🔬 materials science

Optical-Phonon-Enabled Large Lattice Thermal Conductivity Anisotropy in Hexagonal Perovskites Cs $BX_3$ ( $B$ = Mg, Cd; $X$ = Cl, Br, I)

This study reveals that hexagonal perovskites CsBX $_3$ exhibit remarkably large lattice thermal conductivity anisotropy despite near-isotropic elasticity, a phenomenon driven by efficient out-of-plane heat transport via medium-frequency optical phonons rather than the conventional acoustic phonon mechanism.

Original authors: Lingzhi Cao, Ying Song, Zhonghao Xia, Jianye Liu, Jiangang He

Published 2026-09-02

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CC BY 4.0

Original authors: Lingzhi Cao, Ying Song, Zhonghao Xia, Jianye Liu, Jiangang He

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Heat is a constant traveler inside solid materials, moving from hot spots to cooler areas through the tiny, vibrating atoms that make up the crystal. In most solids, this heat travels in all directions roughly equally, but in some special materials, it flows like water down a steep river in one direction while barely trickling across the banks in another. This directional difference, known as thermal conductivity anisotropy, is a highly sought-after trait for engineers who need to manage heat in electronics or convert waste heat into electricity. For decades, scientists have looked for these one-way heat highways in materials that are naturally layered or chain-like, where weak bonds between layers act as a barrier, forcing heat to stay within the strong layers. These materials often rely on a specific type of weak connection between atoms, similar to how a stack of paper sheets might slide easily against each other but resist being pulled apart.

However, a new study suggests that nature has a different trick up its sleeve, one that does not require weak layers or fragile structures. Researchers have discovered that certain solid crystals, which are mechanically strong and uniform in all directions, can still channel heat with surprising efficiency in just one specific direction. By examining a family of crystals made from cesium, magnesium or cadmium, and a halogen like chlorine, bromine, or iodine, the team found that heat can race along the vertical axis of the crystal while being held back in the horizontal plane. What makes this discovery so significant is that these crystals do not have the weak, layered bonds usually thought necessary for such behavior. Instead, the secret lies in how the atoms vibrate at specific speeds, turning a part of the crystal that was previously thought to be a dead end for heat into a superhighway.

The researchers used powerful computer simulations to map out the behavior of these hexagonal perovskite crystals, specifically looking at how heat moves through them at room temperature. They calculated that heat flowing vertically through the crystal can be between 2.6 and 7.5 times faster than heat flowing horizontally. In some cases, the vertical heat flow reaches up to 6.26 watts per meter per kelvin, a respectable speed for a solid, while the horizontal flow drops to as low as 0.13 watts. This is a remarkable contrast, especially because the crystals themselves are not mechanically stiff in one direction and soft in another; their internal stiffness is nearly the same in all directions. This finding challenges the old idea that you need a material that is physically weak in one direction to make heat flow poorly in that same direction.

To understand how this happens, one must look at the tiny vibrations of the atoms inside the crystal. In most materials, heat is carried by sound-like waves called acoustic phonons, which move quickly and efficiently. The researchers found that in these crystals, these sound waves do move faster vertically than horizontally, but not fast enough to explain the huge difference in heat flow on their own. The real surprise came from a different type of vibration: optical phonons. These are higher-frequency vibrations where atoms move against each other, often thought to be too sluggish to carry much heat. The study revealed that in these crystals, a specific range of these optical vibrations acts as a highly efficient transport channel for heat, but only in the vertical direction.

These special vibrations involve the entire chain of atoms linked together in a column, moving in a coordinated way that allows them to zip up and down the crystal structure. At the same time, other vibrations involving the large cesium atoms act like a brake on the horizontal flow. These cesium atoms rattle loosely within their cages, creating a lot of friction that scatters heat and stops it from moving sideways. The result is a crystal that is mechanically solid and uniform, yet thermally selective. The vertical channels remain open for the optical vibrations to carry heat, while the horizontal paths are clogged by the rattling cesium atoms.

The team tested several variations of these crystals, changing the specific atoms used to see how the heat flow changed. They found that as the atoms became heavier and the chemical bonds weaker, the overall heat flow slowed down, but the directional difference remained strong. In the lightest version of the crystal, the vertical heat flow was nearly eight times faster than the horizontal flow. In the heaviest version, the difference was still more than two and a half times. This consistency suggests that the mechanism is robust and not just a fluke of one specific chemical combination. The researchers also confirmed that the crystals are stable and could potentially be made in a lab, as the energy required to form this specific structure is very close to the most stable forms of these materials.

This work opens a new path for designing materials that can manage heat without relying on fragile, layered structures. By focusing on the way atoms vibrate in specific patterns, rather than just how stiff the material is, scientists can now look for heat highways in a much wider range of solids. The face-sharing chains of atoms in these crystals provide a natural framework for guiding heat in one direction while blocking it in another. This discovery suggests that the future of thermal management might not lie in stacking thin sheets of material, but in engineering the internal vibrations of strong, three-dimensional crystals to create precise, one-way thermal paths.

Technical Summary: Optical-Phonon-Enabled Large Lattice Thermal Conductivity Anisotropy in Hexagonal Perovskites CsBX3

Problem and Motivation
Materials exhibiting strongly anisotropic lattice thermal conductivity ( $\kappa_L$ ) are highly desirable for thermal management and thermoelectric applications, where heat must be efficiently transported along one crystallographic direction while being suppressed along another. Conventionally, such large anisotropy is associated with layered or quasi-one-dimensional van der Waals (vdW) crystals, which rely on weak interlayer interactions to create a low-conductivity direction. However, vdW materials often present challenges regarding mechanical properties, growth preferences, and integration into devices, necessitating exfoliation or precise crystallographic alignment. The authors investigate whether strong $\kappa_L$ anisotropy can be achieved in bulk, non-vdW crystals that are mechanically near-isotropic, thereby offering a more robust platform for thermal engineering. Specifically, the study focuses on hexagonal perovskites with the BaNiO $_3$ -type structure (space group $P6_3/mmc$ ), which feature face-sharing octahedral chains extending along the $c$ -axis.

Methodology
The study employs first-principles calculations combined with the unified theory of thermal transport in crystals and glasses. The computational workflow includes:

Electronic Structure: Density Functional Theory (DFT) calculations were performed using the Vienna Ab initio Simulation Package (VASP) with the PBEsol functional and projector-augmented-wave (PAW) method. Structures were fully relaxed.
Harmonic Properties: Second-order interatomic force constants (2ndIFCs) were computed using the finite-displacement method (Phonopy) with non-analytical term corrections for LO-TO splitting.
Anharmonic Properties: Third- and fourth-order force constants were extracted using the ALAMODE software. Temperature effects on 2ndIFCs were incorporated via the self-consistent phonon (SCPH) method. For the specific case of CsCdI $_3$ , which faced numerical convergence issues with SCPH, machine-learning force fields (MLFF) and the temperature-dependent effective potential (TDEP) method were utilized to extract effective force constants.
Thermal Transport: The lattice thermal conductivity was calculated by solving the Peierls-Boltzmann transport equation iteratively using the FourPhonon package, accounting for both three- and four-phonon scattering. Additionally, the coherent contribution to heat transport (off-diagonal heat flux), arising from wave-like phonon transport, was computed following the formalism by Simoncelli et al.
Validation: For CsCdI $_3$ , homogeneous nonequilibrium molecular dynamics (HNEMD) simulations using a trained neuroevolution potential (NEP) were performed to verify consistency with the phonon-gas approach.

Key Results
The study systematically investigates the $P6_3/mmc$ -type compounds CsBX $_3$ (where B = Mg, Cd; X = Cl, Br, I).

Thermal Conductivity Values: At 300 K, the in-plane ( $\kappa^\perp_L$ ) and out-of-plane ( $\kappa^\parallel_L$ ) lattice thermal conductivities range from 0.13–0.83 W m $^{-1}$ K $^{-1}$ and 0.34–6.26 W m $^{-1}$ K $^{-1}$ , respectively.
Anisotropy: The anisotropy ratio ( $\kappa^\parallel_L / \kappa^\perp_L$ ) ranges from 2.6 (for CsCdI $_3$ ) to 7.5 (for CsMgCl $_3$ ). Notably, CsCdI $_3$ exhibits the lowest $\kappa_L$ in both directions, comparable to heavy-atom vdW compounds like BiI $_3$ .
Elastic vs. Thermal Anisotropy: Despite the large thermal anisotropy, the elastic anisotropy is relatively modest, with the ratio of elastic constants $C_{33}/C_{11}$ ranging only from 0.994 to 1.842. This indicates that the thermal anisotropy is not driven by a simple scaling of acoustic phonon group velocities derived from elastic stiffness.
Scattering Mechanisms: Four-phonon scattering reduces the total $\kappa_L$ by 9–21% but does not significantly alter the anisotropy ratio. However, strong four-phonon scattering associated with low-frequency, Cs-dominated "rattling" modes (near 1 THz) substantially suppresses the in-plane particle-like contribution ( $\kappa^\perp_L$ ).
Role of Optical Phonons: The primary driver of the high out-of-plane conductivity is the contribution of medium-frequency optical phonons (3–6 THz). These modes involve extended vibrations of the face-sharing BX $_6$ octahedral chains. Unlike the in-plane direction, where these modes are weakly dispersive, they exhibit significant dispersion along the $\Gamma$ –A direction (out-of-plane), resulting in high group velocities that facilitate efficient heat transport.

Significance and Claims
The paper claims that these findings challenge the conventional view that lattice heat transport is dominated solely by acoustic phonons and that large thermal anisotropy requires highly anisotropic elastic properties or vdW gaps. Instead, the study demonstrates that:

Optical Phonon Contribution: Medium-frequency optical phonons can serve as effective heat carriers if they possess sufficiently large group velocities in a specific direction.
Mechanism of Anisotropy: In these non-vdW, mechanically near-isotropic crystals, strong thermal anisotropy arises from a combination of suppressed in-plane transport (due to rattling-mode scattering) and enhanced out-of-plane transport (via dispersive optical phonons in face-sharing octahedral chains).
Design Strategy: Face-sharing octahedral frameworks offer a promising platform for engineering strong thermal-conductivity anisotropy in bulk materials, expanding design strategies beyond layered or quasi-one-dimensional vdW systems. This suggests a pathway to materials that can separate heat-flow pathways without the mechanical and processing limitations associated with vdW materials.

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