High-Throughput Study of Lattice Thermal Conductivity in Binary Rocksalt and Zinc Blende Compounds Including Higher-Order Anharmonicity

Thermal transport phenomena are ubiquitous and play a critical role in the performance of various microelectronic and energy-conversion devices. Binary rocksalt and zinc blende compounds, despite their rather simple crystal structures, exhibit an extraordinary range of lattice thermal conductivity (${\kappa }_{\mathrm{L}}$) spanning over 3 orders of magnitude. A comprehensive understanding of the underlying heat transfer mechanism through the development of microscopic theories is therefore of fundamental importance, yet it remains elusive because of the challenges arising from explicitly treating higher-order anharmonicity. Recent theoretical and experimental advances have revealed the essential role of quartic anharmonicity in suppressing heat transfer in zinc blende boron arsenide (BAs) with ultrahigh ${\kappa }_{\mathrm{L}}$. However, critical questions concerning the general effects of higher-order anharmonicity in the broad classes and chemistries of binary solids are still unanswered. Using our recently developed high-throughput phonon framework based on first-principles density functional theory calculations, we systematically investigate the lattice dynamics and thermal transport properties of 37 binary compounds with rocksalt and zinc blende structures at room temperature, with a particular focus on unraveling the impacts of quartic anharmonicity on ${\kappa }_{\mathrm{L}}$. Our advanced theoretical model for computing ${\kappa }_{\mathrm{L}}$ embraces current state-of-the-art methods, featuring a complete treatment of quartic anharmonicity for both phonon frequencies and lifetimes at finite temperatures, as well as contributions from off-diagonal terms in the heat-flux operator. We find the impacts of quartic anharmonicity on ${\kappa }_{\mathrm{L}}$ to be strikingly different in rocksalt and zinc blende compounds, owing to the countervailing effects on finite-temperature-induced shifts in phonon frequencies and scattering rates. By correlating ${\kappa }_{\mathrm{L}}$ with the phonon scattering phase space, we outline a qualitative but efficient route to assess the importance of four-phonon scattering from harmonic phonon calculations. Among notable examples, in zinc blende HgTe, we identify an unprecedented sixfold reduction in ${\kappa }_{\mathrm{L}}$ due to four-phonon scattering, which dominates over the three-phonon scattering in the acoustic region at room temperature. We also demonstrate a possible breakdown of the phonon gas model in rocksalt AgCl, wherein the phonon states are significantly broadened due to strong intrinsic anharmonicity, inducing off-diagonal contributions to ${\kappa }_{\mathrm{L}}$ comparable to the diagonal ones. The deep physical insights gained in this work can be used to guide the rational design of thermal management materials.

Popular Summary

Heat conduction plays a critical role in the performance of microelectronic and energy-conversion devices. To meet the cooling demands of microprocessors and the efficiency of energy convertors, researchers are particularly interested in identifying semiconducting materials with extreme thermal conductivities. Surprisingly, these have been discovered in binary cubic compounds—materials composed of just two elements arranged in a simple cubic lattice. A comprehensive understanding of their underlying heat transfer mechanism is therefore of fundamental importance. Here, we compute the thermal transport properties of 37 binary cubic compounds.

In particular, we focus on two classes of binary cubic compounds—rocksalt and zinc blende compounds—and study how their thermal transport properties are affected by quartic anharmonicity, a fourth-order polynomial approximation to the potential energy of atomic vibrations. We find that including quartic anharmonicity always decreases the lattice thermal conductivity in zinc blendes but can either increase or decrease the conductivity in rocksalts. We also assess the importance of scattering events involving three and four phonons, the fundamental quanta of vibrations in a lattice. We show that four-phonon scattering is unprecedentedly strong in the zinc blende mercury telluride, and strong phonon scattering leads to a possible breakdown of the phonon gas model in the rocksalt silver chloride.

Our results pave the way for an in-depth understanding of heat transfer in a broad class of technologically important compounds, which may guide future engineering.

Figure 1

Comparison of theoretically calculated (${\kappa }_{3\mathrm{ph}}^{\mathrm{HA}}$) and experimentally measured (${\kappa }_{\mathrm{L}}^{\exp }$) lattice thermal conductivities in selected binary rocksalts (red disks) and zinc blendes (blue disks) at 300 K. The theoretical values are calculated at the level of $\mathrm{HA}+3\mathrm{ph}$, namely, phonon dispersion from the HA and phonon scattering rates from 3ph scattering processes. The ratio of ${\kappa }_{3\mathrm{ph}}^{\mathrm{HA}}/{\kappa }_{\mathrm{L}}^{\exp }$ is calculated by dividing the theoretical value by the higher end of the experimental values, both listed in Table 1.

Figure 2

Comparison of lattice thermal conductivity calculated using various levels of theories for selected binary rocksalts and zinc blendes at 300 K. (a) Schematic showing the ladder of approximations employed in computing ${\kappa }_{\mathrm{L}}$ based on the phonon gas model, namely, $\mathrm{HA}+3\mathrm{ph}$, $\mathrm{SCPH}+3\mathrm{ph}$, and $\mathrm{SCPH}+3,4\mathrm{ph}$, denoting the level of theory from low to high. HA and SCPH, respectively, denote the harmonic approximation (without anharmonic renormalization) and self-consistent phonon approximation (with anharmonic renormalization from quartic anharmonicity) used for computing phonon frequencies, and 3ph and 4ph indicate three-phonon and four-phonon scattering rates, respectively. The resulting lattice thermal conductivities are denoted as ${\kappa }_{3\mathrm{ph}}^{\mathrm{HA}}$, ${\kappa }_{3\mathrm{ph}}^{\mathrm{SCPH}}$, and ${\kappa }_{3,4\mathrm{ph}}^{\mathrm{SCPH}}$, respectively. (b,c) Calculated ratio of ${\kappa }_{3\mathrm{ph}}^{\mathrm{SCPH}}/{\kappa }_{3\mathrm{ph}}^{\mathrm{HA}}$ (purple bars) and ${\kappa }_{3,4\mathrm{ph}}^{\mathrm{SCPH}}/{\kappa }_{3\mathrm{ph}}^{\mathrm{HA}}$ (yellow bars), respectively. The absolute values of ${\kappa }_{3\mathrm{ph}}^{\mathrm{HA}}$, ${\kappa }_{3\mathrm{ph}}^{\mathrm{SCPH}}$, and ${\kappa }_{3,4\mathrm{ph}}^{\mathrm{SCPH}}$ are listed in Table 1.

Figure 3

Phonon frequency shifts when including quartic anharmonicity in selected binary rocksalts (red disks) and zinc blendes (blue disks). (a) Ratio of the Frobenius norm of the shifts in phonon frequencies $||\mathrm{\Delta }\omega |{|}_F$ to the Frobenius norm of the phonon frequencies $||\omega |{|}_F$ as a function of $||\omega |{|}_F$. The self-consistent phonon calculations including quartic anharmonicity are performed at 300 K. (b) Phonon dispersions of rocksalt PbTe (upper panel) and zinc blende HgTe (lower panel). The dashed gray lines denote the phonon dispersion from the harmonic approximation, and the colored solid lines denote the phonon dispersion including APRN at 300 K. (c) Strength of the four-phonon interaction matrix elements ($I_{\lambda {\lambda }_1}$) associated with the zone-center transverse optical (TO) phonon mode in the self-consistent phonon formalism (definition in Appendix pp1) as a function of phonon frequency for PbTe (red dots) and HgTe (blue dots). The black arrow points to the largest value of $I_{\lambda {\lambda }_1}$ in PbTe, with both $\lambda$ and ${\lambda }_1$ denoting the TO phonon mode. The insets show the PES of the zone-center TO mode for PbTe (upper) and HgTe (lower), respectively. The DFT computed energies are denoted by disks, while the fitted PES using polynomials up to second and fourth order are denoted by red and blue dashed lines, respectively. (d) Ratio of the shift in the frequency of the zone-center TO phonon mode ($\mathrm{\Delta }{\omega }_{\mathrm{TO}}$) to the harmonic value (${\omega }_{\mathrm{TO}}$) as a function of the ratio of quartic on-site interatomic force constants (IFCs) to harmonic IFCs along the $x$ axis. The insets show the atomic displacement vectors associated with the zone-center TO mode for rocksalts (upper) and zinc blendes (lower), respectively.

Figure 4

Correlation between the lattice thermal conductivity and the phonon scattering phase space for the selected binary rocksalts (red disks) and zinc blendes (blue disks). (a) Ratio of ${\kappa }_{3\mathrm{ph}}^{\mathrm{SCPH}}$ to ${\kappa }_{3\mathrm{ph}}^{\mathrm{HA}}$ plotted as a function of the ratio $||\mathrm{\Delta }\omega |{|}_F/||\omega |{|}_F$. (b) Ratio of ${\kappa }_{3\mathrm{ph}}^{\mathrm{SCPH}}$ to ${\kappa }_{3\mathrm{ph}}^{\mathrm{HA}}$ plotted as a function of the ratio of ${\mathrm{\Theta }}_{3\mathrm{ph}}^{\mathrm{SCPH}}$ to ${\mathrm{\Theta }}_{3\mathrm{ph}}^{\mathrm{HA}}$, wherein $\mathrm{\Theta }$ denotes phonon scattering space (see Appendix pp2 for the definition). (c) Ratio of ${\kappa }_{3,4\mathrm{ph}}^{\mathrm{SCPH}}$ to ${\kappa }_{3\mathrm{ph}}^{\mathrm{SCPH}}$ plotted as a function of the ratio of ${\mathrm{\Theta }}_{4\mathrm{ph}}^{\mathrm{SCPH}}$ to ${\mathrm{\Theta }}_{3\mathrm{ph}}^{\mathrm{SCPH}}$. (d) Ratio of ${\kappa }_{3,4\mathrm{ph}}^{\mathrm{SCPH}}$ to ${\kappa }_{\mathrm{L}}^{\exp }$ plotted as a function of experimental lattice thermal conductivity ${\kappa }_{\mathrm{L}}^{\exp }$. The ratio ${\kappa }_{3,4\mathrm{ph}}^{\mathrm{SCPH}}/{\kappa }_{\mathrm{L}}^{\exp }$ is obtained by dividing the theoretical value by the highest experimental value, both listed in Table 1. The gray dashed lines in panels (a)–(c) represent the linear fit of the data, in line with the coordinate system, while in panel (d), the line denotes equal values between theory and experiment. All calculations are performed at 300 K.

Figure 5

Phonon dispersion, scattering rates, and lattice thermal conductivity of HgTe. (a) Calculated phonon dispersions using the PBE functional (solid blue lines) and the PBEsol functional (solid orange lines) compared with experimental measurements (gray disks) [148, 149]. (b) Cumulative (solid lines) and differential (dashed lines) ${\kappa }_{\mathrm{L}}$ calculated with (blue) and without (pink) 4ph scattering. (c) MFP cumulative ${\kappa }_{\mathrm{L}}$ with (blue) and without (pink) 4ph scattering. (d) Comparison of 3ph scattering rates (empty red squares) and 4ph scattering rates (empty blue circles). The solid black line indicates that the scattering rate equals phonon frequency. (e) Decomposed 4ph scattering rates into the splitting ($\lambda \to {\lambda }_1+{\lambda }_2+{\lambda }_3$), redistribution ($\lambda +{\lambda }_1\to {\lambda }_2+{\lambda }_3$), and combination ($\lambda +{\lambda }_1+{\lambda }_2\to {\lambda }_3$) processes, which are denoted as green circles, magneta squares, and brown triangles, respectively. (f) Calculated ratio of ${\kappa }_{3,4\mathrm{ph}}^{\mathrm{SCPH}}$ to ${\kappa }_{3\mathrm{ph}}^{\mathrm{SCPH}}$ as a function of the gap between acoustic and optical phonons (a-o gap). The insets show the phonon dispersions with longitudinal acoustic phonon branch manually scaled by three different constants, in accordance with the three values of a-o gap at the X point. All calculations are performed at 300 K.

Figure 6

(a) The ratio of lattice thermal conductivity from the off-diagonal terms (${\kappa }_{\mathrm{L}}^{\text{off}\text{−}\text{diagonal}}$) to the diagonal terms (${\kappa }_{\mathrm{L}}^{\text{diagonal}}$) in the heat-flux operator as a function of ${\kappa }_{\mathrm{L}}^{\text{diagonal}}$ for selected rocksalts (red disks) and zinc blendes (blue disks). (b) Comparison of 3ph and 4ph scattering rates in AgCl. (c) Same as panel (b), but for RbCl. The solid black lines in panels (b) and (c) denote the scattering rate with the same value of the phonon frequency. (d) Contour plots of the off-diagonal contribution to lattice thermal conductivity associated with various pairs of phonon frequencies (${\omega }_s$ and ${\omega }_{s^{\prime }}$) for AgCl. (e) Same as panel (d), but for RbCl. All calculations are performed at 300 K.