Two-channel model for ultralow thermal conductivity of crystalline Tl3VSe4

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Science  29 Jun 2018:
Vol. 360, Issue 6396, pp. 1455-1458
DOI: 10.1126/science.aar8072

Glass-like and crystal-like

Crystals with glass-like ultralow thermal conductivity are appealing as barrier coatings and thermoelectric materials. Mukhopadhyay et al. developed a class of thallium selenides with glass-like thermal conductivity. These materials may be promising for applications, but they also require the combination of glass-like and crystal-like thermal transport to explain their thermal properties. This two-channel model can be used to identify potential ultralow-thermal-conductivity compounds.

Science, this issue p. 1455


Solids with ultralow thermal conductivity are of great interest as thermal barrier coatings for insulation or thermoelectrics for energy conversion. However, the theoretical limits of lattice thermal conductivity (κ) are unclear. In typical crystals a phonon picture is valid, whereas lowest κ values occur in highly disordered materials where this picture fails and heat is supposedly carried by random walk among uncorrelated oscillators. Here we identify a simple crystal, Tl3VSe4, with a calculated phonon κ [0.16 Watts per meter-Kelvin (W/m-K)] one-half that of our measured κ (0.30 W/m-K) at 300 K, approaching disorder κ values, although Raman spectra, specific heat, and temperature dependence of κ reveal typical phonon characteristics. Adding a transport component based on uncorrelated oscillators explains the measured κ and suggests that a two-channel model is necessary for crystals with ultralow κ.

The search for materials with extreme thermal properties continues because of their importance for thermal management applications. Materials with low κ are used in data storage devices, thermal barrier coatings, and thermoelectrics, whereas high-κ materials are useful for thermal energy transmission and heat dissipation. Energy costs, carbon emissions from fossil fuels, and energy wasted as heat in the world’s energy economy (>60%) drive tremendous interest in advanced processes and materials for thermal energy transmission, storage, and conversion. Toward these goals, discovery of higher-efficiency thermoelectric materials (TEs) for use in waste heat recovery is highly desirable (1). The efficiency of TEs for thermal-to-electric energy conversion is characterized by the figure of merit; Embedded Image, where σ is electrical conductivity, S is the Seebeck coefficient, and κ is thermal conductivity. Numerous schemes for improving the power factor (Embedded Image) (28) and minimizing κ (916) have been devised to achieve enhanced ZT. However, the complex interplay of σ, S, and κ remains a challenging bottleneck toward further advances. Disordered and amorphous materials typically exhibit ultralow κ values owing to lack of lattice periodicity, but these also give low σ. Therefore, theoretical and experimental efforts targeting quality crystalline materials with very low κ and large power factors have been a strong driver in thermoelectric research.

A number of materials, mostly with complex structure, have been proposed for thermoelectric applications, including half-Heuslers (17, 18), skutterudites (19, 20), and clathrates (21, 22) where anharmonic rattling modes give strong intrinsic phonon resistance and suppressed κ. The lowest reported room temperature κ value among bulk crystalline TEs is 0.47 W/m-K in SnSe (23). Thermal conductivity values similar to this are rare in crystalline materials, although lower values are typical in highly disordered or amorphous materials where the phonon picture is not applicable. Disordered transport regimes are not yet fully understood, although great strides have been taken, experimentally and theoretically (24), to gain insights into these. Most relevant to this work, Cahill, Watson, and Pohl (25) put forth a κ model (CWP formula) to describe a lower bound to κ based on random hops of “instantaneously” localized (not to be confused with Anderson localization) vibrational thermal energy among uncorrelated oscillators in glassy and amorphous materials. This was first proposed by Einstein to explain the measured κ of KCl (25, 26) and failed dramatically as phonons carry the majority of heat in that material. In disordered materials, κ monotonically increases with temperature (T) before saturating above the Debye temperature, thus having an effective minimum κ nearly independent of T for a broad range of T (27). In contrast, phonon κ in crystalline materials typically increases with T from T = 0 following a Debye T3 behavior, peaks at intermediate T, often dictated by isotope or defect scattering, and then exhibits a near 1/T  behavior with increasing T above the Debye temperature owing to intrinsic umklapp scattering. For complex crystalline materials (2830) whose phonon mean free paths (λ) are strongly suppressed, approaching lengths on the order of interatomic spacings [Ioffe-Regel limit (31)] similar to those in disordered materials, the CWP formula is often invoked.

We investigated the mechanisms behind the very low κ of a cubic Tl3VSe4 crystal (Fig. 1) using combined measurements and a Peierls-Boltzmann transport (PBT) methodology coupled with interatomic forces from density functional theory (3234). Tl3VSe4 generally has typical phonon behavior from 5 K < T < 300 K. We calculated intrinsic phonon-phonon interactions, and extrinsic phonon scattering, from natural isotope variations and from crystallite boundaries with the length (0.2μm) empirically determined from the measured κ at 6 K. We found that the extrinsic scattering has little impact on the overall κ for T > 50 K as the intrinsic phonon scattering resistance is very strong. For lower temperatures, where extrinsic scattering is more important, our calculated and measured κ are in good agreement.

Fig. 1 Thermal conductivity of Tl3VSe4.

Measured thermal conductivity (κmeasured) and calculated values from the PBT (κphonon). The proposed κtwo-channel values using the CWP formula and the Einstein κ formula give the range of values shown by the shaded area between the red curves. A small “radiation tail” can be seen in the measured Tl3VSe4 κ data near 300 K. For comparison, κ values of low-κ materials at 300 K from the literature are shown: PbTe (47), KSbS2 (48), Cu2Se (49), Bi2Te3 (50), CuCl (51), SnSe (23), Tl9BiTe6 (52); A, Cu3SbSe3 (48) and CuBiS2 (35); B, Gd117Co56Sn112 (53); C, CsAg5Te3 (54). Materials with very complex unit cells (>30 atoms) are designated with purple half-filled circles. We also note that microcrystalline thin-film molecular structures of fullerene derivatives have attained κ values of ~0.03 to 0.07 W/m-K at room temperature (55, 56).

However, for T > 50 K, the κ values begin to diverge as the temperature dependences are different. The calculated κ = 0.16 W/m-K at 300 K is nearly half that of the measured κ = 0.30 W/m-K [also reported in (35)]. The ultralow κ values in Tl3VSe4 are particularly notable for a crystalline material with little disorder, especially given the simple crystal structure (body-centered cubic; space group Embedded Image) with only eight atoms per primitive cell, most with relatively small mass [fig. S6 (36)]. Other low-κ crystals typically have much more complex unit cells with many heavy atoms, which provide more phonon scattering channels and lower sound velocities. The ultralow κ values in Tl3VSe4 prompted us to investigate its origin, phonon characteristics (frequencies and λ), and relation to the CWP formula.

In Tl3VSe4, Tl atoms are very weakly bonded, having nearest-neighbor distances of 3.22 and 3.95 Å with Se and V atoms, respectively, slightly larger than the sum of the individual ionic radii. Additionally, Tl atoms exhibit nonoverlapping symmetric spherical electron densities with little deformation, representative of their 6s2 lone-pair character (fig. S6). The V and Se atoms form Embedded Image tetrahedral units with equal V–Se bond distances of 2.29 Å (see tables S1 and S2 for structural information), and the total charge density between V and Se atoms shows considerable overlap of the electronic densities, representative of hybridization in the Embedded Image tetrahedra. These electronic and structural features suggest ionic bonding between Tl and Se or V atoms and covalent bonding between V and Se atoms. Given the weak atomic bonding to the Tl3VSe4 lattice, Tl atoms have relatively large thermal displacements at room temperature, as shown by the calculated mean square displacements (MSDs) (fig. S7). At 300 K, the calculated MSD for Tl atoms is 0.067 Å2, larger than that of V (0.017 Å2) and Se (0.027 Å2) atoms. MSD data from powder x-ray diffraction measurements give similar ratios but smaller values (also shown in fig. S7). The large MSD values indicate that the Tl atoms reside in a relatively flat potential energy environment and that displacing them from their equilibrium sites costs little energy. This scenario is similar to Zintl-type TlInTe2 (37, 38) where it was argued that Tl atoms act as phonon rattlers responsible for an avoided acoustic branch crossing and low κ (~0.5 W/m-K) (38). In Cu3SbSe3, Cu atoms exhibited similar calculated displacements and were considered as a partly liquid sublattice in a crystalline matrix (30). The low κ values in these systems were partly explained in terms of vibrations of rattling modes. In Tl3VSe4, the Tl atoms are responsible for the soft acoustic branches (Fig. 2A) corresponding with very low longitudinal (LA, 2189 m/s) and transverse (TA, 881 m/s) acoustic group velocities. For comparison, AgSbSe2 (TA: 1362 m/s, 2105 m/s; LA: 3433 m/s) (39) and AgSbTe2 (TA: 1325 m/s, 2469 m/s; LA: 3526 m/s) (39) have higher velocities and higher κ values of 0.48 and 0.68 W/m-K, respectively.

Fig. 2 Validity of the phonon picture.

(A) Calculated phonon dispersion and corresponding projected phonon density of states. Irreducible representations for the measured Raman active modes are shown, while calculated and measured frequencies, linewidths, and irreducible representations for all modes at the Γ point are given in table S3. The uppermost modes at ~360 cm−1 are triply degenerate and correspond to asymmetric bond stretching of the Embedded Image tetrahedral units. The single mode at ~220 cm−1 is the symmetric bond stretching of the Embedded Image tetrahedral units. Phonon frequencies from Raman measurements are marked by green circles. (B) Measured and calculated Raman spectra. Raman measurements were performed at 300 K.

The soft acoustic modes also give large mode Grüneisen parameters (Embedded Image, which characterize the relative change in phonon frequency with change in crystal volume and are often considered a measure of anharmonicity. Near the zone center, γ values for acoustic modes are as large as 10 (fig. S8), exceeding the average value of 7.2 reported for SnSe (23). The physical origin of the large anharmonicity in Tl3VSe4 may be traced to the 6s2 lone electron pair of Tl+, contributing partially to the valence band otherwise composed of V d-orbitals and Se p-orbitals (fig. S9). The Tl+ 6s2 pair is repelled by these p- and d-electrons, making Tl s-orbitals more prone to deformation by lattice vibration, which was reported to result in strong anharmonicity (40) and low κ values in SnSe (23), AgSbSe2 (39), and NaSbTe2 (39). The strong anharmonicity as suggested by the large γ values in Tl3VSe4 manifest themselves in strong scattering of the low-frequency heat-carrying phonons, thus short λ (Fig. 3A) and ultralow κ (Fig. 1). As revealed by the projected phonon density of states (Fig. 2A), the acoustic phonons with short λ are primarily derived from vibration of the heavy Tl atoms. Therefore, we conclude that a combination of very low group velocities and strong anharmonicity governs the ultralow κ behavior in Tl3VSe4. This extends to the other compounds that we investigate in the Tl3XSe4 (X is V, Nb, Ta) family (figs. S10 to S13) and highlights that the Tl atoms and their associated chemical bonding environment dictate the lattice dynamical properties in Tl3XSe4.

Fig. 3 Evaluation of lattice anharmonicity.

(A) Calculated phonon mean free paths λ for low-frequency phonons at 300 K. The purple horizontal line marks the Ioffe-Regel limit (31) defined by the average atomic distance in Tl3VSe4. (B) Phonon lifetimes from linewidths of measured Raman peaks and calculations at 300 K. Minimum lifetimes from the CWP formula and the Einstein κ formula are given by the black curve and the green horizontal dashed line, respectively.

These ultralow-κ semiconducting thallium-based systems may also show promise as efficient low-temperature TEs, provided that the electronic properties and other factors (stability, ability to dope, and so forth) are also beneficial. Our calculations of the electronic structure of Tl3VSe4 reveal a bandgap (Eg) of 1.5 eV (fig. S9) lying within the range of good TEs. The calculated Eg is overestimated as compared to our measurements (0.8 eV) (fig. S2), a discrepancy that may arise from defect states in the samples. Whereas the conduction band minimum is doubly degenerate, the valence band maximum has a degeneracy of three, and many bands have energies close to the Fermi level (Ef). Collectively, these lead to a higher density of states near Ef, which is known to give an enhanced Seebeck coefficient for improved ZT (see fig. S9 for calculations of these parameters).

Our calculated κ and the insights developed above are predicated on the conventional phonon picture, well-defined modes, and diffusive gas-like transport. However, a considerable number of phonons have λ below the scale of the atomic spacing, partly due to small velocities (like those of typical optic phonons) and partly due to small lifetimes. This raises concerns regarding the validity of the PBT methodology as phonons are often considered to be “ill-defined” in this case (41). However, the dependence of κ on T (Fig. 1) suggests phonon-dominated transport as κ increases to a maximum from the lowest T and then decreases as umklapp scattering becomes stronger. This behavior is qualitatively captured by the PBT calculations, even quantitatively for low T. However, at higher T, our measurements have a κ ~ T−0.65 behavior, whereas our calculations give a dependence (κ ~ T−0.85) closer to the typical T−1 expected when umklapp scattering dominates. This difference leads to increasing discrepancies with increasing T. Our Raman spectroscopy measurements on Tl3VSe4 at 300 K demonstrate well-defined, though broadened, phonon peaks, in excellent agreement with calculated frequencies (Fig. 2, A and B). The experimental Raman peaks (Fig. 2B) were fitted with Lorentzians to extract their corresponding widths (table S3) and to estimate the corresponding lifetimes, which also compare favorably with the calculated Γ-point linewidths (table S3) and lower-frequency lifetimes that govern transport (Fig. 3B). Well-defined Raman modes support the calculations and suggest that the phonon transport picture is valid. However, the weaker temperature dependence that we measured for κ strongly suggests that another transport mechanism is at play. Electronic κ contributions can be eliminated as Tl3VSe4 is a semiconductor with a notable electronic bandgap (figs. S2 and S9).

We propose that a phonon conduction channel and a hopping channel of “instantaneously” localized vibrations coexist (κtwo-channel = κphonon + κhop) in crystalline materials, particularly with loosely bound atoms or rattling modes. The localized vibrational energy from “phonon” modes with mean free paths below the Ioffe-Regel limit (Fig. 3A) is conducted in the κhop channel (model described below), and κphonon represents the usual conduction from “well-defined” phonons with mean free paths greater than this limit. The κhop channel only becomes apparent when the phonon contribution to κ is very small, as in Tl3VSe4. The idea of a phenomenological two-channel model has been previously introduced to describe amorphous glasses (from which the “hop” term is derived) (42) and in the interpretation of molecular dynamics simulations of solid argon (43) and disordered materials with so-called propagons, locons, and diffusons (24, 44). Furthermore, guest atoms in weakly bonded systems (e.g., clathrates and filled skutterudites) were previously suggested as Einstein oscillators in host substructures that are treated within the Debye model. This Debye-Einstein combination was used to explain various thermodynamic properties of this class of materials (21, 38).

We calculated κhop via the CWP formula (25) and from the Einstein (26) κ formula from which it is derived (36). (As an interesting side note, Einstein’s original estimation of κ ~ 0.3 W/m-K for KCl is similar to the calculated and measured values that we obtained for Tl3VSe4.) The CWP formula requires sound velocities as input, whereas the Einstein κ formula requires defining a constant frequency/temperature (Embedded Image) for the oscillators (36). This can be estimated from the low-temperature specific heat [22 K (measured) and 18 K (calculated); fig. S3], from the MSD = Embedded Image (ħ is the reduced Planck’s constant, m is the mass of Tl, and Embedded Image is Boltzmann’s constant) for the heavy Tl atom [48 K (measured) and 32 K (calculated) at 300 K (36)] or the Raman lifetimes (τ) using Embedded Image (36 K), all of which are moderately constistent. We note that specific heat from measurements and calculations compare favorably for T > 20K and give similar “Einstein peaks” at low T; however, the purely harmonic phonon theory overpredicts the measured specific heat at very low T (fig. S3). Combining κphonon (excluding phonons with mean free paths below the Ioffe-Regel limit) and κhop from these models gives values that are consistently in better agreement with measured κ and its temperature dependence (Fig. 1), with the CWP formula giving the highest values and the Einstein κ formula with Embedded Image 22 K giving the lowest. Indeed, using Embedded Image 65 K gives excellent agreement with the measured data, though not justified by other measurements. We examined measured and calculated κ in other low-κ materials from the literature (28, 45, 46) and found large improvements (see Table 1 and fig. S14 for T dependence) using this two-channel transport with the CWP formula. This suggests that two vibrational heat-conduction channels coexist in crystalline materials, and the hopping channel becomes evident in materials with small phonon κ.

Table 1 Validity of the two-channel model.

Room-temperature κ values of Tl3VSe4, YbFe4Sb12, CsSnI3, CsPbI3, and CsPbBr3: calculated κphonon, calculated κtwo-channel = κphonon + κhop, and κmeasured. The YbFe4Sb12 calculated and measured values are given in (46) and (45), respectively; measured and calculated values for the CsSnI3, CsPbI3, and CsPbBr3 nanowire systems are given in (28). The CWP formula was used to calculate κhop, and the small contributions from phonons with mean free paths below the Ioffe-Regel limit (<5% in Tl3VSe4) were not subtracted from κphonon here as sufficient information about these is not available.

View this table:

In this study, we found that crystalline Tl3VSe4 has ultralow thermal conductivity (κmeasured = 0.30 W/m-K, κcalculated = 0.16 W/m-K at room temperature), despite lacking disorder. The ultralow κ stems from very small acoustic group velocities from heavy, loosely bound Tl atoms and strong anharmonicity likely arising from Tl s2 lone-pair repulsion. Describing κ as the sum of two separate vibrational transport channels, phonons, and random walk among uncorrelated oscillators gives substantially better agreement with measured κ data and its temperature dependence. The latter channel is only apparent in crystals for which phonons carry little heat and explains some previous discrepancies of measured and calculated κ values in other low-κ materials. The improvement in theory for ultralow-κ crystalline materials provides insight for developing efficient thermoelectrics and thermal barrier coatings, possibly including the Tl3XSe4 family of compounds.

Supplementary Materials

Materials and Methods

Crystal Structure and Atomic Coordinates

Figs. S1 to S14

Tables S1 to S3

References (5766)

References and Notes

  1. Materials and Methods are available as supplementary materials.
Acknowledgments: We acknowledge helpful discussion with D. Cahill, D. Broido, P. Allen, and D. Mandrus. Funding: This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. S.M. was in part supported by the NRC-NRL Research Associateship Program for supplemental calculations of structural and thermal properties. This work used computational resources from the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under contract no. DE-AC02-05CH11231. Raman spectroscopy measurements were conducted as a user proposal at the Center for Nanophase Materials Sciences (CNMS), which is a DOE Office of Science User Facility. Author contributions: D.S.P. and S.M. initiated the work. S.M., D.S.P., and L.L. designed the research and carried out calculations. B.C.S. led the experimental efforts, synthesized the crystal and measured thermal conductivity, heat capacity, and electrical resistivity. M.A.M. measured lattice expansion and displacement parameters. A.A.P. measured the Raman spectrum. All authors contributed to construction of the manuscript. Competing interests: The authors declare no competing interests. Data and materials availability: All measured and calculated numerical data are available in the supplementary materials.

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