0.0) |
|  | c=12.6155 | Tl2 |  16$l$ (0.1476 0.6476 0.1607) |
|  |  | Te1 |  4$a$ (0.5 0.5 0.25) |
|  |  | Te2 |  8$h$ (0.6565 0.1565 0.0) |
CdTl4Te3 | $I4/mcm$ | a=8.5120 | Cd |  4$c$ (0.5 0.5 0.0) |
|  |  $c$=12.2642 | Tl |  16$l$ (0.1452 0.6452 0.1651) |
|  |  | Te1 |  4$a$ (0.5 0.5 0.25) |
|  |  | Te2 |  8$h$ (0.6624 0.1624 0.0) |
HgTl4Te3 | $I4/mcm$ | a=8.6668 | Hg |  4$c$ (0.5 0.5 0.0) |
|  |  $c$=12.1854 | Tl |  16$l$ (0.1460 0.6460 0.1624) |
|  |  | Te1 |  4$a$ (0.5 0.5 0.25) |
|  |  | Te2 |  8$h$ (0.6599 0.1599 0.0) |
Table 2: The calculated energy/formula used in Fig. 2. Compounds | ET(eV) | EF(eV) | $\widetilde{\rm{E}}^{F}$(eV) |
---|---|---|---|---
Cd | -1.5037 |  |  |
Tl | -2.9543 |  |  |
Te | -3.8046 | 0.2237 |  |
Tl2Te3 | -17.9934 |  | -0.6709 |
CdTe | -6.0324 | -0.5004 | -0.7241 |
TlTe | -7.1281 | -0.1455 | -0.3691 |
Tl5Te3 | -28.0154 | -1.1591 | -1.8300 |
CdTlTe2 | -12.8369 | -0.3224 | -0.7697 |
CdTl4Te3 | -26.4372 | -1.0315 | -1.7025 |
Figure 2: (a) The ternary phase diagram of CdTl4Te3 with respect to the
chemical potentials of Cd, Tl2Te3 and Tl,
$\Delta\mu$(Tl)=$\frac{1}{2}$[EF(CTT)-$\Delta\mu$Tl2Te3-$\Delta\mu$Cd]. (b)
The Phonon dispersion curves of CdTl4Te3 without SOC.

To further check the dynamical stability of CdTl4Te3, the phonon spectrum
based on the body centered tetragonal phase with $I$4/$mcm$ symmetry is
calculated. As shown in Fig. 2(b), there is no phonon mode with negative
frequency in the entire BZ, which indicates that CdTl4Te3 is dynamically
stable by adopting the body centered tetragonal structure. The above
thermodynamical and dynamical investigations strongly demonstrate that
CdTl4Te3 is readily synthesized in experiment.

##  IV Electronic properties of CdTl4Te3

The projected density of states (PDOS) of tetragonal CdTl4Te3 are calculated
and plotted. As shown in Fig. 3(a), Tl-6s orbitals mainly contribute to the
states between -8 $\sim$ -3.5 eV, while Tl-6p orbitals mainly contribute to
the states above 0.3 eV. Considering the electronic configuration
$6s^{2}6p^{1}$ of Tl, we conclude that Tl favors +1 valence in CdTl4Te3,
similar to valence of Tl atoms at 16$l$ site in Tl5Te3 APLMaterials.3.041507 ;
PhysicsandChemistryofSolidState.21.492 . The states in energy range of -3.5
$\sim$ 0 eV are approximately from Te-5$p$ orbitals with admixing of Tl-6$p$
and Cd-5$s$ states, which implies that the bonding between Cd, Tl and Te is
not the pure ionic bond but has sizable metal-metal bond characters. Fig. 3(a)
shows that the Cd-5$s$ orbitals are almost empty and mainly contribute the
states between 0 $\sim$ 3 eV. Therefore, we can understand the electron
transfer roughly as follows. Each Tl atom donates one 6$p$ electron, and Cd
atom donates two 5$s$ electrons to the Te-5$p$ orbitals. As a result, CdTl4Te3
is close to the electronic configuration of the atomic insulator with the full
filled sub-shell of Te2-, Tl1+, Cd2+ ions approximately.

However, we notice that the Cd-5$s$ orbitals are very extended, which also
exhibit considerable amplitude under the EF, implying a band inversion between
Cd-5$s$ and Te-5$p$ states. The character is further verified by the projected
band structures in Fig. 3(b), which clearly demonstrates that the Cd-5$s$
states with even parity are lower 1.82 eV than Te-5$p$ states with odd parity
at $\Gamma$ point. The band inversion in CdTl4Te3 is already occurred even
without SOC and can be alternatively viewed as a consequence of the inert pair
effect in chemistry, which is the propensity for the two electrons in the
outermost 5$s$ orbital to remain unionized in heavier elements
narang2021topology , just like that in HgTe science.1133734 ;
PhysRevB.74.085308 . In Fig. 3(b), when SOC is excluded, the band crossing
points between Cd-5$s$ and Te-5$p$ can be protected by time reversal symmetry
($\mathcal{TRS}$) and inversion symmetry ($\mathcal{I}$) symmetry and form
nodal rings as plotted in Fig. S2(a) SM . Since LDA type exchange-correlation
potential usually overestimates band inversion between valence and conduction
bands, then MBJ PhysRevLett.102.226401  is employed. The amplitude of band
inversion in LDA is reduced to 1.10 eV with MBJ calculations as shown in Fig.
S2(b) SM . Therefore, the band inversion in CdTl4Te3 is robust against the
functional potentials, and the more accurate calculation of LDA with MBJ
semilocal exchange functional potential is adopted to investigate the
electronic and topological properties in the following.

When SOC is considered, the nodal rings are all gapped, inducing a 0.13 eV
band gap as shown in Fig. 3(c), which is larger than the energy scale of room
temperature theoretically. For insulators with $\mathcal{I}$, the topological
invariant $\nu_{0}$ based on the Fu-Kane formula PhysRevLett.98.106803  can be
characterized by the parity products ($\xi_{i}$) of the half numbers of the
occupied states at eight time-reversal invariant momentum (TRIM) points
(Kramers pairs have the same parities). As shown in Fig. 1(b), there are one
$\Gamma$ (0.0, 0.0, 0.0), one $Z$ (0.5, 0.5, -0.5), two $X$ (0.0, 0.0, 0.5)
and four $N$ (0.5, 0.0, 0.0) TRIM points in the first BZ. Therefore, only
$\xi_{\Gamma}$ and $\xi_{Z}$ could determine the topological property of the
tetragonal CdTl4Te3, while the other TRIM points always give the trivial
products. Our calculations indicate $\xi_{\Gamma}=-1$ and $\xi_{Z}=1$, and
give rise to $\nu_{0}=1$. These results are consistent with the band inversion
analysis at $\Gamma$ point, and confirm that CdTl4Te3 is a strong TI.

Figure 3:  The electronic properties of CdTl4Te3. (a) The projected DOS
without SOC. (b) The fatted bands with spectral weight of Cd-5$s$(red),
Te-5$p$(green) and Tl-6$p$(blue) orbitals without SOC. (c) The band structures
of with MBJ by including SOC.  Figure 4: (a) The Wilson loops of k2k3 plane
(k1 = 0) and (b) (k1 = $\pi$) plane. (c) Band structures projected onto the
(100) surface. (d) Topological surface states with chemical potential at 0.02
eV and corresponding spin texture on the (100) surface.

##  V Topological properties of CdTl4Te3

We construct the maximally localized Wannier functions of Cd-5$s$, Te-5$p$ and
Tl-6$p$ states to investigate the topological features more explicitly. The
Wilson loop method PhysRevB.84.075119  is used by calculating the evolution of
Wannier charge centers for the occupied bands in k1 = 0 (Fig. 4(a)) and
k${}_{1}=\pi$ planes (Fig. 4(b)). The evolution lines cross the reference
line(red dashed line) once time in Fig. 4(a), indicating that the k1 = 0 plane
corresponds to a quantum spin Hall system with non-trivial 2D topological
invariant. The evolution lines cross the reference line(red dashed line) zero
time in Fig. 4(b), confirming that the k1 = $\pi$ plane is a trivial 2D
system. These results combining with Wilson loops on other surface planes(Fig.
S3) SM , give rise a complete topological index $\mathbb{Z}_{2}$=(1;000),
which further confirm that CdTl4Te3 fall into the strong TI phase. In Fig.
4(c), we plot the surface electronic structures on the (100) surface by the
iterative surface Green’s function method JPhysFMetPhys.14.1205 ;
JPhysFMetPhys.15.851 . Two robust surface states connect the valence and
conduction bands and form a Dirac cone in the bulk gap at $\bar{\Gamma}$ point
due to the requirement of the $\mathcal{TRS}$. In Fig. 4(d), we plot the Fermi
surfaces of the Dirac cone at 0.02 eV and their spin orientation, which
exhibits a left-hand spin texture enclosed a $\pi$ phase like that in Bi2Se3
zhang2009topological , indicating a positive SOC in CdTl4Te3 Nie10596 . Such
kind of spin momentum locking surface states have been reported that have very
highly efficient performance on spin torque equipment Mellnik2014 ; RN37  and
spin device zhang2009topological ; hsieh2009tunable .

##  VI Discussion and Conclusion

Considering that Hg is isoelectronic with Cd in the IIB group of the periodic
table, HgTl4Te3 is naturally expected to be stabilized into the same crystal
structure of CdTl4Te3 with similar electronic structures and topological
property. By using the optimized structure parameters in Table. 1, the
calculated formation energy of HgTl4Te3 through the reaction Hg + 4Tl + 3Te
$\rightarrow$ HgTl4Te3 is -1.459 eV with respect to elemental precursors.
Limited by the stoichiometry, the products $\frac{1}{2}$Tl5Te3 \+ HgTe +
$\frac{1}{2}$TlTe + Tl, Tl2Te3 \+ Hg + 2Tl, 3TlTe + Hg + Tl and 2TlTe + HgTe +
2Tl are taken into account by calculating the formation energy which are equal
to -1.397, -0.671, -1.107, and -1.035 eV, respectively. Obviously, HgTl4Te3
will be formed in the reaction because it is thermodynamically favourable. The
MBJ calculated band structures are presented in Fig. S4 SM . As expected, the
Hg-5$s$ state with even parity is lower 2.26 eV than Te-5$p$ state with odd
parity at $\Gamma$ point. Furthermore, 0.046 eV nontrivial band gap induced by
SOC makes it a strong TI which is similar to CdTl4Te3.

In conclusion, we predict a new class of 3D topological insulators $M$Tl4Te3
($M$ = Cd, Hg) by using first-principles calculations. Our ternary chemical
potential phase diagrams and phonon spectrum calculations demonstrate that
$M$Tl4Te3 ($M$ = Cd, Hg) are both thermodynamically and dynamically stable in
the body centered tetragonal crystal structure with $I$4/$mcm$ symmetry.
Further electronic structures calculations confirm that the nontrivial band
topology stems from the band inversion between $M$-5$s$ and Te-5$p$ orbitals
at $\Gamma$ point, and the SOC induced topological band gap is about 0.13 eV
in CdTl4Te3 which is larger than the energy scale of room temperature in
theory. The isolated Dirac cone type surface states with left-hand helicity of
the spin momentum locking texture is obtained in the (100) surface spectra at
$\bar{\Gamma}$ point. These results suggest that $M$Tl4Te3 are synthesizable
and suitable for the study of efficient spin torque equipment and spin device,
which should stimulate many experimental efforts in the future.

##  VII Acknowledgments

This work was supported by the National Key Research and Development Program
of China (2018YFA0307000), and the National Natural Science Foundation of
China (11874022).

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