Abstract
Understanding the lattice dynamics and low thermal conductivities of IV–VI, V_{2}–VI_{3} and V materials is critical to the development of better thermoelectric and phasechange materials. Here we provide a link between chemical bonding and low thermal conductivity. Our firstprinciples calculations reveal that longranged interaction along the 〈100〉 direction of the rocksalt structure exist in lead chalcogenides, SnTe, Bi_{2}Te_{3}, Bi and Sb due to the resonant bonding that is common to all of them. This longranged interaction in lead chalcogenides and SnTe cause optical phonon softening, strong anharmonic scattering and large phase space for threephonon 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.
Introduction
Most good thermoelectric and phasechange materials are found in group IV–VI, V_{2}–VI_{3} and V materials. For example, PbTe, Bi_{2}Te_{3} and Bi_{1−x}Sb_{x} have been the best thermoelectric materials in the intermediate, room and lowtemperature ranges, respectively^{1,2,3,4}. Alloys of GeTe and Sb_{2}Te_{3} (GST) have been the most popular materials for optical storage technologies such as compact disc and phasechange random access memory^{5,6}. These materials all have low thermal conductivity, which is crucial for thermoelectric and phasechange memory applications.
Usually, group IV–VI, V_{2}–VI_{3} and V materials have low thermal conductivity. This becomes particularly obvious when IV–VI materials are compared with III–V compounds. For example, the thermal conductivity of SnTe is only 4 W/mK while that of InSb, adjacent to SnTe in the periodic table, is 16 W/mK at room temperature^{7,8}. The low thermal conductivities of rocksalt IV–VI materials (hereafter IV–VI materials) compared with 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 conductivity^{9}.
Our firstprinciples 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 Slack^{10}. The formula gives reasonable predictions for many materials with zincblende and rocksalt structures^{11}. According to the formula, thermal conductivity is given by
where B is a numerical coefficient, is the average mass of the basis atoms, n is the number of phonon branches and γ is the Grüneisen parameter. The average volume per atom is denoted by δ^{3}, and θ is the acoustic Debye temperature. Here, harmonic properties include , θ and δ; 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 materials normalized by the harmonic properties, , as a function of the mass ratio of 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 behaviour in the lattice dynamics of IV–VI, V_{2}–VI_{3} 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, V_{2}–VI_{3} and V materials commonly have longranged interactions along the 〈100〉 direction of rocksalt as a result of the resonant bonding. Then, we infer that the significant longranged interaction in IV–VI materials play a key role in their low thermal conductivities.
Results
Resonant bonding in IV–VI, V_{2}–VI_{3} and V materials
Resonant bonding can be understood as resonance or hybridization between different electronic configurations: three valence pelectrons alternate their occupancy of six available covalent bonds that exist between a given atom and its octahedral neighbours^{12}. For example, in PbTe, sphybridization is small and the sband is lower than the pband by 1.5 eV (ref. 13). Therefore, we can consider only pelectrons 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 V_{2}–VI_{3} and many alloys of IV–VI and V_{2}–VI_{3} materials^{14}. In general, the unsaturated covalent bonding by pelectrons with rocksaltlike crystal structures can be regarded as resonant bonding^{15}.
Materials with resonant bonding have several features. First, because of their coordination number of six, they have rocksaltlike crystal structures. Many IV–VI, V_{2}–VI_{3} and V compounds have rocksaltlike crystal structures, as shown in Fig. 2. In addition, these materials have very weak sphybridization and the sbands are well below the pbands. Several past works show that PbTe, Bi_{2}Te_{3} and Bi satisfy this condition^{13,16,17} and our calculations using density functional theory also present weak sphybridization in these materials (see Supplementary Note 1; Supplementary Figs 1–7). One very important feature of resonant 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 charges^{18,19}. For example, the dielectric constants of PbTe and Bi_{2}Te_{3}() are 33 and 50, respectively, while for Si the value is 11.76 (refs 20, 21, 22). In a similar way, certain electronic properties of thermoelectric and phasechange materials have also been explained by invoking resonant bonding^{12,15}. Matsunaga et al.^{23} used experimental observations of low frequency of transverse optical (TO) phonons, socalled 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 GST. Here, from firstprinciples, we provide a detailed lattice dynamicsbased link between resonant bonding and the soft TO modes and low thermal conductivities of a wide range of materials.
Longranged interaction due to the resonant bonding
We calculated the harmonic force constants in typical IV–VI, V_{2}–VI_{3} and V materials using firstprinciples density functional theory and found that a common feature among these compounds is the presence of longranged 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 selfinteracting IFC tensor (see Supplementary Note 2 for details). From Fig. 3a, fourthnearest neighbours, separated by 6 Å (for example, PbPb and TeTe), have interactions which are comparable to those of firstnearest neighbours, spaced 3 Å apart, and are much stronger than second and thirdnearest neighbour interactions. In addition, eighthnearest neighbours, separated by 9 Å, have even positive force constants, giving them the behaviour of ‘antisprings’. Fourteenthnearest neighbours, separated by 12 Å, also have nonnegligible force constants. The force constants at the fourteenthnearest neighbours are clearly distinguished from other force constants nearby when using finer q mesh in the electron response calculation to capture the longranged interaction more accurately. These first, fourth, eighth and fourteenthnearest neighbours 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 behaviours. These behaviours were not captured by earlier works on the lattice dynamics of PbTe and PbS within the shell model^{24,25}. 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 sphybridized covalent bonding materials, respectively. The longranged interactions along the 〈100〉 direction in NaCl are much smaller compared with those in PbTe. In the case of InSb, interactions besides for firstnearest neighbour ones are negligible.
We need to point out that the longranged interactions we observe are different from longranged Coulomb interaction, which cannot explain why fourthnearest neighbour interactions are stronger than second or thirdnearest neighbour interactions. The longranged and nonmonotonically decreasing interaction is due to the longranged electronic polarizability (see Supplementary Note 3 for details). In Fig. 4a,b, we compare electron density distribution at ground state of PbTe and NaCl. From the groundstate 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, electron polarization in NaCl is short ranged and electrons surrounding the fourthnearest neighbours are not perturbed much. However, electron polarization in PbTe in Fig. 4f is long ranged and reaches the fourthnearest neighbour. The electron density distribution surrounding the fourthnearest neighbours is largely disturbed by the displacement of the center atom.
The longranged 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 p_{x} 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 p_{x} orbital^{26}. This perturbation can persist over long ranges owing to the large electronic polarizability and the collinear bonding in resonant bonding materials. The bandbyband electron polarization analysis in Fig. 4 confirms that the longranged polarization is owing to resonantly occupied pelectrons. PbTe has very weak sphybridization and we plot the electronic polarization for sband and pband electrons separately in Fig. 4f–h. From Fig. 4g, the polarization of selectron is short ranged and does not reach the fourthnearest neighbour. However, the pelectron distribution in Fig. 4h exhibits the longranged polarization. This analysis shows that the easily polarized electrons in PbTe are resonantly occupied pelectrons, rather than lone pair Pb s electrons suggested in recent work^{27}.
The resonant bonding picture discussed above also applies to V_{2}–VI_{3} (Bi_{2}Te_{3}) and V (Bi and Sb) materials. Bi_{2}Te_{3} has a rhombohedral structure that can be understood as a deformed rocksalt structure with layer spacing as shown in Fig. 2. This rocksaltlike 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), 3.03 Å, and that of the secondstrongest 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°, respectively, making them similar to the rocksalt structure since the rocksalt has an angle of 180° exactly. Owing to this small structural distortion, the resonant bonding exists in a weakened form. As shown in Fig. 3c, the IFCs of Bi_{2}Te_{3} show a behaviour similar to that of lead chalcogenides, but the longranged interactions are weakened owing to the structural distortions, resulting in weakened resonant bonding. Bi–Bi and Te1–Te2, both spaced about 6 Å apart, have interactions that are equivalent to the fourthnearest neighbour interactions in an exact rocksalt structure, and are less significant compared with the PbTe case. The interactions of Bi–Te1 (at a distance of 9 Å) separated by Te2–Bi, which are equivalent to the eighthnearest neighbour 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 work^{19}. 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 Bi_{2}Te_{3} is that there is no longranged interaction between atoms separated by Te1–Te1. It is well known that Te1–Te1 has van der Waals type of bonding owing to the induced dipole–dipole interaction^{16}, prohibiting the longranged interaction by the longranged electron polarization shown in Fig. 4.
Pure Bi and Sb have rhombohedral crystal structures that are Peierls distortions of simple cubic structure^{28}. 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 firstnearest neighbours in the rocksalt structure become three firstnearest neighbours and three secondnearest neighbours. The distances between the first and the secondnearest neighbours are 3.05 Å and 3.49 Å, respectively. Considering that the distances between the first and the secondnearest neighbours are 3.03 Å and 3.22 Å, respectively, in Bi_{2}Te_{3}, the distortion from the rocksalt structure is larger in Bi and Sb than in Bi_{2}Te_{3}. 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; Supplementary Fig. 8). This structural distortion further weakens the longranged interactions, as shown in Fig. 3d. However, the ninthnearest neighbours, separated by 6 Å, have interactions that correspond to the fourthnearest neighbour interactions in rocksalt and are thus significant.
Effects of longranged interaction on thermal conductivity
The longranged interaction along the 〈100〉 direction is related to the existence of a soft TO mode. This directional longranged interaction was also predicted and considered as a main reason for the ferroelectric behaviour in perovskite BaTiO_{3} and PbTiO_{3} (ref. 29). We used a onedimensional (1D) lattice chain model to explicitly show that the longranged interaction causes the softening of the TO mode (see Supplementary Note 5 for details). In Fig. 5a, as the second and thirdnearest neighbour interactions in the 1D chain (equivalent to fourth and eighthnearest neighbour 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 longranged interactions along 〈100〉 are the main reason for the nearferroelectric behaviour, phonon dispersion is calculated for fictitious PbTe without the fourth and eighthnearest neighbour interactions. The TO mode in the fictitious PbTe is not softened as shown in Fig. 5b. By comparison, Bi_{2}Te_{3}, Bi and Sb have weakened longranged interactions owing to distortion of the structure, and their TO modes are not as soft as those of IV–VI materials (Fig. 5c,d).
The TO phonon softening leads to strong anharmonicity and phonon scattering by the TO modes^{30}. 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. Bi_{2}Te_{3}, 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 observed^{30,31,32,33,34}. In Fig. 6, we show detailed phonon transport characteristics in several III–V and IV–VI materials, calculated from the density functional theory. We calculate phonon transport of SnTe and InSb in addition to PbTe, PbSe and GaAs from literature^{33,35}. It is interesting to directly compare SnTe and InSb as these materials are close to each other in the periodic table, and therefore they have a similar Debye temperature and mass ratio. The thermal conductivity calculation of InSb using firstprinciples was recently reported^{36}. 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 analysing the phonon meanfree path and phonon lifetime. For the phonon meanfree path, we present accumulated thermal conductivity as a function of the phonon meanfree path at 300 K, defined as , where Λ, q, s and χ represent the phonon meanfree path, wavevector, polarization and a step function, respectively^{37,38}. The accumulated thermal conductivity function shows the meanfree 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 meanfree path ranges and phonon lifetime compared with 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 a factor of two, although atomic mass of PbTe is less than that of Bi (refs 39, 40) (see 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 (that is, 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 threephonon scattering phasespace volume of these materials in Fig. 7a. Assuming constant phonon anharmonicity, the inverse of the threephonon scattering phasespace 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 phasespace volume for III–V we calculated is similar to the previously reported data using a bond charge model^{41}. 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 behaviour for many materials that have large atomic mass differences^{42}. 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, lowfrequency acoustic phonons cannot couple to optical phonons and thus have fewer chances to be scattered. Only highfrequency 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 acoustic optical 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 threephonon process.)
The threephonon scattering pathway in IV–VI materials is much different from that of III–V materials since the (a,o,o) channel significantly contributes to the total phasespace 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 phasespace 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 the (a,o,o) channel is a large contributor to total phasespace 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 phasespace volume of the (a,o,o) process can be explained with the wide spectrum of optical modes. The bandwidth of the optical phonons is comparable to the bandwidth 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 the (a,o,o) channel in IV–VI materials. In particular, the (a,o,o) process in PbS contribute more than half of the total phase space. In addition, the (a,a,o) channel also considerably contributes to total phasespace volume in the case of PbS, while it is almost negligible in AlSb. This is owing 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 highfrequency 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 the small mass ratio as shown in Fig. 1.
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, Bi_{2}Te_{3}, Bi and Sb) commonly have longranged interactions along the 〈100〉 direction in rocksalt structure. This longranged interaction is significant in IV–VI materials owing to the strong resonant bonding, and results in the nearferroelectric instability in these materials. However, the longranged interaction is less significant with increasing distortion of the crystal structures as in Bi. The nearferroelectric behaviour caused by the resonant bonding reduces the lattice thermal conductivity through two mechanisms: strong anharmonic scattering and a large scattering phasespace volume, both due to softened optical phonons, resulting in the lower thermal conductivities of IV–VI materials compared with 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 phasechange 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 phasechange materials. These insights help researchers to explore better thermoelectric and phasechange materials.
Methods
Lattice dynamics calculation
We used the density functional perturbation theory to calculate harmonic lattice dynamics properties of the materials^{43,44}. We checked that results are well converged with respect to many parameters such as cutoff 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 longranged interaction is significant in the resonant bonding materials, we used a relatively fine qgrid to capture the longranged interaction. Also, for heavy elements such as Bi, Pb, Te, Sb and Se, spinorbit interaction is included in the calculation. After calculating IFC tensors, we normalize the trace of the IFC tensor to compare many materials with different crystal structures and different strengths 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 materials^{45,46}.
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 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 fourthnearest neighbour 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 the finite difference of two cases to calculate change in electron density distribution by the displacement.
Phasespace volume for threephonon scattering
The phasespace integral for threephonon scattering is the volume satisfying energy and momentum conservation in the threephonon process. Therefore, it can be defined as^{41}
where j, j′ and j′′ represent phonon branches. The two δfunctions 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 δfunctions, we used a qgrid of 20 × 20 × 20 and confirmed the convergence of phase space with respect to the qgrid size. As seen in the expression for phasespace volume above, phasespace volume is integration of δfunctions of phonon frequencies. Therefore, the phasespace volume is inversely proportional to phonon frequency scale. To compare many materials with different phonon frequency scales, we normalized the phasespace volume by the inverse of maximum optical phonon frequency. To validate the phonon dispersion used in the phasespace volume calculation, we present calculated phonon dispersions compared with the experimental data in Supplementary Figs 15 and 16.
Lattice thermal conductivity from firstprinciples
We used the 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 literature^{47}. We calculate secondorder force constants using the density functional perturbation theory and the calculation conditions for the force constants are shown in Supplementary Table 1. The calculated secondorder force constants were validated by comparing the calculated phonon dispersion to the experimental data in Supplementary Figs 15 and 16. For thirdorder force constants, we calculate forces on atoms in large enough supercells (4 × 4 × 4 supercells, 128 atoms) with various atomic displacements. The set of thirdorder force constants, minimally reduced by symmetry considerations, is then fitted to the calculated forcedisplacement data with enforcing translational and rotational invariances^{48}. We considered up to secondnearest neighbours and fifthnearest neighbours for the thirdorder force constants of InSb and SnTe, respectively. The thermal expansion coefficients were calculated using the second and thirdorder force constants, showing reasonable agreement with the experimental values (Supplementary Note 8). The calculated second and thirdorder force constants gives the rate of threephonon 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.
Additional information
How to cite this article: Lee, S. et al. Resonant bonding leads to low lattice thermal conductivity. Nat. Commun. 5:3525 doi: 10.1038/ncomms4525 (2014).
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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 SolarThermal Energy Conversion Center (S3TEC),’ an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number: DESC0001299/DEFG0209ER46577 (for PbTe and Bi_{2}Te_{3}), and partially supported by the US Department of Defense AFOSR MURI via Ohio State (for Bi and Sb). S.L. acknowledges support from Samsung scholarship.
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Affiliations
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
 Sangyeop Lee
 , Jiawei Zhou
 , Zhiting Tian
 & Gang Chen
Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, New Jersey 08901, USA
 Keivan Esfarjani
Institute for Advanced Materials and Devices for Nanotechnology (IAMDN), Rutgers University, Piscataway, New Jersey 08854, USA
 Keivan Esfarjani
Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA
 Tengfei Luo
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Contributions
S.L., K.E., T.L., J.Z. and Z.T. carried out the firstprinciples calculations. S.L., K.E. and G.C. analysed 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 interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Gang Chen.
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