# Accurate estimation of interfacial thermal conductance between silicon and diamond enabled by a machine learning interatomic potential

Ali Rajabpour, Bohayra Mortazavi, Pedram Mirchi, Julien El Hajj, Yangyu Guo, Xiaoying Zhuang, Samy Merabia

<figure><img src="image_1.png" /></figure>

## ► To cite this version:

Ali Rajabpour, Bohayra Mortazavi, Pedram Mirchi, Julien El Hajj, Yangyu Guo, et al.. Accurate estimation of interfacial thermal conductance between silicon and diamond enabled by a machine learning interatomic potential. International Journal of Thermal Sciences, 2025, 214, pp.109876. 10.1016/j.ijthermalsci.2025.109876. hal-05012208

HAL Id: hal-05012208

https://hal.science/hal-05012208v1

Submitted on 31 Mar 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Accurate estimation of interfacial thermal conductance between silicon and diamond enabled by a machine learning interatomic potential

Ali Rajabpour<sup>a,b,*</sup>, Bohayra Mortazavi<sup>c,**</sup> Ⓓ, Pedram Mirchi<sup>a,b</sup> Ⓓ, Julien El Hajj<sup>b</sup>, Yangyu Guo<sup>d</sup>, Xiaoying Zhuang<sup>c</sup>, Samy Merabia<sup>b,***</sup>

## ABSTRACT

Thermal management at silicon-diamond interface is critical for advancing high-performance electronic and optoelectronic devices. In this study, we calculate the interfacial thermal conductance between silicon and diamond using a computationally efficient machine learning (ML) interatomic potential trained on density functional theory (DFT) data. Using non-equilibrium molecular dynamics (NEMD) simulations, we compute the interfacial thermal conductance (ITC) for various system sizes. Our results reveal an extremely close agreement with experimental data than those obtained using traditional semi-empirical potentials such as Tersoff and Brenner which overestimate ITC. In addition, we analyze the frequency-dependent heat transfer spectrum, providing insights into the contributions of different phonon modes to the interfacial thermal conductance. The ML potential accurately captures the phonon dispersion relations and lifetimes, in good agreement with DFT calculations and experimental observations. It is shown that the Tersoff potential predicts higher phonon group velocities and phonon lifetimes compared to the DFT results. Furthermore, it predicts higher interfacial bonding strength, which is consistent with higher interfacial thermal conductance as compared to the ML potential. This study highlights the use of ML interatomic potentials to improve the accuracy and computational efficiency of thermal transport simulations of complex material interface systems.

## 1. Introduction

Diamond-silicon structures have attracted considerable attention as composite systems for applications in electronic and optoelectronic devices [1–5]. Chemical vapor deposition techniques provide a feasible method for synthesizing diamond films on silicon substrates. The increasing power densities and heterogeneity of modern electronic systems make interfacial thermal management critical to the performance and reliability of the main device. Efforts to minimize interfacial thermal resistance have followed various strategies, including nanostructuring of interfaces and enhancing interfacial bonding [6–8]. In the past decades, numerous studies have focused on the phonon transport properties of the diamond/silicon interface [9–14]. Feng et al. comprehensively reviewed the theoretical and experimental works on thermal transport at silicon-diamond interface, with a particular focus on the critical aspects of thermal boundary conductance across wide and ultrawide bandgap semiconductor interfaces [15]. Using nanoscale

graphoepitaxy to grow diamonds on silicon substrates shows tunable thermal transport across diamond membranes and diamond/silicon interfaces [11]. The effect of the amorphous silicon interlayer thickness on thermal transport at silicon/diamond interfaces shows a significant increase in the interfacial thermal conductance with a thinner amorphous layer due to the reduction of the phonon mismatch between materials and subsequent enhancement of the heat transport channels [16]. In addition, studies of the interfacial thermal conductance between diamond and silicon substrates have elucidated the temperature-dependent behavior and the effects of system size on thermal transport, as well as the significant influence of interface defects on phonon scattering and the consequent reduction in thermal conductance [12].

Molecular dynamics (MD) simulation is a powerful tool for studying interfacial thermal transport as it provides detailed insight into atomic-level interactions and transport mechanisms [17–24]. MD simulations can effectively model the complex dynamics of atoms and molecules at the interface with high resolution, capturing the interplay of phonons,

<footnote><sup>a</sup> Mechanical Engineering Department, Imam Khomeini International University, Qazvin, 34148-96818, Iran</footnote>

<footnote><sup>b</sup> Institut Lumière Matière, Université Claude Bernard Lyon 1-CNRS, Université de Lyon, Villeurbanne, 69622, France</footnote>

<footnote><sup>c</sup> Institute of Photonics, Department of Mathematics and Physics, Leibniz Universität Hannover, Welfengarten 1A, 30167, Hannover, Germany</footnote>

<footnote><sup>d</sup> School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001, China</footnote>

<footnote>This article is part of a special issue entitled: Micro/Nanoscale Transport published in International Journal of Thermal Sciences.</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>** Corresponding author. Department of Mathematics and Physics, Leibniz Universität Hannover, Welfengarten 1A, 30167, Hannover, Germany.</footnote>

<footnote>*** Corresponding author.</footnote>

<footnote>E-mail addresses: bohayra.mortazavi@gmail.com (B. Mortazavi), samy.merabia@univ-lyon1.fr (S. Merabia).</footnote>
defects, and interface structures that govern thermal transport [25–28]. By manipulating parameters such as interface roughness, interfacial bond strength, and temperature gradients, MD simulations can reveal how these factors influence thermal conductivity and interfacial thermal conductance (ITC) [29–33]. In addition, MD simulations allow the exploration of various scenarios that may be difficult or not practical to investigate experimentally. However, selecting an appropriate potential in MD simulations is critical as it directly affects the accuracy and reliability of the results. Different types of potentials, such as empirical potentials (e.g., Lennard-Jones), semi-empirical potentials (e.g., Tersoff), and *ab initio* potentials, offer different levels of accuracy and computational efficiency depending on the system being studied and the desired level of detail. Careful consideration must be given to selecting a potential that accurately represents the physical properties of materials while remaining computationally tractable for the intended simulation timescale and system size [34,35].

Machine learning potentials are increasingly recognized for their importance in MD simulations because they capture complex interactions with reasonable accuracy and efficiency [36–43]. Unlike empirical potentials, which rely on predefined functional forms and parameters, machine learning potentials are based on algorithms trained on large datasets of quantum mechanical calculations [44–47]. This data-driven approach enables the representation of atomic interactions, including those involved in interfacial thermal transport [48–51]. In other words, machine learning interatomic potentials (MLIPs) offer significant advantages in terms of accuracy at the level of *ab initio* calculations but with computational times on the order of classical MD simulations [52–57].

Previous simulations of the silicon-diamond interface have mainly relied on semi-empirical potentials such as Tersoff and Brenner, which tend to overpredict the interfacial thermal conductance compared to experimental values [11,12,15,16]. In this study, non-equilibrium molecular dynamics simulations with machine-learning interatomic potentials are employed to accurately calculate the interfacial thermal conductance across bulk silicon and diamond structures. First, we detail the process of generating the potential by training it on density functional theory (DFT) data. We then perform simulations to calculate interfacial heat transfer at various structure lengths, allowing us to evaluate the finite size effects on the thermal conductance. Finally, we analyze the results by presenting the frequency-dependent heat transfer spectrum, which provides insight into the thermal transport mechanisms

at the silicon-diamond interface.

## 2. Methodology

In order to investigate heat transfer at the silicon/diamond interface, two silicon and diamond crystals with lattice constants, respectively 5.43 Å and 3.57 Å, are inserted and their interface is oriented along (1,0,0) direction. The initial distance between two crystals equals silicon and diamond's average atomic bond length. Also, in transverse directions, to minimize the stress at the interface, the value of 5.43 nm is considered for the system size along y and z dimensions. Fig. 1 schematically shows the studied atomic structure with a limited length. Tersoff interatomic potential is traditionally used to describe Si-Si, Si-C, and C-C covalent interactions in silicon, diamond, and their combinations. Instead of focusing on the semi-empirical Tersoff potential in this study, we employ state-of-the-art machine-learning interatomic potential trained on the quantum mechanical data.

Vienna Ab-initio Simulation Package (VASP) [58] was utilized for ab-initio molecular dynamics (AIMD) simulations, employing the generalized gradient approximation (GGA) and revised Perdew-Burke-Ernzerhof (PBE) functional for solids [59]. The plane wave cutoff energy was set as 500 eV with a self-consistency convergence criterion of $10^{-4}$ eV, adopting a 2 × 2 × 2 k-point Monkhorst Pack mesh [60]. AIMD calculations were conducted for a periodic heterostructure with 96 atoms (40:Si and 56:C), pure diamond with 144 atoms, and pure silicon with 64 atoms, each for 3000 steps, with temperatures gradually increased from 1 to 2000 K. First the complete AIMD dataset with 9000 configurations was sampled, and 800 configurations were selected for training the first moment tensor potential (MTP) with 409 parameters and a cutoff distance of 4 Å, using the MLIP package [61]. Following the first training, the MTP accuracy was assessed across the entire AIMD dataset, identifying configurations with the highest extrapolation grades [62], which were added to the original dataset. The second and final MTP was trained from scratch using the updated dataset. This two-step passive fitting [63,64] procedure ensures the optimal usage of the training data for modeling thermal transport. AIMD training data and developed MTP potential can be found in the data availability section.

After obtaining the potential based on machine learning, we calculate the thermal conductance at the interface of the silicon/diamond structure using the non-equilibrium molecular dynamics (NEMD)

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Atomic structure of silicon/diamond interface and the NEMD setup to calculate the interfacial thermal conductance. (For interpretation of the colours in this figure atoms, the reader is referred to the Web version of this article.)</figcaption></figure>
simulation method. For this purpose, as shown in Fig. 1, a prescribed temperature gradient is applied to the system by creating hot and cold temperature baths at both ends of the system with target temperatures, and a heat flux is established across the structure. To create thermal insulating conditions at both ends of the system, atoms located at the left and right boundaries are frozen. Periodic boundary conditions are applied along the y- and z-directions. The time step used throughout the simulation is 0.5 fs. The simulation runs for 50 ps to reach the relaxed state through the Nose-Hoover thermostat and the barostat. It then continues for 0.5 ns after applying the temperature gradient until the system reaches a steady state, and then continues for another 0.5 ns for data collection. All classical molecular dynamics simulations are performed using the LAMMPS package linked to the MLIP machine learning package [65,66]. The NEMD simulations are performed using the appropriate thermostats and coupling parameters, as highlighted in the literature [67], ensuring accurate thermal control during the simulations. In our simulations, the Nosé-Hoover thermostat is used in the whole system during initial equilibration and after equilibration for the dynamics of the thermal reservoirs, as it enables continuous heat exchange without perturbing phonon transport in the central region. To ensure stable temperature control, the thermostat coupling time constant is set to 100 times the simulation time step. During the NPT relaxation phase, a barostat with a coupling time of 10 times the thermostat constant is applied to allow smooth pressure fluctuations without interfering with thermalization. Complete details, including the thermostats, coupling constants, can be found in the data availability section.

Interfacial thermal conductance ($h$) is calculated based on the temperature difference on both sides of the interface through the equation $h = Q/(A \cdot \Delta T)$ where $Q$ is the heat flow across the system, $A$ is the cross-sectional area, and $\Delta T$ is the temperature difference on the two sides of the interface. To calculate $\Delta T$, the extrapolation of the temperature profile on both sides of the interface is used discarding temperature data very close to the interface.

To calculate the frequency-dependent interfacial thermal conductance, the decomposition of the heat flux passing through the interface is considered:

$$Q = \int_{0}^{\infty} q(\omega) \frac{d\omega}{2\pi} = \sum_{i \in I} \sum_{j \in J} \int_{0}^{\infty} q_{i \to j}(\omega) \frac{d\omega}{2\pi} \quad (1)$$

where $\omega$ is the frequency, $I$ and $J$ represent the left and right sides of the interface (with a thickness equal to the interatomic force cutoff). The spectral heat flux can be calculated from the following equation [68–71]:

$$q(\omega) = 2 \operatorname{Re} \left[ \sum_{j \in J} \int_{0}^{\infty} \langle \mathbf{F}_{j}(t) \cdot \mathbf{v}_{j}(0) \rangle \exp(-i\omega t) dt \right] \quad (2)$$

Here, Re is the real part, $\mathbf{F}_j = \sum_{i \in I} \mathbf{F}_{ij}(t)$ is the net force exerted on each atom at one side of the interface ($J$) only by the atoms on the other side of the interface. $\mathbf{v}$ is the atomic velocity, and $\langle \cdot \rangle$ denotes an average over the simulation time. Having the spectral heat flux, the spectral conductance can be calculated from the following equation:

$$h(\omega) = \frac{q(\omega)}{A \Delta T} \quad (3)$$

In classical molecular dynamics, the statistical distribution of kinetic energy obeys the classical Boltzmann distribution. However, at low temperatures (i.e., below the Debye temperature of the material), quantum effects become significant, and the Bose-Einstein distribution is more appropriate to describe the behavior of quasi-particles such as phonons. Therefore, a correction factor is introduced to adjust for the differences between the classical Boltzmann distribution and Bose-Einstein distributions at each frequency to model the system at any temperature. For this purpose, we apply a coefficient equal to the ratio of

the quantum heat capacity ($\hbar\omega\partial f_{BE}/\partial T$) to the classical heat capacity ($k_B$) in a Landauer approach [71]:

$$h = \frac{1}{A \Delta T} \int_{0}^{\infty} q(\omega) \frac{\hbar \omega}{k_B} \frac{\partial f_{BE}}{\partial T} \frac{d\omega}{2\pi} \quad (4)$$

where $f_{BE}$ is the Bose-Einstein distribution function and $\hbar$ is the reduced Planck constant. It should be noted that the current approach of correcting quantum effects by rescaling the specific heat is a simplified approximation. Quantum effects can influence not only the specific heat but also bulk phonon lifetimes, as demonstrated by He et al. [72]. Regarding interfacial transport, two situations may be distinguished: when elastic phonon scattering at the interface is dominant, a given phonon mode is transmitted at the interface without interfering with other modes. The interface conductance may be viewed as the cumulative effect of the energy transmission of independent modes. In this situation, rescaling the specific heat is a reasonable approximation as the energy transmission of a given mode depends only on its population. By contrast, when inelastic scattering processes are important, a given phonon mode will interfere with other modes at the interface giving rise to a merged mode which propagates at the other side of the interface. The transmission of a given mode depends thus not only on the population of the mode considered but also on the population of other modes. In this situation, rescaling the heat capacity is not a good approximation as it cannot capture accurately simultaneously the Bose-Einstein population of different modes interacting at the interface. Therefore, our approximation describes situations where interfacial transport is dominated by elastic scattering and cannot capture accurately the effect of inelastic processes taking place at the interface.

To provide a clear summary of the overall workflow, Fig. 2 illustrates the methodology employed in this study. The process begins with the preparation of structures, including a periodic heterostructure (96 atoms) and pure bulk configurations of diamond (144 atoms) and silicon (64 atoms), optimized to minimize stress. This is followed by ab initio simulations using the VASP software, where AIMD simulations were conducted with specific parameters (GGA-PBE functional, 500 eV energy cutoff, $2 \times 2 \times 2$ k-point mesh, and a temperature range of 1–2000 K) to generate a dataset of 9000 configurations. The exact dimensions of the simulation cell are $16.74 \times 7.30 \times 7.30$ Å along the x-, y-, and z-axes, respectively. Pure diamond has cell dimensions of $10.72 \times 10.72 \times 7.14$ Å, and pure silicon has cell dimensions of $10.87 \times 10.87 \times 10.87$ Å. To ensure comparable stress conditions in AIMD and LAMMPS simulations, the systems are prepared as follows. In AIMD, the structure is fully relaxed, allowing both atomic coordinates and the simulation cell dimensions to adjust freely, minimizing residual stresses.

The dataset was used in the MLIP package to train the potential function through regression analysis, involving initial training with 800 configurations and iterative refinement using high-grade configurations. It is important to stress that here the potential has been trained on both bulk systems and their interface, not solely on the interface. Next, molecular dynamics (MD) simulations were performed with the LAMMPS software to study the system, employing NEMD simulations with a total simulation time comprising 50 ps relaxation, 0.5 ns steady-state, and 0.5 ns data collection phase. Finally, the results include heat flow analysis and the calculation of interfacial thermal conductance, with the temperature profile across the interface.

### 3. Results & discussion

In this section, the results of the interfacial thermal conductance for different lengths of silicon/diamond heterostructures are presented, and the length-independent interfacial thermal conductance calculated by the moment tensor potential (MTP) is compared with the experimental results and also with other MD reports that consider Tersoff or Brenner potentials.

Fig. 3 (top panel) shows the temperature profile along the silicon/
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Workflow of the study, detailing the sequential steps from structure preparation to results. The process includes structure optimization, ab initio simulations with VASP to generate training data, machine learning potential training, nonequilibrium molecular dynamics simulations using LAMMPS, and the final calculation of interfacial thermal conductance.</figcaption></figure>

diamond heterostructure when the temperature of the hot thermostat is 320 K and the temperature of the cold thermostat is 280 K. The length of the considered structure is equal to 32 nm and the thickness of the hot/ cold bath and the thermal insulating regions are 1 nm, individually. As can be seen in Fig. 3 (top panel), a significant temperature jump is observed at the interface. In addition, it is known that the temperature gradient on either side of the interface is proportional to the effective thermal conductivity of silicon and diamond, so the temperature gradient in silicon is higher due to its lower thermal conductivity compared to diamond. Using the extrapolation of the fitted lines to silicon and diamond temperature points, ΔT at the middle of the structure is calculated with a value equal to 28 K. To calculate the heat flux in the system, Fig. 3 (bottom panel) is considered, which shows the accumulated variations of the energy of the hot and cold temperature thermostats with respect to the time in steady state. The slope of the energy change with respect to time in both hot and cold baths is equal to the heat flux established in the system. Also, the good matching of the rate of energy changes in the hot and cold baths (less than 5 % difference) indicates the conservation of energy in the MD simulation with the MTP model. It should be noted that changing the thermostat might affect the ITC result, as it has been shown that different thermostats, such as the Langevin and Nose-Hoover thermostats, can lead to distinct mode-level phonon excitations and thermalization behaviors [73,74]. Our results, obtained using the Nose-Hoover thermostat, show that the ITC variation due to different thermostat sizes falls within the uncertainty range (5 %), indicating a minimal effect of the reservoir for the system studied. To further ensure that the system reaches steady-state, the evolution of temperature was monitored over time at several locations near the interface, and its stabilization indicates the steady-state conditions. This ensures that the difference found between the interfacial thermal conductance of the MTP model and semi-empirical potential are not explained by differences in the simulation time required for steady-state convergence.

Fig. 4(left panel) shows the interfacial thermal conductance for different heterostructure lengths. It can be seen that the interfacial thermal conductance for short heterostructure lengths is a function of

the system size. There are several reasons for this size dependence. First, the thermostats may scatter the phonons traveling across the interfaces in small systems. Second, the contribution of ballistic phonons to thermal transport is more pronounced for system sizes smaller than the phonon mean free path. By considering a functional form $h(L) = aL/(b+L)$, the value of $h_∞$ corresponding to the infinite length of the system can be estimated. The error bars in the left panel of Fig. 4 were obtained by repeating the simulations for each length four times with uncorrelated initial velocities and subsequently calculating the uncertainty of the estimated ITCs. Fig. 4(right panel) shows the results of $h_∞$ calculated by the MTP with and without quantum corrections. The MTP-based results give a closer estimate of the experimental results than the Tersoff potential. However, experimental measurements have revealed a nanometer-thick amorphous carbon layer [11,75], which is significantly larger than the naturally evolved layer in our modeling. This discrepancy highlights that our modeling provides an upper bound for interfacial conductance. Furthermore, it should be noted that in the experimental test reported in Refs. [11,75], the crystal orientation of the diamond is likely (110) rather than (100). To investigate the effect of orientation, we performed additional simulations of the Si(100)/C(110) interface and found that the predicted ITC varies by less than 10 % compared to the Si(100)/C(100) configuration. In addition, it can be seen that Tersoff and Brenner potentials overestimate the interfacial thermal conductance as compared to the trained MTP. We also explored the possibility of thermal rectification by reversing the sign of the temperature gradient, with diamond at 320 K and silicon at 280 K. The predicted interfacial thermal conductance in the reversed configuration was approximately 8 % lower than in the original setup for a sample with a total length of 11 nm. This indicates the marginal presence of thermal rectification, which may hold potential for asymmetric thermal transport at interfaces [76–79].

To interpret the significant difference between the MTP and Tersoff potential results, we calculate and plot the spectral interfacial thermal conductance as a function of the phonon frequency. Fig. 5(left panel) shows the distribution of the interfacial thermal conductance for the two potentials, MTP and Tersoff, using classical and quantum
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Top panel: Temperature profile at silicon/diamond heterostructure using the trained MTP; bottom panel: Accumulated energy variations of the hot and cold thermostats with respect to simulation time. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

approximation. As can be seen, the Tersoff potential overestimates the value of $h(\omega)$ for all frequencies. Fig. 5(right panel) shows the cumulative variation of $h$ with respect to phonon frequency. It shows that for the trained MTP model, the conductance at frequencies below 5 THz and above 17 THz is negligible (comparable to the level of numerical noise), and also that the highest increase of $h$ occurs in the frequency range of 10–15 THz.

Based on the Landauer approach, the interfacial thermal conductance depends on the phonon group velocity and interface bonding, which both affect the phonon transmission at the interface [7,80]. Phonon transmission at interfaces involves also inelastic phonon scattering which depends on the anharmonicities of atomic vibrations in the vicinity of the interface. In the following, we aim to understand why empirical potentials strongly overestimate the interface thermal conductance by separately comparing the accuracy of the group velocities, interfacial bonding, and the anharmonicity as predicted by the MTP and the empirical potentials retained to model silicon, diamond, and the interface. Phonon dispersion relations will be obtained through bulk phonon dispersion curves. Interfacial bonding will be estimated based on the temperature dependence of the mean square displacements (MSD) of interface atoms. To quantify the amount of vibrational anharmonicities, we will consider separately anharmonicities in the bulk of the materials that can be evaluated based on phonon lifetimes calculations. The level of anharmonicity of atomic vibrations at the interface may also be measured by the temperature dependence of the MSD of interfacial atoms. Note that the respective effects of anharmonicities in the bulk and at the interface affect heat transport in different ways: anharmonicities in the bulk tend to decrease the bulk thermal conductivity due to enhanced phonon scattering. By contrast, anharmonicities at the interface tend to increase the interfacial thermal conductance as a result of enhanced inelastic processes involving two phonon modes or more.

First, we concentrate on the predictions of the dispersion relations of the bulk materials. To calculate phonon dispersion, we computed the second-order forces for the Tersoff and MTP models using the LAMMPS and ALAMODE packages. Additionally, the second-order forces for the DFT calculations were obtained using the VASP and Phonopy packages. Fig. 6 shows the phonon dispersion relation of bulk silicon and diamond based on experimental data, DFT calculations, trained MTP, and Tersoff potential computations. For consistency, we have used the initially relaxed GGA-PBE structures for all phonon calculations. Nonetheless, while re-relaxing the structure with each potential may introduce slight

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Left panel: Interfacial thermal conductance (ITC) of silicon/diamond heterostructure calculated for different sizes using the MTP model and commonly used empirical models. The dashed line shows the fitting curve. Right panel: Comparison of interfacial thermal conductance measured in experiments [11,14,15], calculated from MTP-based simulations with classical and quantum-corrected values and obtained from Tersoff and Brenner interatomic potentials [11,12,16]. It should be noted that an amorphous carbon layer of about 1 nm thickness was observed in the experimental results reported in Ref. [11]. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Left panel: Frequency-dependent interfacial thermal conductance of silicon/diamond heterostructure using the trained MTP and Tersoff potential [16]; right panel: Cumulative conductance versus the frequency using the MTP (classical and quantum-corrected). The shaded area are the phonon modes localized at the interface. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Phonon dispersion curves calculated with MTP and Tersoff potential and compared with DFT calculations and experimental data [83–86] for bulk silicon (**top panel**) and bulk diamond (**bottom panel**). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

variations in the lattice constants, our analysis indicates that these changes have a negligible effect on the overall phonon properties. The Phonopy package was used to calculate the phonon dispersion relation using empirical interatomic potentials. As can be seen in Fig. 6, the MTP model predicts the phonon dispersion relation in better agreement with the DFT and experimental results. In addition, the Tersoff potential displays high frequencies not present in the experimental data and DFT results. By calculating the group velocity of the phonons from the dispersion relation shown in Fig. 6, we find that the Tersoff potential

overpredicts the phonon group velocity compared to the trained MTP with respect to the DFT calculations. Moreover, Fig. 7 shows the phonon lifetimes for two bulk silicon and diamond structures. To smooth the data, the lifetimes are averaged over every 5 points. Our data are in good agreement with previous studies on bulk silicon and diamond [81,82]. As can be seen, the agreement between the phonon lifetimes calculated from the DFT and MTP is much better than for the Tersoff potential, which significantly overestimates the lifetimes, particularly for silicon.

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Phonon lifetime calculated with MTP and Tersoff potential and compared with DFT calculations for bulk silicon (**top panel**) and bulk diamond (**bottom panel**) calculated at 300 K. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>
This improvement is likely due to the fact that MTP is specifically trained on DFT data, enabling it to better capture the atomic interactions and phonon properties in silicon.

The slight underestimation of phonon lifetimes by the MTP model is likely due to the fact that the potential has been trained using both bulk and interface systems, not solely on bulk systems. In interface systems, the phonon lifetimes of silicon or diamond systems are reduced with respect to the bulk as a phonon mode may be scattered at the interface. These effects may persist even if we compute the phonon lifetimes of bulk systems. However, Fig. 7 shows that the MTP model still exhibit superior accuracy than the Tersoff counterpart.

It should be noted that the calculations of phonon dispersion and phonon lifetime associated with the MTP and Tersoff potentials have been performed using the ALAMODE software [87]. To compute the lifetimes, we incorporated anharmonic effects by calculating the third-order force constants for DFT using VASP and Phono3py. Similarly, we determined the third-order forces and phonon lifetimes for the MTP and Tersoff potentials using LAMMPS and ALAMODE. It is worth noting that the size of both silicon and diamond structures for force calculations is 2 × 2 × 2.

So far, we have considered the behavior of the bulk systems to interpret why Tersoff potential overestimates the interfacial thermal conductance compared to MTP. We now investigate the interface properties. Our discussion will be focused on the temperature dependence of the mean square displacement MSD $\langle u^2(T) \rangle$ of interfacial atoms. This latter quantity may be written $\langle u^2(T) \rangle \approx aT + bT^2$. The first term corresponds to the temperature behavior of a harmonic oscillator, and the coefficient $a$ is therefore a measure of the bonding of the atom with its neighbours. The second term corresponds to the temperature dependence of an anharmonic oscillator and the coefficient $b$ is a measure of the degree of vibrational anharmonicity at the interface.

To estimate the interfacial bond strength, a region with a thickness of one atomic layer of C on the right side and a region with a thickness of one atomic layer of Si on the left side of the interface are considered. The mean square displacement (MSD) of these two regions is calculated using two potentials, MTP and Tersoff. Fig. 8 shows the results of the MSD versus temperature from 100 K to 400 K. As can be seen, the Tersoff potential predicts a lower value of the MSD of carbon atoms compared to the MTP-based results. This means that the Tersoff potential corresponds to a stronger bonding at the silicon/diamond interface than the MTP model. This finding is consistent with the higher interfacial thermal conductance of the Tersoff potential compared to the MTP. Moreover, the nonlinear behavior of MSD of Si atoms versus temperature above

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Mean-square displacement of two atomic layers of Si and C atoms adjacent to the interface calculated by Tersoff and MTP models at various temperatures.</figcaption></figure>

300 K highlights the importance of anharmonic vibrations in thermal transport as a new channel. The stronger temperature dependence of the MSD displayed by the Tersoff potential indicates a relatively higher level of anharmonicities of vibrations at the interface. This contributes to enhance interfacial thermal transfer through inelastic interface scattering processes and may explain why the Tersoff potential predicts higher interfacial thermal conductance (ITC).

To see which frequencies contribute to heat transport at the silicon/diamond interface, we calculate the vibrational density of states (VDOS) in the four different regions of the heterostructure. First, we consider two regions far from the interface with a thickness of 0.5 nm, and we also consider two regions near both sides of the interface having a thickness equal to the MTP cutoff (0.4 nm). Fig. 9 shows the VDOS for these four regions. As can be seen, new frequencies in the range of 8–12 THz, corresponding to localized phonon modes, have appeared at the interface, which are not present in the bulk regions. Based on the accumulative ITC shown in Fig. 5, these phonon modes give an important contribution to the interfacial thermal conductance of the silicon/diamond heterostructure in which about 40 % of total ITC belongs to this frequency range. Note that the importance of interface modes has been confirmed by simulations and experiments in previous studies of silicon/geranium heterostructures [88–90]. It should be noted that the VDOS shown in Fig. 9 was calculated at room temperature. Moreover, The VDOS was extracted from equilibrium molecular dynamics (EMD) simulations to capture the intrinsic vibrational characteristics of the materials at a well-defined mean temperature. Room temperature may

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Phonon density of states calculated on the basis of trained MTP for two regions far from the interface (upper panel) and for two regions near the interface (lower panel). The shaded areas correspond to interface phonon modes that are localized at the interface.</figcaption></figure>
provide a more realistic representation of the thermal vibrations for the systems under study, and it better matches with the conditions under which the interfacial thermal conductance is being evaluated. This allows us to capture the relevant phonon modes that contribute to thermal transport at temperatures of practical interest [91].

## 4. Concluding remarks

This study demonstrated the performance of machine learning (ML) interatomic potentials to accurately estimate the interfacial thermal conductance (ITC) between silicon and diamond. By using non-equilibrium molecular dynamics (NEMD) simulations trained on density functional theory (DFT) data, we have addressed the limitations of traditional semi-empirical potentials, such as Tersoff and Brenner, which tend to overpredict ITC. Our results show that the passively-fitted ML-based MTP model provides a significantly closer match to experimental ITC values, reducing considerably the discrepancies observed with NEMD simulations based on semi-empirical potentials. This accuracy is achieved while maintaining computational efficiency, demonstrating the practical applicability of MTP in simulating complex thermal transport phenomena at interfaces. The frequency-dependent analysis of heat transfer revealed that phonon interactions play a crucial role in interfacial thermal conductance, and the MTP effectively captures these interactions. The phonon dispersion relations and lifetimes predicted by the MTP align well with both DFT calculations and experimental observations, underscoring the potential of MLIPs methods in enhancing our understanding of interfacial thermal transport. By contrast, we showed that not only empirical potentials do not predict accurately bulk phonon dispersion curves, but they can also do a poor job in describing both interfacial bonding and vibrational anharmonicities of interfacial atoms, which affects the predictions of interface thermal conductance through elastic and inelastic phonon scattering, respectively.

Overall, this study highlights the advantages of ML-based interatomic potentials in providing detailed insights into the atomic-level mechanisms governing thermal transport, which are critical for designing and optimizing high-performance electronic and optoelectronic devices. Future work could extend this approach to other material systems and explore the impact of different interface conditions and configurations, further broadening the applicability and robustness of ML-driven thermal management solutions.

### CRediT authorship contribution statement

**Ali Rajabpour:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Bohayra Mortazavi:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Pedram Mirchi:** Writing – original draft, Software, Methodology, Investigation, Formal analysis, Data curation, Visualization.
**Julien El Hajj:** Writing – review & editing, Methodology, Investigation, Data curation.
**Yangyu Guo:** Methodology, Investigation.
**Xiaoying Zhuang:** Resources.
**Samy Merabia:** Writing – review & editing, Supervision, Methodology, Investigation, Funding acquisition, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgment

This work is based upon research funded by Iran National Science

Foundation (INSF) under project No.4035102. Moreover, this research received support from the French Agence Nationale de la Recherche (ANR) under Grant No. ANR-20-CE05-0021-01 (NearHeat). We also acknowledge the use of computational resources from Raptor at iLM, Université Claude Bernard Lyon 1. Also, B. M. and X. Z. appreciate the funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453).

### Data availability

The data associated with this work, which includes the trained MTP model and corresponding AIMD dataset in CFG and XYZ formats, as well as the VASP, LAMMPS, and ALAMODE input files, can be found at https://doi.org/10.17632/6tjhd74t5r. Additional data presented in this study are available upon request from the first author.

### References

[1] X. Zhu, T. Bi, X. Yuan, Y. Chang, R. Zhang, Y. Fu, J. Tu, Y. Huang, J. Liu, C. Li, H. Kawarada, C-Si interface on SiO2/(1 1 1) diamond p-MOSFETs with high mobility and excellent normally-off operation, Appl. Surf. Sci. 593 (2022) 153368, https://doi.org/10.1016/j.apsusc.2022.153368.

[2] O.M. Küttel, E. Schaller, J. Osterwalder, L. Schlapbach, X-ray photoelectron diffraction on the nickel/diamond, the silicon/diamond and the gold/diamond interface, Diam. Relat. Mater. 4 (1995) 612–616, https://doi.org/10.1016/0925-9635(94)05216-6.

[3] Z.B. Milne, J.D. Schall, T.D.B. Jacobs, J.A. Harrison, R.W. Carpick, Covalent bonding and atomic-level plasticity increase adhesion in silicon–diamond nanocontacts, ACS Appl. Mater. Interfaces 11 (2019) 40734–40748, https://doi.org/10.1021/acsami.9b08695.

[4] M. Citroni, S. Lagomarsino, G. Parrini, M. Santoro, S. Sciortino, M. Vannoni, G. Ferrari, A. Fossati, F. Gorelli, G. Molesini, G. Piani, A. Scorzoni, A novel method of preparation of silicon-on-diamond materials, Diam. Relat. Mater. 19 (2010) 950–955, https://doi.org/10.1016/j.diamond.2010.02.038.

[5] G.N. Yushin, A. Aleksov, S.D. Wolter, F. Okuzumi, J.T. Prater, Z. Sitar, Wafer bonding of highly oriented diamond to silicon, Diam. Relat. Mater. 13 (2004) 1816–1821, https://doi.org/10.1016/j.diamond.2004.04.007.

[6] S. Volz (Ed.), Thermal Nanosystems and Nanomaterials, Springer Berlin Heidelberg, Berlin, Heidelberg, 2009, https://doi.org/10.1007/978-3-642-04258-4.

[7] J. Chen, X. Xu, J. Zhou, B. Li, Interfacial thermal resistance: past, present, and future, Rev. Mod. Phys. 94 (2022) 025002, https://doi.org/10.1103/RevModPhys.94.025002.

[8] G. Chen, Introduction, in: Nanoscale Energy Transport and Conversion, Oxford University PressNew, York, NY, 2005, pp. 3–42, https://doi.org/10.1093/oso/9780195159424.003.0001.

[9] S. Bin Mansoor, B.S. Yilbas, Phonon transport in silicon–silicon and silicon–diamond thin films: consideration of thermal boundary resistance at interface, Phys. B Condens. Matter 406 (2011) 2186–2195, https://doi.org/10.1016/j.physb.2011.03.028.

[10] K.E. Goodson, O.W. Käding, M. Rösler, R. Zachai, Experimental investigation of thermal conduction normal to diamond-silicon boundaries, J. Appl. Phys. 77 (1995) 1385–1392, https://doi.org/10.1063/1.358950.

[11] Z. Cheng, T. Bai, J. Shi, T. Feng, Y. Wang, M. Mecklenburg, C. Li, K.D. Hobart, T. I. Feygelson, M.J. Tadjer, B.B. Pate, B.M. Foley, L. Yates, S.T. Pantelides, B.A. Cola, M. Goorsky, S. Graham, Tunable thermal energy transport across diamond membranes and diamond–Si interfaces by nanoscale graphoepitaxy, ACS Appl. Mater. Interfaces 11 (2019) 18517–18527, https://doi.org/10.1021/acsami.9b02234.

[12] N. Khosravian, M.K. Samani, G.C. Loh, G.C.K. Chen, D. Baillargeat, B.K. Tay, Molecular dynamic simulation of diamond/silicon interfacial thermal conductance, J. Appl. Phys. 113 (2013) 024907, https://doi.org/10.1063/1.4775399.

[13] L. Chen, S. Chen, Y. Hou, Understanding the thermal conductivity of Diamond/ Copper composites by first-principles calculations, Carbon N Y 148 (2019) 249–257, https://doi.org/10.1016/j.carbon.2019.03.051.

[14] V. Goyal, D. Kotchetkov, S. Subrina, M. Rahman, A.A. Balandin, Thermal conduction through diamond - silicon heterostructures, in: 2010 12th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, IEEE, 2010, pp. 1–6, https://doi.org/10.1109/ITHERM.2010.5501369.

[15] T. Feng, H. Zhou, Z. Cheng, L.S. Larkin, M.R. Neupane, Correction to “A critical review of thermal boundary conductance across wide and ultrawide bandgap semiconductor interfaces,” ACS Appl. Mater. Interfaces 16 (2024) 34429–34430, https://doi.org/10.1021/acsami.4c08962.

[16] Y. Qu, J. Yuan, N. Deng, W. Hu, S. Wu, H. Wang, Effect of the thickness of amorphous silicon intermediate layer on the thermal transport of silicon/diamond interface, Results Phys. 52 (2023) 106827, https://doi.org/10.1016/j.rinp.2023.106827.
[17] A. Rajabpour, S. Volz, Universal interfacial thermal resistance at high frequencies, Phys. Rev. B Condens. Matter (2014) 2–5, https://doi.org/10.1103/PhysRevB.90.195444.

[18] M.S. Alborzi, A. Rajabpour, Effect of overlapping junctions on the heat transfer between 2D layered composite materials, Int. Commun. Heat Mass Tran. 109 (2019), https://doi.org/10.1016/j.icheatmasstransfer.2019.104348.

[19] A. Rajabpour, S. Volz, Thermal boundary resistance from mode energy relaxation times: case study of argon-like crystals by molecular dynamics, J. Appl. Phys. 108 (2010) 094324, https://doi.org/10.1063/1.3500526.

[20] S.M. Hatam-Lee, K. Gordiz, A. Rajabpour, Lattice-dynamics-based descriptors for interfacial heat transfer across two-dimensional carbon-based nanostructures, J. Appl. Phys. 130 (2021) 135106, https://doi.org/10.1063/5.0055708.

[21] Y. Guo, C. Adessi, M. Cobian, S. Merabia, Atomistic simulation of phonon heat transport across metallic vacuum nanogaps, Phys. Rev. B 106 (2022) 85403–85410, https://doi.org/10.1103/PhysRevB.106.085403.

[22] H. Han, S. Mérabia, F. Müller-Plathe, Thermal transport at solid-liquid interfaces: high pressure facilitates heat flow through nonlocal liquid structuring, J. Phys. Chem. Lett. 8 (2017) 1946–1951, https://doi.org/10.1021/acs.jpclett.7b00227.

[23] J. El Hajj, C. Adessi, M. de San Feliciano, G. Ledoux, S. Merabia, Enhanced thermal conductance at interfaces between gold and amorphous silicon and between gold and amorphous silica, Phys. Rev. B 110 (2024) 115437, https://doi.org/10.1103/PhysRevB.110.115437.

[24] S. Merabia, K. Termentzidis, Thermal boundary conductance across rough interfaces probed by molecular dynamics, Phys. Rev. B 89 (2014) 054309, https://doi.org/10.1103/PhysRevB.89.054309.

[25] Z. Fan, Y. Wang, P. Ying, K. Song, J. Wang, Y. Wang, Z. Zeng, K. Xu, E. Lindgren, J. M. Rahm, A.J. Gabourie, J. Liu, H. Dong, J. Wu, Y. Chen, Z. Zhong, J. Sun, P. Erhart, Y. Su, T. Ala-Nissila, GPUMD: a package for constructing accurate machine-learned potentials and performing highly efficient atomistic simulations, J. Chem. Phys. 157 (2022) 204114, https://doi.org/10.1063/5.0106617.

[26] C. Li, J. Wang, Y. Sheng, L. Yang, Y. Su, The strain-dependent interfacial thermal resistance at graphene-silicon interface under various deformation conditions, Int. J. Heat Mass Tran. 198 (2022) 123383, https://doi.org/10.1016/j.ijheatmasstransfer.2022.123383.

[27] Y. Liu, L. Qiu, J. Liu, Y. Feng, Enhancing thermal transport across diamond/graphene heterostructure interface, Int. J. Heat Mass Tran. 209 (2023) 124123, https://doi.org/10.1016/j.ijheatmasstransfer.2023.124123.

[28] R. Rabani, M.H. Saidi, A. Rajabpour, L. Joly, S. Merabia, Enhanced heat flow between charged nanoparticles and an aqueous electrolyte, Langmuir 39 (2023) 15222–15230, https://doi.org/10.1021/acs.langmuir.3c01847.

[29] Y. Liu, J. Hao, A. Chernatynskiy, G. Ren, J. Zhang, Effect of period length distribution on the thermal conductivity of Si/Ge superlattice, Int. J. Therm. Sci. 170 (2021) 107157, https://doi.org/10.1016/j.ijthermalsci.2021.107157.

[30] Y. Tang, Z. Zhang, L. Li, J. Guo, P. Yang, Thermal transport enhancement resolution for graphene/Si and graphene/SiC interfaces, Int. J. Therm. Sci. 171 (2022) 107231, https://doi.org/10.1016/j.ijthermalsci.2021.107231.

[31] Q. He, Y. Xu, H. Wang, Z. Li, Y. Zhou, Role of elastic phonon couplings in dictating the thermal transport across atomically sharp SiC/Si interfaces, Int. J. Therm. Sci. 204 (2024) 109182, https://doi.org/10.1016/j.ijthermalsci.2024.109182.

[32] X. Cai, H. Li, J. Zhang, T. Ma, Q. Wang, Mechanism of interfacial thermal resistance variation in diamond/Cu/CNT tri-layer during thermal cycles, Int. J. Therm. Sci. 207 (2025) 109380, https://doi.org/10.1016/j.ijthermalsci.2024.109380.

[33] R. Rabani, M.H. Saidi, L. Joly, S. Merabia, A. Rajabpour, Enhanced local viscosity around colloidal nanoparticles probed by equilibrium molecular dynamics simulations, J. Chem. Phys. 155 (2021), https://doi.org/10.1063/5.0065050.

[34] X. Wu, W. Zhou, H. Dong, P. Ying, Y. Wang, B. Song, Z. Fan, S. Xiong, Correcting force error-induced underestimation of lattice thermal conductivity in machine learning molecular dynamics, J. Chem. Phys. 161 (2024), https://doi.org/10.1063/5.0213811.

[35] K. Xu, Y. Hao, T. Liang, P. Ying, J. Xu, J. Wu, Z. Fan, Accurate prediction of heat conductivity of water by a neuroevolution potential, J. Chem. Phys. 158 (2023), https://doi.org/10.1063/5.0147039.

[36] Z. Fan, Z. Zeng, C. Zhang, Y. Wang, K. Song, H. Dong, Y. Chen, T. Ala-Nissila, Neuroevolution machine learning potentials: combining high accuracy and low cost in atomistic simulations and application to heat transport, Phys. Rev. B 104 (2021) 104309.

[37] H. Dong, Y. Shi, P. Ying, K. Xu, T. Liang, Y. Wang, Z. Zeng, X. Wu, W. Zhou, S. Xiong, S. Chen, Z. Fan, Molecular dynamics simulations of heat transport using machine-learned potentials: a mini-review and tutorial on GPUMD with neuroevolution potentials, J. Appl. Phys. 135 (2024), https://doi.org/10.1063/5.0200833.

[38] A.T. Vu, S. Gulati, P.-A. Vogel, T. Grunwald, T. Bergs, Machine learning-based predictive modeling of contact heat transfer, Int. J. Heat Mass Tran. 174 (2021) 121300, https://doi.org/10.1016/j.ijheatmasstransfer.2021.121300.

[39] R. Hu, S. Iwamoto, L. Feng, S. Ju, S. Hu, M. Ohnishi, N. Nagai, K. Hirakawa, J. Shiomi, Machine-learning-optimized aperiodic superlattice minimizes coherent phonon heat conduction, Phys. Rev. X 10 (2020) 21050.

[40] S. Arabha, Z.S. Aghbolagh, K. Ghorbani, S.M. Hatam-Lee, A. Rajabpour, Recent advances in lattice thermal conductivity calculation using machine-learning interatomic potentials, J. Appl. Phys. 130 (2021) 210903, https://doi.org/10.1063/5.0069443.

[41] B. Mortazavi, A. Rajabpour, X. Zhuang, T. Rabczuk, A. V Shapeev, Exploring thermal expansion of carbon-based nanosheets by machine-learning interatomic potentials, Carbon N Y (2022), https://doi.org/10.1016/j.carbon.2021.10.059.

[42] B. Mortazavi, I.S. Novikov, A. V Shapeev, A machine-learning-based investigation on the mechanical/failure response and thermal conductivity of semiconducting

BC2N monolayers, Carbon N Y 188 (2022) 431–441, https://doi.org/10.1016/j.carbon.2021.12.039.

[43] X. Qian, S. Peng, X. Li, Y. Wei, R. Yang, Thermal conductivity modeling using machine learning potentials: application to crystalline and amorphous silicon, Materials Today Phys. 10 (2019) 100140, https://doi.org/10.1016/j.mtphys.2019.100140.

[44] C. Zhu, E.A. Bamidele, X. Shen, G. Zhu, B. Li, Machine learning aided design and optimization of thermal metamaterials, Chem Rev 124 (2024) 4258–4331, https://doi.org/10.1021/acs.chemrev.3c0078.

[45] C. Cao, S. Cao, Y. Zhu, H. Dong, Y. Wang, P. Qian, Thermal transports of 2D phosphorous carbides by machine learning molecular dynamics simulations, Int. J. Heat Mass Tran. 224 (2024) 125359, https://doi.org/10.1016/j.ijheatmasstransfer.2024.125359.

[46] K.A. Lopes Lima, F.L. Lopes Mendonça, W.F. Giozza, R.T. de Sousa Junior, L. A. Ribeiro Junior, Insights into the DHQ-BN: mechanical, electronic, and optical properties, Sci. Rep. 14 (2024) 2510, https://doi.org/10.1038/s41598-024-52347-2.

[47] A.A. Kistanov, I.V. Kosarev, S.A. Shcherbinin, A.V. Shapeev, E.A. Korznikova, S. V. Dmitriev, Unified approach to generating a training set for machine learning interatomic potentials: the case of BCC tungsten, Mater. Today Commun. 42 (2025) 111437, https://doi.org/10.1016/j.mtcomm.2024.111437.

[48] B. Mortazavi, Recent advances in machine learning-assisted multiscale design of energy materials, Adv. Energy Mater. 15 (2025) 2403876, https://doi.org/10.1002/aemn.202403876.

[49] E.J.A. dos Santos, M.L. Pereira, R.M. Tromer, D.S. Galvão, L.A. Ribeiro, Exploring the electronic and mechanical properties of the recently synthesized nitrogen-doped amorphous monolayer carbon, Nanoscale (2025), https://doi.org/10.1039/D4NR04305G.

[50] K.A.L. Lima, R.A.F. Alves, D.A. da Silva, F.L.L. Mendonça, M.L. Pereira, L. A. Ribeiro, TH-graphyne: a new porous bidimensional carbon allotrope, Phys. Chem. Chem. Phys. (2025), https://doi.org/10.1039/D4CP02923B.

[51] S.V. Dmitriev, A.A. Kistanov, I.V. Kosarev, S.A. Scherbinin, A.V. Shapeev, Construction of machine learning interatomic potentials for metals, Russ. Phys. J. 67 (2024) 1408–1413, https://doi.org/10.1007/s11182-024-03261-7.

[52] S. Arabha, A. Rajabpour, Thermo-mechanical properties of nitrogenated holey graphene (C2N): a comparison of machine-learning-based and classical interatomic potentials, Int. J. Heat Mass Tran. 178 (2021) 121589, https://doi.org/10.1016/j.ijheatmasstransfer.2021.121589.

[53] K.A.L. Lima, F.F. Monteiro, E.J.A. Santos, R.A.F. Alves, W.F. Giozza, L.A. Ribeiro, Dodecanophane: a novel 2D carbon allotrope with untunable metallic behavior under stress, Mater. Today Commun. 40 (2024) 109455, https://doi.org/10.1016/j.mtcomm.2024.109455.

[54] A.A. Kistanov, Characterization of monovacancy defects in vanadium diselenide monolayer: a DFT study, Appl. Sci. 14 (2024) 1205, https://doi.org/10.3390/app14031205.

[55] K.A.L. Lima, D.A. da Silva, G.D.A. Nze, F.L.L. de Mendonça, M.L. Pereira, L. A. Ribeiro, Structural, electronic, and Li-ion adsorption properties of PolyPyGy explored by first-principles and machine learning simulations: A new multi-ringed 2D carbon allotrope, J. Energy Storage 117 (2025) 116099, https://doi.org/10.1016/j.est.2025.116099.

[56] E.A.J. Santos, K.A.L. Lima, F.L.L. Mendonça, D.A. da Silva, W.F. Giozza, L.A. R. Junior, PHOTR-graphene: a new 2D carbon allotrope with low barriers for Li-ion mobility, Sci. Rep. 14 (2024) 9526, https://doi.org/10.1038/s41598-024-59858-y.

[57] E.A.J. Santos, K.A.L. Lima, L.A. Ribeiro Junior, Proposing TODD-graphene as a novel porous 2D carbon allotrope designed for superior lithium-ion battery efficiency, Sci. Rep. 14 (2024) 6202, https://doi.org/10.1038/s41598-024-56312-x.

[58] G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54 (1996) 11169–11186, https://doi.org/10.1103/PhysRevB.54.11169.

[59] J.P. Perdew, A. Ruzinszky, G.I. Csonka, O.A. Vyrov, G.E. Scuseria, L. A. Constantin, X. Zhou, K. Burke, Restoring the density-gradient expansion for exchange in solids and surfaces, Phys. Rev. Lett. 100 (2008) 136406, https://doi.org/10.1103/PhysRevLett.100.136406.

[60] D.J. Chadi, M.L. Cohen, Special points in the brillouin zone, Phys. Rev. B 8 (1973) 5747–5753, https://doi.org/10.1103/PhysRevB.8.5747.

[61] A.S. Ivan Novikov, Konstantin Gubaev, Evgeny Podryabinkin, The MLIP package: moment tensor potentials with MPI and active learning, Mach Learn Sci Technol 2 (2021) 025002.

[62] E. V Podryabinkin, A. V Shapeev, Active learning of linearly parametrized interatomic potentials, Comput. Mater. Sci. 140 (2017) 171–180, https://doi.org/10.1016/j.commatsci.2017.08.031.

[63] B. Mortazavi, E. V Podryabinkin, S. Roche, T. Rabczuk, X. Zhuang, A. V Shapeev, Machine-learning interatomic potentials enable first-principles multiscale modeling of lattice thermal conductivity in graphene/borophene heterostructures, Mater. Horiz. 7 (2020) 2359–2367, https://doi.org/10.1039/D0MH00787K.

[64] B. Mortazavi, X. Zhuang, T. Rabczuk, A. V Shapeev, Atomistic modeling of the mechanical properties: the rise of machine learning interatomic potentials, Mater. Horiz. 10 (2023) 1956–1968, https://doi.org/10.1039/D3MH00125C.

[65] A.P. Thompson, H.M. Aktulga, R. Berger, D.S. Bolintineanu, W.M. Brown, P. S. Crozier, P.J. in 't Veld, A. Kohlmeyer, S.G. Moore, T.D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, S.J. Plimpton, Lammps - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comput. Phys. Commun. 271 (2022) 108171, https://doi.org/10.1016/j.cpc.2021.108171.
[66] I.S. Novikov, K. Gubaev, E. V Podryabinkin, A. V Shapeev, The MLIP package: moment tensor potentials with MPI and active learning, Mach Learn Sci Technol 2 (2021) 25002, https://doi.org/10.1088/2632-2153/abc9fe.

[67] Z. Li, S. Xiong, C. Sievers, Y. Hu, Z. Fan, N. Wei, H. Bao, S. Chen, D. Donadio, T. Ala-Nissila, Influence of thermostatting on nonequilibrium molecular dynamics simulations of heat conduction in solids, J. Chem. Phys. 151 (2019) 234105, https://doi.org/10.1063/1.5132543.

[68] K. Sääskilahti, J. Oksanen, J. Tulkki, S. Volz, Spectral mapping of heat transfer mechanisms at liquid-solid interfaces, Phys. Rev. E 93 (2016) 052141, https://doi.org/10.1103/PhysRevE.93.052141.

[69] K. Sääskilahti, J. Oksanen, S. Volz, J. Tulkki, Frequency-dependent phonon mean free path in carbon nanotubes from nonequilibrium molecular dynamics, Phys. Rev. B 91 (2015) 115426, https://doi.org/10.1103/PhysRevB.91.115426.

[70] K. Sääskilahti, J. Oksanen, J. Tulkki, S. Volz, Role of anharmonic phonon scattering in the spectrally decomposed thermal conductance at planar interfaces, Phys. Rev. B 90 (2014) 134312, https://doi.org/10.1103/PhysRevB.90.134312.

[71] Y. Guo, M. Gómez Viloria, R. Messina, P. Ben-Abdallah, S. Merabia, Atomistic modeling of extreme near-field heat transport across nanogaps between two polar dielectric materials, Phys. Rev. B 108 (2023) 085434, https://doi.org/10.1103/PhysRevB.108.085434.

[72] Y. He, I. Savić, D. Donadio, G. Galli, Lattice thermal conductivity of semiconducting bulk materials: atomistic simulations, Phys. Chem. Chem. Phys. 14 (2012) 16209, https://doi.org/10.1039/c2cp42394d.

[73] S. Hu, C.Y. Zhao, X. Gu, Impact of thermostat on interfacial thermal conductance prediction from non-equilibrium molecular dynamics simulations, Chin. Phys. B 31 (2022) 056301, https://doi.org/10.1088/1674-1056/ac4238.

[74] Y. Hu, T. Feng, X. Gu, Z. Fan, X. Wang, M. Lundstrom, S.S. Shrestha, H. Bao, Unification of nonequilibrium molecular dynamics and the mode-resolved phonon Boltzmann equation for thermal transport simulations, Phys. Rev. B 101 (2020) 155308, https://doi.org/10.1103/PhysRevB.101.155308.

[75] C.L. Jia, K. Urban, X. Jiang, Heteroepitaxial diamond films on silicon (001): interface structure and crystallographic relations between film and substrate, Phys. Rev. B 52 (1995) 5164-5171, https://doi.org/10.1103/PhysRevB.52.5164.

[76] B. Li, J. Lan, L. Wang, Interface thermal resistance between dissimilar anharmonic lattices, Phys. Rev. Lett. 95 (2005) 104302, https://doi.org/10.1103/PhysRevLett.95.104302.

[77] J. Chen, G. Zhang, B. Li, Thermal contact resistance across nanoscale silicon dioxide and silicon interface, J. Appl. Phys. 112 (2012), https://doi.org/10.1063/1.4754513.

[78] L. Yang, J. Chen, N. Yang, B. Li, Significant reduction of graphene thermal conductivity by phononic crystal structure, Int. J. Heat Mass Tran. 91 (2015) 428-432, https://doi.org/10.1016/j.ijheatmasstransfer.2015.07.111.

[79] W. Xu, L. Zhu, Y. Cai, G. Zhang, B. Li, Direction dependent thermal conductivity of monolayer phosphorene: parameterization of Stillinger-Weber potential and molecular dynamics study, J. Appl. Phys. 117 (2015) 214308.

[80] E.T. Swartz, Thermal boundary resistance, Rev. Mod. Phys. 61 (1989) 605-668.

[81] X. Gu, S. Li, H. Bao, Thermal conductivity of silicon at elevated temperature: role of four-phonon scattering and electronic heat conduction, Int. J. Heat Mass Tran. 160 (2020) 120165, https://doi.org/10.1016/j.ijheatmasstransfer.2020.120165.

[82] L. Shi, X. Ma, M. Li, Y. Zhong, L. Yang, W. Yin, X. He, Molecular dynamics simulation of phonon thermal transport in nanotwinned diamond with a new optimized Tersoff potential, Phys. Chem. Chem. Phys. 23 (2021) 8336-8343, https://doi.org/10.1039/D1CP00399B.

[83] J.L. Warren, J.L. Yarnell, G. Dolling, R.A. Cowley, Lattice dynamics of diamond, Phys. Rev. 158 (1967) 805-808, https://doi.org/10.1103/PhysRev.158.805.

[84] M. Schwoerer-Böhning, A.T. Macrander, D.A. Arms, Phonon dispersion of diamond measured by inelastic X-ray scattering, Phys. Rev. Lett. 80 (1998) 5572-5575, https://doi.org/10.1103/PhysRevLett.80.5572.

[85] J.Q. Broughton, X.P. Li, Phase diagram of silicon by molecular dynamics, Phys. Rev. B 35 (1987) 9120-9127, https://doi.org/10.1103/PhysRevB.35.9120.

[86] X.-P. Li, G. Chen, P.B. Allen, J.Q. Broughton, Energy and vibrational spectrum of the Si(111) (7×7) surface from empirical potentials, Phys. Rev. B 38 (1988) 3331-3341, https://doi.org/10.1103/PhysRevB.38.3331.

[87] T. Tadano, Y. Gohda, S. Tsuneyuki, Anharmonic force constants extracted from first-principles molecular dynamics: applications to heat transfer simulations, J. Phys. Condens. Matter 26 (2014) 225402, https://doi.org/10.1088/0953-8984/26/22/225402.

[88] Z. Cheng, R. Li, X. Yan, G. Jernigan, J. Shi, M.E. Liao, N.J. Hines, C.A. Gadre, J. C. Idrobo, E. Lee, K.D. Hobart, M.S. Goorsky, X. Pan, T. Luo, S. Graham, Experimental observation of localized interfacial phonon modes, Nat. Commun. 12 (2021) 6901, https://doi.org/10.1038/s41467-021-27250-3.

[89] R.K. Kelayeh, A. Rajabpour, E. Taheran, Y. Bahari, Optimization of interfacial mixing for thermal transport along Si/Ge heterostructures: a molecular dynamics study, Appl. Surf. Sci. 626 (2023) 157236, https://doi.org/10.1016/j.apsusc.2023.157236.

[90] Y. Guo, Z. Zhang, M. Bescond, S. Xiong, M. Nomura, S. Volz, Anharmonic phonon-phonon scattering at the interface between two solids by nonequilibrium Green's function formalism, Phys. Rev. B 103 (2021) 174306, https://doi.org/10.1103/PhysRevB.103.174306.

[91] N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, B. Li, Colloquium: phononics: Manipulating heat flow with electronic analogs and beyond, Rev. Mod. Phys. 84 (2012) 1045-1066, https://doi.org/10.1103/RevModPhys.84.1045.