Soft anharmonic phonons and ultralow thermal conductivity in Mg3(Sb, Bi)2 thermoelectrics

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Science Advances  21 May 2021:
Vol. 7, no. 21, eabg1449
DOI: 10.1126/sciadv.abg1449


The candidate thermoelectric compounds Mg3Sb2 and Mg3Bi2 show excellent performance near ambient temperature, enabled by an anomalously low lattice thermal conductivity (κl) comparable to those of much heavier PbTe or Bi2Te3. Contrary to common mass-trend expectations, replacing Mg with heavier Ca or Yb yields a threefold increase in κl in CaMg2Sb2 and YbMg2Bi2. Here, we report a comprehensive analysis of phonons in the series AMg2X2 (A = Mg, Ca, and Yb; X = Bi and Sb) based on inelastic neutron/x-ray scattering and first-principles simulations and show that the anomalously low κl of Mg3X2 has inherent phononic origins. We uncover a large phonon softening and flattening of low-energy transverse acoustic phonons in Mg3X2 compared to the ternary analogs and traced to a specific Mg-X bond, which markedly enlarges the scattering phase-space, enabling the threefold tuning in κl. These results provide key insights for manipulating phonon scattering without the traditional reliance on heavy elements.


Thermoelectric (TE) materials directly interconvert electrical and thermal gradients, providing opportunities in waste heat recovery and cooling applications (14). Improving TE conversion efficiency requires minimizing the thermal conductivity κtot = κe + κl (sum of electronic and lattice components) and simultaneously maximizing the electronic power factor (3, 4). While first-principles band structure calculations have enabled a predictive understanding of electronic transport properties and in silico material design, achieving a quantitative microscopic understanding of thermal transport in strongly anharmonic materials has remained challenging (3, 4). Impressive advances in TE performance have been achieved over the past decade at intermediate to high temperature (T > 450 K) (512), enabled in part because κl is strongly suppressed by anharmonic scattering for high phonon occupations (scaling with T). Around room temperature, a relative paucity of material candidates has hampered TE developments, with (Bi, Sb)2(Te, Se)3 alloys remaining the best for decades (13, 14). Recently, however, n-type Mg3(Bi, Sb)2 Zintl compounds were reported to achieve comparable performance near ambient conditions, potentially with much lower cost (1518).

Despite their much lighter weight than conventional TE compounds PbTe or Bi2(Te, Se)3, these promising Zintl compounds exhibit strikingly low values of κl. For instance, κl = 1.4 W m−1 K−1 was achieved in polycrystalline Mg3Sb2 at 323 K (19), despite its low mass and relatively simple crystal structure. This low κl is comparable to those of 2.5× heavier compounds PbTe and Bi2(Te, Se)3 or barium clathrates with complex structures, contradicting the expected trend of κlM−1 [from the inverse relationship between the square of phonon frequency, ω, and mass, M, and the proportionality of group velocity vg and ω. κl of polycrystalline PbTe, Bi2Te3, and Ba8Ga16Ge30 are 2.0, 1.2, and 1.5 W m−1K−1 at room temperature (2022). The mass densities of Mg3Sb2, PbTe, and Bi2Te3 are 4.02, 8.16, and 7.74 g/cm3, respectively.] Further, this surprisingly low κl in Mg3X2 is strongly tunable upon targeted substitutions: Replacing A-site Mg with heavier Ca or Yb results in a striking anomalous increase in κl by a factor of 2 to 3 (23, 24), again directly contradicting the expected suppression from more complex composition and heavier mass, and suggests heretofore unknown mechanisms capable of strongly suppressing phonon propagation in simple, lightweight crystalline compounds.

Experimental and theoretical TE investigations of Mg3X2 have so far mainly focused on the electronic properties, for example, the origin of n-type behavior (25, 26), the multivalley character of the conduction band (2730), and routes to increased carrier mobility (17, 18, 3133). In contrast, the origin of the intrinsically low κl of Mg3X2 and the counterintuitive increase of κl in homologous ternary AMg2X2 have remained puzzling, although the low κl plays an equally important role in achieving high zT. Previous computational studies suggested that the anharmonic interlayer shearing transverse acoustic (TA) modes (23, 24) and anharmonic mid-energy optical phonons (24) suppress κl. Simulations of alloys also predicted that doping 25% Ca or Yb on the A-site reduces κl by reducing the averaged phonon velocity, while the averaged phonon lifetime remains unchanged (29). Measurements of elastic moduli revealed soft shear moduli in Mg3Sb2 and a higher rate of softening upon warming than in ternary AMg2X2, which was interpreted in terms of ionic radii, unit cell volumes, and large negative Gru¨neisen parameters in TA modes (23). Yet, a clear microscopic picture of phonon scattering remains elusive because of the lack of detailed phonon measurements, precluding a clear understanding of the origin of the abnormally low κl in Mg3X2.

Here, we reveal the phononic origins of the unusual thermal conductivity of Mg3X2 using both inelastic neutron scattering (INS) and inelastic x-ray scattering (IXS), supported by computational modeling based on density functional theory (DFT) and ab initio molecular dynamics (AIMD). We uncover a striking phonon softening (shift to lower energy) in Mg3X2 compared with ternary compounds (A= Ca or Yb) and an unusual flat low-frequency TA phonon branch in Mg3X2 that considerably stiffens upon substitution with Ca or Yb. Combining our extensive phonon measurements and simulations, we quantitatively determined the origin of the low κl in Mg3X2. In particular, we identify the importance of a weak A-X nearest-neighbor bond in Mg3X2. This weak bond effectively destabilizes TA phonons, leading to a soft branch along Γ − M, overcoming expected mass trends between Mg and heavier A-site ions (A = Ca and Yb). Concomitant with a suppression in group velocities, we show how the phonon softening markedly increases (fivefold) the weighted scattering phase-space, overcoming an actual suppression (50%) in third-order force constants (FC). These results quantitatively account for the anomalous two- to threefold suppression in κl in binary Mg3X2 compared to ternary AMg2X2 despite the lighter mass and simpler chemistry of the former. Our momentum and energy-resolved phonon measurements thus rationalize the basis of the high thermal performance of Mg3X2 for TE applications.


Soft phonons in Mg3X2

The Zintl compounds AMg2X2 (A = Mg, Ca, and Yb; X = Sb and Bi) belong to the CaAl2Si2-type family that crystallize in a trigonal conventional cell with space group P3¯m1 (no. 164). Their structures consist of alternating [Mg2X2] and A atom layers, as shown in Fig. 1A. Despite the structural anisotropy, the bonding and electronic structure are fairly isotropic (24). In the following, we refer to Mg on the octahedrally coordinated A-site by X atoms as Mg(1) and Mg on the tetrahedrally coordinated sites as Mg(2).

Fig. 1 Soft phonons from inelastic neutron scattering.

(A) Crystal structure of AMg2X2 and illustration of octahedral A-site and tetrahedral Mg(2) atoms. Red, orange, and blue represent A, Mg(2), and X, respectively. d1 and d2 are the nearest- and second-nearest-neighbor Mg(2)-X bonds. The nearest-neighbor A-X bond is d3. Comparison of experimental neutron DOS Ei = 20 meV at 300 K of (B) Mg3Bi2 (blue) and YbMg2Bi2 (purple) and (C) Mg3Sb2 (red) and CaMg2Sb2 (orange). Significant stiffening is observed in ternary compounds despite a heavier mass of Ca or Yb than Mg. Extra shoulders at low energy are observed only in binary compounds. Neutron dynamical structure factor S(∣Q∣, ω) of (D) Mg3Bi2, (E) YbMg2Bi2, (F) Mg3Sb2, and (G) CaMg2Sb2, the same datasets as in (B) and (C). Total intensities are normalized to the value of Mg3Bi2. Much stronger intensities at the low-energy region can be observed in binary compared to ternary compounds.

The phonon density of states (DOS) of six AMg2X2 (A = Mg, Ca, and Yb; X = Sb and Bi) powder samples were measured with INS as a function of temperature, using the wide angular-range chopper spectrometer (ARCS) at the Spallation Neutron Source (34). The contributions from different atoms are spectrally well separated, as seen from the calculated partial DOS (Fig. 2A and fig. S9). For instance, in Mg3Bi2, Mg(2) contributes mainly at energies greater than 20 meV, Bi contributes mostly below 10 meV, and Mg(1) dominates the intermediate region. The lower vibrational frequency of Mg(1) than Mg(2) thus suggests a weaker bonding environment of Mg(1). When replacing A-site Mg(1) with Ca or Yb, the Mg(2) modes at E > 20 meV remain almost unaffected in both experiments and simulations (figs. S5 and S9). This indicates relatively unchanged local environments and bondings for Mg(2) across compositions. Considering that the atomic mass directly shifts phonon frequencies (ω2K/M, where K is the spring constant and M is the mass), all else being fixed, heavier atoms should exhibit lower phonon energies. Naturally, the heavier Yb atoms vibrate at a lower frequency than Mg or Ca (173, 24, and 40 atomic mass units, respectively). However, when comparing Mg(1) and Ca, the partial DOS of Mg(1) contributes more to the lower energy side and is substantially broader than for Ca (fig. S9). Substitution on the A-site also strongly affects the X partial DOS at low energy, reflecting a strong change in the A-X interaction, which we will explain later.

Fig. 2 Anharmonic Mg(1) phonon modes and momentum-resolved measurements from inelastic x-ray scattering.

(A) Phonon DOS of Mg3Bi2 from INS at 300 K (blue) and 600 K (red), compared with VACF calculation at 300 K (black) and renormalized DOS from TDEP (gray). The computed total DOS is NW and convolved with the experimental resolution. (B and C) Momentum-resolved IXS measurements on single crystals (markers), compared with DFT phonon dispersions (solid lines) for Mg3Bi2 and YbMg2Bi2. Blue, green, and red markers correspond to the fitted phonon energy at 80, 300, and 600 K. (D) Brillouin zone (irreducible wedge in red) and high-symmetry points.

A strong phonon stiffening (energy increase) is observed at E < 12 meV when substituting Mg(1) with Ca or Yb. This is clearly seen in the Ei= 20 meV INS data at 300 K, as shown in Fig. 1 (B and C) and fig. S4. Compared with Mg3X2, all the peaks in CaMg2X2 below 12 meV are stiffened, with relative shifts (ΔE/E) as large as +6.4, +15.1, and +4.6% in Bi compounds, and +12.3, +12.7, and +6.2% in Sb compounds (see table S1). On the basis of mass alone, one would predict a softening of −3% in CaMg2X2 from DFT. Even for the heavy YbMg2Bi2, where Yb contributes significantly below 12 meV, stiffer phonon energies are observed compared to Mg3Bi2 for 8 < E < 12 meV (see fig. S4B). The frequency stiffening of the X partial DOS in ternary compounds relative to the binaries again suggests that the interactions between Ca-X and Yb-X are stronger than Mg(1)-X. While a 10% stiffening on average is observed in the DOS, specific modes stiffen significantly more. Our DFT calculations show that the lowest TA mode at the zone boundary M-point stiffens by more than 60% (E = 3.33, 5.40, and 5.53 meV for Mg3Bi2, CaMg2Bi2, and YbMg2Bi2; see fig. S8). Notably, an extra shoulder is seen at low energy around 3.5 meV in Mg3Bi2 (5 meV in Mg3Sb2) that does not exist in the ternaries.

The powder-averaged dynamical structure factor from INS, S(∣Q∣, ω), shown in Fig. 1 (D to G), reveals more details. In all four compounds, vertical intensity streaks disperse out from the elastic line (at 0 meV), corresponding to acoustic phonons. The top of the acoustic phonon and the low-energy optical phonons contribute to the strong intensity between 6 and 10 meV. At 2 to 4 meV, only the binary Mg3X2 show strong intensity, especially visible at higher Q, which originate from the softened acoustic phonons, and give rise to the extra shoulder in the DOS.

The soft phonon and low-energy shoulder seen in Mg3X2 with INS are well captured by our DFT simulations, as shown in fig. S6. We resolve the low-E shoulder as a softened TA branch in binaries, as shown by the red curves in fig. S7 (a peak/shoulder in the DOS arises from a less dispersive branch over an extended volume of the Brillouin zone). The lowest-energy TA branch along Γ − M is drastically less dispersive in Mg3X2 than in CaMg2X2 or YbMg2X2 (highlighted in red in fig. S8). We note that previous studies (24) reported an unphysical unstable phonon branch for Mg3Bi2, in particular around the L-point. Our simulations resolved this issue by using the Perdew-Burke-Ernzerhof revised for solids (PBEsol) functional and including spin-orbit coupling (SOC). The softer dispersion and flat TA branch in Mg3X2 result in suppressed phonon group velocities (vg =∣∇kω∣, where E = ℏω). Simultaneously, the three-phonon scattering phase-space is strongly expanded, as detailed below and in section S4.

We further compare the INS DOS with first-principles simulation including anharmonic effects. We performed temperature-dependent effective potential (TDEP) simulations, which include anharmonic frequency renormalization but no anharmonic broadening, and calculations of the velocity autocorrelation function (VACF) from AIMD, capturing anharmonic mode broadening as well as shifts. As seen in Fig. 2A, peak positions from TDEP (gray) agree well with the INS DOS for Mg(2) and Bi-dominated modes at 300 K (blue). However, a stronger broadening is seen in INS for 10 < E < 20 meV, corresponding to Mg(1) modes, reflecting an anharmonic bonding environment for Mg(1). This broadening is well captured by the VACF at 300 K (black). Additional INS results for all compounds as a function of temperature are shown in figs. S2 and S3. The entire spectrum broadens upon warming as a result of anharmonic lifetime suppression when vibrational amplitudes increase, but the Mg(1) modes broaden the most in panels (A) and (D).

Origin of soft TA modes in Mg3X2

To fully elucidate the origins of the anomalous low-E acoustic modes, we performed Q-resolved dispersion measurements with IXS on small single crystals of Mg3Bi2 (T = 80, 300, and 600 K) and YbMg2Bi2 (T = 300 K), using the high-energy-resolution inelastic x-ray spectrometer (HERIX) beamline (sector 30) at the Advanced Photon Source (35, 36). The IXS spectra show well-defined peaks (see fig. S16), enabling accurate estimates of the phonon energies. The phonon energies from IXS are compared with our DFT simulations in Fig. 2 (B and C). Very good agreement is observed for YbMg2Bi2, which exhibits strongly dispersive TA along Γ − M, Γ − A, and the long-wavelength portion of Γ − L (we track the lowest-energy TA by phonon polarization; see fig. S13). Good agreement is also found in Mg3Bi2. Upon warming Mg3Bi2 from 80 to 600 K, most phonons soften and broaden, but the TA branch at the L-point anomalously stiffens, consistent with the negative Grüneisen parameter from our quasi-harmonic simulations (fig. S10) and those in (23).

The lowest TA branch along Γ − M is significantly softer in Mg3Bi2 than in YbMg2Bi2, which explains the extra shoulder observed in the DOS. To rationalize this soft TA branch, we analyze the modulus of the second-order FC (∣Φ(2)∣) from DFT for Mg3Sb2 (red), CaMg2Sb2 (orange), Mg3Bi2 (blue), and YbMg3Bi2 (purple), respectively, as shown in Fig. 3A and as a function of bond length in fig. S14A. The two strongest bonds are the nearest-neighbor (d1) and second-nearest-neighbor (d2) Mg(2)-X bonds. For these bonds, ∣Φ(2)∣ varies less than 20% between binaries and ternaries. However, for the A-X nearest-neighbor bond (d3), ∣Φ(2)∣ very strongly depends on the A cation: 1.06 and 0.52 eV/Å2 in CaMg2Sb2 and Mg3Sb2 and 1.17 and 0.75 eV/Å2 in YbMg2Bi2 and Mg3Bi2, respectively. This large increase in ∣Φ(2)(d3)∣ by 105% (Sb) and 56% (Bi) in the ternary compounds reveals a strong change in A-X bonding. The much smaller ∣Φ(2)∣ for d3 than d1 or d2 also leads to the lower phonon frequency of Mg(1) than Mg(2) in the DOS. The weak d3 bond directly controls the TA branch along the Γ − M direction. As we show in fig. S14F, artificially replacing Φ(2)(d3) in Mg3Bi2 with that from YbMg2Bi2 leads to a more dispersive TA branch along Γ − M, while changing Φ(2) for d1 or d2 has relatively little effect.

Fig. 3 Weakened A-X bonds in binary Mg3X2 compared with ternary CaMg2X2.

(A) Modulus of second-order FC (∣Φ(2)∣) in Mg3Sb2 (red), CaMg2Sb2 (orange), Mg3Bi2 (blue), and YbMg2Bi2 (purple). Bonds d1 through d5 are defined in (C) and (D). The largest change is found for the d3 bond, which is much weaker in binaries than in ternaries. (B) The negative integrated pCOHP (−IpCOHP) for d1, d2, and d3. The larger absolute value of IpCOHP at the Fermi energy of d3 in CaMg2Sb2 than in Mg3Sb2 is in accordance with the ∣Φ(2)∣. Deformation electron density in (C) Mg3Sb2 and (D) CaMg2Sb2 at an isosurface value of 0.004 e3. Yellow and blue are positive and negative values, respectively.

The change in Φ(2) can be related to the electronic bonding via the crystal orbital Hamiltonian population (COHP), here calculated with the LOBSTER software (3740). Figure S15 shows the projected partial COHP for bonds d1, d2, and d3, comparing Mg3Sb2 (solid lines) and CaMg2Sb2 (dashed lines). Bonding states are negative and antibonding states are positive. All states are bonding below −2 eV. In the valence band near the Fermi energy (EF = 0 eV), Mg3Sb2 and CaMg2Sb2 have similar pCOHP for d1 and d2. However, for d3, Mg(1)-Sb remains nonbonded down to −1 eV, while Ca-Sb has antibonding interaction in the valence band down to −2 eV. The pCOHP for Mg(1)-Mg(2) (d5) in Mg3Sb2 was shown to involve nonbonding states at the valence band maximum but bonding states at the conduction band minimum in (33). The integrated pCOHP (IpCOHP) probes covalency and relative bond strength, where a more negative value corresponds to a stronger covalent bonding (41). We note that all AMg2X2 compounds are isostructural with only A-site substitution, so comparing their pCOHP is justified. As seen in Fig. 3B, the −IpCOHP values at EF for d1 are slightly larger than d2 in both Mg3Sb2 and CaMg3Sb2, and much larger than d3, which matches the trend of the corresponding ∣Φ(2)∣. While a slightly smaller −IpCOHP for d1 and d2 is found in CaMg2Sb2 than in Mg3Sb2, the −IpCOHP for d3 is larger, which supports the observation of weaker Mg(1)-Sb than Ca-Sb bonds. Meanwhile, the deformation charge density (Fig. 3, C and D) shows that despite the larger charge transfer in the case of CaMg2Sb2, the charge density around Mg(2) is similar in both cases. A more ionic character was found in Ca-Mg(2) compared to Mg(1)-Mg(2) (d5) in agreement with a prior study (29).

Differences in Mg(1)-X versus Ca-X bonding character cannot fully explain the surprising disparity in d3 FC. The softer Mg(1)-X bonds likely arise in part from negative chemical pressure experienced by Mg(1). Although Mg is much smaller than Ca (atomic radii: 1.45 versus 1.94; ionic radii: 0.72 versus 1.00 in Å), the difference between the Ca-Sb versus Mg(1)-Sb bond lengths (d3) is only 0.14 Å. In the binary compounds, the Mg(1) coordination environment can thus be described as an oversized pnictide cage. The MgX6 octahedra in the binaries are highly distorted compared with the near-ideal octahedral geometry of the ternary compounds. Both binaries exhibit a high-temperature β-phase in which Mg is tetrahedrally coordinated by X (42). The formation of the Mg3X2 compounds in the CaAl2Si2 structure type at lower temperatures is a testament to the extreme chemical adaptability of Mg (43). The binaries can be seen as a textbook example of driving a system to the edge of structural stability to achieve soft TA modes by negative chemical pressure.

Thermal conductivity analysis

We now rationalize the anomalously low κl in Mg3Sb2 and Mg3Bi2. Replacing Mg(1) with Ca or Yb leads to two- to threefold higher κl in polycrystalline samples, as shown in Fig. 4 (A and B) (filled markers) (24, 44, 45). The κl of Mg3X2 is twice higher in single crystals (46) versus polycrystals owing to the absence of boundary scattering. Our calculated κl using TDEP (details in the Supplementary Materials) matches very well with the single-crystal measurements in Mg3X2 at 300 K (red and blue hollow markers versus solid lines). Further, we find an almost isotropic κl at all temperatures (e.g., 2.38 in-plane, 2.29 out-of-plane for Mg3Sb2 and 6.30 in-plane, 6.05 out-of-plane for CaMg2Sb2 at 300 K), in agreement with previous Mg3Sb2 simulations (29). The calculated κl of CaMg2Sb2 is about 3× higher than for Mg3Sb2 (2× in Bi compounds), matching the experimental trend (only polycrystalline data available).

Fig. 4 Lattice thermal conductivity analysis and origin of suppression in binary Mg3X2.

Reported κl of (A) AMg2Sb2 and (B) AMg2Bi2 from (24, 4446) compared with our TDEP simulations. κl of Mg3X2 is two to three times lower than in the Ca or Yb compounds in both experiments and simulations. Comparison of phonon linewidths from TDEP for (C) AMg2Sb2 and (D) AMg2Bi2. Binary compounds exhibit a factor of 2 to 3 larger linewidths, similar to the ratio in κl. (E) Spectrally decomposed κl in Mg3Sb2 and CaMg2Sb2. Inset shows the corresponding weighted phase-space (W). A three times larger W is observed in Mg3Sb2 below 10 meV. (F) κl comparison of Mg3Sb2 and CaMg2Sb2. Red and orange markers are the same κl in (A). Blue, green, and black hollow markers represent the κl when replacing vg, τ, and Cv with the values from CaMg2Sb2 (details in the main text).

Figure 4E and fig. S18 show the spectrally decomposed thermal conductivity, κl(E), revealing that phonons below 15 meV dominate the overall κl in both Mg3Sb2 and CaMg2Sb2 (12 meV in AMg2Bi2). The peak contribution to κl(E) corresponds to the top of the acoustic branches, around 5 meV in Mg3Sb2 and 7 meV in the CaMg2Sb2. A marked suppression of κl is observed in Mg3Sb2 over the range 1 < E < 15 meV. The κl ratios (2 to 3) between binaries and ternaries are directly reflected in their phonon linewidths (κl1) in both Sb and Bi compounds (Fig. 4, C and D, and fig. S18). The much larger phonon linewidths below 4 meV in Mg3X2 correspond to the flat TA mode (e.g., along Γ − M). These are strongly scattered, as they satisfy the energy conservation rules for phonon-phonon scattering easily. They also have low phonon group velocities and thus contribute little to the total κl. In ternary compounds, these phonon branches are more dispersive, are less scattered, and carry more heat. Zhang et al. (24) emphasized these TA phonons and mid-energy optical phonon (13 to 20 meV) as the main suppression mechanism. However, this is insufficient, as the suppression happens over the entire energy range. The anharmonic scattering rates (∝ linewidth) calculated in (24), using harmonic Φ(2), are almost 10 times larger in Mg3Sb2 than in CaMg2Sb2, leading to an underestimated κl even lower than polycrystalline data. We note that using harmonic phonon dispersions significantly overestimates the scattering phase-space.

We further examine the thermal transport mechanism by isolating the relative contribution of Cv, vg, and τ in κl, and revealing the critical importance of the d3 bond in modulating the scattering phase-space. We use the TDEP κl of Mg3Sb2 and CaMg2Sb2 as references and replace each component in Mg3Sb2 with its counterpart from CaMg2Sb2, as shown in Fig. 4F. For instance, the blue curve labeled as “vg Ca” represents κl=13CVMg3Sb2vgCaMg2Sb22τMg3Sb2. As one would expect, the heat capacity effect is minimal, which almost overlaps with the κl of Mg3Sb2 (black hollow circles). The energy shift observed in INS indicates a strong change in vg. As shown in table S5, at some Q vectors, vg is enhanced by more than 50% in CaMg2Sb2, showing a corresponding increase in mode κl compared to Mg3Sb2. However, the overall increase in κl is only 22% (blue), which is a small part of the actual difference (∼3×) in total κl. On the other hand, replacing τ leads to a 2.4× higher κl (green). The scattering rate (∝τ−1) results from a product of the scattering phase-space, weighted by the phonon occupation factor, and the scattering probability Vλλλ± involving phonon modes λ, λ′, and λ″. The latter can be viewed as the mass-, frequency-, and eigenvector-weighted Fourier transform of the third-order FC (Φ(3)). Φ(2) controls the phonon group velocity, frequency, and eigenvector and determines the scattering phase-space. Φ(3) only determines the scattering probability. By separately replacing Φ(2) and Φ(3) of Mg3Sb2 with those of CaMg2Sb2, we further break down the contributions to κl. As can be seen in fig. S14B, ∣Φ(3)∣ only differs by 10% on average between the two Sb compounds, while weak Mg(1)-X bonds are again found. These weaker d3 bonds result in a 50% smaller scattering probability, and substituting the Ca Φ(3) (blue) leads to 50% decrease in κl (fig. S19). However, substituting the Ca Φ(2) (green) results in a 5× increase in κl, arising from frequency and scattering phase-space renormalization by the d3 bond. The weighted scattering phase-space (W), defined by equation S7, is shown in the inset of Fig. 4E. A large increase in W is observed for Mg3Sb2 compared with CaMg2Sb2, in the same energy range where κl is suppressed. This renormalization is controlled by Φ(2) as shown in fig. S18E.

Through a systematic investigation of phonons and thermal transport in AMg2X2 compounds with INS, IXS, and first-principles modeling including AIMD, we revealed how the superior thermal properties of Mg3X2 for TE applications originate from specific bonding characteristics, especially a weak d3 bond associated with Mg(1)-X. This weak bond plays a critical role in destabilizing TA phonons, leading to a soft flat branch along Γ − M, overcoming expected mass trends between Mg and heavier A-site ions (Ca and Yb). Our thermal conductivity analysis fully accounts for the unexpected threefold suppression in the κl of Mg3X2 compared to heavier ternaries, revealing the impact of this soft d3 bond. While group velocity suppression reduces κl by a modest 20%, the soft anharmonic dispersion enables a threefold increase in phonon scattering primarily by enhancing the scattering phase-space. These findings rationalize the microscopic origins of the outstanding thermal properties of AMg2X2 compounds and provide fundamental insights into means to control thermal transport properties, which will enable the further design of highly efficient TE materials.


For a detailed description of methods, see the Supplementary Materials. Neutron experiments were performed on AMg2X2 powders using the ARCS time-of-flight neutron spectrometer at the Spallation Neutron Source, Oak Ridge National Laboratory. The samples were sealed in a 1.27-centimeter-diameter thin-walled aluminum can and placed in a high-T closed-cycle refrigerator in an inert atmosphere. IXS measurements were performed on small single crystals mounted on standard copper posts with a beryllium dome and heated with high-T closed-cycle refrigerators in vacuum, using the HERIX beamline at Sector 30 at the Advanced Photon Source, Argonne National Laboratory. Powder samples were grown by solid-state reaction and single crystals were synthesized by the self-flux growth method.

Theoretical calculations were performed with the Vienna ab initio simulation package (VASP 5.4) (4749), using PBEsol (5052). SOC was included for Bi compounds. Phonon and thermal conductivity calculations were performed with Phonopy (53), ShengBTE (54), and TDEP (55, 56).


Supplementary material for this article is available at

This is an open-access article distributed under the terms of the Creative Commons Attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Acknowledgments: Funding: We thank O. Hellman for access to and support with the TDEP software package. X-ray and neutron scattering data collection and analysis by J.D. and T.L.-A. and first-principles simulations by J.D. were supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under award no. DE-SC0019299. Sample synthesis by M.C.-C. was supported by the U.S. DOE, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under award no. DE-SC0019252. The use of the Advanced Photon Source was supported by the U.S. DOE, Office of Science, Office of Basic Energy Sciences, under contract no. DE-AC02-06CH11357. The use of Oak Ridge National Laboratory’s Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. DOE. Theoretical calculations were performed using the National Energy Research Scientific Computing Center, a U.S. DOE Office of Science User Facility supported by the Office of Science of the U.S. DOE under contract no. DE-AC02-05CH11231. Author contributions: O.D. and A.Z. designed the research. J.D., T.L.-A., A.B., and D.L.A. performed neutron scattering measurements. J.D., T.L.-A., and A.S. performed inelastic x-ray scattering measurements. M.C.-C. and A.Z. grew and characterized the samples. J.D. performed simulations and analyzed data. J.D. and O.D. wrote the paper. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors. The numerical data for the figures are available from the Harvard Dataverse Repository at

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