Unified first-principles theory of thermal properties of insulators

The conventional first-principles theory for the thermal and thermodynamic properties of insulators is based on the perturbative treatment of the anharmonicity of crystal bonds. While this theory has been a successful predictive tool for strongly bonded solids such as diamond and silicon, here we show that it fails dramatically for strongly anharmonic (weakly bonded) materials, and that the conventional quasiparticle picture breaks down at relatively low temperatures. To address this failure, we present a unified first-principles theory of the thermodynamic and thermal properties of insulators that captures multiple thermal properties within the same framework across the full range of anharmonicity from strongly bonded to weakly bonded insulators. This theory features a new phonon renormalization approach derived from many-body physics that creates well-defined quasiparticles even at relatively high temperatures, and it accurately captures the effects of strongly anharmonic bonds on phonons and thermal transport. Using a prototypical strongly anharmonic material, sodium chloride (NaCl), as an example, we demonstrate that our new first-principles framework simultaneously captures the apparently contradictory experimental observations of large thermal expansion and low thermal conductivity of NaCl on the one hand, and anomalously weak temperature dependence of phonon modes on the other, while the conventional theory fails in all three cases. We demonstrate that four-phonon scattering due to higher-order anharmonicity significantly lowers the thermal conductivity of NaCl and is required for a proper comparison to experiment. Furthermore, we show that our renormalization framework, along with four-phonon scattering, also successfully predicts the measured phonon frequencies and thermal properties of a weakly anharmonic material, diamond, indicating universal applicability for thermal properties of insulators. Our work gives new insights into the physics of heat flow in solids, and presents a computationally efficient and rigorous framework that captures the thermal and thermodynamic properties of both weakly and strongly bonded insulators simultaneously.

Figure 1

(a) Calculated quasiharmonic phonon dispersions compared with measured data from Raunio et al. [32] at 80 K (blue) and 300 K (red). The circled region shows the transverse optic (TO) branch near $\mathrm{\Gamma }$. (b) The rate of thermal expansion calculated in the QHA (dashed black curve) compared with measured data [33, 34, 35, 36]. (c) Three-phonon and 3+4-phonon limited thermal conductivity as a function of temperature calculated using quasiharmonic phonons compared with measured data from Hakansson et al. [37] (extrapolated from low pressure values), McCarthy et al. [38] and Yukutake et al. [39].

Figure 2

Ratio of 3+4-phonon scattering rates to phonon frequency (${\mathrm{\Gamma }}_{\left(3+4\right)}/\omega )$ calculated using quasiharmonic phonons at three different temperatures. ${\mathrm{\Gamma }}_{\left(3+4\right)}/\omega$ approaches 1 above 400 K and even exceeds it beyond $\sim 500\mathrm{K}$, causing the quasiparticle description of the quasiharmonic phonons to break down.

Figure 3

Renormalized phonon dispersions at the fourth-order anharmonic free energy minimum lattice constants (solid curves), and phonon dispersions at the quasiharmonic lattice constants (dashed curves) compared with measured data from Raunio et al. [32] at (a) 80 K and (b) 300 K.

Figure 4

(a) Rate of thermal expansion calculated in the QHA (dashed black curve) and by the phonon renormalization approach (solid black curve) compared with measured data [33, 34, 35, 36]. (b) Ratio of 3+4 phonon scattering rates to phonon frequency (${\mathrm{\Gamma }}_{(3+4)}/\omega$) calculated using renormalized phonons at three different temperatures. The weakened scattering rates after phonon renormalization give a significant reduction to ${\mathrm{\Gamma }}_{(3+4)}/\omega$ and produce well-defined quasiparticles. (c) Three-phonon and 3+4-phonon limited thermal conductivity as a function of temperature calculated using the renormalization approach (solid red curve) compared with measured data from Hakansson et al. [37], McCarthy et al. [38], and Yukutake et al. [39]. The corresponding curves (black) for the quasiharmonic phonons are included for comparison.

Figure 5

Evolution of (a) quasiharmonic and (b) renormalized phonon dispersions between 100 and 600 K. (c) Temperature-dependent phonon frequencies of the TO mode at $\mathrm{\Gamma }$ and the LO modes at $2\pi /a\left(0.1,0.1,0.1\right)$, $2\pi /a\left(0.2,0.2,0.2\right)$, and $2\pi /a\left(0.3,0.3,0.3\right)$ points in the Brillouin zone compared with INS [32, 48] and infrared emission measurements [49]. The quasiharmonic phonons show significant temperature driven softening while the renormalized phonon dispersions exhibit weak temperature dependence, consistent with the measured data [32, 48, 49].

Figure 6

(a) Quasiharmonic (dashed curves) and renormalized phonon (solid curves) dispersions of diamond at 300 K compared with experiments from Warren et al. [50]. (b) Rate of thermal expansion as a function of temperature for diamond compared with measured data [51]. Both QH and renormalized phonons produce nearly identical thermal expansion rate. (c) Three-phonon (dashed curves) and 3+4-phonon (solid curves) limited thermal conductivity of diamond calculated using renormalized phonons and fourth-order free energy minimized lattice constant compared with experiments. The green curves are calculations for synthetic isotopically purified diamond with 0.07% ${\mathrm{C}}^{13}$ and the black curves are for naturally occurring diamond (1.1% ${\mathrm{C}}^{13}$). The experiments are from Olson et al. [52] (green empty triangles: synthetic isotopically purified diamond with 0.07% ${\mathrm{C}}^{13}$, orange filled triangles: naturally occurring diamond with 1.1% ${\mathrm{C}}^{13}$), Onn et al. [53] (green empty circles: synthetic isotopically purified diamond with $<0.05\%$ ${\mathrm{C}}^{13}$, green filled circles: naturally occurring diamond with 1.1% ${\mathrm{C}}^{13}$), Wei et al. [54] (blue filled squares: naturally occurring diamond with 1.1% ${\mathrm{C}}^{13}$), and Berman et al. [55] (red filled inverted triangles: naturally occurring diamond with 1.1% ${\mathrm{C}}^{13}$). (d) Three-phonon (dashed curves) and 3+4-phonon (solid curves) limited thermal conductivity of diamond with quasiharmonic (black) and renormalized (red) phonons.