††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: These authors contributed
equally to this work.

#  Machine learning for predicting ultralow thermal conductivity and high ZT
in complex thermoelectric materials

Yuzhou Hao  State Key Laboratory for Mechanical Behavior of Materials, School
of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an 710049,
China  Yuting Zuo  State Key Laboratory for Mechanical Behavior of Materials,
School of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an
710049, China  Jiongzhi Zheng<EMAIL_ADDRESS>Thayer School of
Engineering, Dartmouth College, Hanover, New Hampshire, 03755, USA  Department
of Mechanical and Aerospace Engineering, The Hong Kong University of Science
and Technology, Clear Water Bay, Kowloon, 999077, Hong Kong  Wenjie Hou
Institute of Fundamental and Frontier Sciences, University of Electronic
Science and Technology of China, Chengdu 610054, China  Taizhou Key Laboratory
of Minimally Invasive Interventional Therapy and Artificial Intelligence,
Taizhou, 317502, China  Hong Gu  Suzhou Laboratory, No.388, Ruoshui Street,
SIP, Jiangsu, 215123, China  Xiaoying Wang  State Key Laboratory for
Mechanical Behavior of Materials, School of Materials Science and Engineering,
Xi’an Jiaotong University, Xi’an 710049, China  Xuejie Li  State Key
Laboratory for Mechanical Behavior of Materials, School of Materials Science
and Engineering, Xi’an Jiaotong University, Xi’an 710049, China  Jun Sun
State Key Laboratory for Mechanical Behavior of Materials, School of Materials
Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China
Xiangdong Ding  State Key Laboratory for Mechanical Behavior of Materials,
School of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an
710049, China  Zhibin Gao<EMAIL_ADDRESS>State Key Laboratory for
Mechanical Behavior of Materials, School of Materials Science and Engineering,
Xi’an Jiaotong University, Xi’an 710049, China

###### Abstract

Efficient and precise calculations of thermal transport properties and figure
of merit, alongside a deep comprehension of thermal transport mechanisms, are
essential for the practical utilization of advanced thermoelectric materials.
In this study, we explore the microscopic processes governing thermal
transport in the distinguished crystalline material Tl9SbTe6 by integrating a
unified thermal transport theory with machine learning-assisted self-
consistent phonon calculations. Leveraging machine learning potentials, we
expedite the analysis of phonon energy shifts, higher-order scattering
mechanisms, and thermal conductivity arising from various contributing factors
like population and coherence channels. Our finding unveils an exceptionally
low thermal conductivity of 0.31 W m-1 K-1 at room temperature, a result that
closely correlates with experimental observations. Notably, we observe that
the off-diagonal terms of heat flux operators play a significant role in
shaping the overall lattice thermal conductivity of Tl9SbTe6, where the
ultralow thermal conductivity resembles that of glass due to limited group
velocities. Furthermore, we achieve a maximum $ZT$ value of 3.17 in the
$c$-axis orientation for p-type Tl9SbTe6 at 600 K, and an optimal $ZT$ value
of 2.26 in the $a$-axis and $b$-axis direction for n-type Tl9SbTe6 at 500 K.
The crystalline Tl9SbTe6 not only showcases remarkable thermal insulation but
also demonstrates impressive electrical properties owing to the dual-
degeneracy phenomenon within its valence band. These results not only
elucidate the underlying reasons for the exceptional thermoelectric
performance of Tl9SbTe6 but also suggest potential avenues for further
experimental exploration.

##  I 1\. Introduction

Thermoelectric materials have the capacity to transform heat into electrical
energy without emitting pollutants Snyder and Toberer (2008); Zebarjadi  _et
al._ (2012). They lack movable components, generate no noise pollution, and
are adjustable in size, rendering them indispensable in a multitude of fields,
including waste heat recovery Baxter  _et al._ (2009) and photovoltaic-
photothermal cogeneration Dallan  _et al._ (2015). The conversion efficiency
of thermoelectric materials can be determined by the figure of merit
$ZT=S^{2}\sigma/(\kappa_{e}+\kappa_{L})$, where $S$, $\sigma$, $\kappa_{e}$
and $\kappa_{L}$ are the Seebeck coefficient, electrical conductivity,
electrical thermal conductivity and lattice thermal conductivity,
respectively. In order to achieve optimal thermoelectric performance, a
promising strategy involves enhancing electrical transport while
simultaneously suppressing thermal transport Nolas  _et al._ (2010).
Nevertheless, it is challenging to achieve high electrical conductivity
($\sigma$) and low electronic thermal conductivity ($\kappa_{e}$)
simultaneously. Consequently, ultralow lattice thermal conductivity
($\kappa_{L}$) becomes a crucial factor contributing to high thermoelectric
efficiency.

Recent studies have identified several materials with relatively complex
structures, including Cu12Sb4S13 Xia  _et al._ (2020a), AgPbBiSe3 Dutta  _et
al._ (2019), Cs2BiAgBr6 Haque and Hossain (2018), KCu5Se3 Li  _et al._ (2023),
and Cu7PS6 Shen  _et al._ (2024). These materials exhibit potential for
thermoelectric applications due to their ability to achieve ultralow thermal
conductivity ($\kappa_{L}$). Therefore, research on complex thermoelectric
materials is not only of fundamental interest but also of crucial importance
for the advancement of thermoelectric technology.

The Tl-Te system contains numerous complex materials with ultralow
$\kappa_{L}$. For instance, the $\kappa$ of Ag8Tl2Te5, Ag9TlTe5, Tl9CuTe5,
Tl2GeTe3, TlInTe2, etc. are all lower than 0.5 W m-1 K-1 at room temperature
Guo  _et al._ (2014); Yamanaka  _et al._ (2006). Moreover, the $\kappa$ of the
promising thermoelectric materials Tl9BiTe6 and Tl9SbTe6 was reported to be
0.5 W m-1 K-1 and 0.7 W m-1 K-1, respectively, at room temperature Guo  _et
al._ (2014). Although experiments have shown that Tl9BiTe6 exhibits lower
thermal conductivity, it also exhibits high resistivity, which significantly
reduces the power factor and constrains its further enhancement as an
additional component. Even with minimal Tl9BiTe6 incorporation, it can
markedly diminish the thermoelectric performance of the material Teubner
(2001).

Compound Tl9SbTe6 represents the most closely related material to Tl9BiTe6
within the Tl5Te3 family. Its thermal conductivity is comparable to that of
Tl9BiTe6, yet its electrical conductivity is markedly superior, as evidenced
by the experimental results Guo  _et al._ (2014). Although there have been
experimental studies on thermoelectric properties Guo  _et al._ (2013, 2014,
2015) and analysis of electronic structure Tao  _et al._ (2011), there is
still a lack of in-depth research on its low $\kappa$ and high thermoelectric
efficiency due to its complex structure. Further comprehensive research will
help us better understand and design excellent thermoelectric materials based
on the complex Tl-Te system.

Despite the significant advancements in measuring ultralow $\kappa$ in
experiments, theoretical prediction and explanation of ultralow $\kappa$ in
complex compounds still remain challenging. Recent studies have explored
anharmonic effects as potential sources of discrepancies of $\kappa$ between
the experiment and theory observed in highly anharmonic crystals Xia  _et al._
(2020b); Wang  _et al._ (2023a); Tadano and Tsuneyuki (2018); Xia  _et al._
(2020a); Zheng  _et al._ (2022a); Yue  _et al._ (2024); Zheng  _et al._
(2024). These studies have revealed that lattice anharmonicity significantly
influences the phonon linewidth and lattice thermal conductivity Tadano and
Tsuneyuki (2015). While the conventional Peierls-Boltzmann framework is
adequate for describing phonon transport in systems with well-defined phonon
modes, it encounters challenges in highly anharmonic systems Tadano and
Tsuneyuki (2015); Zhao  _et al._ (2021). Therefore, it becomes imperative to
consider the temperature-dependent frequencies evaluated by the self-
consistent phonon (SCPH) approach in elucidating thermal transport in complex
compounds Xia  _et al._ (2020c); Kang  _et al._ (2019).

Moreover, a discrepancy exists between the experimentally measured and
theoretically predicted $\kappa$ by the conventional Peierls-Boltzmann
transport equation Simoncelli  _et al._ (2019). Consequently, it becomes
necessary to consider the influence of wavelike interbranch tunneling of
coherence, originating from off-diagonal terms of the heat flux operator
Simoncelli  _et al._ (2019), to comprehensively explain this phenomenon.
Overall, to accurately predict thermal transport properties in complex
thermoelectric materials, the higher-order force constants, i.e., fourth-order
force constants, and coherent phonons are required to explicitly evaluate the
phonon energy shifts and scattering rates. Overall, to accurately predict
thermal transport properties in complex thermoelectric materials, it is
necessary to explicitly evaluate the phonon energy shifts and scattering rates
using the higher-order force constants i.e., fourth-order force constants, and
coherent phonons.

Despite the significant achievements in DFT-based thermal conductivity
calculations Li  _et al._ (2014), the computational expense of calculating
thermal conductivity considering higher-order phonon scatterings is so high
that it is often prohibitive Han  _et al._ (2022); Tong  _et al._ (2023). The
use of machine Learning Potential (MLP) can significantly reduce the cost of
calculating force constants while maintaining considerable computational
accuracy Mortazavi  _et al._ (2021). Machine learning potential (MLP) is a
potential that employs machine learning algorithms to describe the atomic
potential energy surface and predict the energy and force based on the atomic
environment. Unlike machine learning models (ML-Models) Wan  _et al._ (2019),
MLP can be combined with BTE methods to provide detailed phonon information,
thereby enabling a more comprehensive understanding of the mechanism of
thermal transport in the large system. Furthermore, it is capable of
calculating complex systems under a multitude of conditions, including those
involving defects, disorder, anharmonicity, and other complex systems that are
challenging to calculate using DFT methods.

Shapeev et al. developed another machine learning potential function, the
moment tensor potential (MTP) Novikov  _et al._ (2020); Shapeev (2016), which
achieves a good balance between computational accuracy and computational
complexity. MTP can obtain high-precision second-order and third-order force
constants. According to the previous research, MTP can also obtain reliable
fourth-order force constants, which can accurately calculate the $\kappa_{L}$
containing four-phonon scattering Ouyang  _et al._ (2022). Consequently, based
on the fourth-order anharmonicity Han  _et al._ (2022), we have modified and
developed corresponding interface using the MTP to obtain fourth-order force
constants in an efficient manner.

In this study, we conducted a comprehensive investigation of the thermal
transport and thermoelectric properties of the benchmark compound Tl9SbTe6. To
this end, we integrated machine-learning based anharmonic lattice dyanmics
with an unified theory of thermal transport and momentum relaxation time
approximation. This integration included: (1) Construction of a machine-
learning potential with DFT-level accuracy for the complex crystalline
Tl9SbTe6, (2) anharmonically renormalizing phonons using the SCPH approach,
(3) evaluation of thermal transport considering both particle-like and wave-
like phonon transport channels, (4) application of distinct scattering
mechanisms, including acoustic deformation potential (ADP), ionized impurity
(IMP) scattering, and polar optical phonon (POP) scattering, to systematically
study the electrical transport properties, (5) exploration of the trend of the
thermoelectric property $ZT$ value with carrier concentration and temperature
variation.

##  II 2\. Results and discussion

###  II.1 2.1 Crystal structure and machine learning potential

Figure 1: (a) The workflow of the thermoelectric property calculations. (b)
Phonon dispersion of Tl9SbTe6 calculated by Moment tensor potential (MTP) and
DFT. A comparisons of (c) the energy per atom (d) and the atomic force between
MTP and DFT calculations for Tl9SbTe6. (e) The conventional cell structure of
Tl9SbTe6. The red and purple atoms represent Thallium (Tl) atoms in different
chemical environments. The atoms represented by the blue color are Tellurium
(Te) atoms, while the green ones are Antimony (Sb) atoms. Figure 2: (a) The
temperature-dependent phonon spectrum from T = 50 K to 800 K for Tl9SbTe6. (b)
Projected phonon density of states (PDOS) of Tl9SbTe6 at 0 K, 300 K, and 700
K. (c) Phonon populations’ thermal conductivity ($\kappa_{p}$) as a function
of the temperature of Tl9SbTe6. (d) Intrinsic lattice thermal conductivity
($\kappa_{L}$) as a function of the temperature of Tl9SbTe6. The ratio
$\kappa_{c}/\kappa_{p}$ represents $\kappa_{p,SCPH}^{3,4ph}$. The solid lines
represent the $a$-axis direction, and the dashed lines represent the $c$-axis
direction.  Figure 3: (a) Calculated spectral and cumulative $\kappa_{p}$
using the SCPH+3,4ph model at 300 K. (b) Calculated spectral and cumulative
coherence thermal conductivity at 300 K. (c) Three-dimensional visualization
model $\kappa_{c}$($\omega_{i}$, $\omega_{j}$) of the contributions to the
coherence thermal conductivity in the $a$-axis direction at 300 K. (d) The
same as (c) but in $c$-axis direction. (e) The same as (c) but at 700 K. (f)
The same as (d) but at 700 K.

The optimized lattice constants of Tl9SbTe6, calculated using LDA potentials,
are $a$ = $b$ = 8.63 Å, and $c$ = 12.77 Å. These values are consistent with
the experimental values, which are $a$ = $b$ = 8.830 Å and $c$ = 13.013 Å Guo
_et al._ (2014); Tao  _et al._ (2011). As illustrated in Fig. 1(b), the phonon
spectrum of the LDA functional is devoid of imaginary frequencies, indicating
the dynamical stability of Tl9SbTe6. The primitive cell comprises 16 atoms:
one Sb, six Te, and nine Tl. Tl atoms are divided into eight Tl and one Tl*,
based on their deisparate chemical environments and locations, as depicted in
Fig. 1(e).

Fig. 1(a) depicts the workflow of our work. In this work, we obtain the
training dataset from Ab Initio Molecular Dynamics (AIMD) using the isobaric-
isothermal (NPT) ensemble. We then train the MTP and the training principles
are detailed in the Supplementary Information Novikov  _et al._ (2020). After
obtaining the MTP, we test it on the validation dataset.

The MTP is employed to ascertain the energy and atomic forces of each
configuration in the validation dataset, with the resulting root-mean-square
error (RMSE) being presented in Figs. 1(c-d). The RMSE of energy per atom and
atomic force is 0.00029 eV/atom and 0.023 eV/Å, respectively. These values
indicate that the trained MTP is sufficiently accurate for calculating
thermodynamic properties Novikov  _et al._ (2020); Ouyang  _et al._ (2022).
Furthermore, as shown in Fig. 1(b), the phonon spectrum calculated by MTP is
in good agreement with the DFT results.

###  II.2 2.2 Anharmonic lattice dynamics and thermal transport properties

The impact of forth-order anharmonicity on the phonon spectra at finite
temperatures is next examined, as depicted in Figs. 2(a-b), using the
interatomic force constants (IFCs) obtained from MLP. It is evident that
overall phonon hardening phenomenon is observed in crystalline Tl9SbTe6 as the
temperature increases from 50 to 800 K. This observation suggests the presence
of strong lattice anharmonicity in Tl9SbTe6 and highlights the importance of
properly treating lattice anharmonicity in lattice dynamics Zheng  _et al._
(2022a); Tadano and Tsuneyuki (2015); Wang  _et al._ (2023a); Feng  _et al._
(2024). In particular, the low-frequency phonons with a frequency below 1.5
Thz are predominantly influenced by the Tl atoms, as evidenced by the
projected phonon density of states (PDOS) in Fig. 2(b). The pronounced phonon
hardening observed in the Tl-dominated modes can be attributed to the
significant atomic displacements resulting from their loose bonding, as
illustrated in Fig. S1 in the supplementary information. Furthermore, it can
be observed that Tl atoms contribute to the low-lying flattened phonon bands
in the frequency range of 0.7 to 1.0 THz. This suggests that strong phonon
scattering is occurring, which suppresses the primary heat carriers, i.e.,
acoustic phonon modes Li and Mingo (2015); Zheng  _et al._ (2022b); Wang  _et
al._ (2024a). In contrast, the optical modes with higher energies exhibit less
increase in stiffness corresponding to the majority of Te and Sb vibrations.

With the zero-K/finite-temperature IFCs at hand, we proceed to calculate the
temperature-dependent $\kappa_{p}$ using three different levels of theory, as
shown in Figs. 2(c-d). It is observed that the lattice thermal conductivity
($\kappa_{L}$) is anisotropic, with the $\kappa_{p}$ approximately 10% smaller
in the $a$-axis or $b$-axis direction than in the $c$-axis direction. The
lowest level of thermal transport theory, namely the HA+3ph model, yields the
following values for $\kappa_{p,HA}^{3ph}$ in the $a$-axis at 50, 300, and 800
K: 0.88, 0.14, and 0.053 W m-1 K-1, respectively. As previously stated,
anharmonic phonon renormalization is a significant factor in the prediction of
finite-temperature $\kappa_{L}$ in complex compounds. With the additional
effect of phonon energy shifts, we advance the HA+3ph model to a more accurate
SCPH+3ph model, which gives $\kappa_{p,SCPH}^{3ph}$ values of 1.02, 0.24, and
0.15 W m-1 K-1 at 50, 300, and 800 K, respectively, in the $a$-axis direction.
These values are 16%, 71%, and 183% larger than $\kappa_{p,HA}^{3ph}$,
indicating significant anharmonic phonon renormalization in Tl9SbTe6, with a
substantial increase as temperature rises. This trend is also confirmed by
Fig. 2(a), which shows significant phonon frequency shifts with increasing
temperature.

In addition, the quartic anharmonicity not only induces significant shifts in
phonon frequencies but also results in pronounced 4ph scatterings.
Consequently, additional 4ph scatterings were included to obtain
$\kappa_{p,SCPH}^{3,4ph}$ values of 0.96, 0.19, and 0.093 W m-1 K-1 at 50,
300, and 800 K, respectively, in the $a$-axis direction (see Fig. 2(c)). These
values represent decreases of 6%, 21%, and 38% compared with
$\kappa_{p,HA}^{3ph}$, indicating the importance of the 4ph scattering
processes Wang  _et al._ (2023a, 2024a, 2024b). It is noteworthy that the
discrepancy between the $\kappa_{p,HA}^{3ph}$ and $\kappa_{p,HA}^{3,4ph}$ is
6%, 26%, and 46% greater than that between the anharmonic phonon renormalized
ones. This implies that phonon renormalization and 4ph scattering induced by
quartic anharmonicity exert an inverse influence on thermal transport.

Nevertheless, $\kappa_{p,SCPH}^{3,4ph}$ remains considerably lower than the
experimentally measured $\kappa_{L}$, as illustrated in Fig. 2(d), indicating
that coherence contributions are not negligible. By further considering the
off-diagonal terms of heat flux operators, the total $\kappa_{L}$
significantly enhanced by incorporating both $\kappa_{p,SCPH}^{3,4ph}$ and
$\kappa_{c}$ Simoncelli  _et al._ (2019). In this context, $\kappa_{c}$ can be
expressed as,

$\displaystyle\kappa_{c}=$
$\displaystyle\frac{\hbar^{2}}{k_{B}T^{2}VN_{0}}\sum_{\boldsymbol{q}}\sum_{s\neq
s^{\prime}}\frac{\omega_{\boldsymbol{q}}^{s}+\omega_{\boldsymbol{q}}^{s^{\prime}}}{2}\boldsymbol{v}_{\boldsymbol{q}}^{s,\boldsymbol{s}^{\prime}}\boldsymbol{v}_{\boldsymbol{q}}^{\boldsymbol{s}^{\prime},s}$
(1)
$\displaystyle\times\frac{\omega_{\boldsymbol{q}}^{s}n_{\boldsymbol{q}}^{s}\left(n_{\boldsymbol{q}}^{s}+1\right)+\omega_{\boldsymbol{q}}^{s^{\prime}}n_{\boldsymbol{q}}^{s^{\prime}}\left(n_{\boldsymbol{q}}^{s^{\prime}}+1\right)}{4\left(\omega_{\boldsymbol{q}}^{s^{\prime}}-\omega_{\boldsymbol{q}}^{s}\right)^{2}+\left(\Gamma_{\boldsymbol{q}}^{s}+\Gamma_{\boldsymbol{q}}^{s^{\prime}}\right)^{2}}$
$\displaystyle\times\left(\Gamma_{\boldsymbol{q}}^{s}+\Gamma_{\boldsymbol{q}}^{s^{\prime}}\right),$

where $\hbar$ is the reduced Planck constant, $k_{B}$ is Boltzmann constant,
$V$ is the primitive cell volume, and $N_{0}$ is the total number of sampled
phonon vectors.

As illustrated in Fig. 2(d), the ratio $\kappa_{c}/\kappa_{p}$ is greater than
0.5 at 300 K and exhibits a pronounced increase with temperature.
Specifically, in the $a$-axis direction, he ratio of $\kappa_{c}/\kappa_{p}$
reaches 1.60 at 800 K, which is 2.44 times larger than the 0.65 observed at
300 K. This observation suggests that $\kappa_{c}$ is the dominant
contribution to $\kappa_{L}$ in crystalline Tl9SbTe6 at temperatures above 600
K. The dominant role of coherence contribution in thermal transport was also
observed in other complex compounds such as Cs2AgBiBr6 Zheng  _et al._ (2024),
Cu12Sb4S13 Xia  _et al._ (2020a), CsPbBr3 He  _et al._ (2018), and even simple
cubic CsCl Wang  _et al._ (2024b). Although the heat transport phenomena in
Tl9SbTe6 exhibit similarities to those in glasses, the strong phonon
anharmonicity is the key factor contributing to coherent thermal conductivity.
In contrast, in glasses, the off-diagonal contributions arise from structural
disorder.

Figure 4: (a) Squared phonon velocities of Tl9SbTe6 in the harmonic
approximation and anharmonic renormalization approximation at 300 K. (b)
Phonon scattering rates of 3ph and 4ph for Tl9SbTe6 at 300 K and 700 K. (c)
Weighted phonon scattering phase space of 3ph and 4ph for Tl9SbTe6 at 300 K
and 700 K.  Figure 5: The temperature-dependent (a) Seebeck coefficient S, (b)
electrical conductivity $\sigma$, (d) electronic thermal conductivity
$\kappa_{e}$, and (e) the power factor ($PF=S^{2}\sigma$) of the Tl9SbTe6 were
calculated as a function of carrier concentration (holes and electrons) along
different axes at 300 K, 500 K and 700 K, respectively. (c) and (d) represent
the electronic relaxation time for distinct scattering mechanisms, including
acoustic deformation potential (ADP), ionized impurity (IMP) scattering, and
polar optical phonon (POP) scattering, respectively.  Figure 6: Electron
Localization Function (ELF) of Tl9SbTe6 in (a) 3D and (b) 2D plots. ELF=1
corresponds to complete localization and ELF=0 corresponds to complete
delocalization of electrons. (c) Calculated Fermi surfaces for the valence
band structure minimum of Tl9SbTe6 in the first Brillouin zone around the
Fermi level. (d) Electronic band structure and (e) electronic density of
states (DOS) of Tl9SbTe6. (f) Normalized Lorenz number as a function of
carrier concentration at temperatures of 300 K, 500 K and 700 K, respectively.
The Lorenz number, denoted by $L_{0}$, is a constant that is closely
approximately by the Sommerfeld value, $L_{0}=2.44\times 10^{-8}\ W\ \Omega\
K^{-1}$.

After incorporating $\kappa_{c}$, the total $\kappa_{L}$ is approximately 0.27
W m-1 K-1 and 0.26 W m-1 K-1 at 400 K and 500 K, respectively. This is
consistent with experimental measurements, where $\kappa_{L}$ is approximately
0.34 and 0.30 W m-1 K-1 at 400 K and 500 K, respectively Guo  _et al._ (2014).
In previous experiment, the $\kappa_{L}$ is the the difference between total
thermal conductivity ($\kappa$) and electronic thermal conductivity
($\kappa_{e}$), where $\kappa_{e}$ is estimated via the Wiedemann Franz
relationship $\kappa_{e}=L\sigma T$. The Lorenz number $L$ was calculated by
utilizing the single parabolic band and elastic carrier scattering assumption
Guo  _et al._ (2014). This implies that the little difference between our
$\kappa_{L}$ and the experimental predicted ones is negligible. Additionally,
we observed that the dependence of $\kappa$ on temperature approximately
follows $T^{-1/3}$, which indicates the glass-like behaviour of crystalline
Tl9SbTe6.

To gain a deeper insight into the thermal transport in complex crystalline
Tl9SbTe6, we calculate the spectral and cumulative $\kappa_{L}$ for both
populations’ and coherences’ contributions, as illustrated in Figs. 3(a) and
(b). Fig. 3(a) shows that the majority of $\kappa_{p}$ in Tl9SbTe6 is carried
by phonons with a frequency below 2 THz at 300 K. Phonons with frequencies
below 0.8 THz at 300 K contribute 57% and 62% of $\kappa_{p}$ in the $a$-axis
and $c$-axis directions, respectively. In contrast, as depicted in Fig. 3(b),
the majority of $\kappa_{c}$ from the wave-like tunneling channel is carried
by phonons with a frequency less than 2.4 THz at 300 K. However, phonons with
frequencies less than 0.8 THz contribute only 7% of $\kappa_{c}$, due to the
good particle-like nature of these low-frequency phonons He  _et al._ (2018).
Overall, the data presented in Fig. 3(a) and (b) indicate that low-frequency
phonons predominantly contribute to $\kappa_{p}$, while high-frequency phonons
primarily contribute to $\kappa_{c}$ Zheng  _et al._ (2024); Wang  _et al._
(2023a). This observation can be attribted to the fact that phonon scattering
rates increase with increasing temperature in Tl9SbTe6, as illustrated in Fig.
4(c).

Furthermore, we observe a dip in $\kappa_{p}$ and a peak in $\kappa_{c}$
around 0.85 THz. This can be attributed to the low-lying flattened modes,
which are dominated by Tl atoms with strong phonon-phonon scattering. Due to
the strong anharmonicity of these modes, they significantly contribute to the
coherence conductivity at this frequency range. Figs. 3(c) and (d) illustrate
the contributions to $\kappa_{c}$ in Tl9SbTe6, calculated using the SCPH+3,4ph
model at 300 K and 700 K, respectively, and resolved in terms of pair phonon
energies. From Figs. 3(c-e), it is evident that quasi-degenerate phonons
dominate $\kappa_{c}$, with optical phonons (frequencies larger than 0.8 THz)
contributing the most. Additionally, it can be observed that the contribution
of the couplings between phonons with large frequency differences to
$\kappa_{c}$ in Tl9SbTe6 increases as the temperature rises from 300 K to 700
K in both the $a$ and $c$ directions. This phenomenon can be attributed to the
observed increase in phonon scattering rates with temperature.

To gain further insight into microscopic mechanism of the ultralow
$\kappa_{L}$ in Tl9SbTe6, we conducted a detailed study of several parameters
closely related to $\kappa_{L}$, including phonon velocities, weighted phonon
scattering phase space, and phonon scattering rates. These findings are
presented in Figs. 4(a-c). Since $\kappa_{L}$ is proportional to the square of
the phonon group velocity ($v^{2}$), we calculate frequency-dependent $v^{2}$
where $v$ is the group velocity at 300 K, as shown in Fig. 4(a). It can be
seen that the $v^{2}$ of Tl9SbTe6 is extremely low, with a maximum value of 5
km${}^{2}\ $s-2. In all directions, the maximum value of $v^{2}$ is lower than
that of PbTe, which is approximately 14 km${}^{2}\ $s-2 Tian  _et al._ (2012),
with $\kappa_{L}$ of 2 W m-1 K-1 at 300 K. Furthermore, for the ultralow
$\kappa_{L}$ Bi4O4SeCl2, with a thermal conductivity of 0.1 W m-1 K-1 at 300
K, exhibits the highest velocity of 8 km${}^{1}\ $s-1 Tong  _et al._ (2023),
which is higher than the maximum velocity of Tl9SbTe6. Consequently, the
ultralow $\kappa$ in Tl9SbTe6 is attributed primarily to the low phonon group
velocities.

Furthermore, we observe the pronounced scattering rates in crystalline
Tl9SbTe6, as illustrated in Fig. 4(b). In particular, Fig. 4(b) clearly shows
a strong peak in phonon scattering rates, originating from the low-lying
flattening modes dominated by Tl atoms [see Fig. 2(a)]. In general, the low-
lying flattening modes contribute to strong phonon-phonon scattering rates,
particularly four-phonon scattering rates, thereby suppressing thermal
transport Li and Mingo (2015); Zheng  _et al._ (2022b); Wang  _et al._ (2024b,
a). Therefore, the strong phonon scattering rates arising from the loose
bonding of Tl atoms are another factor contributing to the ultralow $\kappa$
in Tl9SbTe6.

To obtain a more comprehensive understanding of phonon scattering rates, we
also calculated the phonon scattering phase space, as depicted in Fig. 4(c).
As illustrated in Fig. 4(c), the weighted phase space for both three-phonon
(3ph) and four-phonon (4ph) scattering processes increases as the temperature
rises from 300 K to 700 K. It is evident that the phase space of 4ph processes
is larger than that of 3ph processes, resulting in a significant source of 4ph
scattering rates in Tl9SbTe6. As anticipated, the peak region around 0.75 THz
of the 4ph phase space in Tl9SbTe6 is in alignment with the partial density of
states (PDOS) of Tl atoms, as illustrated in Fig. 2(b). As previously
discussed, the low-lying flattening modes typically result in elevated 4ph
scattering rates by enhancing the 4ph scattering channels. Consequently, this
will further suppress the thermal transport in Tl9SbTe6.

###  II.3 2.3 Electronic transport properties

The calculated electronic transport coefficients including the Seebeck
coefficient $S$, electrical conductivity $\sigma$, electronic thermal
conductivity $\kappa_{e}$, and the power factor ($PF=S^{2}\sigma$) of Tl9SbTe6
based on the Onsager coefficients are presented shown in Fig. 5. For non-polar
crystals, acoustic deformation potential scattering is the main contributor to
electric conductivity Bardeen and Shockley (1950). Furthermore, we also
consider the effects of polar optical phonon scattering Fröhlich (1954) and
ionized impurity scattering Ganose  _et al._ (2021).

Fig. 5 presents a depiction of these parameters as functions of carrier
concentration at distinct temperatures of 300 K, 500 K, and 700 K,
respectively. Notably, Fig. 5(a) reveals that both p- and n-type doped
Tl9SbTe6 manifest notably elevated Seebeck coefficients $S$ across all
investigated temperatures, auguring well for the attainment of enhanced $ZT$
values. For instance, at a temperature of 300 K, the maximal Seebeck
coefficients for p- and n-doped Tl9SbTe6 materialize at 528 and 393 $\mu$V/K
respectively. Our Tl9SbTe6 has a Seebeck coefficient that is of a similar
order to that of SnSe, which has a favorable 530 $\mu$V/K at room temperature
Zhao  _et al._ (2014).

The Mott formula Mahan and Sofo (1996) indicates that the Seebeck coefficient,
a crucial parameter in elucidating the interplay between thermal and
electrical characteristics in three-dimensional (3D) materials, is expressed
as,
$S=\frac{2\pi^{2}k_{B}^{2}T}{3q\hbar^{2}}(\frac{1}{3\pi^{2}n})^{\frac{2}{3}}m_{DOS,F}^{*}$,
where “$n$” and “$m^{*}$” denote the carrier concentration and the the
effective mass of electron states around the Fermi level. Furthermore, the
density of state effective mass is also a function of band effective mass. The
latter is assumed to be a rigid band model, and its dispersion relationship is
a standard parabola. They differ by a factor of $N_{v}^{\frac{2}{3}}$:
$m_{DOS}^{*}=N_{v}^{\frac{2}{3}}m_{band}^{*}$, in which $N_{v}$ is the valley
degeneracy Pei  _et al._ (2011). The valley degeneracy of the Tl9SbTe6 valence
band minimum is two. One is at the $M$ point and the other one is between the
$\Gamma$ and $X$ points, as shown in Fig. 6. Therefore, a favorable
$m_{DOS}^{*}$ will result in a potential high thermoelectric performance for
the Tl9SbTe6 material.

Moreover, it can be gleaned from the Mott formula that the Seebeck coefficient
for both p\- and n-type exhibits a dependence on temperature and doping
concentration. As depicted in Fig. 5(a), at lower carrier concentrations, the
value of $S$ at 300 K reaches its highest point. However, as the carrier
concentration increases beyond a certain threshold, a sequential decline in
the Seebeck coefficient from 300 K to 700 K is observed, aligning with the
anticipated result.

This nonlinear relationship between $S$ and doping concentration stems from
the bipolar effect Zhu  _et al._ (2021); Gong  _et al._ (2016). When there are
significant number of both electrons and holes contributing to charge
transport (bipolar charge transport) the thermoelectric properties are greatly
affected. This occurs when electrons are excited across the band gap producing
minority charge carriers (e.g., holes in an n-type material) in addition to
majority charge carriers (e.g., the electrons in an n-type material). Bipolar
effects are observed in small band gap materials at high temperatures.
Furthermore, it is notable that the Seebeck coefficient of Tl9SbTe6 exhibits
significant anisotropy, regardless of whether the material is doped with p\-
or n-type ions.

Fig. 5(b) and (e) show the direct correlation between electrical conductivity
$\sigma$ and electronic thermal conductivity $\kappa_{e}$, as well as carrier
concentrations and temperatures both p\- and n-type doping of Tl9SbTe6. It can
be observed that as temperature increases, conductivity decreases. Conversely,
as concentration increases, conductivity increases. Furthermore, Tl9SbTe6
exhibits pronounced anisotropy in both electrical conductivity and electronic
thermal conductivity, with distinct behaviors observed for p\- and n-type
doping scenarios. In the context of electron doping, it is evident that the
values along the $c$-axis (dashed lines) are notably lower than those along
the $a$-axis (solid lines). For example, at a temperature of 300 K and a
carrier concentration of 1020 cm-3, the values of $\sigma$ and $\kappa_{e}$
along the $a$-axis are 6640.35 S cm-1 and 4.18 W m-1 K-1, which are
approximately three times the values observed along the $c$-axis with 2254.63
S cm-1 and 1.57 W m-1 K-1. Conversely, under hole doping conditions, the
values along the $c$-axis are slightly higher than the corresponding $a$-axis.

Fig. 5(c) and (d) exhibit the electronic relaxation time ($\tau_{e}$) for
distinct scattering mechanisms, including acoustic deformation potential
(ADP), ionized impurity (IMP) scattering, and polar optical phonon (POP)
scattering, respectively. It can be seen that $\tau_{e}$ of IMP is greater
than that of ADP, and $\tau_{e}$ of ADP is greater than POP, indicating that
POP has the strongest scattering. As the temperature increases, the strength
of electron scattering increases, resulting in a corresponding decrease in
$\tau_{e}$. This is illustrated in Fig. 5(d), where the total electronic
relaxation time is observed to be on the order of 10-14 cm${}^{-}3$. As the
temperature increases from 300 K to 700 K, the electron relaxation time
decreases from 2.4 $\times$ 10-14 cm${}^{-}3$ to 1.0 $\times$ 10-14
cm${}^{-}3$ for p\- and n-type doping Tl9SbTe6.

As illustrated in Fig. 5(f), the power factor $(S^{2}\sigma)$ was computed
based on the Seebeck coefficient and electrical conductivity, under varying
temperatures and carrier concentrations. Our findings reveal that Tl9SbTe6
exhibits substantial anisotropy in its power factor. Across intermediate
temperature and carrier concentration regimes, the power factor for electron-
doped Tl9SbTe6 along the $a$-axis consistently surpasses that for hole doping.
This indicates that, in order to enhance the thermoelectric performance of
Tl9SbTe6, electron doping is more advantageous than hole doping. In contrast,
the effect of hole doping along the $c$-axis is notably more pronounced and
superior in enhancing the power factor. The variation of power factor with
concentration and temperature is not monotonic as a function of electrical and
electronic thermal conductivity. The power factor exhibits a characteristic
trend whereby it initially rises with increasing carrier concentration and
subsequently declines, with a peak concentration range between 1020 and 1021
cm-3 for Tl9SbTe6. Below this peak concentration, higher temperatures are
associated with lower power factors. It is noteworthy that at a temperature of
500 K and a carrier concentration of 2×1020 cm-3, the power factor along the
$c$-axis attains its maximum value, reaching 8.57 mW m-1 K-2.

Fig. 6(a) and (b) show the 3D and 2D Electron Localization Function (ELF) of
Tl9SbTe6. ELF values of 1 and 0 represent complete localization and
delocalization of electrons. The Te and Sb elements exhibit a stronger
tendency to attract electrons than the Tl elements, suggesting a greater
degree of delocalization for thallium atoms. The delocalized electrons show a
stronger electron-phonon coupling Gao  _et al._ (2021); Qin  _et al._ (2022).
This is evident in Fig. 2(b), where the thallium element exhibits a very
pronounced peak in the low-frequency region (<1.0 Thz) for the phonon density
of states Gao and Wang (2020); Gao  _et al._ (2018).

Fig. 6(d) and (e) present the electronic band structure and the electronic
density of state (DOS). Tl9SbTe6 emerges as an indirect bandgap semiconductor,
characterized by its conduction band minimum (CBM) positioned between the
$\Gamma$ and $X$ points, and its valence band maximum (VBM) located between
the $N$ and $\Gamma$ points, resulting in a discernible bandgap of 0.279 eV.
The contributions to the bottom of the conduction band and the top of the
valence band predominantly arise from the $p$-orbitals of Te atoms and Tl
atoms respectively. Furthermore, the band structure reveals a dual-degeneracy
phenomenon within the valence band at the $M$ point. Importantly, this
observation holds profound significance for enhancing the thermoelectricity
$ZT$ of the material, as the insights from band engineering studies Pei  _et
al._ (2011); Liu  _et al._ (2012).

It is well established that electrical transport is not solely dependent on
the movement of electrons, but also on the transfer of heat. This phenomenon
is commonly described by the Wiedemann-Franz (WF) law, which states that the
ratio of electrical conductivity to the conductivity of heat
($\kappa_{e}$/$\sigma$) is directly proportional to the absolute temperature
$T$. This relationship can be expressed as $\kappa_{e}/\sigma=LT$, where $L$
is a constant ratio known as the Lorenz number. The value of $L$ is believed
to be close to the Sommerfeld value $L_{0}$, which is approximately

$2.44\times 10^{-8}\ W\ \Omega\ K^{-1}$

Lee  _et al._ (2017); Wang  _et al._ (2023b). It is notable that most metals
that considered to be “good” in terms of their conductivity satisfy this WF
law. However, the Lorenz number of transition metals such as palladium (Pd),
nickel (Ni), cobalt (Co), and platinum (Pt) exhibits larger deviations from
the Sommerfeld value $L_{0}$ Tong  _et al._ (2019); Zhang  _et al._ (2022).
Recently, a $\kappa_{e}$ that is an order of magnitude lower than expected
from WF law was observed in metallic VO2 near its insulator-metal transition.
This phenomenon was explained in terms of the absence of quasiparticles in a
strongly correlated electron fluid in which heat and charge diffuse
independently Lee  _et al._ (2017).

The calculated doping-dependent ratio is shown in Fig. 6(f). It can be seen
that when the doping concentration is high, especially when it exceeds 1020
cm-3, $L$ is near equivalent to $L_{0}$, exhibiting metallic properties. The
lower limit of the doping concentration for the Lorenz number being close to
$L_{0}$ increases with temperature. For p-type Tl9SbTe6, the lower limit of
the doping concentration for the Lorenz number to be close to $L_{0}$ at 300 K
is around 1018 cm-3, while at 700 K it is 6$\times$1019 cm-3. This suggests
that in order to achieve enhanced thermoelectric performance under high
temperature conditions, it may be necessary to adjust the doping concentration
within an optimal range.

A higher $L$ value represents a higher ratio of electronic thermal
conductivity to electrical conductivity, which is less conducive to improving
the thermoelectric properties of the material. At lower concentrations, a
substantial deviation of $L$ from the ideal value $L_{0}$ is observed,
particularly when the concentration is below 5$\times$1019 cm-3. Furthermore,
the deviation of $L$ from $L_{0}$ is severer with higher temperature than
lower temperature. This is consistent with the trend shown in Fig. 7, where
the lower the doping concentration, the closer the $ZT$ peak value is to the
low temperature region. This also makes it difficult to use the Wiedemann-
Franz law to predict the $\kappa_{L}$ of the material by considering the
experimental values of $\kappa$ and electrical conductivity, as considering
$L$ as $L_{0}$ will underestimate $\kappa_{e}$, thus overestimate
$\kappa_{L}$. Consequently, is is necessary to calculate the Lorenz number $L$
as a function of temperature and doping concentration, which is the
prerequisite for accurately estimating thermoelectric performance.

###  II.4 2.4 Thermoelectric performance

Figure 7: The calculated figure of merit ($ZT$) of (a) p-type and (b) n-type
Tl9SbTe6 as a function of temperature along the $a-$ and $c-$ axes at
different carrier concentrations, ranging from 1.0$\times$1018 cm-3 to
1.0$\times$1020 cm-3.

The calculated $ZT$ of Tl9SbTe6 at different temperatures and doping
concentrations is presented in Fig. 7. As the doping concentration increases
from 1.0$\times$1018 cm-3 to 1.0$\times$1020 cm-3, the peak value of the $ZT$
value will vary from 300 K to 700 K and higher, with a greater bias towards
higher temperatures. Concurrently, the peak value will initially increase and
subsequently decrease, reaching a maximum value at a doping concentration
between 5$\times$1018 cm-3 and 1$\times$1019 cm-3. Due to the orientation-
dependent properties of phonons and electrons, the $ZT$ value will exhibit a
significant anisotropy.

In the case of p-type doping, the $ZT$ value in the $c$-axis direction is
significantly higher than that in the $a$-axis direction, which is the
opposite in the case of n-type. At the same temperature, the $ZT$ values of
p\- and n-type vary significantly. The highest $ZT$ value of the p-type can
reach a peak of 3.17 at 600 K, with a doping concentration of 5$\times$1019
cm-3. The highest $ZT$ value of the n-type also reaches a peak of 2.26 at 500
K, with a doping concentration of 1.0$\times$1019 cm-3. It can be concluded
that the Tl9SbTe6 is a highly promising thermoelectric material in the medium
temperature zone (300 K-600 K), particularly for the p-type doping.

##  III 3\. Conclusion

In conclusion, our study examines the thermal and electrical transport
properties of Tl9SbTe6 through the integration of machine-learning-potential-
based self-consistent phonon (SCPH) theory, first-principles calculations, and
a unified theory of thermal transport that incorporates both coherence and
population conductivities. The Moment Tensor Potential, trained with Machine
Learning Inter-atomic Potential (MLIP), achieves DFT-level accuracy in
predicting energy and atomic forces, thereby accurately producing the phonon
dispersions for crystalline Tl9SbTe6. The force constants derived from the
MLIP and SCPH theory indicate a phonon hardening phenomenon across the entire
frequency range, which suggests significant lattice anharmonicity in Tl9SbTe6.
Subsequently, an ultralow $\kappa_{L}$ for Tl9SbTe6 is predicted, with a value
of 0.3 W m-1K-1 at room temperature. This value is consistent with the
experimentally measured $\kappa_{L}$ and exhibits a glass-like temperature
dependence.

Our research indicates that four-phonon interaction processes and off-diagonal
elements of heat flux operators have a significant impact on thermal transport
in Tl9SbTe6. Specifically, the four-phonon interaction processes influence
phonon energy shifts, which increase thermal conductivity. However, they also
significantly enhance overall scattering rates, thereby decreasing thermal
conductivity. Furthermore, we observe that in crystalline Tl9SbTe6 at
temperatures above 600 K, the coherent component $\kappa_{c}$ represents the
dominant contributor to the lattice $\kappa_{L}$, with a
$\kappa_{c}/\kappa_{p}$ ratio exceeding 1. By analyzing the electronic band
structure, we find that Tl9SbTe6 also exhibits excellent electrical transport
properties, attributed to the dual-degeneracy phenomenon within the valence
band.

Our investigation has revealed that Tl9SbTe6, with a maximum figure of merit
(ZT) of up to 3.17, is a promising candidate for thermoelectric applications.
This is due to the ultralow $\kappa_{L}$ and favorable electrical transport
properties. We have also elucidated the origins of the ultralow $\kappa_{L}$
and relatively high electrical conductivity in Tl9SbTe6. This has involved
confirming its optimal charge carrier concentration and operating temperature
range. Our study offers insights into potential improvements and application
directions for the Tl5Te3 family of materials and paves the way to accelerate
the accurate prediction of thermoelectric materials.

##  IV ACKNOWLEDGMENTS

We acknowledge the support from the National Natural Science Foundation of
China (No.12104356 and No.52250191, No.12104236), the Opening Project of
Shanghai Key Laboratory of Special Artificial Microstructure Materials and
Technology (No.Ammt2022B-1), and the Fundamental Research Funds for the
Central Universities. H. Gu acknowledges the support from Gusu Leading Talent
(No.ZXL2021383). This work is sponsored by the Key Research and Development
Program of the Ministry of Science and Technology (No.2023YFB4604100). We also
acknowledge the support by HPC Platform, Xi’an Jiaotong University.

## References

  * Snyder and Toberer (2008) G. Snyder and E. Toberer, “Complex thermoelectric materials,” Nat. Mater. 7, 105–114 (2008).
  * Zebarjadi  _et al._ (2012) M. Zebarjadi, K. Esfarjani, M. S. Dresselhaus, Z. F. Ren, and G. Chen, “Perspectives on thermoelectrics: from fundamentals to device applications,” Energy Environ. Sci. 5, 5147–5162 (2012).
  * Baxter  _et al._ (2009) Jason Baxter, Zhixi Bian, Gang Chen, David Danielson, Mildred S. Dresselhaus, Andrei G. Fedorov, Timothy S. Fisher, Christopher W. Jones, Edward Maginn, Uwe Kortshagen, Arumugam Manthiram, Arthur Nozik, Debra R. Rolison, Timothy Sands, Li Shi, David Sholl, and Yiying Wu, “Nanoscale design to enable the revolution in renewable energy,” Energy Environ. Sci. 2, 559–588 (2009).
  * Dallan  _et al._ (2015) Bruno S. Dallan, Joseph Schumann, and Frederic J. Lesage, “Performance evaluation of a photoelectric-thermoelectric cogeneration hybrid system,” Sol. Energy 118, 276–285 (2015).
  * Nolas  _et al._ (2010) George S. Nolas, Jeffrey Sharp, and H. Julian Goldsmid, “Thermoelectrics: Basic principles and new materials developments,” (Springer Berlin, 2010) Chap. 6, p. 177–207, 1st ed.
  * Xia  _et al._ (2020a) Yi Xia, Vidvuds Ozoliņš, and Chris Wolverton, “Microscopic Mechanisms of Glasslike Lattice Thermal Transport in Cubic Cu12Sb4S13 Tetrahedrites,” Phys. Rev. Lett. 125, 085901 (2020a).
  * Dutta  _et al._ (2019) Moinak Dutta, Koushik Pal, Umesh V. Waghmare, and Kanishka Biswas, “Bonding heterogeneity and lone pair induced anharmonicity resulted in ultralow thermal conductivity and promising thermoelectric properties in n-type AgPbBiSe3,” Chem. Sci. 10, 4905–4913 (2019).
  * Haque and Hossain (2018) Enamul Haque and M. Anwar Hossain, “Origin of ultra-low lattice thermal conductivity in Cs2BiAgX6 (X=Cl, Br) and its impact on thermoelectric performance,” J. Alloys Compd. 748, 63–72 (2018).
  * Li  _et al._ (2023) Fan Li, Xin Liu, Ni Ma, Yi-Chang Yang, Jian-Ping Yin, Ling Chen, and Li-Ming Wu, “Overdamped Phonon Diffusion and Nontrivial Electronic Structure Leading to a High Thermoelectric Figure of Merit in KCu5Se3,” J. Am. Chem. Soc. 145, 14981–14993 (2023).
  * Shen  _et al._ (2024) Xingchen Shen, Niuchang Ouyang, Yuling Huang, Yung-Hsiang Tung, Chun-Chuen Yang, Muhammad Faizan, Nicolas Perez, Ran He, Andrei Sotnikov, Kristin Willa, Chen Wang, Yue Chen, and Emmanuel Guilmeau, “Amorphous-Like Ultralow Thermal Transport in Crystalline Argyrodite Cu7PS6,” Adv. Sci. , 2400258 (2024).
  * Guo  _et al._ (2014) Quansheng Guo, Meghan Chan, Bryan A. Kuropatwa, and Holger Kleinke, “Thermoelectric properties of Sn- and Pb-doped Tl9BiTe6 and Tl9BiTe6,” J. Appl. Phys. 116, 183702 (2014).
  * Yamanaka  _et al._ (2006) Shinsuke Yamanaka, Ken Kurosaki, Atsuko Kosuga, Keita Goto, and Hiroaki Muta, “Extremely low thermal conductivity substances as novel thermoelectric materials,” Mater. Res. Soc. Symp. Proc. 886, 337 (2006).
  * Teubner (2001) Jens Thilo Teubner,  _Optimization of high efficiency thermoelectrics based on Tl 5Te3_, Ph.D. thesis, Universität Konstanz, Fachbereich Physik (2001).
  * Guo  _et al._ (2013) Quansheng Guo, Meghan Chan, Bryan A. Kuropatwa, and Holger Kleinke, “Enhanced Thermoelectric Properties of Variants of Tl9SbTe6 and Tl9BiTe6,” Chem. Mater. 25, 4097–4104 (2013).
  * Guo  _et al._ (2015) Li Bin Guo, Lingyun Ye, Yuan Xu Wang, Jue Ming Yang, Yu Li Yan, and Feng Zhu Ren, “The electronic structure and thermoelectric properties of BiTl9Te6 and SbTl9Te6: First-principles calculations,” J. Appl. Phys. 118, 235703 (2015).
  * Tao  _et al._ (2011) Xiaoma Tao, Philippe Jund, Romain Viennois, and Jean-Claude Tedenac, “Physical properties of thallium-tellurium based thermoelectric compounds using first-principles simulations,” J. Phys. Chem. A 115, 8761–8766 (2011).
  * Xia  _et al._ (2020b) Yi Xia, Koushik Pal, Jiangang He, Vidvuds Ozoliņš, and Chris Wolverton, “Particlelike Phonon Propagation Dominates Ultralow Lattice Thermal Conductivity in Crystalline Tl3VSe4,” Phys. Rev. Lett. 124, 065901 (2020b).
  * Wang  _et al._ (2023a) Xiaoying Wang, Zhibin Gao, Guimei Zhu, Jie Ren, Lei Hu, Jun Sun, Xiangdong Ding, Yi Xia, and Baowen Li, “Role of high-order anharmonicity and off-diagonal terms in thermal conductivity: A case study of multiphase ${\mathrm{CsPbBr}}_{3}$,” Phys. Rev. B 107, 214308 (2023a).
  * Tadano and Tsuneyuki (2018) Terumasa Tadano and Shinji Tsuneyuki, “Quartic anharmonicity of rattlers and its effect on lattice thermal conductivity of clathrates from first principles,” Phys. Rev. Lett. 120, 105901 (2018).
  * Zheng  _et al._ (2022a) Jiongzhi Zheng, Dongliang Shi, Yuewang Yang, Chongjia Lin, He Huang, Ruiqiang Guo, and Baoling Huang, “Anharmonicity-induced phonon hardening and phonon transport enhancement in crystalline perovskite BaZrO3,” Phys. Rev. B 105, 224303 (2022a).
  * Yue  _et al._ (2024) Jincheng Yue, Jiongzhi Zheng, Junda Li, Siqi Guo, Wenling Ren, Han Liu, Yanhui Liu, and Tian Cui, “Ultralow Glassy Thermal Conductivity and Controllable, Promising Thermoelectric Properties in Crystalline o-CsCu5S3,” ACS Appl. Mater. Interfaces 16, 20597–20609 (2024).
  * Zheng  _et al._ (2024) Jiongzhi Zheng, Changpeng Lin, Chongjia Lin, Geoffroy Hautier, Ruiqiang Guo, and Baoling Huang, “Unravelling ultralow thermal conductivity in perovskite Cs2AgBiBr6: dominant wave-like phonon tunnelling and strong anharmonicity,” npj Comput. Mater. 10, 4941–4948 (2024).
  * Tadano and Tsuneyuki (2015) Terumasa Tadano and Shinji Tsuneyuki, “Self-consistent phonon calculations of lattice dynamical properties in cubic SrTiO3 with first-principles anharmonic force constants,” Phys. Rev. B 92, 054301 (2015).
  * Zhao  _et al._ (2021) Yinchang Zhao, Shuming Zeng, Geng Li, Chao Lian, Zhenhong Dai, Sheng Meng, and Jun Ni, “Lattice thermal conductivity including phonon frequency shifts and scattering rates induced by quartic anharmonicity in cubic oxide and fluoride perovskites,” Phys. Rev. B 104, 224304 (2021).
  * Xia  _et al._ (2020c) Yi Xia, Vinay I. Hegde, Koushik Pal, Xia Hua, Dale Gaines, Shane Patel, Jiangang He, Muratahan Aykol, and Chris Wolverton, “High-throughput study of lattice thermal conductivity in binary rocksalt and zinc blende compounds including higher-order anharmonicity,” Phys. Rev. X 10, 041029 (2020c).
  * Kang  _et al._ (2019) Joon Sang Kang, Huan Wu, Man Li, and Yongjie Hu, “Intrinsic low thermal conductivity and phonon renormalization due to strong anharmonicity of single-crystal tin selenide,” Nano Lett. 19, 4941–4948 (2019).
  * Simoncelli  _et al._ (2019) Michele Simoncelli, Nicola Marzari, and Francesco Mauri, “Unified theory of thermal transport in crystals and glasses,” Nat. Phys. 15, 809–813 (2019).
  * Li  _et al._ (2014) Wu Li, Jesús Carrete, Nebil A. Katcho, and Natalio Mingo, “Shengbte: A solver of the boltzmann transport equation for phonons,” Comput. Phys. Commun. 185, 1747–1758 (2014).
  * Han  _et al._ (2022) Zherui Han, Xiaolong Yang, Wu Li, Tianli Feng, and Xiulin Ruan, “Fourphonon: An extension module to shengbte for computing four-phonon scattering rates and thermal conductivity,” Comput. Phys. Commun. 270, 108179 (2022).
  * Tong  _et al._ (2023) Zhen Tong, Alessandro Pecchia, ChiYung Yam, Traian Dumitrică, and Thomas Frauenheim, “Glass-like Transport Dominates Ultralow Lattice Thermal Conductivity in Modular Crystalline Bi4O4SeCl2,” Nano Lett. 23, 9468–9473 (2023).
  * Mortazavi  _et al._ (2021) Bohayra Mortazavi, Evgeny V. Podryabinkin, Ivan S. Novikov, Timon Rabczuk, Xiaoying Zhuang, and Alexander V. Shapeev, “Accelerating first-principles estimation of thermal conductivity by machine-learning interatomic potentials: A MTP/ShengBTE solution,” Comput. Phys. Commun. 258, 107583 (2021).
  * Wan  _et al._ (2019) Xiao Wan, Wentao Feng, Yunpeng Wang, Haidong Wang, Xing Zhang, Chengcheng Deng, and Nuo Yang, “Materials discovery and properties prediction in thermal transport via materials informatics: A mini review,” Nano Lett. 19, 3387–3395 (2019).
  * Novikov  _et al._ (2020) Ivan S Novikov, Konstantin Gubaev, Evgeny V Podryabinkin, and Alexander V Shapeev, “The MLIP package: moment tensor potentials with MPI and active learning,” Mach. Learn. Sci. Technol. 2, 025002 (2020).
  * Shapeev (2016) Alexander V. Shapeev, “Moment tensor potentials: A class of systematically improvable interatomic potentials,” Multiscale Model. Simul. 14, 1153–1173 (2016).
  * Ouyang  _et al._ (2022) Yulou Ouyang, Cuiqian Yu, Jia He, Pengfei Jiang, Weijun Ren, and Jie Chen, “Accurate description of high-order phonon anharmonicity and lattice thermal conductivity from molecular dynamics simulations with machine learning potential,” Phys. Rev. B 105, 115202 (2022).
  * Feng  _et al._ (2024) Minxuan Feng, Xiaoying Wang, Guimei Zhu, Cheng He, Jun Sun, Xiangdong Ding, Junichiro Shiomi, Yi Xia, Baowen Li, and Zhibin Gao, “The relation between the atomic mass ratio and quartic anharmonicity in alkali metal hydrides,” Mater. Today Phys. 44, 101423 (2024).
  * Li and Mingo (2015) Wu Li and Natalio Mingo, “Ultralow lattice thermal conductivity of the fully filled skutterudite YbFe4Sb12 due to the flat avoided-crossing filler modes,” Phys. Rev. B 91, 144304 (2015).
  * Zheng  _et al._ (2022b) Jiongzhi Zheng, Dongliang Shi, Sijing Liu, Yuewang Yang, Chongjia Lin, Zheng Chang, Ruiqiang Guo, and Baoling Huang, “Effects of high-order anharmonicity on anomalous lattice dynamics and thermal transport in fully filled skutterudite YbFe4Sb12,” Phys. Rev. Mater. 6, 093801 (2022b).
  * Wang  _et al._ (2024a) Yang Wang, Zhibin Gao, Xiaoying Wang, Jinping Sun, Minxuan Feng, Yuzhou Hao, Xuejie Li, Yinchang Zhao, and Xiangdong Ding, “Anomalous thermal conductivity in 2D silica nanocages of immobilizing noble gas atom,” Appl. Phys. Lett. 124, 122205 (2024a).
  * Wang  _et al._ (2024b) Xiaoying Wang, Minxuan Feng, Yi Xia, Jun Sun, Xiangdong Ding, Baowen Li, and Zhibin Gao, “Revisiting lattice thermal conductivity of CsCl: The crucial role of quartic anharmonicity,” Appl. Phys. Lett. 124, 172201 (2024b).
  * He  _et al._ (2018) Yihui He, Liviu Matei, Hee Joon Jung, Kyle M. McCall, Michelle Chen, Constantinos C. Stoumpos, Zhifu Liu, John A. Peters, Duck Young Chung, Bruce W. Wessels, Michael R. Wasielewski, Vinayak P. Dravid, Arnold Burger, and Mercouri G. Kanatzidis, “High spectral resolution of gamma-rays at room temperature by perovskite CsPbBr3 single crystals,” Nat. Commun. , 1609 (2018).
  * Tian  _et al._ (2012) Zhiting Tian, Jivtesh Garg, Keivan Esfarjani, Takuma Shiga, Junichiro Shiomi, and Gang Chen, “Phonon conduction in PbSe, PbTe, and PbTe1-xSex from first-principles calculations,” Phys. Rev. B 85, 184303 (2012).
  * Bardeen and Shockley (1950) J. Bardeen and W. Shockley, “Deformation potentials and mobilities in non-polar crystals,” Phys. Rev. 80, 72–80 (1950).
  * Fröhlich (1954) Herbert Fröhlich, “Electrons in lattice fields,” Adv. Phys. 3, 325–361 (1954).
  * Ganose  _et al._ (2021) Alex M. Ganose, Junsoo Park, Alireza Faghaninia, Rachel Woods-Robinson, Kristin A. Persson, and Anubhav Jain, “Efficient calculation of carrier scattering rates from first principles,” Nat. Commun. 12, 2222 (2021).
  * Zhao  _et al._ (2014) Li-Dong Zhao, Shih-Han Lo, Yongsheng Zhang, Hui Sun, Gangjian Tan, Ctirad Uher, Christopher Wolverton, Vinayak P Dravid, and Mercouri G Kanatzidis, “Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals,” Nature 508, 373–377 (2014).
  * Mahan and Sofo (1996) G D Mahan and J O Sofo, “The best thermoelectric.” Proc. Natl. Acad. Sci. 93, 7436–7439 (1996).
  * Pei  _et al._ (2011) Yanzhong Pei, Xiaoya Shi, Aaron LaLonde, Heng Wang, Lidong Chen, and G. Jeffrey Snyder, “Convergence of electronic bands for high performance bulk thermoelectrics,” Nature 473, 66–69 (2011).
  * Zhu  _et al._ (2021) Rui Zhu, Zhibin Gao, Qijie Liang, Junxiong Hu, Jian-Sheng Wang, Cheng-Wei Qiu, and Andrew Thye Shen Wee, “Observation of anisotropic magnetoresistance in layered nonmagnetic semiconducting PdSe2,” ACS Appl. Mater. Interfaces 13, 37527–37534 (2021).
  * Gong  _et al._ (2016) JJ Gong, AJ Hong, J Shuai, L Li, ZB Yan, ZF Ren, and J-M Liu, “Investigation of the bipolar effect in the thermoelectric material CaMg2Bi2 using a first-principles study,” Phys. Chem. Chem. Phys. 18, 16566–16574 (2016).
  * Gao  _et al._ (2021) Zhibin Gao, Tao Zhu, Kangtai Sun, and Jian-Sheng Wang, “Highly anisotropic thermoelectric properties of two-dimensional As2Te3,” ACS Appl. Electron. Mater. 3, 1610–1620 (2021).
  * Qin  _et al._ (2022) Feiyu Qin, Xiaoying Wang, Lei Hu, Ning Jia, Zhibin Gao, Umut Aydemir, Jun Chen, Xiangdong Ding, and Jun Sun, “Switch of thermal expansions triggered by itinerant electrons in isostructural metal trifluorides,” Inorg. Chem. 61, 21004–21010 (2022).
  * Gao and Wang (2020) Zhibin Gao and Jian-Sheng Wang, “Thermoelectric penta-silicene with a high room-temperature figure of merit,” ACS Appl. Mater. Interfaces 12, 14298–14307 (2020).
  * Gao  _et al._ (2018) Zhibin Gao, Gang Liu, and Jie Ren, “High thermoelectric performance in two-dimensional tellurium: An ab initio study,” ACS Appl. Mater. Interfaces 10, 40702–40709 (2018).
  * Liu  _et al._ (2012) Wei Liu, Xiaojian Tan, Kang Yin, Huijun Liu, Xinfeng Tang, Jing Shi, Qingjie Zhang, and Ctirad Uher, “Convergence of Conduction Bands as a Means of Enhancing Thermoelectric Performance of $n$-Type ${\mathrm{Mg}}_{2}{\mathrm{Si}}_{1-x}{\mathrm{Sn}}_{x}$ Solid Solutions,” Phys. Rev. Lett. 108, 166601 (2012).
  * Lee  _et al._ (2017) Sangwook Lee, Kedar Hippalgaonkar, Fan Yang, Jiawang Hong, Changhyun Ko, Joonki Suh, Kai Liu, Kevin Wang, Jeffrey J. Urban, Xiang Zhang, Chris Dames, Sean A. Hartnoll, Olivier Delaire, and Junqiao Wu, “Anomalously low electronic thermal conductivity in metallic vanadium dioxide,” Science 355, 371–374 (2017).
  * Wang  _et al._ (2023b) Wen O. Wang, Jixun K. Ding, Yoni Schattner, Edwin W. Huang, Brian Moritz, and Thomas P. Devereaux, “The Wiedemann-Franz law in doped Mott insulators without quasiparticles,” Science 382, 1070–1073 (2023b).
  * Tong  _et al._ (2019) Zhen Tong, Shouhang Li, Xiulin Ruan, and Hua Bao, “Comprehensive first-principles analysis of phonon thermal conductivity and electron-phonon coupling in different metals,” Phys. Rev. B 100, 144306 (2019).
  * Zhang  _et al._ (2022) Xinyu Zhang, Shouhang Li, Ao Wang, and Hua Bao, “Pressure-dependent thermal conductivity in Al, W, and Pt: Role of electrons and phonons,” Phys. Rev. B 106, 094313 (2022).