Understanding the flat thermal conductivity of ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ at ultrahigh temperatures

Many crystals, such as lanthanum zirconate $\left({\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7\right)$, exhibit a flat temperature dependence of thermal conductivity at elevated temperatures. This phenomenon has recently been attributed to the interband phonon tunneling (or diffuson) contribution using different formalisms. However, the contributions of finite-temperature corrections (e.g., higher-order phonon scattering, phonon renormalization, and phonon-scattering cross-section softening effects) and radiation at high temperatures remain unclear. In this work, we predict and compare the thermal conductivity of ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ using three distinct first-principles methods. The first method is Green-Kubo molecular dynamics (MD) based on temperature-dependent machine-learning interatomic potentials trained from ab initio MD simulations, which successfully predict the flat trend at ultrahigh temperatures. The second method is the Peierls Boltzmann transport equation (BTE), within the phonon particle framework, using phonon lifetime that includes all the finite-temperature corrections. Four-phonon scattering is found large but is canceled by the phonon-scattering cross-section softening effect. As a result, BTE with temperature corrections does not reproduce the flat thermal conductivity. The third method is Wigner formalism, which includes both phonon particle and wave contributions, which successfully reproduce the flat thermal conductivity. Diffuson and phonon contribute about 67 and 27% of thermal conductivity at 1800 K, respectively. The radiation contribution to thermal conductivity is calculated by the Rosseland model and found to be around 6%. The scaling laws of the phonon, diffuson, radiation, and total thermal conductivity are found to be $\sim T^{-0.97}$, $\sim T^{0.43}$, $\sim T^{2.01}$, and $\sim T^{-0.40}$, respectively. This work clarifies the thermal transport mechanisms in ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ at ultrahigh temperatures from different aspects.

Figure 1

Methodology employed in this work.

Figure 2

(a) Temperature-dependent relative lattice constant $\left[a\left(T\right)/a_{300\mathrm{K}}\right]$ of ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ predicted in this work compared to the experimental data. (b) Linear thermal expansion coefficient $\left[{\alpha }_L\left(T\right)\right]$. Experimental data: Lehmann et al., 2003 [57], Wang et al., 2009 [58], and Zhang et al., 2012 [56].

Figure 3

(a) MTP level test. (b) Phonon dispersion calculated by MTP and DFT. (c) Energy and (d) forces predicted for configurations in the testing database from MTP and DFT.

Figure 4

(a) Thermal conductivity of ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ obtained using GKMD at 300 and 1200 K as a function of correlation time. (b) Thermal conductivity of ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ obtained from GKMD at 300 K as a function of supercell size.

Figure 5

(a) Thermal conductivity of ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ using GKMD with MTP. Experimental data are provided for reference. (b) Specific heat capacity and (c) thermal diffusivity of ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$. References of experimental data: Zhang et al. [8], Xiang et al. [10], Wang et al. [11], Wan et al. [12], Vassen et al. [6], and Suresh et al. [9].

Figure 6

Phonon total SED data for a randomly picked $q$ point (1/3, 0, 0) at (a) 300 K and (b) 1200 K. SED data for a single randomly picked phonon mode, i.e., the third branch of $q$ point (1/3, 0, 0), at (c) 300 K, (d) 750 K, and (e) 1200 K.

Figure 7

Phonon-scattering rates extracted using SED analysis and BTE (3ph and 4ph) at (a) 300 K and (b) 1200 K. The $q$-mesh sizes are all $6\times 6\times 6$. (c) Thermal conductivity of ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ within the framework of BTE using the phonon lifetimes obtained from SED analysis and BTE (3ph and 3+4ph). (d) The comparison of thermal conductivity between BTE (3ph) and experimental data. Since 3ph thermal conductivity agrees with SED thermal conductivity, this figure can also be seen as the compassion between SED and experimental data. References of experimental data: Zhang et al. [8], Xiang et al. [10], Wang et al. [11], Wan et al. [12], Vassen et al. [6], and Suresh et al. [9].

Figure 8

(a) Temperature-dependent thermal conductivity of ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ obtained using Wigner formalism. References of experimental data: Zhang et al. [8], Xiang et al. [10], Wang et al. [11], Wan et al. [12], Vassen et al. [6], and Suresh et al. [9]. (b) Phonon lifetime at 300 and 1200 K by solving BTE with $16\times 16\times 16$ q mesh for 3ph. (c) Percentage cumulative phonon thermal conductivity with respect to mean free path at 300, 1200, and 1800 K. (d) Phonon dispersion obtained from SED analysis at 300 and 1200 K, compared to the ground-state dispersion.

Figure 9

(a) Thermal conductivity obtained using Wigner formalism as a function of $\mathbf{q}$-mesh density. (b) Lifetime of phonons in the framework of BTE with different $\mathbf{q}$-mesh densities.

Figure 10

The impact of nonanalytical correction on (a) temperature-dependent thermal conductivity, (b) phonon lifetime at 300 K, (c) group velocities ($x$ component) at 300 K, and (d) phonon dispersion.

Figure 11

Spectral radiative thermal properties of ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ calculated from first principles. Ground-state force constants are used in the calculations. (a) Real part and (b) imaginary part of the refractive index calculated from the Lorentz oscillator model using the phonon linewidths calculated from the 3ph and 4ph scattering. (c) Penetration depth of photons. (d) Radiation thermal conductivity as a function of temperature. $T^{2.01}$ is a fitted power law using the calculated ${\kappa }_{\mathrm{rad}}$ data.