# How does chemical functionalization affect the thermal transport of antimony
films?

Tian Zhang1 Yuan-Yuan Qi2 Xiang-Rong Chen1<EMAIL_ADDRESS>Ling-Cang Cai3
1Institute of Atomic and Molecular Physics, College of Physical Science and
Technology, Sichuan University, Chengdu 610065, China
2College of Science, Henan University of Technology, Zhengzhou 450001, China
3National Key Laboratory for Shock Wave and Detonation Physics Research,
Institute of Fluid Physics, Chinese Academy of Engineering Physics, Mianyang
621900, China

(June 26, 2016)

###### Abstract

Chemical functionalization is an effective means to tune electronic and
crystal structure of two-dimensional material, but very little is known
regarding its correlation with thermal transport. Based on the first-principle
calculation and an iterative solution of Boltzmann transport equation, we find
that antimony films are potential excellent thermoelectrical materials with
rather low thermal conductivities $k$ ($<$ 2.5 W/mK), and chemical
functionalization can induce the reduction in $k$ to some extent, which is
mainly due to the reduction of phonon lifetimes limited by the anharmonic
scattering. More interesting, the origin of the reduction in $k$ is not the
anharmonic interaction but the harmonic interaction from the depressed phonon
spectrum mechanism, and for some chemical functional atom in halogen, flat
modes appearing in the low frequency range play also a key factor in the
reduction of $k$ by significantly increasing the three-phonon scattering
channels. Our work provides a new view to adjust the thermal transport which
can benefit thermoelectric material design, and analyzes the reduction
mechanism in $k$ from the chemical functionalization for antimonene.

###### pacs:

66.70.Df, 63.20.dk, 63.20.kg

Two-dimensional (2D) materials have been a focus of intense researchTW1 ; TW2
; TW4  since the discovery of graphene with high electron mobilityAJ1 , and
very high thermal conductivityAJ2 ; AJ3 . More 2D materials, such as silicene,
phosphorene, and transition metal dichalcogenides, are now considered for
various practical usages due to their distinguished properties. Recently, a
new and promising 2D semiconductor, antimonene in the same group of the
periodic table with phosphorene, was proposed on the basis of first-principles
calculationsan1 . Its isolation was expected to be achieved by exfoliating
layered Sb crystal, which is due to the weak interlayer interaction. Similar
to silicene, antimonene has also a non-planar honeycomb-like structure, and
its electronic structure can also show the topological character under
external strainto1 . The character of topological insulator for antimonene is
useful to thermoelectrical application with perfectly conducting channels for
electrons, thanks to its non-dissipative edge statestd2 . A high
thermoelectric performance requires the system to be a bad conductor for
phonon but a good conductor for electron. Compared to other widely studied 2D
materials, the thermal transport properties of monolayer antimonene is still
not well-understood.

Generally both electrons and phonons contribute to the total thermal
conductivity $k$ in nonmagnetic crystalep1 , with phonons dominating in
semiconductor and insulators. Electron transport can be optimized by adjusting
the geometry size to maximize the contribution of non-dissipative edge states
for 2D topological insulators, and phonon transport is usually be tuned by
external pressurepre2 ; pre3 ; pre4 . The theory of Liebfried and
Schlömann(LS)LS  have typically been approved for the pressure dependence of
$k$ in nonmetallic crystalline materials, which predicts that $k$ must
increase with $P$kp , i.e., $dk/dP$ $>$ $0$. Filler is also excellent method
to adjust phonon transport for 3D materialsst ; st1 , and some relative
theories have also been raised, such as a rattling mechanism postulated by
Slack and Tsoukala to explain the reduction of $k$ in filled skutteruditesst ,
which implies the fillers move independently from the host matrix (Einstein
motion) and effectively scatter phonons. Like to filler, the chemical
functionalization is also a universal way to tune mechanical, electronic and
other properties for 2D materials. Furthermore, the chemical functionalization
can even induce the topological phase transition, which is found by Xu $et$
$al$XU . However, the influence for the chemical functionalization and
corresponding mechanism in thermal transport properties are still
uninvestigated.

Figure 1: (Color online) Top and side view of (a) the antimonene structure and
(b) the decorated antimonene structure. $X$ represents the chemical functional
group. (c) The calculated lattice thermal conductivities $k$ for antimonene
and decorated antimonene.

In this work, we calculate the lattice thermal conductivities $k$ of antimony
films using the first-principle and an iterative solution of Boltzmann
transport equation (BTE) for phononsBTE . The rather low $k$ ($<$ 2.5 W/mK)
are found in these antimony films, which is very favorable to realize high
thermoelectrical efficiency. More interesting, the chemical functionalization
can reduce the value of $k$ compared with the antimonene to some extent, and
the $k$ becomes lower and lower along with the mass increase of chemical
functional element . We make a detailed discussion for this difference in $k$,
and find that the anharmonic third-order interatomic force constants play a
trivial role. This difference can be due to the depressed phonon spectrum by
analyzing the weighted phase space $W$, and the flat phonon modes is also a
key factor in the reduction of $k$ by increasing more scattering channels when
the chemical functional atom is in halogen.

The first-principle calculations based on the density functional theory are
performed using the Vienna ab simulation package (VASP)vasp1 ; vasp2  . We use
the generalized gradient approximations (GGA) of Perdew-Burke-Ernzerhof
(PBE)PBE  for electron-electron interactions and the projector-augmented-wave
(PAW)PAW  pseudo-potentials in the plane-wave basis with an energy cutoff of
700 eV. The Brillouin zone (BZ) was sampled using a 13$\times$13$\times$1
gamma-centered Monkhorst-Pack gridMK  during structural relaxation for the
unit cell. A slab model, together with a vacuum layer larger than 30 Å, was
employed to avoid spurious interactions due to the nonlocal nature of the
correlation energyslab .

Thermal conductivity of Sb monolayer is calculated using BTE with relaxation
time approximation as implemented in ShengBTE packageBTE , where thermal
conductivity is given bytermal

$k_{\alpha}=1/N_{q}V\sum\limits_{q,j}C_{q,j}\upsilon^{2}_{q,j,\alpha}\tau_{q,j},$
(1)

where $N_{q}$ is the number of uniformly spaced q points in the Brillouin
zone, $V$ is the volume of the unit cell, $C_{q,j}$, $\upsilon_{q,j,\alpha}$
and $\tau_{q,j}$ are special heat, group velocity along transport direction
$\alpha$, and relaxation time of the phonon mode with wave vector q and
polarization $j$, respectively. In the calculation of phonon dispersion, the
harmonic second-order IFCs are obtained using density functional perturbation
theory (DFPT) as implemented in the PHONOPY codeFP  with 5$\times$5$\times$1
k-mesh. The ShengBTE package is used to obtain the anharmonic third-order
IFCs, and solve the BTE which needs harmonic second- and anharmonic third-
order interatomic force constants (IFCs) as inputs. The ShengBTE package is
completely parameter-free and based only on the information of the chemical
structure. Full solution of BTE is also obtained by using an iterative scheme,
and it is found that the relaxation time approximation (RTA) has a very good
accuracy.

The optimal lattice structures of antimonene monolayer and decorated
antimonene monolayer (denoted as Sb and SbX) are shown in Fig. 1(a) and 1(b),
respectively, where X represents the chemical functional group. The lattice
constant is 4.139 Å for Sb, slightly longer than other theoretical resultsan1
; zhao1 , 5.279 Å for SbH, 5.144 Å for SbF and 5.245 Å for SbBr. Compared to
Sb, in SbX the Sb-Sb bond length slightly increases, and interestingly,
different from the low-buckled configuration for the 2D Sb, the chemical
functionalization of the buckled Sb induces its structure to transition into
an extremely 2D sheet [see Fig. 1(b)], which is similar to the phenomenon in
fully hydrogenated arseneneAS . Fig. 1(c) shows the obtained lattice thermal
conductivities $k$ for all antimony films based on the iterative solution at
300 K. The $k$ of these antimony films are universally low, which is very
favorable to realize high thermoelectrical efficiency, especially as
topological insulator under external strainto1 . The $k$ about 2.258 W/mK for
Sb is small compared with the bulk Sb ($\sim$ 10-60 W/mK for 50-300 KBULK ).
This is reasonable because phonons are more sensitive to boundary scattering
when the thickness of the Sb thin film decreases, and thus further reduce $k$.
In order to check the reliability of the $k$ for Sb, we change the supercell
size from 3$\times$3$\times$1 to 5$\times$5$\times$1 to obtain more accurate
anharmonic third-order IFCs, and simultaneously reexamine the harmonic second-
order IFCs within density functional perturbation theory using the QUANTUM
ESPRESSO package (QE)QE1 ; QE2 . Results indicate that $k$ is not affected
highly by the supercell size with error lower than 0.1 W/mK, and the error of
$k$ is about 0.4 W/mK between VASP and QE packages. We make also a comparison
with other 2D hexagonal materials, and their $k$ are as follows: MoS2
$\approx$ 4-8 W/mKMOS2 , blue phosphorene $\approx$ 78 W/mKbp , silicene
$\approx$ 7-11 W/mKsi , stanene $\approx$ 11.6 W/mKsn and arsenene $\approx$
7.8 W/mK (armchair direction)as . Although the obtained $k$ for Sb is smaller
than those for other same group (blue phosphorene and arsenene), this
discrepancy can be accepted because the larger mass usually leads to the lower
$k$ based on LS relationLS .

Figure 2: (Color online) Phonon spectrum $\omega(\vec{k})$ of (a) the
antimonene Sb. ZA and TA are the transverse acoustic phonon branches, and LA
is the longitudinal acoustic phonon branch. G, M and K refer to special points
in the first Brillouin zone. (b) The enlarged view of phonon spectrums for the
antimonene Sb and decorated antimonene SbH. (c) The phonon density of states
(PDOS) for SbH, SbF and SbBr.

An interesting result is found in Fig. 1(c) that the $k$ values for all SbX
films are lower than that for Sb, and the $k$ becomes lower and lower along
with the mass increase of chemical functional element. Phonon spectrums and
phonon density of states (PDOS) of the antimony films are presented in Fig. 2,
where the phonon spectrum of the Sb is well consistent with that by Wang $et$
$al$wang . All lattice structures are thermodynamically stable and their
stability do not depend on the substrates. It is noted that the low frequency
range has a less partial PDOS of H atoms, and the high frequency range is
dominated by the vibrations of H atoms. However, along with the mass increase
of chemical functional element, the partial PDOS of chemical functional atoms
are more and more in the low frequency range, and them become even the
dominant ingredients for SbBr, as shown in Fig. 2(c). Hence, next we will
elucidate the role of the chemical functionalization in the reduction of $k$
by considering conventional anharmonic scattering along.

As we know, $C_{\nu}$, $\upsilon_{\lambda}$ and $\tau_{\lambda}$ can influence
the value of $k$. For most of the heat conductive modes $C_{\nu}$ approaches
the classic value $k_{B}$ at room temperature, and thus the $C_{\nu}$ can not
influence $k$ value although the more number of phonon modes may lead to the
total heat capacity increase slightly. We make a particular comparison for
$\upsilon_{\lambda}$ between Sb and SbH in low frequency phonon dispersion
range, as shown in Fig. 2(b). The calculated speed of sound along the G-M
direction in the Sb, $\upsilon_{LA}$ =$7.5$ Km/s, is higher about 1.42 time
than that value of 5.3 Km/s for SbH, and the values of $\upsilon_{\lambda}$
for remainder two acoustic branches are also higher slightly for Sb than those
for SbH. The difference of $\upsilon_{\lambda}$ is so small to explain the
reduction of $k$. Hence we believe that $\tau_{\lambda}$ should be the main
factor for the decrease of $k$ in all SbX, and thus make a detailed discussion
about their anharmonic scattering.

Figure 3: (Color online) (a) The normalized cumulative $k$ for the Sb and SbH
monolayers. (b) Anharmonic scattering rates SRs of the Sb and SbH monolayers
in the frequency range of three acoustic branches. (c) Weighted phase space
$W$, defined in Eq. (4) and (5), vs angular frequency for the Sb and SbH
monolayers.

We neglect the isotope scattering and only take into account the anharmonic
scattering for simplicity sake. The total scattering rate is equal to the
inversion of the phonon lifetime $\tau_{\lambda}$, which can be obtained as
the sum of three-phonon transition probabilities
$\Gamma^{\pm}_{\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}$.
$\Gamma^{\pm}_{\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}$ can
be expressed ast1 ; t2 ; t3

$\Gamma^{\pm}_{\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}=\left(\begin{array}[]{c}f_{\lambda^{{}^{\prime}}}-f_{\lambda^{{}^{\prime\prime}}}\\\
f_{\lambda^{{}^{\prime}}}+f_{\lambda^{{}^{\prime\prime}}}+1\\\
\end{array}\right)\frac{h\pi\delta(\omega_{\lambda}+\omega_{\lambda^{{}^{\prime}}}-\omega_{\lambda^{{}^{\prime\prime}}})}{4\omega_{\lambda}\omega_{\lambda^{{}^{\prime}}}\omega_{\lambda^{{}^{\prime\prime}}}}|V^{\pm}_{\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}|^{2},$
(2)

where $\Gamma^{+}_{\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}$
corresponds to absorption processes,
$\Gamma^{-}_{\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}$
describes emission processes, and $\delta$ function represents the energy
conservation.
$V^{\pm}_{\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}$ is the
scattering matrix elements, which depends the anhamonic third-order IFCs
$\phi^{\alpha\beta\gamma}_{ijk}$ t3 as

$V^{\pm}_{\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}=\sum\limits_{i\in
u.c.}\sum\limits_{j,k}\sum\limits_{\alpha\beta\gamma}\phi^{\alpha\beta\gamma}_{ijk}\frac{e^{\alpha}_{\lambda}(i)e^{\beta}_{p^{{}^{\prime}},\pm
q^{{}^{\prime}}}(j)e^{\gamma}_{p^{{}^{\prime\prime}},-q^{{}^{\prime\prime}}}(k)}{\sqrt{M_{i}M_{j}M_{k}}},$
(3)

where $M_{i}$ is the mass of atom $i$, and $e^{\alpha}_{\lambda}(i)$ is the
$\alpha$ component of the eigenfunction of mode $\lambda$ at the $i$th atom.
The contributions from different phonon branches to $k$ for Sb and SbH are
analyzed. Like to the case in graphene, the total contribution of acoustic
branches is more than 99$\%$ for Sb, implying that the contribution of optic
phonons to $k$ is negligible. The contribution of LA is more than 57$\%$ and
the contribution of ZA is less than 13$\%$, which is completely opposite
compared with graphene (8$\%$ LA and 80$\%$ ZA) but is very similar to that in
stanenesn . This especial phenomenon in graphene is due to a symmetry
selection rule arising from the reflection symmetry perpendicular to the
graphene plane, which strongly restricts the participation of odd number of ZA
phonons in three-phonon processes, e.g. ZA+ZA$\longleftrightarrow$ZA,
ZA+LA/TA$\longleftrightarrow$LA/TA. However, the reflection symmetry in Sb is
broken, which means that the selection rule does not apply. It is also not
suited in SbH, and thus the contribution of TA to the $k$ is still little ($<$
17$\%$). The normalized cumulative $k$ for the Sb and SbH, which is the
fraction of heat conducted by the phonons with frequencies below a given
value, is shown in Fig. 3(a). The normalized cumulative $k$ in the two cases
are quite similar in low frequency range below 10 rad/ps. These phonons below
10 rad/ps contribute almost whole of the $k$ for Sb, and the phonons below 20
rad/ps contribute the most of $k$. Thus, next we need only to analyze phonon
scattering rates (SRs) in low frequency range to look into the effect of
$\tau_{\lambda}$ to $k$.

The SRs, $\tau_{\lambda}^{-1}$, is plotted in Fig. 3(b) versus
$\omega_{\lambda}$ for Sb and SbH. It is obvious that the SRs for SbH is
nearly larger than that for Sb in whole low frequency range, roughly by a
factor of $\sim$ 6.0 in all range (not shown). This confirms that the
reduction of $\tau_{\lambda}$ is the main reason for the reduced $k$ in SbH.
The overall reduction of $\tau_{\lambda}$ and the main PDOS of H atom beyond
60 rad/ps in SbH overturns the resonant scattering theoryrest , which
indicates that $\tau$ is reduced in a narrow frequency range i.e. the
frequency of the guest atom vibrational mode. Anharmonic scattering arises
mainly from the anharmonic interaction between atoms, and thus the additional
anharmonic interaction introduced by the guest atom (H atom) may be the main
reason for the reduction of $\tau_{\lambda}$. Here the H-related anharmonic
interaction will not affect low frequency phonons below 15 rad/ps, because
these vibrations do not involve H atoms observed by PDOS in Fig. 2(c). In
additional, the harmonic second-order IFCs of Sb and SbH have not an excellent
transferability and we do not interchange the anharmonic third-order IFCs to
investigate whether the change of SRs comes from the anharmonic third-order
IFCs. Hence, we can only research the contribution of the harmonic interaction
described by harmonic second-order IFCs to the reduction in SRs. Weighted
phase space $W$ provide by Li $et$ $al$st1  is the sum of frequency-containing
factors in the expression of transition probabilities, where
$V^{\pm}_{\lambda\lambda^{{}^{\prime}}\lambda^{{}^{\prime\prime}}}$ is assumed
unchange in decorated antimonene films, written asst1

$W^{\pm}=\sum\limits_{\lambda^{{}^{\prime}}p^{{}^{\prime\prime}}}\left(\begin{array}[]{c}f_{\lambda^{{}^{\prime}}}-f_{\lambda^{{}^{\prime\prime}}}\\\
f_{\lambda^{{}^{\prime}}}+f_{\lambda^{{}^{\prime\prime}}}+1\\\
\end{array}\right)\frac{\delta(\omega_{\lambda}+\omega_{\lambda^{{}^{\prime}}}-\omega_{\lambda^{{}^{\prime\prime}}})}{\omega_{\lambda}\omega_{\lambda^{{}^{\prime}}}\omega_{\lambda^{{}^{\prime\prime}}}},$
(4)

$W=W^{+}+W^{-}.$ (5)

$W$ is highly sensitive to the change of phonon spectrum, i.e. harmonic
second-order IFCs, and the harmonic second-order IFCs can affect indirectly
the SRs by the frequencies and the eigenvectors. Hence, $W$ is an excellent
indicator to assessment the contribution to SRs from harmonic interaction. The
calculated $W$ for Sb and SbH is plotted in Fig. 3(c). Almost for the whole
low frequency range below 15 rad/ps the $W$ is apparently higher for SbH than
that in Sb. Moreover, the difference in $W$ is well similar the difference in
SRs for SbH with a similar factor of $\sim$ 5.9 on average. The agreement of
the difference in $W$ and in $\tau^{-1}$ suggests that the change in frequency
indeed accounts for the decrease of $\tau$. When the external temperature is
much higher than the Debye temperature $\Theta_{D}$ ($\Theta_{D}$ for all SbX
is $<$ 193 K), $f_{\lambda}$ in Eq. (4) can be approximated by
$\frac{1}{exp(\hbar\omega_{\lambda}/k_{B}T)-1}$ $\approx$
$k_{B}T/\hbar\omega_{\lambda}$, and thus the $W$ is $\sim$
$\omega_{\lambda}^{-4}$. If the whole phonon spectrum is scaled by a factor of
$c$, then $W$ for all modes are scaled by a factor of $c^{-5}$. We choose the
reduction proportion of low phonon frequency nearby gamma point as the value
of $c$, i.e. the ratio of $\upsilon_{LA}$, which is reliable because the set
value of $c$ $\sim$ 1.13 in other work is very close to the ratio of group
velocity $\sim$ 1.1st1 . From the previous discussion, the phonon spectrum of
Sb is about 1.42 (c=1/1.42) times for that in SbH, and thus we have
$c^{-5}$=5.77, which agrees with the difference of the $W$ observed in Fig.
3(c). This result makes us believe that the depressed phonon spectrum is the
main origin of the increase of scattering in SbH.

Figure 4: (Color online) (a) The phonon spectrum and (b) the cumulative $k$ of
the SbBr monolayer. The green shadowed area makes the flat modes. G, M and K
refer to special points in the first Brillouin zone. The scale of ordinate in
(b) is magnified 100 times. (c) Anharmonic scattering rates SRs of the SbBr
and SbH monolayers compared with those of SbH monolayers calculated by using
the anharmonic third IFCs of SbBr monolayer vs angular frequency.

Based on the discussion in the PDOS of the antimony films [Fig. 2(c)], we know
that the distribution of PDOS and the phonon spectrum for SbH are enormous
different from those in SbX (X=F and Br). We may guess whether the effect of
the chemical functional atom in halogen to the reduction in $k$ is different
to that in the H atom, and thus make a specific comparison between SbH and
SbBr. The phonon spectrum of SbBr is plotted in Fig. 4(a). Unlike to SbH, the
SbBr displays noticeably flat modes in the frequency range from 0.2 to 0.6
THz. Furthermore, the cumulative $k$ for SbBr is also highly different from
those in Sb and SbH, as shown in Fig. 4(b). The curve of SbBr shows a
inflection point and begins to slower increase in the intermediate frequencies
range from 1.0 to 7.0 rad/ps, and in those frequency range the SRs have the
obvious enhancing, as shown in Fig. 4(c). The contributions of different
phonon branches are also calculated and we can find that the contribution of
optical branches between 20 to 30 rad/ps is unexpectedly large ($\sim$
55$\%$), which exceeds the value of 21$\%$ in three acoustic branches. The
especial phenomenon in the SbBr monolayer can be due to the unique flat modes,
which is dominated by the vibrations of guest atoms.

Some works11 ; YB  indicated that this flat modes might cause an increase in
phonon scattering, and used a conceptual diagram of phonon dispersion
relations to prove it. When the frequency of the guest modes ($E_{guest}$) is
larger than the gap frequency between the acoustic and upper-lying optical
phonon modes ($\Delta$), the Umklapp scattering can be one of important
processes in the suppression of thermal conductivity11 . The value of $\Delta$
here is much less than the value of $E_{guest}$, and the $E_{guest}$ about 0.4
THz is also a small value which can increase the number of phonons
contributing to the process. Based on these unique conditions, the process of
acoustic + optical $\rightarrow$ optical phonon transition is easily found in
a very wide area for SbBr, and thus the three-phonon channels are increased in
acoustic phonons. We also consider the contribution of the anharmonic third-
order IFcs to the difference (SRs or $k$) between SbBr and SbH. The $k$ of
SbBr with third-order IFCs of SbH is obtained about 0.103 W/mK, and the $k$ of
SbH with third-order IFCs of SbBr is obtained about 0.275 W/mK, which shows
that their anharmonic third-order IFCs are very similar, and can not be the
reason for the very different SRs or $k$. The SRs of SbBr is larger than those
of SbH roughly by a factor of $\sim$ 3.0 and the obtained $c^{-5}$ is about
1.93, which indicates that the depressed phonon spectrum mechanism also play a
role in the low $k$ for SbBr and the remained difference is mainly from this
flat modes. For the SbF, its phonon spectrum is very similar to that for SbBr,
and its depressed ratio of phonon spectrum to that for SbH are obtained about
-1.1 (negative represents lift). Because the special flat modes in the SbF can
significantly increase the three-phonon scattering channels, and further
reduce $k$ to seme extent compared to SbH. Hence, it is reasonable that the
$k$ of SbF is comparative with that for SbH monolayer.

The reduction mechanism of the chemical functionalization on $k$ established
above is similar to that from the filler in 3D materialsst1 ; YB . F (light
element) and Br (heaven element) atoms are the two typical halogen elements,
and their phonon spectrums of SbX (X in halogen) are also very similar. Thus
we believe that this reduction mechanism of the chemical functionalization on
$k$ is not special for the studied decorated antimonene here but a common
feature.

In summary, the thermal conductivities $k$ of antimonene Sb and decorated
antimonene SbX have been predicted based on $ab$ $initio$ Boltzmann transport
equation method. The values of $k$ for these antimony films are low compared
with other 2D hexagonal structures, which is very favorable to realize high
thermoelectrical efficiency and be a promising candidate for next-generation
thermoelectrical devices. We have found that the chemical functionalization
can induce the reduction of $k$ for the antimony films, and proven that this
change is mainly from the anharmonic scattering rates SRs, and comes minority
from group velocity. The anharmonic third-order interatomic force constants is
not the origin of different SRs but the depressed phonon spectrum by analyzing
the weighted phase space $W$. In additional, the flat phonon modes is also a
key factor in the reduction of $k$ by increasing more scattering channels when
the chemical functional atom is in halogen. Our works prove that antimony
films are excellent thermoelectric materials, and provide a new means to
adjust the $k$ which can benefit materials design.

The work was supported by the NSAF Joint Fund Jointly set up by the National
Natural Science Foundation of China and the Chinese Academy of Engineering
Physics (Grant Nos. U1430117, U1230201).

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