1. Introduction

The relentless miniaturization and increasing integration density of electronic devices are leading to continuously rising power densities, exacerbating challenges in thermal management^[1-4]. Accordingly, tailoring thermal transport and improving heat-dissipation capability have become important design strategies for advanced thermal-management materials^[5-9]. Consequently, materials exhibiting intrinsic anisotropic thermal conductivity (κ) have emerged as a focus of research, given their ability to conduct heat efficiently along one crystallographic direction while acting as a thermal insulator in the perpendicular direction^[10,11]. This capability to directionally channel heat flow presents a viable approach to mitigating localized hotspots and enhancing the efficacy of thermal management systems in modern electronics. Although these applications are promising, significant gaps persist in the systematic understanding of thermal transport anisotropy and the underlying physical mechanisms. Concerted efforts to deepen the exploration of these materials and develop effective strategies for modulating their thermal properties are essential, as such advances could yield critical insights for tackling thermal management challenges in next-generation electronics.

Capitalizing on versatile bonding configurations, carbon materials give rise to diverse 2D allotropes. The successful exfoliation of graphene facilitated the first experimental investigation of thermal transport properties in strictly two-dimensional crystals^[12]. Measurements revealed that its thermal conductivity exceeds the graphite bulk limit^[13,14], sparking significant interest in both the thermal properties of graphene and heat conduction in low-dimensional carbon materials^[15]. 2D polymeric fullerenes exhibit remarkable stability^[16-18], suitable bandgaps, and abundant surface-active sites, endowing them with excellent electronic and optical properties^[19-21]. These characteristics make them promising candidates for applications such as hydrogen storage^[22-26] and photocatalytic water splitting^{[16,19,27,28]}. However, a significant research gap exists in the exploration of anisotropic thermal conductivity in 2D carbon materials, primarily due to the historical focus on graphene, known for its isotropic heat transport. This gap is particularly critical given the urgent requirement for materials capable of directional heat dissipation in next-generation electronic devices.

Notably, the neuroevolution potential (NEP), a leading machine learning potential (MLP) developed by Fan et al.^[29], is central to anisotropic thermal transport research. It merges the precision of first-principles methods with the speed of classical potentials, leading to its widespread application^[30]. This capability enables the rational design of novel 2D carbon materials through atomic-level engineering, allowing for the accurate simulation and optimization of their thermal transport processes, thereby opening a new pathway for developing high-performance thermal management materials.

Atomic-scale manufacturing, a technology that enables precise manipulation and construction of material structures at the atomic level, has become a cornerstone of modern nanotechnology. The mainstream techniques for achieving atomic-scale precision control in thin-film fabrication are molecular beam epitaxy (MBE)^[31], atomic layer deposition (ALD)^[32], and chemical vapor deposition (CVD)^[33]. Their fundamental distinctions arise from unique growth mechanisms. These atomic-scale manufacturing techniques have transcended the limitations of conventional methods, enabling the precise construction of materials at the atomic level. Despite the extreme technical complexity involved, they are driving revolutionary advancements in cutting-edge fields such as semiconductor chips, quantum computers, and high-efficiency energy devices. Recent studies have successfully synthesized multiple covalently bonded monolayer^[21], few-layer^[34,35] and quasi-tetragonal phase (qTP) and quasi-hexagonal phase (qHP) fullerene networks^[21] in 2D systems. Among fullerene clusters, the C₂₄ cage represents one of the smallest stable closed-cage clusters^[27,36].

Previous studies have revealed distinctive structural and electronic characteristics of C₂₄, including its electron-affinity behavior^[37,38], and its pronounced curvature and low symmetry may favor anisotropic thermal transport. These unique properties open new possibilities for applications in quantum devices and directional thermal management systems. Notably, Li et al. recently investigated the elastic and thermal transport properties of qTP and qHP C₂₄ monolayers using a NEP-based molecular dynamics framework^[39], demonstrating nearly isotropic thermal transport in qTP C₂₄ and pronounced in-plane anisotropy in qHP C₂₄ and relating these behaviors primarily to their distinct bonding topologies, phonon group velocities, and mean free paths. Building on this foundation, several important aspects remain to be clarified. In particular, an independent first-principles validation of the thermal-transport trends, a more comprehensive understanding of the roles of phonon anharmonicity, mode localization, acoustic-optical coupling, and electronic orbital hybridization, and the translation of the intrinsic thermal anisotropy into device-level directional heat spreading have not yet been systematically established. Here, we address these issues by combining density functional theory-Boltzmann transport equation (DFT-BTE) and NEP-homogeneous nonequilibrium molecular dynamics (HNEMD) calculations, mode-resolved phonon and electronic-structure analyses, and finite-element device simulations, thereby establishing a multiscale relationship from atomic spatial arrangement to directional thermal-management performance.

In this study, we systematically explore the thermal transport behaviors of the qTP C₂₄ and qHP C₂₄ monolayers by integrating high-accuracy NEP with first-principles calculations. The research focuses on key parameters such as atomic structural features, phonon dispersion relations, lattice thermal conductivity, and spectral thermal conductivity distribution, elucidating the modulation mechanisms of different atomic arrangements on phonon group velocities, relaxation times, and scattering processes. Through detailed phonon transport analysis and mechanical property characterization, the intrinsic relationship between crystalline structural differences and anisotropic thermal conductivity is revealed. Furthermore, orbital-projected electronic structures and electron localization functions are examined to clarify the relationship between orbital hybridization and phonon transport. Finally, finite-element simulations are performed to assess the device-level heat-spreading capability of C₂₄ monolayers. This combined atomistic-to-device investigation provides a comprehensive understanding of how atomic spatial arrangement governs intrinsic thermal anisotropy and offers guidance for designing carbon-based materials for directional thermal management.

2. Methods

2.1 First-principles calculations

The first-principles calculations were carried out employing density functional theory (DFT) as implemented in the Vienna ab initio Simulation Package (VASP)^[40,41]. The PBEsol^[42] under the generalized gradient approximation (GGA)^[43] framework was utilized to describe exchange-correlation interactions, combined with the projector augmented-wave (PAW) method^[44]. An energy cutoff of 800 eV was set for the plane-wave basis set. Geometry optimizations were performed until the total energy difference between consecutive steps fell below 10^-6 eV in total energy and the maximum residual force on any atom was less than 10^-2 eV Å^-1. For Brillouin-zone integrations, Γ-centered k-point grids of 5 × 5 × 1 and 3 × 5 × 1 were employed for the structural relaxation and electronic structure calculations of the qTP and qHP C₂₄ monolayers, respectively. Phonon thermal transport properties were assessed by solving the phonon Boltzmann transport equation with the Phonopy package^[45] and ShengBTE code^[46]. Both second- and third-order interatomic force constants (IFCs) were derived via supercell calculations, where the adopted 2 × 2 × 1 cells correspond to 96-atom and 192-atom configurations for the qTP and qHP C₂₄ monolayer structures, respectively. Comprehensive convergence tests were conducted for both nearest neighbor interactions and Q-grid mesh to guarantee the reliability of the predicted lattice thermal conductivity values. Based on the convergence results shown in Figure S2, the interatomic interactions up to the eighth nearest neighbor and a 27 × 27 × 1 Q-grid were ultimately selected for subsequent calculations. To further characterize the chemical-bonding interactions, crystal orbital Hamilton population (COHP) and integrated crystal orbital Hamilton population (ICOHP) analyses were performed using the Local Orbital Basis Suite Towards Electronic-Structure Reconstruction (LOBSTER) code^[47,48].

The effective thickness was defined as the vertical distance between the outermost carbon atoms plus twice the van der Waals radius of carbon, following a commonly adopted treatment for buckled two-dimensional materials^[49,50]. This definition gives effective thicknesses of approximately 0.6 nm for qTP C₂₄ and 0.8 nm for qHP C₂₄. The same thicknesses were consistently employed throughout the DFT-BTE, NEP-HNEMD, and finite element method (FEM) calculations. Therefore, the thermal-conductivity anisotropy and the comparison between different calculation methods remain unaffected.

2.2 Machine-learned potential

For the accurate characterization of interatomic interactions, the state-of-the-art NEP methodology was adopted in our work^[29,51-53]. This framework, which falls into the category of MLP, is constructed based on a standalone neural network (NN) and trained using the separable natural evolution strategy (SNES) optimization algorithm^[54]. In this approach, the local potential energy associated with a specific atom i is formulated as a functional mapping of a descriptor vector comprising N_des elements, denoted as $U_i(q)=U_i\left({\left\{q_v^i\right\}}_{v=1}^{N_{des}}\right)$. To approximate this mapping, a feedforward neural network architecture with a single hidden layer is utilized. The hidden layer consists of N_neu neurons, and the network outputs the site energy via the following relation:

(1)$U_i=\sum _{\mu =1}^{N_{\text{neu}}}{\omega }_{\mu }^{(1)}\tanh (\sum _{v=1}^{N_{\text{des}}}{\omega }_{\mu v}^{(0)}{q}_v^i-b_{\mu }^{(0)})-b^{(1)}$

In this expression, ω⁽⁰⁾, ω⁽¹⁾, b⁽⁰⁾, and b⁽¹⁾ correspond to the weight matrices and bias vectors that are iteratively refined during model training.

The primary dataset was constructed through ab initio molecular-dynamics (AIMD) simulations and structural distortions of crystalline unit cells. For every carbon allotropic form, AIMD runs were carried out in the canonical ensemble (NVT), implementing a linear temperature ramp from 100 K to 1,000 K across a 200-ps simulation span. Configurations were extracted at 1 ps intervals, producing 200 snapshots per phase. Each system was further augmented with 200 structurally perturbed configurations, generated by imposing randomized lattice deformations (5% magnitude) and atomic displacements (0.1 Å), culminating in a combined dataset of 800 configurations for both qTP and qHP C₂₄ monolayers. A test subset comprising 100 configurations (50 from AIMD and 50 from perturbed structures) was randomly drawn for each allotrope, leaving the remaining 600 configurations for training the NEP potential. The radial and angular cutoff radii were set to 7 and 4 Å, respectively, while 10 radial and 8 angular descriptor basis functions were employed to represent the local atomic environments. The feedforward neural network contained a single hidden layer with 30 neurons. The complete set of NEP training hyperparameters is provided in Table S2.

The evaluation of anisotropic lattice thermal conductivities for qTP and qHP C₂₄ monolayers was conducted using the highly efficient HNEMD methodology. This approach introduces a minimal external perturbative force F_e to drive the system out of thermal equilibrium, thus allowing for the calculation of the thermal conductivity based on^[55]:

(2)$\kappa (t)=\frac{1}{t}{\int }_0^t\frac{⟨J\left(t^{\prime }\right)⟩}{TV{F}_e}d{t}^{\prime }$

where t is the simulation time, V denotes the simulation box volume, T is the system temperature (300 K), and $⟨J\left(t^{\prime }\right)⟩$ represents the average non-equilibrium heat current. To maintain the linear response regime, external driving forces of F_e = 0.000005 Å^-1 were applied along both the x and y crystallographic directions for the qTP and qHP C₂₄ monolayers. The validity of the selected driving force was further confirmed by additional HNEMD simulations with different F_e values, where the calculated thermal conductivities remained nearly unchanged within the tested range, as demonstrated in Figure S5. The simulations employed orthorhombic supercells containing 38,400 atoms for qTP (244 Å × 244 Å × 23 Å) and 31,920 atoms for qHP (218 Å × 216 Å × 25 Å). All HNEMD simulations were conducted in the NVT ensemble using a Nosé-Hoover thermostat with a total sampling time of 5 ns.

The finite-temperature phonon dispersions and linewidths were extracted from the spectral energy density (SED) obtained by projecting the atomic velocity trajectories into reciprocal space and Fourier transforming them in time, where the SED peak positions and linewidths characterize the phonon frequencies and spectral broadening, respectively^[56,57].

2.3 FEM simulations

Finite-element simulations were carried out using the Heat Transfer in Solids module in COMSOL Multiphysics 6.4. A three-dimensional steady-state thermal model was constructed to evaluate the heat-spreading performance of qTP and qHP C₂₄ monolayers at the device level. The model consists of a substrate with dimensions of 100 μm × 100 μm × 10 μm, a central silicon processor core with dimensions of 26 μm × 30 μm × 2 μm, and six silicon memory blocks with dimensions of 10 μm × 10 μm × 2 μm. The silicon regions were placed on the top surface of the substrate. The C₂₄ monolayers were modeled using the thin-layer approximation as thermal-spreading coatings on the top surfaces of the silicon core and memory blocks. Notably, the C₂₄ thin layer was assumed to be in ideal thermal contact with the underlying silicon surface, and no interfacial thermal resistance was introduced. This ideal-contact approximation was adopted to isolate the influence of the intrinsic in-plane thermal conductivity and thermal anisotropy of the C₂₄ monolayers on device-level heat spreading. Consistent with the DFT-BTE and NEP-HNEMD calculations, an effective thickness of 0.6 nm was adopted for the isotropic qTP C₂₄ monolayer, and the in-plane thermal conductivity was set to k_x = k_y = 234 W m^-1 K^-1. For the anisotropic qHP C₂₄ monolayer, an effective thickness of 0.8 nm was adopted, with k_x = 130 W m^-1 K^-1 and k_y = 216 W m^-1 K^-1. These thermal conductivity values were obtained from NEP-based molecular dynamics calculations. A heat rate of 1 mW was applied to the processor core, while the memory regions were assigned a heat-rate setting of 0.1 mW. The bottom surface of the substrate was fixed at 293.15 K, and the remaining outer boundaries were treated as thermally insulated. Three steady-state studies were performed: the first without the C₂₄ monolayer, the second with the qTP C₂₄ monolayer, and the third with the qHP C₂₄ monolayer. The temperature distribution was solved using a swept mesh with quadratic temperature elements. The temperature profile was extracted along the central line across the device, while the average temperatures of the processor core and memory blocks were monitored using domain-averaged temperature probes.

3. Results and Discussion

Both the qTP and qHP C₂₄ monolayers crystallize in orthorhombic lattices, as shown in Figure 1a, with the optimized lattice parameters listed in Table S1. The qTP phase exhibits nearly square in-plane symmetry with a = b = 6.105 Å, whereas the qHP phase has strongly unequal lattice constants of a = 11.479 Å and b = 6.178 Å, consistent with previous results^[27]. In qTP C₂₄, periodically arranged molecular units are connected by three non-coplanar intercluster bonds, forming a relatively balanced two-dimensional covalent network analogous to C₆₀-derived carbon frameworks. By contrast, qHP C₂₄ features a quasi-one-dimensional chain-like topology: C₂₄ units are strongly connected along the b-direction by three non-coplanar bonds, while adjacent chains are coupled along the a-direction through weaker diagonal single bonds. This spatially asymmetric bonding arrangement may give rise to thermal anisotropy of qHP C₂₄.

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Figure 1. The crystal structures, NEP model validation, and lattice-dynamical properties of qTP and qHP C₂₄ monolayers. (a) Side and top views of the optimized atomic structures of qTP and qHP C₂₄ monolayers; (b,c) Phonon dispersion relations of (b) qTP and (c) qHP C₂₄ monolayers calculated using the NEP potential, compared with DFT benchmark results; (d) Evolution of the total loss function and individual loss components during NEP training, including energy, force, virial, and regularization terms; (e-h) Parity plots comparing NEP-predicted and DFT-calculated (e) energy; (f) virial; (g) force; (h) stress for the training and testing datasets. qTP: quasi-tetragonal; qHP: quasi-hexagonal; NEP: neuroevolution potentials; DFT: density functional theory.

Then, a NEP model was developed for the qTP and qHP C₂₄ monolayers to enable reliable large-scale atomistic simulations. The dynamical stability of the two monolayers was first examined through phonon dispersion calculations. As shown in Figure 1b,c, no imaginary phonon modes appear throughout the investigated Brillouin-zone paths, confirming the dynamical stability of both qTP and qHP C₂₄. The thermal stability was further assessed by AIMD simulations at 300, 500, and 700 K, as presented in Figure S1. The total energy fluctuations remain bounded without any apparent structural reconstruction or bond breaking throughout the simulations, indicating that both qTP and qHP C₂₄ monolayers maintain good thermal stability up to 700 K. These results collectively demonstrate that both C₂₄ phases are dynamically and thermally stable under ambient conditions.

The training behavior of the developed NEP model is shown in Figure 1d, where the total loss and the individual energy, force, virial, and stress losses decrease steadily and approach convergence after 10⁵ generations, indicating stable and balanced model training. The trained NEP was then benchmarked against DFT reference data, as shown in Figure 1e,f,g,h. The test-set RMSEs are 0.960 meV/atom for energy, 13.149 meV/atom for virial, 145.834 meV/Å for force, and 58.412 MPa for stress, closely matching the corresponding training errors of 0.877 meV/atom, 12.690 meV/atom, 145.051 meV/Å, and 56.729 MPa. Together with the nearly ideal parity correlations, these results demonstrate that the NEP model achieves high accuracy without evident overfitting and possesses strong generalization capability for unseen configurations. Moreover, the NEP-calculated phonon spectra closely reproduce the DFT benchmarks over the entire frequency range, further validating the ability of the NEP model.

Notably, the phonon dispersion shows that the acoustic branches display relatively steep slopes in qTP C₂₄, particularly along the Γ-X direction, implying large phonon group velocities and efficient phonon propagation. By contrast, qHP C₂₄ exhibits a more pronounced directional dependence in its acoustic phonon dispersion. The acoustic branches along the Γ-Y direction are evidently steeper than those along Γ-X, indicating higher phonon group velocities along the y-direction. Since the lattice thermal conductivity is strongly governed by phonon group velocity, this anisotropic dispersion suggests more efficient heat transport along y- than along x-direction in qHP C₂₄.

To quantitatively evaluate the in-plane thermal transport properties of qTP and qHP C₂₄ monolayers, their lattice thermal conductivities were calculated using the trained NEP potential combined with the HNEMD method, with DFT-BTE calculations performed for comparison. To ensure statistical reliability, four independent HNEMD simulations were carried out for each crystallographic direction at 300 K, as shown in Figure 2a,b, and the final thermal conductivities were obtained by averaging the converged values. For the DFT-BTE calculations, convergence tests with respect to the nearest-neighbor cutoff and Q-grid density were carefully performed, as shown in Figure S2.

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Figure 2. The anisotropy analysis of the thermal transport properties and the elastic properties of qTP and qHP C₂₄ monolayers. The time-dependent thermal conductivity of (a) qTP and (b) qHP C₂₄ monolayers in x- and y-directions at 300 K, calculated using the HNEMD method with the NEP potential; (c) Comparison of converged thermal conductivities obtained from DFT calculations and the NEP model; (d) Cumulative lattice thermal conductivity with respect to the phonon MFP; (e) Spectral thermal conductivity versus vibrational frequency of qTP and qHP C₂₄ monolayers; (f) The percentage contribution to κ from different branches including the acoustic phonon branches (ZA, TA, LA) and optical phonon branches; Orientation-dependent (g) Young’s modulus E(θ); (h) Shear modulus G(θ); (i) Poisson’s ratio ν(θ) of monolayer qHP and qTP C₂₄ from DFT and NEP calculations, where the dashed lines represent NEP results and the solid lines represent DFT results. DFT: density functional theory; NEP: neuroevolution potentials; qTP: quasi-tetragonal; qHP: quasi-hexagonal; HNEMD: homogeneous nonequilibrium molecular dynamics; MFP: phonon mean free path; ZA: out-of-plane acoustic; TA: transverse acoustic; LA: longitudinal acoustic.

After establishing the computational reliability of the NEP-HNEMD and DFT-BTE approaches, the obtained thermal conductivities were further compared to clarify the intrinsic differences in heat transport between qTP and qHP C₂₄ monolayers, as summarized in Figure 2c. The qTP C₂₄ monolayer exhibits nearly identical thermal conductivities along the x- and y-directions, with DFT-BTE and NEP-HNEMD values of 373 and 234 W m^-1 K^-1, respectively, indicating essentially isotropic in-plane heat transport. In contrast, qHP C₂₄ displays a lower overall thermal conductivity and a pronounced in-plane anisotropy. The DFT-BTE calculations yield κ_x = 181 W m^-1 K^-1 and κ_y = 301 W m^-1 K^-1, while the corresponding NEP-HNEMD values are κ_x = 130 W m^-1 K^-1 and κ_y = 216 W m^-1 K^-1. Both methods give a consistent anisotropy ratio of κ_y/κ_x ≈ 1.7, demonstrating preferential thermal transport along the y-direction. This directional preference agrees well with the steeper acoustic phonon dispersion along Γ-Y than along Γ-X, as discussed in Figure 1. By contrast, the nearly isotropic thermal transport in qTP C₂₄ originates from its higher structural symmetry and more spatially uniform bonding network. It is worth noting that the NEP-HNEMD thermal conductivities of qTP and qHP C₂₄ monolayers are systematically lower than those obtained from DFT-BTE calculations, with reductions ranging from approximately 28% to 37%. This difference should not be attributed to a single mechanism without a direct mode-resolved decomposition. Unlike conventional DFT-BTE calculations based primarily on three-phonon scattering, molecular dynamics samples the anharmonic potential-energy surface directly and therefore naturally incorporates higher-order anharmonic interactions. Previous MLP-MD studies have demonstrated that such higher-order phonon scattering can substantially reduce the thermal conductivity relative to three-phonon BTE predictions in strongly anharmonic or high-κ materials^[58]. In addition, systematic investigations have shown that residual force errors in MLPs, including NEP, can introduce additional low-frequency phonon scattering and consequently lead to an underestimation of lattice thermal conductivity in materials such as Si, GaAs, graphene, PbTe, and BAs^[59,60]. Therefore, the lower NEP-HNEMD values observed in qTP and qHP C₂₄ are consistent with these established methodological effects. Despite the difference in absolute values, both approaches predict the same relative trends, namely isotropic thermal transport in qTP C₂₄ and y-preferred anisotropic thermal transport in qHP C₂₄. It should be noted that the NEP-HNEMD thermal conductivities obtained in this work differ from those reported by Li et al.^[39]. Li et al. reported approximately 272 W m^-1 K^-1 for qTP C₂₄ and 233 and 341 W m^-1 K^-1 for qHP C₂₄ along the x- and y-directions, respectively, whereas the corresponding values in the present work are 234, 130, and 216 W m^-1 K^-1. Nevertheless, both studies consistently predict nearly isotropic heat transport in qTP C₂₄ and preferential transport along the y-direction in qHP C₂₄. The difference in absolute values can be partly attributed to the distinct first-principles reference descriptions used to construct the NEP models. Li et al. employed the Perdew−Burke−Ernzerhof functional (PBE)-based reference calculations, whereas the present work uses the Perdew−Burke−Ernzerhof functional revised for solids (PBEsol) reference data, consistent with the original first-principles characterization of the qTP and qHP C₂₄ structures^[27]. Since a machine-learning potential reproduces the potential-energy surface defined by its reference data, different exchange-correlation functionals can modify interatomic forces, phonon anharmonicity, and phonon lifetimes and consequently lead to appreciable differences in thermal conductivity. Such functional sensitivity has also been demonstrated in other two-dimensional materials. For example, PBE and PBEsol predict substantially different thermal conductivities for phosphorene^[61], while exchange-correlation-dependent variations in phonon lifetimes and Grüneisen parameters have been reported for graphene^[62]. In addition, Li et al. adopted effective thicknesses of 0.568 and 0.629 nm for qTP and qHP C₂₄, respectively, compared with 0.6 and 0.8 nm used in our calculations, which were determined from the out-of-plane geometrical extent of each C₂₄ monolayer, defined as the vertical distance between the outermost carbon atoms plus twice the van der Waals radius of carbon. Such a definition has been commonly adopted for other two-dimensional materials^[49,50]. These methodological differences explain why independently trained NEP models can yield different absolute values while maintaining consistent thermal-transport trends.

Beyond the methodological comparison discussed above, qHP C₂₄ was further compared with other anisotropic two-dimensional materials, as summarized in Table S3. Compared with the previously reported qHP C₆₀ monolayer^[30], qHP C₂₄ exhibits a higher thermal conductivity in both crystallographic directions while maintaining a comparable anisotropy ratio (κ_y/κ_x ≈ 1.7 for qHP C₂₄ and ≈1.3 for qHP C₆₀). In comparison with black phosphorus^[61], qHP C₂₄ possesses thermal conductivities that are more than one order of magnitude higher while exhibiting a similarly pronounced in-plane anisotropic thermal transport. These results demonstrate that the C₂₄ monolayer combines relatively high thermal conductivity with significant in-plane anisotropy, making it a promising candidate for directional thermal management applications.

To further elucidate the microscopic origin of the different thermal conductivities, the cumulative thermal conductivity as a function of phonon mean free path (MFP) and the frequency-resolved spectral thermal conductivity were analyzed, as shown in Figure 2d. For both qTP and qHP C₂₄ monolayers, the cumulative thermal conductivity gradually approaches the saturated value as the phonon mean free path increases, where qTP C₂₄ maintains the largest cumulative thermal conductivity over nearly the entire MFP range. For qHP C₂₄, the cumulative contribution along the y-direction is consistently larger than that along the x-direction, further confirming its anisotropic thermal transport. The frequency-resolved spectral thermal conductivity κ(ω), shown in Figure 2e, provides further insight into the microscopic phonon transport mechanism. The dominant contribution to lattice thermal conductivity arises from phonons below approximately 10 THz, indicating that low-frequency phonons govern thermal transport in both phases. qTP C₂₄ exhibits a larger spectral weight in the low-frequency region than qHP C₂₄, consistent with its higher overall thermal conductivity. For qHP C₂₄, the integrated spectral contribution along the y-direction is larger than that along the x-direction, again confirming the preferential thermal transport along the y-direction.

Because the total thermal conductivity is determined by the collective contribution of different phonon modes, the branch-resolved thermal conductivity was further examined to identify the dominant heat carriers in qTP and qHP C₂₄ monolayers, as presented in Figure 2f. For qTP C₂₄, the x- and y-directions show almost identical modal distributions, with acoustic phonons dominating the thermal conductivity and optical phonons contributing only about 10%. In contrast, qHP C₂₄ exhibits pronounced direction-dependent modal contributions. Optical phonons contribute as much as 41% to κ_x and 30% to κ_y, indicating that optical modes become important heat carriers and play a key role in shaping the anisotropic thermal transport of qHP C₂₄. The enhanced optical contribution in qHP C₂₄ can be attributed to its complex molecular framework and anisotropic bonding topology, which induce strong acoustic-optical phonon hybridization in the low-frequency region, as evidenced by the phonon dispersion. Such hybridization allows low-lying optical modes to acquire appreciable group velocities and participate effectively in thermal transport. Therefore, the thermal anisotropy of qHP C₂₄ arises from the interplay between direction-dependent optical-phonon participation and anisotropic acoustic phonon transport. This behavior differs from conventional crystals, where thermal transport is usually governed almost exclusively by acoustic phonons, and highlights the important role of low-frequency optical modes in complex carbon monolayers.

Since phonon transport is closely related to lattice stiffness and bonding characteristics, the elastic properties of qTP and qHP C₂₄ monolayers were further investigated. The mechanical behaviors of fullerene-based carbon networks have attracted increasing attention, with previous studies revealing anisotropic fracture characteristics in monolayer fullerene networks^[63] and interfacial friction behaviors in fullerene-network/graphene systems^[64]. These studies highlight the critical role of network topology and bonding configurations in determining mechanical responses and provide a broader mechanical context for understanding the stiffness and bonding characteristics of C₂₄ monolayers. The four independent elastic constants obtained from NEP and DFT calculations, together with previously reported values^[39], are listed in Table S4. For qTP C₂₄, the constants C₁₁, C₂₂, C₁₂, and C₆₆ are 248.6, 248.6, -11.9, and 80.8 N/m from DFT, which closely match the NEP values of 236.1, 236.1, -11.1, and 84.3 N/m. For qHP C₂₄, the DFT (NEP) values of C₁₁, C₂₂, C₁₂, and C₆₆ are 225.2 (216.7), 274 (266.2), 21.3 (26.0), and 105.0 (108.5) N/m, respectively. The good agreement between the present results and literature data confirms the reliability of our calculations, while the consistency between NEP and DFT further validates the capability of the constructed NEP model to describe the elastic response of both phases. Based on these elastic constants, the orientation-dependent Young’s modulus E(θ), shear modulus G(θ), and Poisson’s ratio ν(θ) were calculated using established analytical expressions^[65,66], as shown in Figure 2g,h,i. qHP C₂₄ exhibits clear elastic anisotropy, with E(θ) increasing from the x-direction toward the y-direction. This behavior is consistent with its bonding topology: compared with the relatively weak diagonal interchain single bonds along the x-direction, the three non-coplanar bonds along the y-direction provide stronger mechanical connectivity. In contrast, owing to the structural symmetry and balanced bonding network of qTP C₂₄, its in-plane elastic response is essentially isotropic.

To further clarify the microscopic origin of the thermal transport trends observed in Figure 2, we analyzed the phonon group velocity, participation ratio, relaxation time, Grüneisen parameter, three-phonon scattering phase space, and spectral energy density (SED), as summarized in Figure 3. Figure 3a,b show the frequency-dependent phonon group velocities. In qTP C₂₄, large phonon group velocities are mainly concentrated in the low-frequency region, which is also the dominant frequency window for lattice thermal transport. For qHP C₂₄, a clear directional difference is observed: phonons propagating along the y-direction possess higher phonon group velocities than those along the x-direction, particularly below approximately 10 THz. This feature directly supports the thermal conductivity results in Figure 2, where qHP C₂₄ exhibits κ_y > κ_x. Therefore, the directional thermal transport in qHP C₂₄ is strongly governed by anisotropic phonon group velocities, which originate from its chain-like bonding framework and stronger connectivity along the y-direction.

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Figure 3. The phonon transport mechanism underlying the thermal conductivity difference and anisotropy of C₂₄ monolayers. The phonon group velocity as a function of frequency for (a) qTP C₂₄ monolayer along the x direction and (b) qHP C₂₄ monolayer along the x and y directions at 300 K; (c) Frequency-dependent phonon participation ratio of qTP and qHP C₂₄ monolayers; The comparison of (d) phonon relaxation time; (e) Grüneisen parameter; (f) phonon scattering phase space as a function of frequency. SED of (g) qTP C₂₄ along Γ-X; (h) qHP C₂₄ along Γ-X; (i) qHP C₂₄ along Γ-Y at 300 K. The color scale denotes the logarithmic magnitude of the spectral energy density for each phonon mode. qTP: quasi-tetragonal; qHP: quasi-hexagonal; SED: spectral energy density.

However, solely by phonon group velocity analysis, the difference in the thermal conductivity anisotropy between qTP and qHP C₂₄ cannot be explained. Although qHP C₂₄ along the y-direction contains phonons with relatively high group velocities, its thermal conductivity remains lower than that of qTP C₂₄. This indicates that phonon scattering and lifetime play a decisive role in determining the overall magnitude of κ.

As shown in Figure 3c, qTP C₂₄ generally exhibits higher participation ratios than qHP C₂₄ in the low- and intermediate-frequency regions, indicating more delocalized phonon modes. In contrast, the reduced participation ratios in qHP C₂₄ suggest stronger phonon localization, which is associated with its lower structural symmetry and anisotropic molecular arrangement. Such localized vibrational modes are less effective in transporting heat and are more susceptible to scattering, thereby suppressing the lattice thermal conductivity of qHP C₂₄. Moreover, the phonon relaxation time distribution in Figure 3d shows that qTP C₂₄ possesses overall longer phonon relaxation times than qHP C₂₄, indicating weaker phonon scattering and more persistent phonon propagation in qTP C₂₄. Since lattice thermal conductivity is positively correlated with phonon relaxation time, the longer-lived phonons in qTP C₂₄ contribute substantially to its higher κ. The origin of this phonon relaxation time difference can be further understood from the Grüneisen parameter and scattering phase space. As shown in Figure 3e, qHP C₂₄ exhibits much larger absolute Grüneisen parameters, especially in the low-frequency region, indicating stronger intrinsic anharmonicity and enhanced phonon-phonon interactions. By contrast, qTP C₂₄ shows smaller Grüneisen parameters over most of the frequency range, implying weaker anharmonic scattering. Strong anharmonicity has also been shown to play a critical role in regulating phonon transport and thermoelectric performance in low-dimensional materials^[67]. Interestingly, Figure 3f shows that qTP C₂₄ possesses a larger three-phonon scattering phase space than qHP C₂₄. This result indicates a competition between the number of allowed scattering channels and the scattering strength of each channel. In qTP C₂₄, although more scattering channels are available, the weaker anharmonicity reduces the effective scattering strength, leading to longer phonon relaxation time and higher thermal conductivity. In qHP C₂₄, the smaller scattering phase space is outweighed by much stronger anharmonicity and stronger phonon localization, resulting in shorter phonon relaxation time and lower κ.

To further identify which optical modes are responsible for the unusually large optical contribution in qHP C₂₄, the mode- and frequency-resolved thermal conductivities were further decomposed, as shown in Figure S3. For qTP C₂₄, acoustic, low-frequency optical, and mid-to-high-frequency optical modes contribute 90.2%, 7.8%, and 2.0% of the total κ, respectively. In contrast, the corresponding contributions are 59.5%, 38.4%, and 2.1% for qHP C₂₄ along the x-direction and 69.3%, 29.0%, and 1.7% along the y-direction. Therefore, the enhanced optical contribution in qHP C₂₄ originates almost entirely from low-frequency optical modes below approximately 10 THz, whereas the mid-to-high-frequency optical branches make only a minor contribution. These low-lying optical branches overlap spectrally with the dominant acoustic heat-carrying modes and exhibit pronounced acoustic-optical hybridization, which allows some optical modes to acquire appreciable dispersive character and participate effectively in heat transport^[68]. Notably, the larger relative optical contribution along the x-direction does not imply a larger total κ_x; instead, it reflects the stronger suppression of the acoustic channel along x due to the weaker interchain connectivity. Along the y-direction, the stronger bonding connectivity produces larger acoustic phonon group velocities and a larger acoustic contribution, resulting in κ_y > κ_x. Moreover, although the participation-ratio analysis indicates stronger overall phonon localization in qHP C₂₄, localization and optical-mode transport represent competing effects rather than mutually exclusive behaviors. Stronger localization suppresses the overall thermal conductivity of qHP C₂₄, whereas a subset of numerous low-frequency optical modes can still acquire finite group velocities and lifetimes through acoustic-optical hybridization and collectively contribute substantially to κ. Similar large contributions from low-frequency optical phonons have previously been reported in 2H-MoS₂ and monolayer Ga₂O₃^[69,70]. These results demonstrate that the anisotropic thermal transport of qHP C₂₄ arises from the combined effects of direction-dependent acoustic transport, low-frequency acoustic-optical hybridization, and phonon localization.

The SED maps in Figure 3g,h,i further verify the phonon transport mechanism. The phonon branches extracted from the SED spectra agree well with the harmonic phonon dispersions, confirming the reliability of the calculated lattice dynamics. For qTP C₂₄, the low-frequency acoustic branches along Γ-X exhibit relatively steep slopes and narrow linewidths, indicating large phonon group velocities, weak scattering, and long phonon lifetimes, which contribute to its high thermal conductivity. In qHP C₂₄, the Γ-Y direction shows steeper acoustic branches and narrower spectral linewidths than Γ-X, suggesting faster phonon propagation and weaker scattering along the y-direction. This directional difference is consistent with the higher κ_y than κ_x in qHP C₂₄. In addition, the dense low-frequency optical branches and their interaction with acoustic modes in qHP C₂₄ further reveal acoustic-optical phonon coupling, which modifies the phonon scattering channels and contributes to its anisotropic thermal transport. Therefore, the SED results demonstrate that the anisotropic heat transport in qHP C₂₄ originates from the combined effects of direction-dependent phonon group velocity and scattering strength.

Notably, the electronic structure properties are closely correlated with the physical properties of materials^[71-74]. To elucidate the electronic origin of the distinct in-plane thermal-transport behavior of qTP and qHP C₂₄ monolayers, the orbital-projected electronic band structures, projected density of states (pDOS), and electron localization function (ELF) distributions were calculated, as shown in Figure 4. Both qTP and qHP C₂₄ exhibit semiconducting characteristics, with band gaps of 2.02 and 1.67 eV, respectively. For qTP C₂₄, the valence band maximum (VBM) and conduction band minimum (CBM) are mainly located around the M point, indicating a nearly direct band-gap feature. In contrast, qHP C₂₄ possesses a smaller band gap, with both the VBM and CBM primarily appearing near the Γ point. The reduced band gap of qHP C₂₄ suggests stronger orbital hybridization and a more pronounced anisotropic electronic response compared with qTP C₂₄.

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Figure 4. Orbital-projected band structures and pDOS of (a) qTP and (b) qHP C₂₄ monolayers. The inset shows the ELF viewed from the top for qTP and qHP C₂₄ monolayers. qTP: quasi-tetragonal; qHP: quasi-hexagonal; pDOS: partial density of states; ELF: electron localization function.

For qTP C₂₄, the p_x and p_y orbitals are nearly degenerate and therefore presented as p_x,y, reflecting the high in-plane structural symmetry of this phase. Around the VBM, the electronic states are mainly contributed by C-p_x,y and C-p_z orbitals, while the CBM also shows mixed p_x,y/p_z orbital character. The pDOS further confirms the comparable contribution of the in-plane p_x and p_y orbitals, suggesting that the electronic states along different in-plane directions are nearly equivalent. Consistently, the ELF map displays a highly symmetric and spatially uniform electron localization pattern around the carbon framework. Such symmetric orbital hybridization and electron localization indicate nearly equivalent C-C bonding strengths and bond stiffnesses along different in-plane crystallographic directions, which is responsible for the relatively isotropic phonon transport and thermal conductivity of qTP C₂₄. In contrast, qHP C₂₄ shows a clear separation among the p_x, p_y, and p_z orbital contributions. The orbital-projected band structure reveals that the VBM near the Γ point is dominated by strongly hybridized C-p_x, C-p_y, and C-p_z states, whereas the CBM also contains evident p_z contribution together with unequal in-plane p_x and p_y components. The pDOS shows that the p_x and p_y states are no longer degenerate and contribute differently over a wide energy range, demonstrating the inequivalence of electronic states along different in-plane directions. Moreover, unlike the highly symmetric ELF pattern of qTP C₂₄, qHP C₂₄ exhibits an elongated and spatially nonuniform electron localization feature, implying direction-dependent C-C bonding characteristics. This anisotropic electronic localization leads to different bond strengths along different crystallographic directions. Therefore, from the electronic-structure perspective, the intrinsic thermal anisotropy of qHP C₂₄ originates from its reduced lattice symmetry, inequivalent p_x/p_y orbital contributions, and direction-dependent p_z-related orbital hybridization. In contrast, the nearly degenerate p_x/p_y orbitals and symmetric electron localization in qTP C₂₄ produce a more uniform bonding network, thereby explaining its weakly anisotropic or nearly isotropic thermal-transport behavior.

To quantitatively examine how the orbital anisotropy is reflected in the C-C bonding network, COHP and ICOHP analyses were further performed, as shown in Figure S4. Because of the complex fullerene-based bonding topology, representative inter-cage C-C bonds along different crystallographic directions were selected for comparison. In qTP C₂₄, the ICOHP values of the x-short and y-short bonds are -7.886 and -7.881 eV, respectively, while those of the x-long and y-long bonds are -6.860 and -6.864 eV. The nearly identical values of the corresponding x- and y-direction bonds quantitatively confirm an approximately isotropic bonding environment, consistent with the nearly degenerate (p_x/p_y) orbitals, symmetric ELF distribution, elastic constants, and isotropic thermal transport. In contrast, qHP C₂₄ exhibits markedly inequivalent bonding configurations, with ICOHP values of -7.548, -9.024, -7.909, and -6.457 eV for the representative x-short, x-diagonal, y-short, and y-long bonds, respectively. This broad distribution demonstrates that the (p_x/p_y) orbital inequivalence is translated into a heterogeneous and direction-dependent C-C bonding network. Since the macroscopic stiffness is collectively determined by bond strength, orientation, connectivity, and network topology, rather than by a single bond, this bonding heterogeneity is consistent with the larger C₂₂ relative to C₁₁ and the higher low-frequency phonon group velocities along the y-direction. Together, these results establish a consistent microscopic relationship from anisotropic orbital hybridization and bonding topology to elastic anisotropy, direction-dependent phonon propagation, and ultimately the observed thermal-conductivity anisotropy.

To evaluate the chip-level thermal management potential of C₂₄ monolayers, steady-state simulations based on the finite-element method (FEM) were performed using a simplified processor-memory device model, as shown in Figure 5a. The model consists of a central processor core region surrounded by six memory blocks on a substrate, where the C₂₄ monolayer was applied as an ultrathin thermal-spreading coating on the top surfaces of the heat-generating silicon regions. Three cases were considered: the device without a C₂₄ monolayer, the device coated with an isotropic qTP C₂₄ monolayer, and the device coated with an anisotropic qHP C₂₄ monolayer. The temperature profiles extracted along the central line, shown in Figure 5b, indicate that both C₂₄ coatings effectively suppress the temperature rise in the processor core region and neighboring memory blocks. Without the C₂₄ monolayer, the processor core region exhibits the highest temperature, with an average temperature of 120.46 °C, while the average temperature of the memory blocks reaches 95.72 °C. After introducing the isotropic qTP C₂₄ monolayer, these values decrease to 112.70 °C and 92.17 °C, respectively. The anisotropic qHP C₂₄ monolayer further reduces the average temperatures to 112.39 °C for the processor core region and 91.19 °C for the memory blocks. Compared with the bare device, the isotropic qTP C₂₄ monolayer lowers the average temperatures of the processor core region and memory blocks by 7.76 and 3.55 °C, respectively, whereas the anisotropic qHP C₂₄ monolayer achieves slightly larger reductions of 8.07 and 4.53 °C.

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Figure 5. Device-level finite-element simulations of heat dissipation regulated by C₂₄ monolayer thermal-spreading coatings. (a) Schematic of the processor-memory device model, where a thermal-spreading layer is coated on the top surfaces of the heat-generating silicon regions; (b) Temperature profiles along the central x-direction for devices without a C₂₄ monolayer, with isotropic qTP C₂₄, and with anisotropic qHP C₂₄; (c-e) Surface temperature distributions for (c) the bare device; (d) the qTP C₂₄-coated device; (e) the qHP C₂₄-coated device; (f-h) Corresponding isothermal contour plots. qTP: quasi-tetragonal; qHP: quasi-hexagonal.

The temperature maps in Figure 5c,d,e and the corresponding isothermal contour plots in Figure 5f,g,h further confirm the heat-spreading effect of the C₂₄ monolayers. In the absence of a thermal-spreading coating, heat is strongly localized around the processor core region, resulting in a broader high-temperature region. In contrast, both the isotropic qTP C₂₄ and anisotropic qHP C₂₄ coatings redistribute heat more effectively over the chip surface and alleviate thermal accumulation near the heat source. Although the qTP C₂₄ monolayer exhibits isotropic and higher in-plane thermal conductivity, the qHP C₂₄ monolayer provides an additional degree of freedom for directional heat-flow regulation owing to its anisotropic thermal transport, which enables directional heat spreading and more effective regulation of thermal crosstalk. The slightly lower average temperatures obtained with the qHP C₂₄ monolayer suggest that, when the high-thermal-conductivity direction is aligned with the dominant heat-spreading pathway, anisotropic heat transport can be advantageous for hotspot mitigation and thermal crosstalk suppression in integrated electronic devices.

Before closing, it should be noted that the present FEM simulations assume ideal thermal contact between the C₂₄ monolayer and the underlying silicon regions and therefore represent an idealized assessment of its heat-spreading capability. In experimentally integrated two-dimensional material/substrate systems, finite thermal boundary resistance is generally present. For example, room-temperature thermal boundary conductances of approximately 50 MW m^-2 K^-1 for graphene/SiO₂ and approximately 25 MW m^-2 K^-1 for Au/Ti/graphene/SiO₂ interfaces have been experimentally reported^[75,76], and interfacial thermal conductance has been shown to affect hotspot temperatures in graphene-based electronic devices^[77]. For a practical C₂₄-coated device, such an interfacial resistance would restrict heat transfer from the silicon regions into the C₂₄ spreading layer and consequently reduce the predicted temperature decrease relative to the ideal-contact case. The actual magnitude of this effect would depend on the interfacial bonding, conformity, cleanliness, and fabrication process.

From a practical perspective, although monolayer quasi-hexagonal polymeric C₆₀ networks have been experimentally synthesized through an interlayer-bond-cleavage strategy^[21], the qTP and qHP C₂₄ monolayers considered here remain theoretically predicted structures^[27], and their experimental synthesis and integration with device substrates have yet to be demonstrated. Moreover, integrating an atomically thin C₂₄ network with a semiconductor device surface while preserving both its intrinsic structure and efficient interfacial heat transfer remains challenging. Experimental studies of other two-dimensional materials provide encouraging precedents: monolayer graphene can be transferred onto various target substrates^[78]. Nevertheless, transfer-induced residues and interfacial contamination have been observed in graphene/substrate systems^[79], illustrating the sensitivity of atomically thin materials to fabrication and interface quality. Practical implementation of C₂₄ as a thermal-spreading coating would therefore require controlled synthesis or transfer processes that preserve the fullerene-cage network while achieving clean, conformal, and mechanically stable contact with the underlying device layer. Imperfect adhesion, interfacial contamination, or structural perturbation could increase the thermal boundary resistance and diminish the heat-spreading advantage predicted under idealized conditions. These issues represent important subjects for future experimental validation and C₂₄/substrate interface engineering.

4. Conclusion

In summary, we have demonstrated that atomic-level structural design provides an effective route to engineering thermal anisotropy in C₂₄ monolayers. By combining first-principles calculations, NEP, phonon transport analysis, electronic-structure characterization, and finite-element simulations, we systematically investigated the structural origin, microscopic mechanism, and device-level implication of heat conduction in qTP and qHP C₂₄ monolayers. The two phases exhibit distinct in-plane thermal transport behaviors. qTP C₂₄ possesses a nearly square lattice and a balanced covalent bonding network, resulting in nearly isotropic and high thermal conductivity. In contrast, qHP C₂₄ features a chain-like bonding framework, where stronger non-coplanar bonds along the y-direction and weaker diagonal interchain bonds along the x-direction give rise to pronounced anisotropic thermal transport. Both DFT-BTE and NEP-HNEMD calculations consistently reveal preferential thermal transport along the y-direction in qHP C₂₄, with an anisotropy ratio of approximately 1.7. Further phonon analysis shows that this anisotropy originates from direction-dependent phonon group velocities, scattering strengths, and acoustic-optical phonon coupling, whereas the lower overall thermal conductivity of qHP relative to qTP is mainly associated with stronger phonon localization, enhanced anharmonicity, and shorter phonon relaxation times. Electronic structure analyses further link the anisotropic thermal response to direction-dependent orbital hybridization and nonuniform electron localization in qHP C₂₄. Finally, finite-element simulations demonstrate that C₂₄ monolayers can serve as ultrathin thermal-spreading coatings for chip-level heat dissipation. In particular, anisotropic qHP C₂₄ offers an additional degree of freedom for directing heat flow and suppressing thermal crosstalk, highlighting C₂₄ monolayers as promising platforms for directional thermal management in next-generation high-power electronics.

More importantly, this work demonstrates that atomic-scale spatial arrangement regulation provides an intrinsic strategy for tailoring thermal transport without relying on external perturbations such as strain engineering or defect introduction. As a complementary approach to these conventional modulation strategies, atomic-scale structural regulation enables deterministic tuning of thermal conductivity anisotropy through intrinsic structural design, providing a general framework for the rational design of high-performance two-dimensional thermal management materials.

Supplementary materials

The supplementary material for this article is available at: Supplementary materials.

Acknowledgements

The numerical calculations in this paper have been done on the supercomputing system of the E.T. Cluster and the National Supercomputing Center in Changsha and Zhengzhou. The authors declare that ChatGPT (OpenAI) was used solely for language polishing during the manuscript preparation process. All scientific content, including study design, numerical calculations, data generation and analysis, figures and tables, interpretation of results, and conclusions, was independently conducted and prepared by the authors and was not generated using AI tools. The authors reviewed and approved the final manuscript and take full responsibility for its content.

This work is dedicated to the memory of Mr. Zhang Xuefeng, a 2003 alumnus majoring in water supply and drainage engineering at Zhengzhou University, in tribute to his sincere dedication to supporting and empowering young students.

Authors contribution

Tian Q, Li R: Investigation, writing-original draft, writing-review & editing.

Zhang H, Zhang E: Formal analysis, writing-review & editing.

Zheng X, Wang H, Qin Z: Writing-review & editing.

Qin G: Conceptualization, supervision, writing-review & editing.

Conflicts of interest

The authors declare no conflicts of interest.

Ethical approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Availability of data and materials

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Funding

This work is supported by the Postgraduate Research and Innovation Project of Hunan Province (Grant No. CX20250532), the National Natural Science Foundation of China (Grant No. 52006057), the National Key R&D Program of China (Grant No. 2023YFB2408100), the Fundamental Research Funds for the Central Universities (Grant No. 531119200237), the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2025A1515012237), the State Key Laboratory of Robotics and Systems at Harbin Institute of Technology (Grant No. SKLRS-2025-KF-10),the National Major Science and Technology Project of China (Grant No. 2025ZD0717301), the State Key Laboratory of Advanced Design and Manufacturing Technology for Vehicle (Grant No.72675007), the Open Research Fund of State Key Laboratory of Materials for Integrated Circuits (Grant No. SKLJC-K2026-04), and the Open Project of Shaanxi Key Laboratory of Gear Transmission (Grant No. SKLGT-2024-005).

Copyright

© The Author(s) 2026.