Interfacial Thermal Transport of Carbon Nanotube on the Substrate

Exploring the thermal transport properties of the interface structure of low-dimensional nanomaterials contributes to a deeper understanding of the interface phonon modes and may provide theoretical support for efficient chip cooling devices. In this manuscript, we have simulated in detail the effects of system temperature, substrate location, and heat flow density on the thermal transport at the SWCNT/Si interface using the Non-equilibrium Molecular Dynamics approach and predicted the interfacial thermal conductance of SWCNT/Si for second, third and fourth order phonon using the Anharmonic Inelastic model. The results show that the low-frequency acoustic branch of SWCNT is suppressed by phonon scattering from the substrate, and the low-frequency phonon branch of SWCNT is boosted by about 1 THz. The anharmonic channels and inelastic phonon scattering significantly affect the interface phonon modes at higher temperatures, and the anharmonic interactions could increase additional thermal transport channels, which result in an increased number of additional phonon peaks. The increase in temperature gradually consumes the phonons incident at the interface in SWCNT, which enhances the anharmonic scattering and weakens the nonlinear characteristics of the material heterostructure, while the weakening of the acoustic branch accompanied by the temperature increase makes the LA phonon branch thermal conduction rate slower and the interfacial thermal conductance gradually stabilizes, which leads to the weakening of the thermal rectification effect.


Introduction
With the trend of miniaturization becoming prominent [1], the influence of the interface formed by nano-contacts on the overall thermal performance of the material is becoming increasingly significant, and the thermal transport between the thermal interface material (TIM) and the heat-producing unit, as well as the heat sink, has become the focus of thermal management and thermal design research in the development of miniaturized micro-and nanostructures and devices [2].Nanostructures are usually prepared at critical length scales on the order of the carrier mean free range, and in nanostructures, Fourier's law does not hold even at length scales larger than the mean free range [3], leading to a greater role for interface thermal conductivity (ITC) in the thermal transport of low-dimensional structures such as carbon nanotubes (CNT) [4].CNT-TIM with excellent thermal transport properties and mechanical properties has been widely used in field-effect transistors applications [5].Despite the ultra-high thermal conductance of Single-Walled carbon nanotubes (SWCNTs) [6], the presence of surrounding media or substrates in the devices significantly affects their thermal transport properties [7][8][9][10].The existence of strong interfacial phonon scattering in CNT-TIM structures and high heat flow density up to 10 8 W•m −2 [11] with local hot spot phenomenon severely limit the practical applications of microelectronic devices, and the high interface thermal resistance largely limits the applications of CNT.Phonons act as the main thermal carriers in semiconductors.Although the importance of phonon-electron coupling at semiconductor interfaces is still controversial [12][13][14], phonon-phonon coupling at the interface is considered to be the main channel for heat energy transfer through semiconductor interfaces [12,15].Therefore, studying the thermal transport properties of carbon nanotubes, especially the phonon-phonon transport between carbon nanotubesemiconductor interfaces, is of great research value and can provide theoretical support for efficient chip cooling devices.
Recent theoretical and experimental studies have found that the thermal conductance of low-dimensional nanomaterials decreases significantly when they are supported on a substrate [16][17][18][19].Changes in physical parameters such as system temperature and interface structure all change the inherent thermal transport properties of low-dimensional nanomaterials, and the anharmonic phonon and interfacial phonon modes play a key role in the thermal transport process.A combination of Kapitza microscopic theory and Molecular Dynamics (MD) has been used to successfully capture the frequency and wave vector correlations of the equilibrium atomic uplift transport, and the results show that the interfacial scattering leads to the emergence of a 13 THz to 14 THz weak dispersion in the Si/Ge interface region and opens additional transport channels [20].By performing spectral energy density (SED) analysis of the interface structure, the results show that the low-frequency acoustic branch is lifted and the phonons are suppressed by the substrate phonon scattering [21].In addition, the heat flow inside the SWCNT-TIM is highly directional because the SWCNT acts as a quasi-onedimensional material, maintaining essentially the normal heat transport capacity in one direction, while the thermal transport in the opposite direction is extremely suppressed.This behavior is similar to that of electric diodes, except that in SWCNT and Si semiconductors the regulated carriers are no longer electrons but phonons, i.e., the phenomenon known as thermal rectification (TR) occurs [22,23].The TR effect caused by the nonlinear interface structure is of great application in the field of regulated thermal transport.The above phonon calculations are all limited to the bulk material system or the interface region on both sides of the bulk material, which has a different atomic structure and lattice vibration from the bulk material.From the theoretical side, many methods have been generated to investigate phonon scattering at interfaces, such as the acoustic mismatch model (AMM) [24], the diffusion mismatch model (DMM) [25], and equilibrium molecular dynamics (EMD) [20] to explain the intrinsic mechanism of phonon scattering at interfaces; however, these methods only consider harmonic scattering and ignore anharmonic scattering.Anharmonic phonon scattering at the interface is critical for thermal transport properties, with the interface phonon modes providing higher-order scattering channels at the interface [26,27] but hindering harmonic scattering thermal transport in bulk nanomaterials [28].
When the thermal boundary is dominated by phonon scattering, there is a limit to harmonic transport in the interfacial thermal transport, in which case anharmonic effects must be taken into account.MD simulations can directly describe the inelastic effects through the anharmonic interatomic forces used in the integration of the classical equations of motion, so we selected the MD simulation method to study the interfacial thermal transport processes of SWCNT/Si.In addition, we predicted the ITC using the anharmonic inelastic model (AIM) [29] with anharmonic phonon coupling and inelastic scattering processes, which separates the contribution of harmonic phonons from anharmonic phonons to the ITC.The multi-order phonon contribution to the ITC obtained by AIM enables the quantification of the thermal transport strength of interface anharmonic phonons, and the analysis paired with the MD results allows a more comprehensive study of the joint effect of anharmonic and interface phonons on the thermal transport properties of interfacial structures constructed of one-dimensional materials at high temperatures.
As the phonon is the main energy carrier of semiconductor and thermal interface materials, the study of phonon dynamics of thermal transport at interfaces would help us to understand and analyze the underlying mechanism more deeply.Although it has been shown that the substrate material interferes with carrier transport inside low-dimensional nanomaterials, there is a lack of characterization of interfacial phonons that are interfered with by the direction of temperature and thermal heat flow.Therefore, in this paper, we first use Non-equilibrium Molecular Dynamics (NEMD) [30] simulations to investigate the effects of system temperature, substrate position, and heat flow density on the thermal transport at the SWCNT/Si interface at a given heat flow density, and compare and analyze the spectral energy density (SED) [31,32] of phonons of suspended and substrate-supported SWCNT to explore the phonon dispersion relationship of the interfacial phonon mode interference system.In addition, we used the AIM [29] that considered the inelastic phonon scattering events between the two materials to predict the ITC of SWCNT/Si.And to analyze the contribution of different phonon modes to the interfacial thermal transport, we 136 Page 4 of 22 decomposed the ITC into TA and LA acoustic branches.The combination of NEMD and AIM methods provides both a comprehensive description of the thermal transport behavior of the interface and an accurate extraction of higher-order phonon information, which provides us with theoretical data support to reveal the anharmonicity of the interface phonons.

Simulation Details in NEMD
The two interface structures constructed by SWCNT and Si substrate are simulated and shown in Fig. 1.In this work, we develop a physical model based on SWCNT with chirality of (6,6).The lattice constants of the single-cell of ( 6 Both axial ends of SWCNT are in perpendicular contact with the Si substrate denoted as SWCNT⊥Si and the left substrate is denoted as Si-left and the right substrate is denoted as Si-right.The SWCNT is placed horizontally on the Si substrate and noted as SWCNT∥Si.The (6,6) armchair crystal cell chosen to construct SWCNT is selected in both interface structures, and the 320 protocells are taken along the axial direction (z) with a length of about 78.70 nm.In the SWCNT⊥Si structure, both sides of Si adopt 8 × 8 × 8 (x × y × z) diamond structure of supercell, i.e., the height and width along the Si matrix and the length of the axial direction are 4.34 nm.In the SWCNT∥Si structure, the crystal cell length of Si is 8 × 8 × 38 (x × y × z), i.e., the height and width length of the matrix are 4.34 nm and the axial length is about 20.64 nm.The simulation uses two different directions of applied heat flow for the SWCNT/ Si structure to investigate the TR effect.As the SWCNT⊥Si structure is shown in Fig. 1a, the 0.5 nm region at each end of the Si substrate is selected as the fixed walls, the 0.5 nm region near the fixed walls is set as the heat sink/heat source, and the 0.5 nm region at 1/2 of the SWCNT axial direction is selected as the heat source/ heat sink.In addition, the x and y directions of the simulated system are set as periodic boundaries, and the z direction is set as a fixed boundary condition.As the SWCNT∥Si structure shown in Fig. 1c, the 0.2 nm region at both ends of the axial direction is set as the fixed wall, the Si substrate immediately adjacent to the fixed wall at 0.3 nm is set as the heat sink/heat source, and the 0.3 nm region at the top of the SWCNT radial (y) is selected as the heat source/heat sink.The x-direction is set as a free boundary condition, the y and z direction is set as a fixed boundary condition in the simulation.The direction of heat flow from the SWCNT to the Si along the axial length of the system is referred to as the forward direction, which is denoted as Method 1.The direction of heat flow from the Si to the SWCNT is referred to as the reverse direction, which is denoted as Method 2.
In this work, all Molecular Dynamics calculations were performed using a largescale atomic/molecular parallel simulator LAMMPS [33].The Tersoff potential function is used to describe the C-C interatomic and Si-Si interatomic interactions within the carbon nanotubes in the model, the expressions of which are shown in Eqs. 1 and 2. Therefore, the Tersoff potential function is used to describe the interaction forces of SWCNT, Si in the system, and the required parameters in the equation are derived from Ref. [34].
where i, j, and k are atomic numbers, E is the total energy of the system, and E i is the energy of atom i;V ij is the bond energy between atoms i and j, and rij is the bond length of atoms i and j; f C and f R are the truncation function and interatomic repulsion energy, respectively, f A is the interatomic attraction energy.b ij is the bond order parameter, which is used to adjust the attractive term in the function and it reflects the effect of the local environment of the atom on the strength of the covalent bond.
The interfacial interaction between SWCNT and Si is described by the LJ potential function as shown in Eq. 3.
where r represents the distance between two atoms, ε is the energy that reflects their interaction strength, σ denotes the zero potential distance, and χ is a scaling factor.The standard C-Si Lennard-Jones parameters (χ = 1) are adopted based on the universal force field (UFF) model [35] with ε C-Si = 8.9092 meV, σ C-Si = 3.6286 Å.The cutoff distance of the L-J potential is set as 2.8 σ max .
(1) Before the simulation starts, the conjugate gradient algorithm is used to minimize the energy of the system so that the atomic structure is fully relaxed.The simulation time step was set to 0.25 fs.To tune the simulated system to 300 K temperature and atmospheric pressure conditions, the Nosé-Hoover heat and pressure bath method is used and this state is maintained for 1 × 10 6 steps, then the simulated system is simulated in a Langevin heat bath for 1 × 10 6 steps.The simulated system is transferred to the microcanonical (NVE) ensemble system after reaching equilibrium under the canonical (NVT) ensemble system synthesis and runs continuously for 5 × 10 5 steps.After the simulation reaches equilibrium, the Jund and Jullien algorithm is used to apply the heat flow and run 4 × 10 6 steps, introducing external heat flow perturbations in the process.The kinetic energy of the atoms is changed by applying a certain amount of external energy to the heat source region, i.e., by adding heat flow, and the same energy is withdrawn from the heat sink region at the same moment, i.e., by subtracting heat flow, which is intended to make the whole system energy conserved.It is worth noting that the simulations set the x-direction as a free boundary condition for the SWCNT∥Si structure, acting the Nosé-Hoover pressure bath only in the y and z directions, and limiting the velocity of CNT in the x-direction to 0 to ensure its no-slip on the substrate surface.
For the SWCNT⊥Si structure, G = Q/(AΔT) is used to calculate the ITC, where Q is the heat flow through the interface per unit of time.And A = (r 0 + r vdw ) 2 − r 0 2 is defined as the interface area, SWCNT radius r 0 = 0.69 nm, and r vdw = 0.17 nm as the van der Waals radius of C atoms [36].For the SWCNT∥Si structure, the ITC of the interface is defined as G = Q/(LΔT) using the unit length method, where Q is the heat flow through the interface per unit time, L represents the axial length of the carbon nanotube in the simulation, ΔT is the interface temperature difference.The vibrational density of states (VDOS) obtained by MD takes into account the full phonon scattering behavior and therefore provides a convenient description of the carrier states of the crystal.By performing a fast Fourier transform on the Velocity Autocorrelation Function (VACF) of the atoms is used to obtain the VDOS of the crystal, which is calculated as shown below [37].
where ω is the phonon frequency, �� ⃗  i (0) is the velocity of atom i at the initial moment, �� ⃗  i (t) is the velocity of atom i at moment t.
To obtain more insight into the effect of interface structure and temperature on the thermal transport of SWCNT/Si, we carry out SED [31,32] analysis of phonons in carbon nanotubes which is defined as shown in Eq. 6.The SED calculations ( 4) allow the extraction of dispersion relations and phonon relaxation times, that contain information on interface phonons affected by the substrate and multi-phonon scattering processes.The phonon SED (Φ) is given as: where k is the wave vector, ω is the angular frequency, τ 0 is the integration time, α is the Cartesian index, B is the total number of atoms in a unit cell, b is the atom index in each unit cell, m, and v is the atomic mass and velocity, respectively.N x , N y , N z denote the number of the unit cell in x, y, and z, respectively.n x , n y , n z 1, 2, 3 are the cells in x, y, z directions respectively.Here we consider a one-dimensional Brillouin zone along the length (z).After structure relaxation with Langevin heat bath, we carried NVE MD simulation to the whole system for 1 ns for the suspended SWCNT and supported SWCNT with different temperatures.

Anharmonic Phonon Interactions at Interfaces
The team of Abel Carreras [38] developed a computational code that allows the extraction of phonon quasiparticles in MD trajectories by using this technique to project atomic velocities onto phonon eigenvectors.The technical route of the code is to use MD simulations to analyze the anharmonicity of the crystal as a function of temperature and to fit the power spectrum of the mass-weighted velocity to a model spectral function to calculate the quasiparticle phonon frequencies and linewidths.By employing PHONOPY software [38], we extracted microscopic anharmonic phonon properties from MD simulations using the normal mode decomposition technique and plotted the phonon dispersion curves of single-cell crystalline Si with single-cell (6,6) armchair SWCNT from Γ point to X point.The phonon information such as phonon group velocity, truncation frequency, and density of states can be obtained from the dispersion relation between Si and SWCNT, and the temperature effect of the ITC of multi-order phonons and the ITC contribution of the decomposed phonon modes could be solved by bringing the phonon information into the AIM model [39].
The accurate prediction of interfacial thermal transport must take into account anharmonic phonon coupling with inelastic scattering processes when the thermal boundary is dominated by phonon scattering.The AIM supports the theory that inelastic phonon scattering adds a mechanism for phonon interfacial transport [39].The AIM model improves on the higher harmonic inelastic model (HHIM) model [40] by considering phonon number conservation but calculating the transport probability by different fluxes since all inelastic scattering is taken into account.The AIM model considering the interaction of phonons in a specific frequency range compared with other models makes it possible to distinguish the contribution of multiple orders of phonons to the interfacial thermal conductance, and we can realize a visual analysis of the intrinsic mechanism of the interfacial thermal transport from the phonon perspective by comparing the interfacial thermal conductance of different orders of phonons.The ( 6) 136 Page 8 of 22 system is assumed to be isotropic in this model, and the phonon dispersion relations and kinetic parameters are shown in Eqs.7-10 [39].
where a is the lattice spacing, v(ω) is the phononic group velocity, D(ω) is the phononic density of states.The the n-phonon transmission coefficient (ξ (n) ) and the thermal boundary conductance (h (n) k ) of the n-phonon inelastic scattering are defined as follows [39]: where ℏ is the reduced Planck's constant, ω is phonon frequency, ω c is the maximum phonon frequency in Material 1, j is the phonon mode (vertical or horizontal), D(ω) is the volumetric phonon density of states per unit frequency, f(ω) is the phonon distribution function, v(ω) is the group velocity of phonons with frequency ω.For the convenience of the following discussion, the subscript "1" refers to the Si substrate, and "2" refers to SWCNT. (n)

Effect of Substrate Position on Interfacial Thermal Transport
To obtain accurate results of the interfaces temperature difference, the temperature and temperature difference of SWCNT and Si near the interface region with simulation time are calculated separately for both contact methods, as shown in Fig. 3a and  b.The temperature and temperature difference between the SWCNT and Si nearinterface regions stabilize after 0.6 ns and 0.5 ns in the vertical contact and parallel states, respectively, which also indicates that the simulated system gradually reaches the steady state, so the temperature difference between these two moments is selected as the final interface temperature difference, respectively.For T = 300 K, the calculated ITC for SWCNT∥Si is 0.014 W•mK −1 (diameter normalized result) versus 9.46 MW•m 2 K −1 .This result coincides with the literature values [41,42].
Our results are lower than the ITC of horizontal CNT/SiO 2 (about 0.1 W•mK −1 )) [43], and the strength of interfacial interactions of C-C is weaker than those of Si-C and O-C.The differences in model diameter and size as well as in the interatomic potentials chosen in the simulations can contribute to this discrepancy.Meanwhile, in order to verify the harmony and accuracy of the simulation session setup in this work, the dependence of ITC on length for (6, 6) SWCNT was simulated and compared with the literature values, as shown in Fig. 2 below [41].Although the literature demonstrates the dependence of ITC on axial length for (10, 10) SWCNT, the literature also calculates the ITC for (6, 6) SWCNT and states that ITC converges after a positive correlation with diameter size.In comparison to our data, the ITC of (6,6) SWCNT is lower than (10,10), and the dependence of ITC on axial length shows a tendency to increase and then stabilize.These data comparisons and analysis provide support for the reliability of our data.
The ITC is redefined as G K = Q/(N B △T) to quantify the comparison of the ITC under the two contact modes of SWCNT⊥Si and SWCNT∥Si, where Q is the heat flow through the interface per unit time; N B is the number of C-Si bonds at the interface; △T is the temperature difference at the interface.According to the above equation, the ITC of SWCNT⊥Si is 662.90 ± 0.10 GW•1 K −1 ; the ITC of SWCNT∥Si is 480.61 ± 0.13 GW•1 K −1 ) when simulated at the system temperature of 300 K.There are differences in the ITC between the two approaches, but the order of magnitude is the same.
VDOS at the interface of crystalline materials could reflect the thermal transport properties between SWCNT and Si substrate.For the interface structures formed under the two contact modes, the phonon density of states of C and Si atoms at the same position at the interface is selected for comparison, as shown by observing Fig. 3c, the SWCNT phonon density of states cutoff frequency is about 80 THz and the Si density of states cutoff frequency is about 20 THz.For C atoms, more low-frequency phonon modes are involved in thermal transport in the low-frequency region (< 20 THz) for the SWCNT⊥Si than for the SWCNT∥Si approach.The VDOS in the SWCNT∥Si model has peaked at 50 THz, which shows a shift to the higher frequency band compared to the peaks in SWCNT⊥Si.The peak of the Si atom VDOS in the SWCNT⊥Si mode is higher in the low-frequency region than in the parallel placement mode, and more low-frequency phonons are involved in the thermal transport at the substrate interface.The VDOS peak of Si atoms is one order of magnitude higher than that of SWCNT.The thermal transport at the interface between SWCNT and Si is dominated by low-frequency phonon modes, and more low-frequency phonon modes are excited when placed vertically, resulting in a better matching of the atomic density of states between Si and SWCNT, which is more conducive to the interface thermal transport.
Figure 4a shows the SED spectra of the suspended SWCNT at 300 K, 600 K, and 900 K.Although all phonon dispersion relations of the suspended SWCNT remain constant and the linewidth of the SED broadens with increasing temperature, the main peak frequency remains essentially identical.This is mainly because the increase in temperature enhances the anharmonic scattering of phonons inside SWCNT, reflecting the typical temperature effect.Figure 4b demonstrates the SED of the SWCNT supported by the Si substrate, and the phonon peak is significantly wider in the supported condition compared to the suspended condition, which is important, especially in the acoustic mode.This suggests that both the interaction with the substrate and the increase in temperature enhance the interface phonon scattering and thus shorten the phonon lifetime.The SEDs of SWCNT with substrates show a new dispersionless phonon state phonon mode in comparison with the dispersion curves of suspended SWCNT [44], which becomes increasingly obvious as the temperature increases.While anharmonic channels and inelastic phonon scattering significantly affect the interface phonon modes at higher temperatures, anharmonic interactions allow additional thermal transport channels to open up when harmonic interactions limit the accumulation to the cutoff frequency of the thermal interface material, thus allowing phonon modes with different frequencies to pass through.The phonon modes of these dispersionless phonon states are assumed to be the result of vibrations around the SWCNT/Si interface.In addition, the degree of SED dispersion is enhanced for SWCNT with the substrate at the same temperature, which indicates stronger coupling between SWCNT and substrate and higher anharmonic interactions within the interface and SWCNT caused by phonon scattering.The strong scattering will inevitably reduce the average free range and phonon lifetime of phonons, and the difference in phonon dispersion curves due to the varying strength of interaction between the substrate and SWCNT is the fundamental reason for the reduced interface thermal transfer capability.
The increase in temperature for the dispersion profile of SWCNT covered with substrate does not change the structure of the profile and the frequency of phonons but only increases the rate of phonon scattering internally.Due to the suppression of the interfacial vibrations of the SWCNT by the substrate, the acoustic mode of the Si-covered SWCNT is displaced at the long wavelength limit with a displacement of about 1 THz, which is most pronounced at a temperature of 900 K.The interface phonon modes disturb the phonon dispersion relationship of SWCNT, and the displacement of the dispersion spectrum further reduces the group velocity of phonon modes and decreases the intensity of interface thermal transport [45].Comparing the SED spectra of parallel and perpendicularly placed SWCNTs at the same temperature, it is observed that the line width of the SED of parallel SWCNTs is narrower and the dispersion is enhanced compared with the perpendicular contact of SWCNTs with the substrate, both of which indicate that the structure of SWCNT∥Si is disturbed by more interface phonon modes, resulting in a lower phonon lifetime.Figure 5 shows the corresponding phonon SED profiles given along the wave vector k = 0 in order to have a better illustration of the difference in phonon information due to temperature and substrate.The position of the peak indicates the eigenfrequency of the phonon, and the half-width at the half-maximum (FWHW) is proportional to the phonon lifetime.
Observing Fig. 5a, the peak of phonon is significantly wider in the case of SWCNT supported by the substrate compared with the suspended case, and there is a large number of additional small peaks in the phonon vibration and a significant blue shift in the phonon peak.The appearance and increase of additional peaks are a result of the coupling of phonons within the interface with the strong scattering effect of interfacial phonons creating a stronger constraint on the axial motion of SWCNT, and the new phonon vibration mode provides an additional channel for interface thermal transfer.As the temperature increases, the enhancement of the anharmonic phonon scattering process leads to an increase in the number of additional phonon peaks and a decrease in the number of interfacial transport phonon modes.Therefore, we analyze the mechanism of TR in terms of the material thermal transfer characteristics of the nonlinear interface structure, which is formed by the dual material interface structure and causes the TR effect due to the difference in Fig. 5 (a) SED profiles of SWCNT suspended and supported by the Si substrate system at different temperatures at k = 0, (b) SED profiles of 0-4 THz for SWCNT suspended and supported by the Si substrate at 300 K, (c) SED profiles of SWCNT supported by substrate at 300 K, 600 K and 900 K their respective heat transfer capabilities.From Fig. 5b, it is observed that the ultralow frequency (< 1 THz) phonons are affected by the interfacial phonon scattering resulting in a significant blueshift of the phonon peaks.Moreover, the comparison of the phonon peaks of the two interfacial structures shows that the phonon peaks of the SWCNT supported vertically by the substrate are more blueshifted than those supported parallel to the substrate.And the half-width heights of the vibrational and phonon peaks of the phonon modes in perpendicular contact with the substrate are higher than those of the parallel contact condition at the same temperature.All these behaviors indicate that there is more interference of interfacial phonon modes without dispersive states in the SWCNT∥Si structure, which leads to a lower phonon lifetime.Figure 5c compares the temperature effect of SED for both contact modes of SWCNT.When the system temperature increases, the lattice vibration is enhanced and the anharmonic phonon scattering at the SWCNT/Si interface is enhanced, which results in a decrease in the average free range of phonons.Simultaneously, the temperature dependence of SWCNT and Si is inconsistent, leading to a weakening of the TR effect with increasing temperature.

The Thermal Rectification Effect
The thermal rectification effect caused by the change in the direction of heat flow is determined by the difference in the thermal transport capacity of the materials on either side of the composition of the interface structure, and the difference in TR capacity depends on the magnitude of the thermal resistance in both directions.Thus the TR effect can be defined by the interfacial thermal conductance: γ = (G C→Si − G Si→C )/G Si→C × 100%, where γ is the TR coefficient, which indicates the intensity of the TR effect, G C→Si is the ITC when the forward heat flow (Method1) is applied; G Si→C is the ITC when the reverse heat flow (Method2) is applied.The temperature changes along the axial direction of the SWCNT⊥Si structure by applying forward and reverse heat flow are shown in Fig. 6, it can be observed that there is a significant temperature difference at the interface between SWCNT and the Si substrate on both sides, and the temperature difference at the interface is about 165 K in the case of Method1 and 130 K in the case of Method2.Substituting into the above equation, After the system was brought into equilibrium, contour clouds of the heat flow distribution in the X-Y cross-section of the Si substrate when forward and reverse heat flow have been applied were plotted, as shown in Fig. 6c and d.Compared with the reverse heat flow, when the forward heat flow is applied to the system, i.e., the heat flow direction is from SWCNT to Si, the heat flow distribution at the interface between SWCNT and Si matrix is more regular, and each equal heat flow region is axisymmetric concerning the carbon nanotube diameter, especially the high heat flow region is symmetrically distributed at the carbon ring interface where the carbon nanotubes are in contact with the matrix.The regularized distribution of heat flow regions weakens the resistance effect of the interface during heat transport, resulting in a well-defined channel for thermal transport.This also confirms from a macroscopic point of view that the interface structure formed by dual materials 136 Page 14 of 22 induces TR effects due to the disparity of their respective thermal transport capacities, as well as the temperature dependence of the materials.
Applying heat flow in different directions to the nonlinear interface structure causes significant differences in the ITC and induces TR effects.As seen in Fig. 7a, the TR coefficient decreases significantly as the system temperature increases, a trend that is consistent with the temperature-dependent characteristics of other thermal rectifiers.The scale or temperature dependence of the thermal conductance of the interface material is the most likely cause of the TR effect.As the temperature increases, anharmonic effects are evident as the thermal conductance contributed by the anharmonic phonons at the interface gradually increases.The interface phonon scattering channels have been increased, but the phonon transport process in the substrate Si is hindered, so the thermal rectification effect continues to decrease with increasing temperature.
The phonon serves as an energy carrier in crystalline materials, while the confinement of nanomaterials on the transverse scale directly affects the thermal properties of the materials.Therefore, the VDOS of SWCNT and Si at the interface are calculated and compared under the applied Model1 and Model2 heat flow at temperatures of 300 K and 900 K, as shown in Fig. 7b.Comparing the morning VDOS match between SWCNT and Si substrate under forward and reverse heat flow, it is obvious that the VDOS match is large when the heat flow direction is C → Si.The peak VDOS of SWCNT and Si in Model1 and Model2 conditions at a temperature of 300 K are matched, but the VDOS match appears significantly different as the temperature increases.At 900 K, the VDOS peak at 15 THz and 16 THz substrates in the Model2 condition are both higher compared to Model1, and the VDOS peak in the low frequency region (5-6 THz) is shifted to the right.When the reverse heat flow is applied, the mid-high frequency phonon modes (11-15 THz) of the substrate are reduced and the interface phonon modes are suppressed.The VDOS of SWCNT shows a significant peak near 14 THz, indicating that the high temperature makes the excitation of more low and middle-frequency phonons at the SWCNT interface, which also indicates that the difference in the dependence of phonon vibration characteristics on temperature causes TR effects.The increase in temperature enhances the phonon inelastic scattering process near the interface while weakening the phonon elastic scattering and gradually dominating the interfacial thermal transport.The phonon inelastic scattering process masks the asymmetry drawback of the material structure to a certain extent, which reduces the intensity of the TR effect at the SWCNT-Si interface.
Subsequently, we have compared and analyzed the intensity of the interface TR effect of SWCNT⊥Si and SWCNT∥Si.It is calculated that the TR coefficient of SWCNT∥Si is about 59%, which far exceeds the intensity of the TR effect of the SWCNT⊥Si structure.Fig. 8a and b show the VDOS of SWCNT and Si atoms at different positions, and for the convenience of description, the VDOS at the interface is denoted as EDOS, and the VDOS of the non-interface atoms is denoted as IDOS.The results show that the frequencies corresponding to the peaks of VDOS for the SWCNT∥Si structure are the same for both SWCNT and Si atoms.Observing the VDOS of Si in Fig. 8a, it can be seen that the vibration of phonons at low frequencies (0-5 THz) at the interface of SWCNT⊥Si is higher than the phonons away from the interface, but the peak of VDOS at 15 THz is much lower than that of IDOS.In addition, the forward and reverse heat flow interfere little with the phonon modes, while the vibrations of EDOS and IDOS are identical.The EDOS is significantly lower than the IDOS when reverse heat flow is applied to the SWCNT∥Si structure, while Model2's EDOS is higher than Model1's EDOS.Setting the Si substrate end to a heat source will affect the vibration amplitude of Si atoms to a greater   8b shows the VDOS of SWCNT for both contact modes.The VDOS of SWCNT⊥Si can be observed that the heat flow direction strongly interferes with the intensity of phonon vibrations in the low to mid-frequency region (<15 THz), and the VDOS under reverse heat flow is significantly higher than that under forward heat flow.It is shown by observing the VDOS of the SWCNT∥Si structure that the peak of VDOS in the low-frequency region (<10 THz) shifts to the right when the reverse heat flow is applied, as well as the peak is much lower than that of the SWCNT⊥Si structure.It indicates that the atoms at the interface are laterally constrained by other particles when the SWCNT is placed flat on the Si substrate, which results in the decrease of EDOS.From 40 to 60 THz in the high-frequency region, the frequencies corresponding to the peaks of EDOS and IDOS are the same, which indicates that a large number of high-frequency phonons are excited at the interface and the phonon mode develops toward the high-frequency mode.
The SWCNT⊥Si and SWCNT∥Si structures both have TR effects owing to the asymmetry of the structures.The overlap of the phonon density of states of Si and C atoms is higher than that of the forward heat flow when the reverse heat flow is applied, which makes the SWCNT∥Si TR effect more obvious.Then we simulated the TR effect of SWCNT⊥Si at heat flow densities of 5 W•m −1 and 20 W•m −1 , and Fig. 9c shows the VDOS of SWCNT and Si.The differences in phonon vibration between the forward and reverse heat flow conditions are small when the heat flow density is 5 W•m −1 , but the presence of VDOS at different heat flow directions is significantly evident when the heat flow density increases to 20 W•m −1 .It is observed that the VDOS of Si atoms has a much higher peak around 30 THz than the forward heat flow when the reverse heat flow is applied, but the VDOS decreases in the low-frequency region.In addition, the cutoff frequency of Si atoms elongates from about 20 THz to 35 THz as the heat flow density increases, the low-frequency phonon peak decreases, and the peak shifts to the right.The energy of the substrate increases when the heat flow direction is directed from Si to SWCNT, which causes the vibration of low-frequency phonons to be enhanced and a large number of lowfrequency phonons are converted to mid-high frequency phonons.The reverse heat flow reduces the high frequency phonon mode (> 20 THz) in the atoms compared to the forward heat flow, while the low-frequency phonon mode is enhanced, reducing the degree of acoustic phonon mismatch between SWCNT and Si atoms and enhancing the interface thermal transport.As the heat flow density increases, the lattice vibrations at the interface between the SWCNT and the Si substrate to which the reverse heat flow is applied are better matched; the excitation of more low-frequency phonon modes at large heat flow densities is the main reason for the enhanced TR effect when forward heat flow is applied.

Multiple-Order Phonon Scattering
Figure 9a shows the dispersion spectrum of Si.The two branches near the point of Γ are acoustic modes, while one is a longitudinal (LA) mode and one is a double-simplified transverse (TA) mode.Figure 9b shows the dispersion relation of SWCNT with the same longitudinal and transverse wave eigenfrequencies in the Brillouin 136 Page 18 of 22 zone at the Γ point.The first four phonon branches around the Γ point is acoustic modes, including one rotational (TW) mode, one double-simplified transverse (TA) mode, and one longitudinal (LA) mode, which is consistent with the results of Dubay [46] et al.We predicted the temperature dependence of the ITC for the 2nd, 3rd, and 4th order phonons of the SWCNT⊥Si interface structure by using the AIM model, as shown in Fig. 9c.The contribution of 2nd, 3rd, and 4th order phonons to the ITC increases monotonically with increasing temperature.The growth rate of hk for 2nd, 3rd, and 4th order phonons decreases as the temperature increases beyond 300 K, and the growth rate of h k for 2nd order phonons is the slowest.In addition, the scattering of 2nd-order phonons dominates the interfacial thermal transport of SWCNT/Si by about twice the value of 3rd-order phonons and four times the value of 4th-order phonons.The ITC at 300 K obtained from NEMD simulations is 1209.52MW•m −2 •K −1 , and the ITC predicted by AIM is 404.03MW•m −2 •K −1 .The AIM model does not incorporate the effects of phonon scattering and quantum effects, and the vibrational spectra of atoms on both sides of the SWCNT-Si structure interface are highly mismatched, which predicts the interfacial thermal conductivity differs from the actual value.the MD simulation considers the elastic and inelastic scattering processes of phonons, and the default material is the ideal material.Therefore, the MD simulations have higher values of the ITC than the results of the AIM model.The phonons at the interface between SWCNT and Si carry higher energy as the temperature increases, thus allowing higher frequency phonon modes to be excited and the number of coupled phonon modes at the interface to increase, and increasing the interface phonon transmission rate.Meanwhile, there are large numbers of 4th-order phonon modes depleted, which causes the transmission coefficient to converge to a constant value and the ITC to be stable.In addition, the contribution of 2nd and 3rd-order phonon scattering to the interfacial thermal transport is higher than that of 4th-order phonons.Because the AIM model considers all inelastic scattering processes, including higher harmonics and non-harmonics, the predicted inelastic processes can have a significant impact on the overall h k .As the temperature increases, the phonon modulus of the substrate Si increases at any frequency, so the phonon transmission increases, and the elastic and inelastic anharmonic phonon modes increase.Eventually, as the temperature continues to increase, the phonons incident to the interface in SWCNT are gradually consumed, the phonon transmission coefficient stabilizes, and the multi-order phonon contribution to the thermal conductance of the interface converges.
The mode decomposition of acoustic phonons was performed for ITC with multiple orders of phonons in order to analyze the respective contributions of different modes of phonons to the interface thermal transport, and Fig. 9d shows the temperature dependence of the ITC of the phonon branches of LA and TA.Observing the change of LA acoustic branching shows that the ITC of multi-order phonons is enhanced with increasing temperature; the contribution of 2nd-order phonons is large until 450 K, and the 3rd-order phonons inversely overtake the 2nd-order phonons as the most contributing phonon mode after 450 K. Analysis of the temperature dependence of the TA acoustic branch reveals that the 2nd-order phonon contributes least to the ITC at high temperatures, while the 3rd-order phonon plays a dominant role in the thermal transport process, followed by the 4th-order phonon.With the continued increase in temperature, the ITC arising from the 2nd and 3rd-order phonon scattering of the LA phonon branch tends to stabilize.By comparing the variations of LA and TA acoustic branches, it is shown that the ITC of the 2nd order phonon modes of the LA acoustic branch is greater than the 2nd order action of the TA phonon branch.However, the 3rd-order phonons of the TA acoustic branch have a very different trend, with a consistently higher effect than the 3rd-order phonons of the LA acoustic branch, although the difference between them is minor.The effect of fourth order phonons of LA and TA phonon branches on the interfacial thermal transport is not much different until 350 K, and then, the effect of the LA phonon branch gradually catches up with the TA phonon branch as the temperature continues to increase.The thermal conductance of SWCNT and Si substrate has significant quantum characteristics at lower temperatures, and the quantum transport of phonons mainly relies on the phonon branch, thus the difference of ITC is more obvious for lower-order phonons.While the contribution of the optical branch to the thermal conductance gradually increases with the continued increase of temperature, weakening the contribution of the phonon branch, which makes the LA phonon branch's thermal conductance rate slower and the value of the ITC gradually stabilizes.

Conclusions
In summary, the effects of substrate placement and temperature on the thermal transport and TR effects at the SWCNT/Si interface were investigated by NEMD method simulations, and the ITC of multi-order phonons was calculated by the AIM method.The results show that both the effect of the substrate and the increase in temperature enhances the nonharmonic interfacial phonons.the SWCNT⊥Si mode excites more low-frequency phonon modes than the SWCNT∥Si mode, and the line widths of the SEDs of the SWCNTs placed in parallel become narrower and the degree of dispersion is enhanced.Such phenomena indicate that the SWCNT∥Si structure is disturbed by more interfacial phonon modes, resulting in a lower phonon lifetime.The acoustic modes of Si-supported SWCNT undergo a shift of about 1 THz in the long wavelength limit and a large number of additional peaks in the phonon peaks, indicating that the interfacial phonon modes interfere with the phonon dispersion relationship of SWCNT, and the shift of the dispersion spectrum further reduces the group velocity of the phonon modes and decreases the interfacial thermal transport intensity.Anharmonic phonon channels and inelastic phonon scattering significantly affect the interface phonon modes at higher temperatures, and anharmonic interactions can open additional thermal transport channels, leading to an increased number of additional phonon peaks.Due to the nonlinearity of the structure, both SWCNT⊥Si and SWCNT∥Si structures have TR effects.When the reverse heat flow is applied, the overlap of the phonon density of states of Si and C atoms is higher than that of the forward heat flow, making the SWCNT∥Si TR effect more obvious.The TR effect has an obvious temperature effect, and the TR coefficient decreases to 6% as the temperature rises from 300 K to 1500 K.There is a significant red shift of the phonon peak with increasing temperature and the half-width height of the 136 Page 20 of 22 phonon peak becomes narrower.The temperature increase enhances the nonharmonic scattering and weakens the nonlinear characteristics of the heterostructure structure, which leads to the weakening of the TR effect.(b) With increasing heat flow density, the lattice vibrations between the carbon nanotubes and the silicon substrate are better matched when the reverse heat flow is applied.The excitation of more low-frequency phonon modes at large heat flow densities when forward heat flow is applied is the main reason for the enhanced TR effect.In addition, by decomposing the interfacial thermal conductance of 2nd, 3rd, and 4th-order phonons, it can be seen that 2nd-order phonons play a dominant role in the interfacial thermal transport process.At lower temperatures, the quantum transport of SWCNT and Si phonons mainly relies on the acoustic branch for their ITC differences are more obvious.As the temperature continues to increase, the contribution of the optical branch to the TC gradually increases, weakening the contribution of the acoustic branch, making the LA phonon branch's thermal transport rate slower and the ITC gradually stable.In this work, the mechanism of hightemperature nonharmonic phonons influencing TR effects is discussed, and the related results and analysis provide theoretical insights into the management of thermal interface transport that will be based on low-dimensional nanomaterials.
, 6) SWCNT are l a = 11.4830Å, l b = 11.4830Å, l c = 2.4595 Å.The lattice constant of the single crystal cell Si are l a = l b = l c = 5.4307 Å.

Fig. 1
Fig. 1 (a) 3D main view and (b) top view of the SWCNT⊥Si physical model, and (c) main top view and (d) left view of the SWCNT∥Si physical model.The arrow of the same color in the figure indicates the direction of the applied heat flow

Fig. 2 Fig. 3
Fig. 2 Dependence of ITC on length at temperature 300 K

Fig. 4
Fig. 4 (a) SED spectra of the suspended SWCNT at 300 K, 600 K, and 900 K, (b) SED of SWCNT supported vertically and parallel to the substrate at different temperatures

Fig. 6
Fig. 6 Temperature distribution of SWCNT⊥Si along the axial direction with (a) forward heat flow applied and (b) reverse heat flow applied.Contour cloud map of heat flow distribution of X-Y section with (c) the forward heat flow and (d) the reverse heat flow

Fig. 7
Fig. 7 (a) Temperature dependence of TR effect, (b) VDOS of SWCNT and Si on the interface under at 300 K, 900 K under different heat flow directions

Fig. 8
Fig. 8 VDOS of (a) Si and (b) SWCNT under two contact modes.(c) VDOS of SWCNT and Si of SWCNT⊥Si interface under forward and reverse heat flow

Fig. 9
Fig. 9 The phonon dispersion spectrum of (a) Si and (b) (6, 6) SWCNT.(c) AIM predictions of the interfacial thermal conductance as a function of temperature for SWCNT⊥Si for n = 2-4.(d) Temperature effect of ITC for different acoustic phonon modes.The solid line is LA acoustic branch, and the dotted line is TA acoustic branch