Resonant bonding leads to low lattice thermal
conductivity
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Citation
Lee, Sangyeop, Keivan Esfarjani, Tengfei Luo, Jiawei Zhou,
Zhiting Tian, and Gang Chen. “Resonant Bonding Leads to Low
Lattice Thermal Conductivity.” Nature Communications 5 (April
28, 2014).
As Published
http://dx.doi.org/10.1038/ncomms4525
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Nature Publishing Group
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Thu May 26 10:33:13 EDT 2016
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http://hdl.handle.net/1721.1/91682
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Detailed Terms
Resonant bonding leads to low lattice thermal conductivity
Sangyeop Lee1, Keivan Esfarjani2,3, Tengfei Luo4, Jiawei Zhou1, Zhiting Tian1 and Gang Chen11
1
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,
MA 02139, USA
2
Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick,
NJ 08901, USA
3
Institute for Advanced Materials and Devices for Nanotechnology (IAMDN), Rutgers
University, Piscataway, NJ 08854, USA
4
Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre
Dame, IN 46556, USA
1
Corresponding author: gchen2@mit.edu
1
Abstract
Understanding the lattice dynamics and low thermal conductivities of IV-VI, V2-VI3,
and V materials is critical to the development of better thermoelectric and phase change
materials. In this work, we provide a link between chemical bonding and low thermal
conductivity. Our first-principles calculations reveal that long-ranged interaction along the
<100> direction of rocksalt structure exist in lead chalcogenides, SnTe, Bi2Te3, Bi and Sb due to
the resonant bonding that is common to all of them. This long-ranged interaction in lead
chalcogenides and SnTe cause optical phonon softening, strong anharmonic scattering, and large
phase space for three-phonon scattering processes, which explain why rocksalt IV-VI
compounds have much lower thermal conductivities than zincblende III-V compounds. The new
insights on the relationship between resonant bonding and low thermal conductivity will help in
the development of better thermoelectric and phase change materials.
2
Introduction
Most good thermoelectric and phase change materials are found in group IV-VI, V2-VI3,
and V materials. For example, PbTe, Bi2Te3, and Bi1-xSbx have been the best thermoelectric
materials in the intermediate, room, and low temperature ranges, respectively1, 2, 3, 4. Alloys of
GeTe and Sb2Te3 (GST) have been the most popular materials for optical storage technologies
such as compact disc and phase change random access memory5, 6. These materials all have low
thermal conductivity, which is crucial for thermoelectric and phase change memory applications.
Usually, group IV-VI, V2-VI3, and V materials have low thermal conductivity. This
becomes particularly obvious when IV-VI materials are compared to III-V compounds. For
example, the thermal conductivity of SnTe is only 4 W/m-K while that of InSb, adjacent to SnTe
in the periodic table, is 16 W/m-K at room temperature7, 8. The low thermal conductivities of
rocksalt IV-VI materials (hereafter IV-VI materials) compared to those of zincblende III-V
materials (hereafter III-V materials) have been attributed to their differences in the crystal
structure. While III-V materials have tetrahedral bonding, many IV-VI materials have octahedral
bonding. The bond length is usually longer in octahedral structures than in tetrahedral structures,
resulting in weaker bonding and lower thermal conductivity9.
Our first principles calculations show that there are more reasons for the low thermal
conductivities of IV-VI materials than that discussed above. To compare the thermal
conductivities of many different III-V and IV-VI materials, we normalized their thermal
conductivities by harmonic properties using the formula suggested by Slack10. The formula gives
reasonable predictions for many materials with zincblende and rocksalt structures11. According to
3
the formula, thermal conductivity is given by
𝜅=𝐵∙
̅ 𝑛1/3 𝛿𝜃3
𝑀
(1)
𝛾2
̅ is the average mass of the basis atoms, 𝑛 is the number of
where 𝐵 is a numerical coefficient, 𝑀
phonon branches, and 𝛾 is the Grüneisen parameter. The average volume per atom is denoted by
̅ , 𝜃, and 𝛿;
𝛿 3 , and 𝜃 is the acoustic Debye temperature. Here, harmonic properties include 𝑀
these three properties determine the average group velocities of acoustic phonons and they
reflect the bonding stiffness. In Fig. 1 we plot the thermal conductivities, 𝜅, of IV-VI and III-V
̅ 𝑛1/3 𝛿𝜃 3 , as a function of the mass ratio of
materials normalized by the harmonic properties, 𝑀
the basis atoms. There are two distinct differences between the thermal conductivities of IV-VI
and III-V materials: 1) overall, IV-VI compounds still have much lower thermal conductivities
than III-V compounds even after normalization, and 2) the thermal conductivity difference
between IV-VI and III-V materials is amplified when the mass ratio is small.
In this work, we present an unusual behavior in the lattice dynamics of IV-VI, V2-VI3,
and V materials and the origin of the two aforementioned differences in the thermal
conductivities of III-V and IV-VI compounds. We first show that seemingly different IV-VI, V2VI3, and V materials commonly have long-ranged interactions along the <100> direction of
rocksalt as a result of the resonant bonding. Then, we infer that the significant long-ranged
interaction in IV-VI materials play a key role in their low thermal conductivities.
4
Results
Resonant bonding in IV-VI, V2-VI3, and V materials
Resonant bonding can be understood as resonance or hybridization between different
electronic configurations: three valence p-electrons alternate their occupancy of six available
covalent bonds that exist between a given atom and its octahedral neighbors12. For example, in
PbTe, sp-hybridization is small and the s-band is lower than the p-band by 1.5 eV (Ref. 13).
Therefore, we can consider only p-electrons for valence states and each atom has three valence
electrons on average. Given PbTe’s octahedral structure and its three valence electrons per atom,
the choice of bond occupation is not unique. This leads to hybridization between all the possible
choices for the three electrons forming the six bonds. This description of resonant bonding is
based on IV-VI compounds and their isoelectronic V elements for simplicity, but the resonant
bonding exists in even more complicated materials such as V2-VI3 and many alloys of IV-VI and
V2-VI3 materials14. In general, the unsaturated covalent bonding by p-electrons with rocksalt-like
crystal structures can be regarded as resonant bonding15.
Materials with resonant bonding have several features. First, because of their
coordination number of six, they have rocksalt-like crystal structures. Many IV-VI, V2-VI3, and
V compounds have rocksalt-like crystal structures, as shown in Fig. 2. In addition, these
materials have very weak sp-hybridization and the s-bands are well below the p-bands. Several
past works show that PbTe, Bi2Te3, and Bi satisfy this condition13, 16, 17 and our calculations using
density functional theory also present weak sp-hybridization in these materials (see
Supplementary Note 1 and Supplementary Fig. 1 – 7). One very important feature of resonant
5
bonding is that the electron density distribution is highly delocalized. As a result, the materials in
this group have large electronic polarizability, dielectric constants, and Born charges18, 19. For
example, the dielectric constants of PbTe and Bi2Te3(∥) are 33 and 50, respectively, while for Si
this value is 11.76 (Ref. 20, 21, 22). In a similar way, certain electronic properties of
thermoelectric and phase change materials have also been explained by invoking resonant
bonding12, 15. Matsunaga et al. used experimental observations of low frequency of transverse
optical (TO) phonons, so called soft TO mode (a feature that is clear in the phonon dispersion to
be shown later), to make a connection between resonant bonding and the small differences in the
thermal conductivities of crystalline and amorphous phases of GST23. Here, from first principles,
we provide a detailed lattice dynamics-based link between resonant bonding and the soft TO
modes and low thermal conductivities of a wide range of materials.
Long-ranged interaction due to the resonant bonding
We calculated the harmonic force constants in typical IV-VI, V2-VI3, and V materials
using first-principles density functional theory and found that a common feature among these
compounds is the presence of long-ranged interactions along the <100> direction of rocksalt
structure. To compare these different materials regardless of crystal structures and bonding
stiffness, traces of interatomic force constant (IFC) tensors are normalized by the trace values of
the self-interaction IFC tensor. (see Supplementary Note 2 for details) From Fig. 3a, fourthnearest neighbors, separated by 6 Å (e.g. Pb-Pb and Te-Te), have interactions which are
comparable to those of first-nearest neighbors, spaced 3 Å apart, and are much stronger than
6
second- and third-nearest neighbor interactions. In addition, eighth-nearest neighbors, separated
by 9 Å, have even positive force constants, giving them the behavior of “anti-springs”.
Fourteenth-nearest neighbors, separated by 12 Å, also have non-negligible force constants. The
force constants at the fourteenth-nearest neighbors are clearly distinguished from other force
constants nearby when using finer q mesh in the electron response calculation to capture the
long-ranged interaction more accurately. These first-, fourth-, eighth-, and fourteenth-nearest
neighbors form a chain along the <100> direction in rocksalt structures, as shown in Fig. 2.
Other rocksalt IV-VI materials, such as PbSe, PbS, and SnTe, exhibit very similar behaviors.
These behaviors were not captured by earlier works on the lattice dynamics of PbTe and PbS
within the shell model24, 25. To compare lead chalcogenides and SnTe with other materials, in Fig.
3b we show that the IFCs of NaCl and InSb decrease with distance. NaCl and InSb are chosen as
prototypes of ionic and sp-hybridized covalent bonding materials, respectively. The long-ranged
interactions along the <100> direction in NaCl are much smaller compared to those in PbTe. In
the case of InSb, interactions besides for first-nearest neighbor ones are negligible.
We need to point out that the long-ranged interactions we observe are different from
long-ranged Coulomb interaction, which cannot explain why fourth-nearest neighbor interactions
are stronger than second- or third-nearest neighbor interactions. The long-ranged and nonmonotonically decreasing interaction is due to the long-ranged electronic polarizability. (see
Supplementary Note 3 for details) In Fig. 4a and Fig.4b, we compare electron density
distribution at ground state of PbTe and NaCl. From the ground state electron density
distribution, it is clear that PbTe has largely delocalized electron density distribution due to the
resonant bonding, but electrons in NaCl are highly localized due to the ionic bonding. In Fig. 4e,
7
electron polarization in NaCl is short-ranged and electrons surrounding the fourth-nearest
neighbors are not perturbed much. However, electron polarization in PbTe in Fig. 4f is longranged and reach fourth-nearest neighbor. The electron density distribution surrounding the
fourth-nearest neighbors is largely disturbed by the displacement of the center atom.
The long-ranged polarization in PbTe can be explained by the resonant bonding. In
resonant bonding, if one atom is displaced along the +x direction, it perturbs the px orbital of the
adjacent atom. In other words, the bonding electrons on the –x side of the adjacent atom easily
move to the +x side since both sides are in the same px orbital26. This perturbation can persist
over long ranges due to the large electronic polarizability and the collinear bonding in resonant
bonding materials. The band-by-band electron polarization analysis in Fig. 4 confirms that the
long-ranged polarization is due to resonantly occupied p-electrons. PbTe has very weak sphybridization and we plot the electronic polarization for s-band and p-band electrons separately
in Fig. 4f-h. From Fig. 4g, the polarization of s-electron is short-ranged and does not reach the
fourth-nearest neighbor. However, the p-electron distribution in Fig. 4h exhibits the long-ranged
polarization. This analysis shows that the easily polarized electrons in PbTe are resonantly
occupied p-electrons, rather than lone pair Pb s electrons suggested in recent work27.
The resonant bonding picture discussed above also applies to V2-VI3 (Bi2Te3) and V (Bi
and Sb) materials. Bi2Te3 has a rhombohedral structure which can be understood as a deformed
rocksalt structure with layer spacing as shown in Fig. 2. This rocksalt-like structure contains five
resonantly bonded layers of atoms (Te1-Bi-Te2-Bi-Te1), and is separated from the next quintuple
layer by weak Te1-Te1 van der Waals interactions. The structural deviation from the exact
rocksalt structure within the quintuple layer is small. The lengths of the strongest bond (Te1-Bi),
8
3.03 Å, and that of the second-strongest bond (Bi-Te2), 3.22 Å, are similar. In addition, the
angles of Te1-Bi-Te2 and Bi-Te2-Bi are 174.6 and 180, making them similar to rocksalt in
structure since the rocksalt has an angle of 180 exactly. Due to this small structural distortion,
the resonant bonding exists in a weakened form. As shown in Fig. 3c, the IFCs of Bi2Te3 show a
behavior similar to that of lead chalcogenides, but the long-ranged interactions are weakened due
to the structural distortions, resulting in weakened resonant bonding. Bi-Bi and Te1-Te2, both
spaced about 6 Å apart, have interactions which are equivalent to the fourth-nearest neighbor
interactions in an exact rocksalt structure, and are less significant compared to the PbTe case.
The interactions of Bi-Te1 (at a distance of 9 Å) separated by Te2-Bi, which are equivalent to the
eighth-nearest neighbor interactions in PbTe, also have positive force constants, but their
magnitudes are smaller than those of the PbTe case. However, it is noticeable that the force
constants of Te2 atoms are very similar to those of the Te atom in PbTe, as predicted in previous
work19. This is because the resonant bonding around the Te2 atom is well maintained: the Bi-Te2
and Te2-Bi bond lengths are same and they make an angle of 180, as shown above. Another
noticeable observation in Bi2Te3 is that there is no long-ranged interaction between atoms
separated by Te1-Te1. It is well known that Te1-Te1 has van der Waals type bonding due to the
induced dipole-dipole interaction16, prohibiting the long-ranged interaction by the long-ranged
electron polarization shown in Fig. 4.
Pure Bi and Sb have rhombohedral crystal structures which are Peierls distortions of
simple cubic structure28. With the two basis atoms, the structure can be understood as a rocksalt
structure stretched along the <111> direction. Because of the distortion, the six first-nearest
neighbors in the rocksalt structure become three first-nearest neighbors and three second-nearest
9
neighbors. The distances between the first- and the second-nearest neighbors are 3.05 Å and 3.49
Å, respectively. Considering that the distances between the first- and the second-nearest
neighbors are 3.03 Å and 3.22 Å, respectively, in Bi2Te3, the distortion from the rocksalt
structure is larger in Bi and Sb than in Bi2Te3. Therefore, the resonant bonding is further
weakened in Bi and Sb and this is clearly visible in the electron density distributions of PbTe and
Bi (Supplementary Note 4 and Supplementary Fig. 8). This structural distortion further weakens
the long-ranged interactions, as shown in Fig. 3d. However, the ninth-nearest neighbors,
separated by 6 Å, have interactions which correspond to the fourth-nearest neighbor interactions
in rocksalt and which are thus significant.
Effects of the long-ranged interaction on lattice thermal conductivity
The long-ranged interaction along the <100> direction is related to the existence of a
soft TO mode. This directional long-ranged interaction was also predicted and considered as a
main reason for the ferroelectric behavior in perovskite BaTiO3 and PbTiO3 (Ref. 29). We used a
1D lattice chain model to explicitly show that the long-ranged interaction causes the softening of
the TO mode. (see Supplementary Note 5 for details) In Fig. 5a, as the second- and third-nearest
neighbor interactions in the 1D chain (equivalent to fourth- and eighth-nearest neighbor
interactions in rocksalt) increased, the zone center TO phonon frequency decreased. The
pronounced softening of the TO modes in IV-VI materials, shown in Fig. 5b, is consistent with
this 1D model. To further confirm that the long-ranged interactions along <100> are the main
reason for the near ferroelectric behavior, phonon dispersion is calculated for fictitious PbTe
10
without the fourth- and eighth-nearest neighbor interactions. The TO mode in the fictitious PbTe
is not softened as shown in Fig. 5b. By comparison, Bi2Te3, Bi, and Sb have weakened longranged interactions due to distortion of the structure, and their TO modes are not as soft as those
of IV-VI materials (Fig. 5c and 5d).
The TO phonon softening leads to strong anharmonicity and phonon scattering by the
TO modes30. To show the anharmonicity of the modes, we plot the calculated mode Grüneisen
parameters of these materials in Fig. 5e. The TO modes in lead chalcogenides and SnTe have
remarkably large mode Grüneisen parameters. Bi2Te3, Bi, and Sb also have increasing Grüneisen
parameters as the zone center is approached, but the magnitude is not as large as in IV-VI
materials. The strong anharmonic TO modes in IV-VI materials lead to their low lattice thermal
conductivities and this was predicted and experimentally observed30, 31, 32, 33, 34. In Fig. 6, we
show detailed phonon transport characteristics in several III-V and IV-VI materials, calculated
from density functional theory. We calculate phonon transport of SnTe and InSb in addition to
PbTe, PbSe and GaAs from literature33,35. It is interesting to directly compare SnTe and InSb as
these materials are closed to each other in the periodic table, and therefore they have similar
Debye temperature and mass ratio. The thermal conductivity calculation of InSb using first
principles was recently reported36. The calculated and experimental thermal conductivity values
in Fig. 6a show large contrast between III-V and IV-VI materials. We further contrast the two
different material groups by analyzing phonon mean free path and phonon life time. For the
phonon mean free path, we present accumulated thermal conductivity as a function of phonon
mean free path at 300 K, defined as 𝜅acc (Λ) = ∑𝐪𝑠 𝜅𝐪𝑠 𝜒(Λ) where Λ, 𝐪, s, and 𝜒 represent
phonon mean free path, wavevector, polarization, and a step function37, 38. The accumulated
11
thermal conductivity function shows the mean free path ranges of phonon modes that
significantly contribute to thermal conductivity. From Fig. 6b-c, it is clearly seen that the IV-VI
materials exhibit much shorter mean free path ranges and phonon lifetime compared to the III-V
materials. The significant effect of soft TO mode on lattice thermal conductivity is also
substantiated by the comparison between PbTe and Bi. The thermal conductivity of PbTe is
smaller than that of Bi by factor of two, even though atomic mass of PbTe is less than that of Bi
(Ref.39, 40). (Supplementary Note 6 for details).
In addition to the phonon anharmonicity, the phonon lifetimes also depend on the threephonon scattering phase space available that meets energy and momentum conservation
requirements. The difference in the volume of the scattering phase space explains the second
difference between III-V and IV-VI materials in Fig. 1 (i.e., the thermal conductivity difference
between III-V and IV-VI materials is much increased when the mass difference between the two
atoms is large). To compare IV-VI and III-V materials, we plot the inverse of the three-phonon
scattering phase space volume of these materials in Fig. 7a. Assuming constant phonon
anharmonicity, the inverse of the three-phonon scattering phase space volume relates linearly to
phonon lifetime and hence thermal conductivity. Since the scattering phase space is inversely
proportional to the frequency scale, the integrated volume of the phase space of each material is
normalized by the inverse of maximum optical phonon frequency to compare many materials
with different phonon frequency scale. The phase-space volume for III-V we calculated is similar
to the previously reported data using a bond charge model41. From Fig. 7a, we can see that III-V
materials with dissimilar atomic masses such as AlSb and InP have much smaller phase spaces
than other III-V materials. This is a common behavior for many materials which have large
12
atomic mass differences42. The large mass difference causes a large gap to appear between the
acoustic and optical phonon bands as shown in Fig. 7b. With such a large gap, low frequency
acoustic phonons cannot couple to optical phonons and thus have fewer chances to be scattered.
Only high frequency acoustic phonons, which are localized in a small region of reciprocal space,
can be coupled with optical phonons. As the mass ratio is reduced from unity, the acousticoptical phonon gap becomes larger and the scattering phase space is further reduced, leading to
longer lifetimes and larger thermal conductivities. As can be seen from Fig. 7c, the phase space
in AlSb is mostly due to (a,a,a) and the phase spaces of (a,a,o) and (a,o,o) are significantly
suppressed. (‘a’ and ‘o’ in (a,a,o) indicate acoustic and optical phonons in the three phonon
process.)
The three-phonon scattering pathway in IV-VI materials is much different from that of
III-V materials since (a,o,o) channel significantly contribute to the total phase space volume in
IV-VI materials. The difference in the scattering pathway between III-V and IV-VI materials can
be easily observed in Fig. 7a; the phase space volume of IV-VI materials increases slightly as the
mass ratio decreases, an opposite trend to the case of III-V materials. The reason for this opposite
trend is that (a,o,o) channel is a large contributor to total phase space volume in IV-VI materials,
and this scattering channel is affected by the overlap between acoustic and optical bands. (see
Supplementary Note 7 for detailed discussions) The large phase space volume of (a,o,o) process
can be explained with the wide spectrum of optical modes. The bandwidth of the optical phonons
is comparable to the band width of the acoustic phonons. As a result, most acoustic phonons
from very low frequencies to high frequencies can participate in the (a,o,o) process, widening
(a,o,o) channel in IV-VI materials. In particular, the (a,o,o) processes in PbS contribute more
13
than half of the total phase space. Additionally, (a,a,o) channel also considerably contributes to
total phase space volume in the case of PbS, while it is almost negligible in AlSb. This is due to
the reduced phonon band gap by soft TO mode; with the reduced phonon band gap, most
acoustic phonons, regardless of frequency, can be scattered by a TO mode contributing to the
(a,a,o) process, while only high frequency acoustic phonons can participate in the (a,a,o) process
when the gap is large. The large (a,o,o) and (a,a,o) scattering phase space of PbSe and PbS
suggests that acoustic phonons are effectively scattered by optical phonons, exhibiting low
thermal conductivity values despite of the small mass ratio as shown in Fig. 1.
14
Discussion
We have presented the effects of resonant bonding on lattice dynamics characteristics
and thermal conductivity. We revealed that materials with resonant bonding (lead chalcogenides,
SnTe, Bi2Te3, Bi and Sb) commonly have long-ranged interactions along the <100> direction in
rocksalt structure. This long-ranged interaction is significant in IV-VI materials due to the strong
resonant bonding, and results in the near ferroelectric instability in these materials. However, the
long-ranged interaction is less significant with increasing distortion of the crystal structures as in
Bi. The near ferroelectric behavior caused by the resonant bonding reduces the lattice thermal
conductivity through two mechanisms: strong anharmonic scattering and a large scattering phase
space volume, both due to softened optical phonons, resulting in the lower thermal conductivities
of IV-VI materials compared to III-V compounds. Thus, the low thermal conductivities of these
materials are traced back to their crystal structure and electronic occupation (the resonant nature
of the bonding). Our findings have significant implications for designing better thermoelectric
and phase change materials. The fundamental understanding of lattice dynamics from chemical
bonding points to the potential to search for good thermoelectric materials through the resonant
bonding picture. Also, we showed a deep connection among ferroelectric, thermoelectric and
phase change materials. These insights help researchers to explore better thermoelectric and
phase-change materials.
15
Methods
Lattice dynamics calculation
We used density functional perturbation theory to calculate harmonic lattice dynamics properties
of the materials43, 44. We checked that results are well converged with respect to many parameters
such as cut off energy of plane wave basis and reciprocal space sampling size (k- and q- grid).
We provide detailed calculation conditions for each material in Supplementary Table 1. Since the
long-ranged interaction is significant in the resonant bonding materials, we used relatively fine
q-grid to capture the long-ranged interaction. Also, for heavy elements such as Bi, Pb, Te, Sb,
and Se, spin-orbit interaction is included in the calculation. After calculating IFC tensors, we
normalize the trace of IFC tensor to compare many materials with different crystal structure and
different strength of bonding. (Supplementary Note 2 for details) To calculate mode Grüneisen
parameter, we calculated phonon frequencies for two cases: equilibrium volume and expanded
volume by 1%. Then, we took differences in phonon frequencies of two cases and calculate
Grüneisen parameter by its definition, γ = − 𝑉𝑑𝜔⁄𝜔𝑑𝑉 . The calculations were performed with
the QUANTUM ESPRESSO for lead chalcogenides and SnTe and the ABINIT package for other
materials45, 46.
Electron density distribution
To confirm that electron polarization is long-ranged in PbTe, but short-ranged in NaCl, we
calculated electron density distributions in PbTe and NaCl using density functional theory for
16
two different cases: without any displacement and with small displacement of one atom. We
displaced a Pb atom and a Na atom for PbTe and NaCl, respectively. The displacement is 2% of
the fourth-nearest neighbor distance in –x direction in Fig. 4. Since a periodic boundary
condition is used in the electron distribution calculation, there is an effect from the periodic
images of the displaced atom. Therefore, we calculated large enough supercells (24 atoms) to
minimize this effect. Other calculation conditions are same as the phonon calculation conditions
in Supplementary Table 1. After calculating electron density of two cases, we took finite
difference of two cases to calculate change in electron density distribution by the displacement.
Phase space volume for three phonon scattering
The phase space integral for three phonon scattering is the volume satisfying energy and
momentum conservation in the three phonon process. Therefore, it can be defined as41
𝐪
𝐪′
𝐪+𝐪′ −𝐆
𝛿 (𝜔𝑗 + 𝜔𝑗′ − 𝜔𝑗′′
)
Phase space volume = ∑𝑗 ∫ d𝐪 ∑𝐺,𝑗′ ,𝑗′′ ∫ d𝐪′ ( 1
)
𝐪
𝐪′
𝐪−𝐪′ −𝐆
+ 2 𝛿 (𝜔𝑗 − 𝜔𝑗′ − 𝜔𝑗′′
)
(2)
where 𝑗, 𝑗′, and 𝑗′′ represent phonon branches. The two 𝛿-functions are for energy conservation
of coalescent and decay processes. The superscripts, 𝐪 + 𝐪′ − 𝐆 and 𝐪 − 𝐪′ − 𝐆 denote
momentum conservation. The factor of half in front of the second term (decay process) is to
prevent counting twice when summing up all the possible scattering pathways. For the
integration of the 𝛿-functions, we used a q-grid of 20×20×20 and confirmed the convergence of
phase space with respect to the q-grid size. As seen in the expression for phase space volume
17
above, phase space volume is integration of 𝛿-functions of phonon frequencies. Therefore, the
phase space volume is inversely proportional to phonon frequency scale. To compare many
materials with different phonon frequency scale, we normalized the phase space volume by the
inverse of maximum optical phonon frequency. To validate the phonon dispersion used in the
phase space volume calculation, we present calculated phonon dispersions compared to the
experimental data in Supplementary Fig. 15 and 16.
Lattice thermal conductivity calculations using density functional theory
We used density functional theory to calculate lattice thermal conductivity of InSb and SnTe.
Here we briefly present the method we used and more details are available in other literature47.
We calculate second-order force constants using density functional perturbation theory and the
calculation conditions for the force constants are shown in Supplementary Table 1. The
calculated second-order force constants were validated by comparing the calculated phonon
dispersion to the experimental data in Supplementary Fig. 15 and 16. For third-order force
constants, we calculate forces on atoms in large enough supercells (4×4×4 supercells, 128 atoms)
with various atomic displacements. The set of third-order force constants, minimally reduced by
symmetry considerations, is then fitted to the calculated force-displacement data with enforcing
translational and rotational invariances48. We considered up to second-nearest neighbors and
fifth-nearest neighbors for the third-order force constants of InSb and SnTe, respectively. The
thermal expansion coefficients were calculated using the second- and third-order force constants,
18
showing reasonable agreement with the experimental values. (Supplementary Note 8) The
calculated second- and third-order force constants gives the rate of three phonon scatterings via
Fermi’s golden rule and also other properties such as phonon frequency and group velocity.
Then, with all these information, the lattice thermal conductivity was calculated by solving
Boltzmann equation under relaxation time approximation.
19
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23
Acknowledgements
We thank C. Carlton, M. Dresselhaus, M. Luckyanova, V. Chiloyan, D. Singh, L.
Lindsay and S. Billinge for useful discussion. This work was partially supported by “Solid State
Solar-Thermal Energy Conversion Center (S3TEC),” an Energy Frontier Research Center funded
by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under
Award Number: DE-SC0001299/DE-FG02-09ER46577 (for PbTe and Bi2Te3), and partially
supported by U.S. Department of Defense AFOSR MURI via Ohio State (for Bi and Sb). S.L.
acknowledges support from Samsung scholarship.
Author Contributions
S.L., K.E., T.L., J.Z., and Z.T. carried out the first principles calculations. S.L., K.E., and
G.C. analyzed the calculation data. S.L. and G.C. wrote the manuscript. G.C. supervised the
research. All authors commented on, discussed and edited the manuscript.
Competing financial interest statement
The authors declare no competing financial interests.
24
Figure Legends
Figure 1. Normalized thermal conductivities of binary III-V and IV-VI compounds at 300
K. The solid lines are for eye-guide.
Figure 2. Rocksalt-like crystal structures of PbTe, Bi2Te3, and Bi. The number on each atom
indicates the shell number. Bi2Te3 , Bi and Sb have distorted rocksalt structures and have
different numbers for shells than the exact rocksalt case. The numbers on the Bi2Te3 and Bi
atoms indicate the equivalent shell numbers as for a rocksalt structure in the absence of lattice
distortion.
Figure 3. Normalized trace of interatomic force constant tensors versus atomic distances.
(a) lead chalcogenides and SnTe (group IV-VI), (b) NaCl and InSb, (c) Bi2Te3 (group V2-VI3),
and (d) Bi and Sb (group V). The element in the parenthesis indicates interaction between the
corresponding atom and other atoms. For example, ‘PbTe(Pb)’ means interaction between Pb and
other atoms in PbTe.
Figure 4. Electron density distribution and polarization in NaCl and PbTe. (a-d) electron
density distribution at ground state. (e-h) electron density distribution change by the
displacement of the center atom. The plot is on (100) plane and each black dot represents an
atom. The unit is Å-3.
25
Figure 5. Near ferroelectric behavior due to resonant bonding. (a) Optical phonon dispersion
in a model 1D atomic chain, showing the softening of the optical phonons due to the long-ranged
interactions. Three numbers in the legend represent relative interaction strength of first, second
and third-nearest neighbors in the 1D chain. (b-d) Soft TO phonon modes along trigonal
direction for lead chalcogenides, SnTe, Bi2Te3, and Bi and Sb, respectively calculated based on
first principles. Lines and circles are calculation and experimental data, respectively. The
experimental data are from Ref. 19, 24, 25, 49, 50, 51. The red dotted line in b is after removing
the fourth-, eighth-, and fourteenth-nearest neighbors interactions in PbTe, which does not show
the soft TO mode. (b-d) are plotted in the same scale of y-axis. (e) Calculated mode Grüneisen
parameters of TO mode, showing strongly anharmonic behavior of TO phonons of lead
chalcogenides and SnTe. The red dotted line is Grüneisen parameters of longitudinal acoustic
mode in PbTe for comparison.
Figure 6. Analysis of phonon transport in IV-VI and III-V materials by first principles
calculation. (a) Calculated and experimental lattice thermal conductivities. Lines and squares
are results by experiments and calculations, respectively. (b-c) Phonon mean free path
distributions and phonon lifetime, showing significant three-phonon scattering in IV-VI
materials. The accumulated thermal conductivity is normalized by the thermal conductivity value
of each material. The data in (b-c) are for 300 K case. The experimental thermal conductivity
data in (a) are from Ref. 7, 8, 39, 52 and the calculation results for PbTe, PbSe and GaAs are
from Ref. 33, 35.
26
Figure 7. Phase space volume for three-phonon scattering. (a) Phase space volumes for three
phonon scattering of IV-VI and III-V, showing the large scattering phase space of PbSe and PbS.
(b) Comparison of the phonon dispersion of PbS and AlSb, showing significantly dispersed
optical phonons of PbS. (c) Contribution of each scattering channel to total scattering phase
space volume. The scattering phase space and phonon dispersion data are normalized by the
inverse of the largest optical phonon frequency of each material for comparison.
27
Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
Supplementary Figures
Supplementary Figure 1. Weak sp-hybridization in PbTe. (a) Electronic structure along high
symmetry lines, (b) Projected density-of-states, and (c) k-resolved projected density-of-states
1
Supplementary Figure 2. Weak sp-hybridization in PbSe. (a) Electronic structure along high
symmetry lines, (b) Projected density-of-states, and (c) k-resolved projected density-of-states
2
Supplementary Figure 3. Weak sp-hybridization in PbS. (a) Electronic structure along high
symmetry lines, (b) Projected density-of-states, and (c) k-resolved projected density-of-states
3
Supplementary Figure 4. Weak sp-hybridization in SnTe. (a) Electronic structure along high
symmetry lines, (b) Projected density-of-states, and (c) k-resolved projected density-of-states
4
Supplementary Figure 5. Weak sp-hybridization in Bi2Te3. (a) Electronic structure along high
symmetry lines, (b) Projected density-of-states, and (c) k-resolved projected density-of-states
5
Supplementary Figure 6. Weak sp-hybridization in Bi. (a) Electronic structure along high
symmetry lines, (b) Projected density-of-states, and (c) k-resolved projected density-of-states
6
Supplementary Figure 7. Weak sp-hybridization in Sb. (a) Electronic structure along high
symmetry lines, (b) Projected density-of-states, and (c) k-resolved projected density-of-states
7
Supplementary Figure 8. Weak resonant bonding in Bi compared to PbTe. Electron density
distribution at ground state in (a) PbTe and (b) Bi shown on the plane equivalent to (100) in
rocksalt structure. The unit is Å-3.
8
Supplementary Figure 9. Diatomic 1D chain. The numbers on the atoms indicate the shell
number.
9
4
Frequency (THz)
PbTe
Bi
3
2
1
0
L

X
Supplementary Figure 10. Phonon dispersions of PbTe and Bi showing lower group velocity
in Bi
10
PbTe (exp)
PbTe (cal)
Bi (exp)
Bi (cal)
-1
-1
Lattice thermal conductivity (Wm K )
25
20
15
10
5
0
50
100
150
200
250
300
Temperature (K)
Supplementary Figure 11. Lattice thermal conductivity of PbTe and Bi
11
Phase space
PbTe
Bi
aaa
aoo
aao
total
Supplementary Figure 12. Normalized phase space volume for three-phonon scattering of
PbTe and Bi
12
Supplementary Figure 13. Channel-by-channel analysis of phase space volume for three
phonon scattering in III-V and IV-VI materials
13
Supplementary Figure 14. Phonon dispersions of SnTe, PbTe, PbSe, and PbS. Phonon
frequency is normalized by the highest frequency value of each material for comparison.
14
Supplementary Figure 15. Phonon dispersions of III-V materials compared to the
experimental values. The experimental data are from Ref. 1, 2, 3, 4, 5, 6
15
Supplementary Figure 16. Phonon dispersions of IV-VI materials compared to the
experimental values. The experimental data are from Ref. 7, 8, 9, 10
16
Supplementary Figure 17. Calculated mode Grüneisen parameters of (a) InSb and (b)
SnTe. Lines are from Supplementary Equation 10 and circles are from the finite difference in
phonon frequencies calculated in crystals with different volumes. Acoustic and optical modes are
shown in black and red, respectively.
17
Supplementary Figure 18. Calculated volumetric thermal expansion coefficients of (a) InSb
and (b) SnTe compared to the experimental values. The experimental data are from Ref. 11,
12.
18
Supplementary Tables
PbTe
Cut
off k-mesh
energy (Ry) (for electrons)
60
16×16×16
q-mesh
(for phonons)
8×8×8
PbSe
60
16×16×16
8×8×8
PbS
60
16×16×16
8×8×8
SnTe
60
16×16×16
8×8×8
Bi2Te3
40
10×10×10
6×6×6
Bi
30
16×16×16
8×8×8
Sb
30
16×16×16
8×8×8
NaCl
30
8×8×8
AlSb
40
InP
40
InSb
50
GaP
40
GaSb
60
GaAs
40
8×8×8
(with 4 shifts)
10×10×10
(with 4 shifts)
10×10×10
(with 4 shifts)
10×10×10
(with 4 shifts)
10×10×10
(with 4 shifts)
8×8×8
(with 4 shifts)
10×10×10
(with 4 shifts)
8×8×8
8×8×8
8×8×8
8×8×8
8×8×8
8×8×8
XC-functional
and
Pseudopotential
Perdew-Zunger13
Norm conserving
Perdew-Zunger13
Norm conserving
Perdew-Zunger13
Norm conserving
Perdew-Zunger13
Norm conserving
Hartwigsen-GoedeckerHutter14
Hartwigsen-GoedeckerHutter14
Hartwigsen-GoedeckerHutter14
Hartwigsen-GoedeckerHutter14
Hartwigsen-GoedeckerHutter14
Hartwigsen-GoedeckerHutter14
Hartwigsen-GoedeckerHutter14
Hartwigsen-GoedeckerHutter14
Hartwigsen-GoedeckerHutter14
Hartwigsen-GoedeckerHutter14
Supplementary Table 1. Calculation conditions for the density functional theory
calculation.
19
Supplementary Notes
Supplementary Note 1. Electronic structure of the resonant bonding materials
The definition of the resonant bonding is unsaturated (or half-filled) covalent bonding by pelectrons as discussed in other literatures15, 16 From this definition, the prerequisite for resonant
bonding is weak sp-hybridization to exclude s-electrons for chemical bonds. In Supplementary
Figures 1 to 7, we present the density-of-states that projected onto s- and p-orbitals using density
functional theory calculations, confirming that the resonant bonding materials we study – PbTe,
PbSe, PbS, SnTe, Bi2Te3, Bi, and Sb – have very weak sp-hybridization.
The example we present here shows that density functional theory calculation may be useful to
explore more materials with resonant bonding. The resonant bonding is usually explained with
rocksalt IV-VI or V materials for simplicity15, 16. However, the resonant bonding is not limited to
rocksalt-like IV-VI and their isoelectronic group V elements, but exists in even more
complicated materials with seemingly different crystal structures such as Bi2Te3 and many alloys
of GeTe and Sb2Te3 (GST) as discussed in our paper and other literatures17, 18 This implies that
resonant bonding may exist in many materials in general if they have rocksalt-like crystal
structure and weak sp-hybridization according to the definition of resonant bonding we
mentioned above. These two conditions can be easily checked using density functional theory in
the material design process.
Supplementary Note 2. Normalization of interatomic force constants (IFC)
To compare many materials with different crystal structures and different strength of bonding, we
calculated normalized trace of IFC tensors. The IFC tensor from first-principles calculation is
20
𝜕2 𝐸
𝜕𝑅𝐼 𝜕𝑅𝐽
=
𝜕2 𝐸
𝜕2 𝐸
𝜕2 𝐸
𝜕𝑅𝑥 𝜕𝑅𝑥
𝜕𝑅𝑥 𝜕𝑅𝑦
𝜕𝑅𝑥 𝜕𝑅𝑧
𝜕2 𝐸
𝜕2 𝐸
𝜕2 𝐸
𝜕𝑅𝑦 𝜕𝑅𝑥
𝜕𝑅𝑦 𝜕𝑅𝑦
𝜕𝑅𝑦 𝜕𝑅𝑧
𝜕2 𝐸
𝜕2 𝐸
𝜕2 𝐸
[ 𝜕𝑅𝑧 𝜕𝑅𝑥
𝜕𝑅𝑧 𝜕𝑅𝑦
𝜕𝑅𝑧 𝜕𝑅𝑧 ]
(1)
where 𝐸 and 𝑅𝛼 represent energy and atomic position along α direction, respectively. By taking
trace of IFC tensors, we can assess bonding stiffness regardless of crystal structure or coordinate
system.
𝜕2 𝐸
trace of IFC = 𝜕𝑅
𝜕2 𝐸
𝑥 𝜕𝑅𝑥
+ 𝜕𝑅
𝑦 𝜕𝑅𝑦
𝜕2 𝐸
+ 𝜕𝑅
(2)
𝑧 𝜕𝑅𝑧
Finally, we need to normalize the trace values of many materials to compare their interaction
ranges as in Fig. 3, since different materials have slightly different bonding stiffness. The
normalization is done by the trace value of self-interaction force constants.
normalized trace of IFC =
𝜕2 𝐸
𝜕2 𝐸
𝜕2 𝐸
+
+
𝜕𝑅0,𝑥 𝜕𝑅𝑛,𝑥 𝜕𝑅0,𝑦 𝜕𝑅𝑛,𝑦 𝜕𝑅0,𝑧 𝜕𝑅𝑛,𝑧
𝜕2 𝐸
𝜕2 𝐸
𝜕2 𝐸
+
+
(3)
𝜕𝑅0,𝑥 𝜕𝑅0,𝑥 𝜕𝑅0,𝑦 𝜕𝑅0,𝑦 𝜕𝑅0,𝑧 𝜕𝑅0,𝑧
In the above expression, 𝜕𝑅
𝜕2 𝐸
0,𝑥 𝜕𝑅𝑛,𝑥
represents the second-order force constant along x-direction
between the origin atom (described as “0” in 𝑅0,𝑥 ) and the n-th neighbor atom (described as “n”
in 𝑅𝑛,𝑥 ). The
𝜕2 𝐸
𝜕𝑅0,𝑥 𝜕𝑅0,𝑥
represents the self-interaction force constant along x-direction, which
means the force constant of one specific atom when the atom itself is displaced. By taking trace
values and normalizing them, we could compare the interatomic interaction ranges of many
materials with different crystal structures and different bonding stiffness.
Supplementary Note 3. Origin of the long-ranged interaction: long-ranged polarization
The second-order force constant can be expressed by using the Hellman-Feynman theorem19, 20.
21
𝜕2 𝐸
𝜕𝑅𝐼 𝜕𝑅𝐽
𝜕𝑛 𝜕𝑉𝑒−𝑛
= ∫ 𝜕𝑅
𝐽
𝜕𝑅𝐼
𝜕2 𝑉
𝜕2 𝐸
𝑒−𝑛
𝑛−𝑛
𝑑𝑟 + ∫ 𝑛 𝜕𝑅 𝜕𝑅
𝑑𝑟 + 𝜕𝑅 𝜕𝑅
𝐼
𝐽
𝐼
𝐽
(4)
where, 𝑅 and 𝑛 are atomic position and electron density distribution as a function of distance, 𝑟.
𝑉𝑒−𝑛 and 𝐸𝑛−𝑛 refer to potentials of electron-nucleus and nucleus-nucleus interactions,
respectively. In the above expression, the second term on the right hand is non-zero only for the
self-interaction terms (𝐼 = 𝐽) and the third term decreases monotonically with distance. Hence,
the long-ranged and non-monotonically decreasing interatomic force constants (IFC) with
distance along <100> cannot be attributed to the second and third terms. Since 𝜕𝑉𝑒−𝑛 ⁄𝜕𝑅 in the
first term is also decreasing with distance, the 𝜕𝑛⁄𝜕𝑅, electron distribution change due to atomic
displacement, must be the only reason of the long-ranged IFC.
Supplementary Note 4. Comparison of electron density distribution in PbTe and Bi
The weakened resonant bonding in Bi compared to PbTe is observed in its electron density
distribution at ground state. We compare electron density distribution at ground state in PbTe and
Bi in Supplementary Figure 8. The distribution is shown on the plane equivalent to (100) plane
of rocksalt. Comparison between AA’ of PbTe and BB’ of Bi in Supplementary Figure 8
explicitly shows the weakened resonant bonding of Bi. The perfect resonant bonding is a
hybridized state of all possible choices for three p-electrons forming six bonds with equal
weight. Therefore, electron density distribution around one atom should be symmetric in the case
of strong resonant bonding. The A and A’ in Supplementary Figure 8 are identical in PbTe,
indicating the strong resonance. However, B and B’ are not identical and B’ is stronger bonding
than B, which means the weaker resonance.
22
Supplementary Note 5. Phonon dispersions of simple 1-dimensonal chain and fictitious
PbTe without the long-ranged interaction
To show that the long-ranged interaction along <100> makes the TO mode soft, we calculated
phonon dispersions of a simple 1D chain and fictitious PbTe without the long-ranged interaction
along <100>. Here, we present details of the calculations.
We would like to start from dynamical matrix, 𝐃(𝐪) to explain the TO softening since phonon
frequency is roughly proportional to √𝐃(𝐪)⁄𝑀, where 𝑀 is atomic mass. The dynamical matrix
in diatomic crystal with basis atoms A and B is
0𝑅 𝑖𝐪∙𝐑
∑ 𝜙𝐴𝐴
𝑒
𝐃(𝐪) = [
0𝑅
∑ 𝜙𝐵𝐴 𝑒 𝑖𝐪∙𝐑
0𝑅 𝑖𝐪∙𝐑
∑ 𝜙𝐴𝐵
𝑒
]
0𝑅
∑ 𝜙𝐵𝐵 𝑒 𝑖𝐪∙𝐑
(5)
0𝑅 𝑖𝐪∙𝐑
where each sub-matrix ∑ 𝜙𝐴𝐴
𝑒
is 3×3 matrix including x, y, and z direction. Here, as q
approaches zone center, the phase term (𝑒 𝑖𝐪∙𝐑 ) becomes unity. Therefore, in the limit of 𝐪 → 𝟎,
the dynamical matrix becomes a simple sum of force constants without the phase term.
0𝑅
∑ 𝜙𝐴𝐴
𝐃(𝐪 → 𝟎) = [
0𝑅
∑ 𝜙𝐵𝐴
0𝑅
∑ 𝜙𝐴𝐵
0𝑅 ]
∑ 𝜙𝐵𝐵
(6)
0𝑅 ∑ 0𝑅 ∑ 0𝑅
In the above expression, the off-diagonal terms in each sub-matrix (∑ 𝜙𝐴𝐴
, 𝜙𝐴𝐵 , 𝜙𝐵𝐴 ,
0𝑅
and ∑ 𝜙𝐵𝐵
), which show dynamical coupling between transverse directions (i.e., x-y, y-z, and x-z
directions), become exactly zero due to the cubic symmetry. Then, non-zero terms in the
dynamical matrix are only diagonal terms in each sub-matrix and the most significant
contribution to the diagonal terms is from interatomic interactions along <100> direction as we
show in Fig. 3.
It can be shown using 1D diatomic chain model that the significant force constants at fourthnearest neighbor and eighth-nearest neighbor soften TO mode. For the simple 1D chain,
Supplementary Figure 9 shows the system. The chain consists of two basis atoms with the equal
spacing. One atom is twice heavier than the other atom. We cut-off the interatomic interaction at
the third-nearest neighbors and assume the interactions with atoms beyond this range are
23
negligible. Also we assume that the basis atom A and B are identical in terms of the force
constants. Therefore, the force constants can be expressed as
First-nearest neighbor force constant: 𝜙1 = −𝛼𝜙
Second-nearest neighbor force constant: 𝜙2 = −𝛽𝜙
Third-nearest neighbor force constant: 𝜙3 = −𝛾𝜙
where 𝜙 is a constant and 𝛼, 𝛽, and 𝛾 represent relative strength of each force constant. The selfinteraction term is decided by the acoustic sum rule,
Self-interaction force constant: 𝜙0 = 2(𝛼 + 𝛽 + 𝛾)𝜙
For Fig. 5a, we varied 𝛼, 𝛽, and 𝛾 to show the effects of the long-ranged interaction on optical
phonon dispersion. (Three numbers in the legend of Fig. 5a are 𝛼, 𝛽, and 𝛾 values.) The selfinteraction force constant is kept constant.
Using the force constants above, in the limit of zero wave vector, a dynamical matrix in
Supplementary Equation 6 can be written as
𝐃(𝐪 → 𝟎) = [
𝜙0 + 2𝜙2
2𝜙1 + 2𝜙3
2𝜙1 + 2𝜙3
]
𝜙0 + 2𝜙2
(7)
In the above dynamical matrix, 𝜙2 and 𝜙3 are negligible in tight binding materials, but
significant in resonant bonding materials. The significant 𝜙2 and 𝜙3 values in the resonant
bonding materials have the opposite sign to 𝜙0 and 𝜙1 , respectively, reducing the magnitude of
each term of the dynamical matrix and thereby softening TO phonon mode.
In order to check the effects of the long-ranged interaction on TO softening in 3D, we assumed
fictitious PbTe without the long-ranged interaction along <100> and show its phonon dispersion
in Fig. 5b. For the fictitious PbTe, we set force constants of the fourth-, eighth-, and fourteenthnearest neighbors to zero. In addition, we increased first-nearest neighbor force constants (Pb-Te)
by the sum of the force constants of fourth, eighth, and fourteenth-nearest neighbors to satisfy
the acoustic sum rule and keep the self-interaction term same.
24
Supplementary Note 6. Lattice thermal conductivity comparison between PbTe and Bi
To further confirm that the soft TO mode due to the resonant bonding reduces lattice thermal
conductivity, we compare lattice thermal conductivity of PbTe and pure Bi. The average mass is
less in PbTe (168) than in Bi (209). Accordingly, from the acoustic phonon dispersions shown in
Supplementary Figure 10, PbTe has higher frequency and group velocity. Therefore, if we
consider only harmonic properties, PbTe should exhibit higher lattice thermal conductivity than
Bi.
However, our first principles calculation and experimental data from literatures show that PbTe
has lower lattice thermal conductivity than Bi. Since Bi is semimetal and has large electronic
thermal conductivity, we calculated lattice thermal conductivity of Bi from first principles. The
calculation method is well presented in the previous work for Si21, 22. As Bi has long-ranged
interaction, we included up to fourteenth-nearest neighbors for the second-order force constants
and tenth-nearest neighbors for the third-order force constants. The lattice thermal conductivities
of PbTe and Bi are shown in Supplementary Figure 11. Both experimental and theoretical results
for PbTe are adopted from the literature23, 24, 25. The experimental results for Bi are also from the
literature, and it is measured under high magnetic field to suppress electronic contribution to
total thermal conductivity and large thermomagnetic effect is corrected26. The lattice thermal
conductivity of Bi is almost isotropic and difference between binary and trigonal direction is less
than 10%. In Supplementary Figure 11, for Bi, only binary direction value is plotted. The
calculations agree well with the experimental data. From Supplementary Figure 11, it is clear
that PbTe has much lower thermal conductivity than Bi, which is opposite to the expectation
from the comparison of average masses.
The lower thermal conductivity of PbTe compared to Bi strengthens the connection between
resonant bonding and lattice thermal conductivity. As discussed in the main manuscript, Bi has
distorted cubic structure (rhombohedral) and its TO softening is much weaker than PbTe case.
25
This causes large difference in mode Grüneisen parameter in Fig. 5e, which is the main reason of
the thermal conductivity difference. In addition, the distortion also leads to slight difference in
phase space for three-phonon scattering, as shown in Supplementary Figure 12. The significant
TO softening in PbTe provides wider scattering channel for the (a,a,o) and (a,o,o) processes.
Therefore, these features from the strong resonant bonding in PbTe cause the lower thermal
conductivity of PbTe compared to Bi, even though PbTe has smaller atomic mass.
Supplementary Note 7. Detailed analysis of phase space volumes in III-V and IV-VI
materials
The increasing phase space volume of IV-VI materials as the mass ratio decreases in Fig. 7a is
interesting in a sense that it is opposite to the well-known behavior of many materials such as IIIV materials. This opposite trend implies that the three phonon scattering pathway in IV-VI
materials is much different from other materials such as III-V materials. Shortly speaking, the
interesting trend of IV-VI materials is due to (a,o,o) scattering channel, which is negligible in IIIV materials. (‘a’ and ‘o’ refer to acoustic and optical modes in three phonon scattering.)
First, we show in Supplementary Figure 13 the contributions from each scattering pathway to the
total phase space volume of three phonon scattering. For III-V materials, it can be seen that the
major scattering channels are (a,a,a) and (a,a,o) channels, and the (a,o,o) channel is almost
prohibited due to the very narrow frequency bands of optical branches. The (a,a,o) channel of IIIV materials is sensitive to the value of band gap between acoustic and optical modes as shown in
Fig. 6 and Supplementary Figure 13 and also as discussed in other literature27.
The phase space of IV-VI materials is much different from that of III-V materials in that (a,o,o)
channels remarkably contribute to the total phase space volume in IV-VI materials. From
Supplementary Figure 13, the (a,o,o) channel is particularly significant in PbS and PbSe, but less
significant in PbTe and SnTe. It can be seen from Supplementary Figure 13 that the (a,o,o)
channel is the main reason of the decreasing phase space volume of IV-VI materials as the mass
26
ratio increases. The decreasing (a,o,o) channel with increasing mass ratio can be explained with
the overlap between acoustic and optical bands. The optical bandwidth for four different IV-VI
materials are similar as shown in Supplementary Figure 14, but the overlap between acoustic and
optical bands is quite different. In PbS, the overlap is almost negligible due to the large mass
contrast, making high frequency optical modes far above from acoustic modes. This high
frequency optical phonon mode is particularly important for wide (a,o,o) channel, because the
final state optical phonon mode in the process that one acoustic and one optical modes coalesce
and produce one optical mode needs high frequency to satisfy energy conservation; the
frequency of the final state optical mode should be same as the sum of frequencies of one
acoustic and one optical modes. Therefore, the small overlap in PbS and PbSe leads to the wide
(a,o,o) channel. However, in PbTe and SnTe, the overlap between acoustic and optical bands is
large due to the small mass contrast, and the frequency of optical modes is similar to that of
acoustic modes in overall. In this case, it is difficult to find optical modes large enough
frequency equaling the sum of frequencies of two phonon modes in the (a,o,o) process. As such,
the large overlap between acoustic and optical bands in PbTe and SnTe reduce the (a,o,o) channel
width, resulting in decreasing total phase space volume as the mass ratio increases.
Supplementary Note 8. Mode Grüneisen parameters and thermal expansion coefficients of
SnTe and InSb
Thermal expansion coefficients for SnTe and InSb were calculated to validate the calculated
second- and third-order force constants. The thermal expansion coefficient, α, is
α=
𝛾𝑐v
3𝐵
(8)
where 𝛾 , 𝑐v , and 𝐵 represent overall Grüneisen parameter, specific heat, and bulk modulus,
respectively. The overall Grüneisen parameter is the average of mode Grüneisen parameter
weighted by specific heat;
27
γ=
∑𝐪𝑠 γ(𝐪𝑠)𝑐v (𝐪𝑠)
∑𝐪𝑠 𝑐v (𝐪𝑠)
(9)
where 𝐪 and 𝑠 represent phonon wavevector and polarization, respectively. The mode Grüneisen
parameter can be calculated using second- and third-order force constants.
1
𝛼𝛽𝛾
γ(𝐪𝑠) = − 6𝜔2 (𝐪𝑠) ∑ Ψ𝟎𝑏,𝐑𝟏 𝑏1 ,𝐑𝟐 𝑏2
𝑒 𝑖𝐪∙(𝐑𝟐 −𝐑𝟏 )
√𝑀𝑏1 𝑀𝑏2
𝛼
𝑋𝐑𝑏
𝑒(−𝐪𝑠, 𝑏1 𝛽)𝑒(𝐪𝑠, 𝑏2 𝛾)
(10)
where 𝜔, Ψ, and 𝑀 represent phonon frequency, third-order force constant, and atomic mass.
Here, 𝛼𝛽𝛾, 𝐑 𝟏 , and 𝑏1 mean polarization, translational vector, and basis atom. Also, 𝑋 and 𝑒 are
atomic position and eigenvector, respectively. The calculated mode Grüneisen parameter from
Eq. S10 is shown as lines in Supplementary Figure 17. The circles in the figure are mode
Grüneisen parameter calculated by another way; phonon dispersions in two different crystals
with different volumes are calculated from density functional perturbation theory, and then the
mode Grüneisen parameter is calculated by its definition, γ = − 𝑉𝑑𝜔⁄𝜔𝑑𝑉 . From
Supplementary Figure 17, two different approaches give similar mode Grüneisen parameters,
confirming the consistency in the calculations.
To calculate thermal expansion coefficient by Supplementary Equation 8, other properties,
specific heat and bulk modulus, were calculated from density functional theory; specific heat
was calculated from the calculated phonon dispersion, and bulk modulus was calculated by
applying the Murnaghan equation of state to the ground state energy values with various lattice
constants. Using the mode Grüneisen parameter, specific heat, and bulk modulus, the thermal
expansion coefficients were calculated and are plotted in Supplementary Figure 18. The
calculated thermal expansion coefficients show reasonable agreement with the experimental
values.
28
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