Anharmonic lattice dynamics and the origin of intrinsic ultralow thermal conductivity in AgI materials

Ionic conductors such as AgI with ultralow thermal conductivities $\left({\kappa }_l\right)$ are of increasing interest because of their excellent thermoelectric properties. However, the origin of their intrinsic low ${\kappa }_l$ values remain elusive. In this study, comprehensive theoretical calculations of the lattice dynamics and the thermal transport properties of $\gamma \text{−}\mathrm{AgI}$ (zinc-blende structure) and $\beta \text{−}\mathrm{AgI}$ (wurtzite structure) as functions of temperature were carried out based on many-body perturbation theory and phonon Boltzmann transport theory. First, the mean-squared displacements (MSDs) of ${\mathrm{Ag}}^{+}$ were significantly larger than those of ${\mathrm{I}}^{-}$ in both $\gamma \text{−}$ and β-phases below the order-disorder phase transition temperature $\left(T_{\mathrm{c}}\right)$, which led to a characteristic “rattling” feature and low-frequency, nearly flat local phonon vibrations. According to our previous work [Xie et al., Phys. Rev. Lett. 125, 245901 (2020)], such nondispersive flat phonon band structures are expected to give rise to four-phonon resonance and result in a dramatic increase in the four-phonon scattering over the conventional three-phonon scattering. For $\gamma \text{−}\mathrm{AgI}$, similar four-phonon resonance behavior was also discovered for the low-lying transverse acoustic phonon branches, and it was found that their four-phonon scattering rates were an order of magnitude larger than the corresponding three-phonon scattering rates. Considering the four-phonon scattering, the theoretical ${\kappa }_l$ of $\gamma \text{−}\mathrm{AgI}$ was predicted to be $\sim 0.32$ W/m K at 300 K, which was in good agreement with the value deduced from our experiments ($\sim 0.36$ W/m K at 300 K). Compared to $\gamma \text{−}\mathrm{AgI}$, the acoustic phonons in $\beta \text{−}\mathrm{AgI}$ were more dispersive, and they intertwined with low-energy optical phonons at the zone boundaries. It was found that three-phonon resonance became as important as four-phonon resonance for the nearly flat longitudinal phonon band. The theoretical ${\kappa }_l$ for $\beta \text{−}\mathrm{AgI}$ was determined to be around $\sim 0.32$ W/m K at room temperature, closely reproducing our measurement value $\sim 0.29$ W/m K. Our results for AgI demonstrate the strong quartic anharmonicity in materials characterized by the rattling of weak bonding atoms as well as dispersionless phonon band structures. It is believed that this intimate relationship between the low-${\kappa }_l$ and flat phonon dispersion can be employed as a good indicator when searching for material systems with ultralow ${\kappa }_l$ values, e.g., cagelike rattling structures, quasi-two-dimensional structures, and chainlike structures.

Figure 1

Helmholtz free energies of γ- and β-AgI, with the wurtzite and zinc-blende structures drawn in the insets.

Figure 2

Phonon dispersions of (a) $\gamma \text{−}\mathrm{AgI}$ and (b) $\beta \text{−}\mathrm{AgI}$ in the temperature range of 100–400 K. Atom-projected densities of states for (c) $\gamma \text{−}$ and (d) $\beta \text{−}\mathrm{phase}$ AgI at 300 K.

Figure 3

Scattering phase space of (a) $\gamma \text{−}\mathrm{AgI}$ and (b) $\beta \text{−}\mathrm{AgI}$ at 300 K. Dark-gray triangles represent the phase space available for three-phonon scattering processes and blue squares represent the phase space available for four-phonon scattering processes.

Figure 4

Phonon spectral functions including (a)–(d) only three-phonon scattering and (e)–(h) both three-phonon and four-phonon scattering processes in the temperature range of 100–400 K for $\gamma \text{−}\mathrm{phase}$ AgI.

Figure 5

Phonon spectral functions including (a)–(d) only three-phonon scattering and (e)–(h) both three-phonon and four-phonon scattering processes in the temperature range of 100–400 K for $\beta \text{−}\mathrm{phase}$ AgI.

Figure 6

Spectral functions $S_{k,s}\left(\omega \right)$ at the (a)–(d) $U$ point for $\gamma \text{−}\mathrm{AgI}$ and (e)–(h) $M$ point for $\beta \text{−}\mathrm{AgI}$. Dashed lines are the results calculated with the three-phonon interactions ${\sum }_{\vec{k},s}^{{}^{\prime }}\left(\omega \right)$ alone, while the solid lines are the results calculated with both three- and four-phonon interactions ${\sum }_{\vec{k},s}^{{}^{\prime }}\left(\omega \right)+{\sum }_{\vec{k},s}^{{}^{\prime \prime }}\left(\omega \right)$. Colored vertical bars denote the phonon eigenfrequency ${\omega }_{\vec{k},s}$. Linewidth broadening and frequency shift due to the four-phonon interactions are evident.

Figure 7

Imaginary part ${\mathrm{\Gamma }}_{\vec{k},s}^{{}^{\prime }}\left(\omega \right)$ (with three-phonon interactions shown by dashed lines) and ${\mathrm{\Gamma }}_{\vec{k},s}^{{}^{\prime \prime }}\left(\omega \right)$ (with three- and four-phonon interactions shown by solid lines) of the acoustic phonon self-energy functions for the (a)–(d) γ phase at the zone-boundary point $U$ and the (e)–(h) β phase at the point $M$ at different temperatures.

Figure 8

Experimental and theoretical thermal conductivities of (a) the γ phase and (b) β phase as functions of temperature. Blue spheres represent our measured thermal conductivity values. Measured ${\kappa }_l$ values in (b) are comparable to the literature values [39] (gray empty circles). Considering the boundary size, the theoretical ${\kappa }_{l,3ph}$ with a boundary size range from 500 nm to ∞ is shown in the purple-shaded area, along with the ${\kappa }_{l,3ph}$ of 1 $\mu \mathrm{m}$ inside (purple triangles). The theoretical ${\kappa }_{l,3ph}$ of $\gamma \text{−}$ and $\beta \text{−}\mathrm{AgI}$ was in good agreement with the previous theoretical work (orange asterisks) in Ref. [40]. The calculated ${\kappa }_{l,3ph+4ph}$ (black squares) were consistent with our measured thermal conductivities.