⇦Chinese Physics Letters, 2022, Vol. 39, No. 11, Article code 116301⇨ Lattice Thermal Conductivity of MgSiO$_3$ Perovskite and Post-Perovskite under Lower Mantle Conditions Calculated by Deep Potential Molecular Dynamics Fenghu Yang (杨凤虎), Qiyu Zeng (曾启昱), Bo Chen (陈博), Dongdong Kang (康冬冬), Shen Zhang (张珅), Jianhua Wu (吴建华), Xiaoxiang Yu (余晓翔)*, and Jiayu Dai (戴佳钰)* Affiliations Department of Physics, National University of Defense Technology, Changsha 410003, China Received 19 September 2022; accepted manuscript online 12 October 2022; published online 19 October 2022 ^*Corresponding authors. Email: xxyu@nudt.edu.cn; jydai@nudt.edu.cn Citation Text: Yang F H, Zeng Q Y, Chen B et al. 2022 Chin. Phys. Lett. 39 116301 Abstract Lattice thermal conductivity ($\kappa_{\rm lat}$) of MgSiO$_3$ perovskite and post-perovskite is an important parameter for the thermal dynamics in the Earth. Here, we develop a deep potential of density functional theory quality under entire thermodynamic conditions in the lower mantle, and calculate the $\kappa_{\rm lat}$ by the Green–Kubo relation. Deep potential molecular dynamics captures full-order anharmonicity and considers ill-defined phonons in low-$\kappa_{\rm lat}$ materials ignored in the phonon gas model. The $\kappa_{\rm lat}$ shows negative temperature dependence and positive linear pressure dependence. Interestingly, the $\kappa_{\rm lat}$ undergos an increase at the phase boundary from perovskite to post-perovskite. We demonstrate that, along the geotherm, the $\kappa_{\rm lat}$ increases by 18.2% at the phase boundary. Our results would be helpful for evaluating Earth's thermal dynamics and improving the Earth model.

DOI:10.1088/0256-307X/39/11/116301 © 2022 Chinese Physics Society Article Text The Earth can be regarded as a massive thermal engine with heat flowing from the hot irony core to the cold rocky crust.[1] Thermal conduction of mantle minerals determines the magnitude of heat flux across the core-mantle boundary (CMB) and is related to the thermal dynamics and evolution of the core and mantle, the formation and stability of the mantle plumes, and the generation of the magnetic field.[1,2] MgSiO$_3$ perovskite, the most abundant mineral in the Earth's lower mantle, would undergo a phase transition into MgSiO$_3$ post-perovskite under the lowermost mantle conditions, i.e., the D$''$ region.[3,4] As a non-metallic minerals, the electronic thermal conductivity of MgSiO$_3$ is negligible. The presence of Fe in the lower mantle will greatly reduce the radiative contribution to the thermal conductivity of MgSiO$_3$,[5] making lattice vibrations the dominant factor in heat transport. Therefore, the lattice thermal conductivity $\kappa_{\rm lat}$ of MgSiO$_3$ is an important parameter in determining the heat budget of the Earth. Direct experimental measurement of $\kappa_{\rm lat}$ of MgSiO$_3$ under lower mantle conditions is a non-trivial task. In 1991, Osako and Ito conducted measurements of perovskite at ambient pressure–temperature ($P$–$T$) conditions.[6] With the development of the diamond anvil cell technique, Manthilake et al. improved the pressure to 26 GPa and the temperature to 1073 K using the multi-anvil press,[7] and Ohta et al. achieved pressure up to 144 GPa for perovskite and 141 GPa for post-perovskite at room temperature.[8,9] The result of Ohta et al. (8.1 W$\cdot$m$^{-1}\cdot$K$^{-1}$ at 300 K and 0 GPa) is higher than that of Osako and Ito (5.1 W$\cdot$m$^{-1}\cdot$K$^{-1}$ at 300 K and 0 GPa) but nearly half the value (15.4 W$\cdot$m$^{-1}\cdot$K$^{-1}$ at 473 K and 26 GPa) of Manthilake et al. The discrepancies between different experimental results may stem from the different grain sizes of samples.[10] Moreover, a recent review paper[11] stated that experimental measurement of $\kappa_{\rm lat}$ at high pressure and high temperature remains a challenge. To cover the whole $P$–$T$ range under lower mantle conditions (i.e., 23–136 GPa, 2000–4000 K),[12] long extrapolations are still indispensable, which will introduce more uncertainty based on different physical models.[6-8,13,14] Theoretical methods by combining the phonon gas model (PGM) with ab initio anharmonic lattice dynamics or phonon Boltzmann transport equation (BTE) have been used to predict the $\kappa_{\rm lat}$ of MgSiO$_3$ under lower mantle conditions.[10,14-17] Under the framework of PGM, in which phonon is considered as a quasiparticle, $\kappa_{\rm lat}$ is calculated based on heat capacity, group velocity, and mean free path or lifetime. However, in low-$\kappa_{\rm lat}$ materials, a number of phonons, whose quasiparticle picture of phonon is suspected to fail, are referred to as ill-defined phonons,[18,19] for example, when phonon mean free path is shorter than the Ioffe–Regel limit.[20] Especially, previous work demonstrated the subminimal mean free path of phonons in MgSiO$_3$ perovskite.[17] To deal with the ill-defined phonons, the molecular dynamics (MD) method is an alternative way to obtain thermal conductivities.[21] It takes into account the full-order anharmonicity, which is significant at high temperatures.[22,23] First-principles MD is accurate but suffer from formidable computation cost and size effect.[24] Classical MD is efficient. However, traditional empirical potential used in classical MD cannot accurately capture the interatomic interactions over a wide $P$–$T$ range under lower mantle conditions. In comparison, machine learning method can address the dilemma of accuracy versus efficiency[25-27] and has been successfully applied in the MD simulations under extreme conditions.[28,29] In this work, we train a deep neural network potential model from density functional theory (DFT) datasets. After a set of careful tests about energy, force, stress, equation of state, and phonon dispersion, we calculate $\kappa_{\rm lat}$ of MgSiO$_3$ perovskite and post-perovskite under lower mantle conditions. We report the increase of $\kappa_{\rm lat}$ at the perovskite–post-perovskite phase boundary. We finally discuss the variation of $\kappa_{\rm lat}$ along the geotherm and its impact on the thermal dynamics of the Earth. Computation Details. To prepare the ab initio datasets, DFT-MD simulations were performed using the Quantum ESPRESSO package.[30] The generalized gradient approximation in the Perdew–Burke–Ernzerhof parametrization[31] was used to treat the electron exchange-correlation functional. A projector augmented wave pseudopotential[32,33] was used with planewave cutoff energy up to 100 Ry. Based on the ab initio datasets, we build deep potential (DP) models of MgSiO$_3$ perovskite and post-perovskite with DeePMD-kit packages.[34] To cover Earth's lower mantle conditions and minimize the computational consumption, DP-GEN[35] was adopted to sample broader datasets to guarantee the uniform accuracy of DP in the explored configurations space. Deep potential molecular dynamics (DPMD) simulations were performed with the LAMMPS package.[36] The Nose–Hoover thermostat[37,38] was employed in the NVT ensemble simulation. Phonon dispersion was calculated by the finite-displacement method[39] using ALAMODE[40,41] to test the accuracy of DP. The $\kappa_{\rm lat}$ is obtained by the Green–Kubo method based on the linear response theory,[42,43] which links $\kappa_{\rm lat}$ to the fluctuations in thermodynamic equilibrium via the fluctuation-dissipation theorem. More details can be found in the Supplementary Material. Validation of the Machine Learning Potential. To confirm the accuracy of machine learning potential, we compare the energies, atomic forces, stresses, and radial distribution functions from the DP with those from DFT calculations. Figure S1 in the Supplementary Material shows that the root-mean-square errors of the energies, atomic forces, and stresses are 8.0822 meV/atom, 0.0531 eV/Å, and 0.8676 GPa, respectively. Figure S2 shows the radial distribution functions of MgSiO$_3$ perovskite at 4000 K, 112 GPa from DFT and DP. Moreover, the equation of state (EOS) under the wide $P$–$T$ conditions is validated. As can be seen in Fig. 1, the EOSs of perovskite and post-perovskite predicted by our DP are in fair agreement with the previous DFT calculations using the same exchange-correlation functional.[10] However, both of them overestimate the pressure at the same volume compared with the experimental values.[44-51] This is because of the exchange-correlation functional used in the DFT calculations. More accurate exchange-correlation functional may give more consistent results, and this is beyond the scope of this work. Furthermore, phonon dispersion is important property for the investigation of $\kappa_{\rm lat}$. In Fig. 2, we show the comparison of the phonon dispersion at 120 GPa predicted by the DP and DFT. Except for a few high-frequency optical branches that show slight shifts, almost all phonon branches are consistent. Therefore, the robustness of DP is illustrated to realize the basic phonon properties such as heat capacity and group velocity.

Fig. 1. Equation of state of MgSiO$_3$ (a) perovskite and (b) post-perovskite at 300 K and 2000 K. Solid curves are obtained using DP, dashed lines are the DFT results by Zhang et al.,[10] and experimental data are taken from Refs. [44,45,46,47,48,49,50,51].

Fig. 2. Phonon dispersions of MgSiO$_3$ perovskite at 120 GPa calculated by DP and DFT.

Then, we come to compare the $\kappa_{\rm lat}$ of perovskite calculated by the DP with other results from MD, PGM, and experiments. The scarcity of high-temperature experimental data only enables us to test our DP results at 300 K and at different pressures. The experimental data (grey dots) are plotted in Fig. 3. Manthilake et al.[7] measured a $\kappa_{\rm lat}$ of 15.6 W$\cdot$m$^{-1}\cdot$K$^{-1}$ at 26 GPa and 473 K, and extrapolated it to 19.2 W$\cdot$m$^{-1}\cdot$K$^{-1}$ at 26 GPa and 300 K. Ohta et al. [9] reported a $\kappa_{\rm lat}$ of 8.1 W$\cdot$m$^{-1}\cdot$K$^{-1}$ under ambient conditions. Although the values of $\kappa_{\rm lat}$ are not the same, they should be close to each other, since an increase in temperature (thus a reduction of thermal conductivity) is partially compensated by an increase in pressure (thus an increase of thermal conductivity). According to the statement in Ref. [10], the grain sizes of measured samples by Manthilake et al. are 10–15 µm, while those by Ohta et al. are 1 µm. In the samples with larger grain sizes, weaker phonon-boundary scatterings lead to longer mean free paths, so Manthilake et al. obtained larger $\kappa_{\rm lat}$. Also, the presence of lattice defects always reduces thermal conductivity. Therefore, higher $\kappa_{\rm lat}$ in experimental measurement typically corresponds to higher-quality samples and is more appropriate for comparisons with our simulations of ideal crystals. In general, as shown in Fig. 3 (red dots and solid line), DPMD results are in the middle region among all the results. A $\kappa_{\rm lat}$ of 23.6 W$\cdot$m$^{-1}\cdot$K$^{-1}$ at 26 GPa and 300 K is obtained by extrapolation. Given that the ideal crystal structures are used in our DPMD simulations, it is reasonable that our result is slightly higher than the experiment data of Manthilake et al.[7] Haigis et al.[13] conducted classical MD using the aspherical ion model potential,[52] and obtained a $\kappa_{\rm lat}$ of 27.0 W$\cdot$m$^{-1}\cdot$K$^{-1}$ at 26 GPa and 300 K, which is larger than the data of Manthilake et al. and our DPMD results. Stackhouse et al.[24] employed the ab initio non-equilibrium MD method and extrapolated $\kappa_{\rm lat}$ to the thermodynamic limit. The results are much lower than the data of Manthilake et al.

Fig. 3. Lattice thermal conductivity of MgSiO$_3$ perovskite versus pressures at 300 K. Filled circles in red are our DPMD results, red solid lines are the linear fitting line, and red shading indicates the standard error. The MD results from Haigis et al.[13] and Stackhouse et al.[24] are given. Grey dots are measured data ($\blacktriangle$ Osako and Ito,[6] $\bigstar$ Manthilake et al.,[7] $\blacksquare$ Ohta et al.,[8] and $\blacktriangledown$ Ohta et al.[9]). Other lines are calculated by the PGM approach from Dekura et al.,[14,53] Tang et al.,[15] Ghader et al.,[16] Zhang et al.,[17] and Zhanget al.[10] For interpretation of the references to color in this figure legend, the reader can refer to the web version of this article.

There are six previous works using the PGM approach. Dekura et al.,[14,53] Tang et al.,[15] and Ghader et al.[16] used the BTE method to calculate the phonon lifetime by considering three-phonon scatterings. Tang et al. and Ghader et al. used denser $k$-mesh than Dekura et al. Ghader et al. used iteratively solved phonon lifetimes while Dekura et al. and Tang et al. employed the relaxation time approximation. Zhang et al.[17] and Zhang et al.[10] considered full-order phonon anharmonicity based on ab initio MD trajectory. Zhang et al.[10] reported a closer value to the experimental data of Manthilake et al.[7] With the development of calculation method and the improvement of precision in recent decades, after careful scrutiny, we favor the work of Zhang et al.[10] as a reference in following discussions about $\kappa_{\rm lat}$ under lower mantle conditions. Lattice Thermal Conductivity under Lower Mantle Conditions. Lattice thermal conductivities of MgSiO$_3$ in a wide $P$–$T$ range from DPMD simulations are summarized in Fig. 4. Both perovskite and post-perovskite phases show the expected decrease in $\kappa_{\rm lat}$ with increasing temperature. Higher temperature leads to stronger phonon scatterings, thus shorter phonon mean free path and lower $\kappa_{\rm lat}$.[54] Meanwhile, the increase in $\kappa_{\rm lat}$ with increasing pressure originates from the compressed bonds and enhanced atomic interactions, thus modifying the phonon dispersions and enhancing the group velocities. The experimental data of $\kappa_{\rm lat}$ at 26 GPa and 1000 K (violet square dot) was extracted from the work of Manthilake et al.[7] Both our DPMD results and the PGM results of Zhang et al.[10] are smaller than the results of Manthilake et al. under the same condition. However, the DPMD value is closer to the experiment. The linear pressure dependence of $\kappa_{\rm lat}$ is a typical feature, which is similar to other experiments and ab initio calculations for other minerals in lower mantle, such as MgSiO$_3$ perovskite,[8,16,17] MgO,[55,56] and cubic $\rm{CaSiO_3}$ perovskite.[57]

Fig. 4. Lattice thermal conductivity of MgSiO$_3$ perovskite and post-perovskite versus pressures at various temperatures (blue for 1000 K, green for 2000 K, yellow for 3000 K, and red for 4000 K). Filled circles are the DPMD results, solid lines are the linear fitting line, and shading indicates the standard error. Dashed lines are the PGM results from Zhang et al.[10] Square dot is the measured data of Manthilake et al.[7] Line width decreases as the temperature increases. For interpretation of the references to color in this figure legend, the reader can refer to the web version of this article.

As one can see, our results are larger than the results of Zhang et al.[10] especially at low pressure and high temperature. The difference becomes larger as temperature rises while decreases with the increase of pressure. We attribute this discrepancy to the difference in the calculation method. It has been confirmed that the mean free paths of a certain number of phonons are shorter than the lattice constants in MgSiO$_3$ under lower mantle conditions.[17] This means that there are many ill-defined phonons in MgSiO$_3$ at elevated temperatures under the framework of PGM. Luo et al.[18] pointed out that the contribution of ill-defined phonons in low-$\kappa_{\rm lat}$ materials, which could be described by the diffusion theory, is underestimated by PGM. The DPMD results inherently contain the contributions of both well-defined and ill-defined phonons, thus averting the trouble of phonon quasiparticle picture. Low pressure results in a low group velocity, meanwhile, high temperature leads to a short phonon lifetime. Therefore, at low pressure or high temperature, the contribution from ill-defined phonons neglected in the PGM approach increases due to its increased proportion. Due to the transition from perovskite into post-perovskite of MgSiO$_3$ near the D$''$ region, it is necessary to demonstrate the change of $\kappa_{\rm lat}$ near the phase boundary at different temperatures. We refer to the phase boundary from the work of Belonoshko et al.[58] Figure 5 shows the $\kappa_{\rm lat}$ of MgSiO$_3$ at the corresponding phase along the isotherm line (green solid line). Non-negligible gaps of $\kappa_{\rm lat}$ at different temperatures are found when MgSiO$_3$ transits from perovskite into post-perovskite. The enhancement at 2000 K and 116 GPa is 13.9%, which is smaller than 18% predicted by Zhang et al.[10] At 4000 K and 136 GPa, the enhancement by DPMD is 21.1%, much lower than the 50% enhancement reported by Dekura and Tsuchiya.[53] The different crystal structure and vibration intensity between perovskite and post-perovskite result in dissimilar phonon property. Zhang et al.[10] showed that post-perovskite has higher phonon velocities and longer lifetime than perovskite at high pressure, which leads to a larger $\kappa_{\rm lat}$. More detailed discussions can be found elsewhere.[10]

Fig. 5. Lattice thermal conductivity of perovskite and post-perovskite along the isotherm of 2000 K, 3000 K, and 4000 K. Phase boundary is obtained from Belonoshko et al.[58]

Fig. 6. Color contour plot of lattice thermal conductivity of perovskite and post-perovskite along the geotherm in the lower mantle. Phase boundary is obtained from Belonoshko et al.[58] and geotherm is found from Stacey and Davis.[4]

Finally, we discuss the thermal dynamics of the Earth based on our $\kappa_{\rm lat}$ results. We plot the $\kappa_{\rm lat}$ map of MgSiO$_3$ along the geotherm[58] under lower mantle conditions in Fig. 6. The $\kappa_{\rm lat}$ increases along the geotherm at low pressure, enhances 18.2% from 6.6 W$\cdot$m$^{-1}\cdot$K$^{-1}$ to 7.8 W$\cdot$m$^{-1}\cdot$K$^{-1}$ across the phase boundary, and decreases near the core where temperature plays a major role. In ancient times, the mantle temperature was higher and the transition of MgSiO$_3$ to post-perovskite near the CMB was suppressed. As the Earth became cooler, a thick post-perovskite layer above the core appeared and enhanced the convection in the mantle and core. According to two-dimensional models of mantle convection,[59,60] the increase of $\kappa_{\rm lat}$ across the phase transition by DPMD would give higher velocity downwellings and larger asymmetry of the convective planform. Based on the geodynamic models,[61-63] compared with the results of Zhang et al.,[10] the larger $\kappa_{\rm lat}$ by DPMD implies a more stable and thicker boundary layer and results in the increases of temperature and size of the mantle plumes from the CMB. In summary, we have trained a machine learning potential model with DFT quality for MgSiO$_3$ perovskite and post-perovskite under Earth's mantle conditions. We reproduce the EOS and phonon dispersion to validate the accuracy of DP. The calculated $\kappa_{\rm lat}$ at 300 K and 26 GPa is 23.6 W$\cdot$m$^{-1}\cdot$K$^{-1}$, close to the experiment value.[7] The $\kappa_{\rm lat}$ of MgSiO$_3$ up to 4000 K and 160 GPa is obtained by combining the DPMD and the Green–Kubo method. The $\kappa_{\rm lat}$ shows negative temperature dependence and positive linear pressure dependence. The $\kappa_{\rm lat}$ from DPMD is found to be larger than that from PGM in the previous work,[10] especially at low pressure and high temperature. The difference is attributed to the failure of phonon quasiparticle picture. The contribution of ill-defined phonons is underestimated in PGM but contained in MD simulations. Furthermore, we show the increase of $\kappa_{\rm lat}$ at the boundary of the perovskite to post-perovskite phase transition and find smaller gaps than those in the previous work.[8,10,53] We demonstrate that the $\kappa_{\rm lat}$ of MgSiO$_3$ increases by 18.2% from 6.6 W$\cdot$m$^{-1}\cdot$K$^{-1}$ to 7.8 W$\cdot$m$^{-1}\cdot$K$^{-1}$ across the phase boundary along the geotherm. The DPMD's higher results and the increase gap at the phase boundary show larger $\kappa_{\rm lat}$ at the CMB, which implies a more thicker boundary layer and stable mantle plumes. Our results would provide basic thermophysical data for evaluating the thermal dynamics and have benefits for improving the Earth model. Moreover, our deep potential model can be used to investigate thermodynamic and transport properties of more complex systems such as polycrystals and interfaces. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant No. U1830206), the National Key R&D Program of China (Grant No. 2017YFA0403200), the National Natural Science Foundation of China (Grant Nos. 11874424, 11904401, 11974423, and 12104507), and the Science and Technology Innovation Program of Hunan Province (Grant No. 2021RC4026). References Core–mantle boundary heat flowAbnormal Elastic Changes for Cubic-Tetragonal Transition of Single-Crystal SrTiO₃Post-Perovskite Phase Transition in MgSiO3Radiative conductivity in the Earth’s lower mantleThermal diffusivity of MgSiO₃ perovskiteLattice thermal conductivity of lower mantle minerals and heat flux from Earth’s coreLattice thermal conductivity of MgSiO3 perovskite and post-perovskite at the core–mantle boundaryThermal diffusivities of MgSiO3 and Al-bearing MgSiO3 perovskitesAb initio lattice thermal conductivity of

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