# Thermal Transport Study in a Strained Carbon Nanotube and Graphene Junction Using Phonon Wavepacket Analysis

## Abstract

**:**

## 1. Introduction

## 2. Simulation Methods

_{jα}(k) is the eigenvector for the corresponding wave vector. r

_{j,l}is the position vector for jth atom in the lth unit cell. For a monatomic crystalline structure with a unidirectional wave vector such as SWCNT, the dynamical matrix is expressed as

_{jlα,j’l’β}is obtained by measuring the α direction force on the jth atom in the lth unit cell when the j’th atom in the l’th unit cell is displaced into β direction by a small distance (small enough not to cause any anharmonicity). To obtain force constants with better accuracy, the crystalline structure is well-equilibrated and energy-minimized at 0 K. Force constants are measured to build the dynamical matrix after displacing basis atoms one by one in the selected reference unit cell into x, y, z directions. Then, eigenvalues and eigenvectors are induced from the constructed dynamical matrix. Using the information in the calculated phonon dispersion relations, initial displacements and velocities of atoms are determined to form a wavepacket. For a monatomic crystalline structure such as SWCNT, the α direction displacement of the jth atom in the lth unit cell is expressed as

_{l}denotes the position of the lth unit cell and z

_{0}denotes the position of the reference unit cell. The amplitude Q is replaced by a normal distribution function with the selected mean wavenumber k

_{0}and a standard deviation σ.

## 3. Results and Discussions

^{2}/sp

^{3}bonds and a 50 nm pillar height and 15 nm inter-pillar distance was 3.1 × 10

^{−10}K-m

^{2}/W. The thermal resistance of SWCNT–graphene with pure sp

^{2}bonds, with a 200 nm pillar height and 5 nm interpillar distance, was reported to be 1.56 × 10

^{−10}K-m

^{2}/W. Phonons passing through a SWCNT–graphene junction must alter their momentum direction to flow over the graphene sheet, which is expected to result in substantial phonon scattering. Additionally, Park et al. [42] calculated the phonon density of states in a SWCNT–graphene junction and found that the frequency and density of populated phonons undergo significant changes when they travel from a SWCNT to a graphene floor. Efficient phonon modes such as LA phonons in a SWCNT must be transformed into significant phonon modes in graphene, such as in-plane phonons, and then reconverted into other significant phonons in a SWCNT. This phonon conversion process is expected to result in a significant amount of energy reflection.

## 4. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The schematic of phonon wavepacket analysis. Interplanar spacing between (6,6) SWCNT–graphene junctions is fixed to be 15 nm. (

**b**) Junction structures with undeformed and deformed junctions and a 5–7 defect in the junction are illustrated. The junction structure is deformed uniaxially into z direction until the distance between the two SWCNT–graphene junctions reaches 3 nm.

**Figure 2.**Interfacial thermal resistance estimation for a SWCNT–graphene junction structure under mechanical deformation using RNEMD.

**Figure 3.**(

**a**) Phonon dispersion relations in 1000 unit cells of (6,6) SWCNT. Five important phonon modes, i.e., LA, TA, TW, RB, FO modes are selected in this research. q denotes wave number and a denotes lattice constant. (

**b**) Exaggerated illustration of LA, TA, TW, RB, FO vibrational modes.

**Figure 4.**Percent energy transmission through a SWCNT–graphene junction in deformed/undeformed junction structures. Group velocity (dotted dark gray line) is also plotted together. (

**a**) LA phonon mode. (

**b**) TA phonon mode. (

**c**) TW phonon mode. (

**d**) RB phonon mode. (

**e**) FO phonon mode. (

**f**) Illustration of deformed and undeformed junctions.

**Figure 5.**Interfacial thermal resistance as a function of elongation in SWCNT–graphene junction structure at different temperatures. The SWCNT pillar height is 200 nm and the graphene floor size between junctions is 15 nm.

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**MDPI and ACS Style**

Park, J.
Thermal Transport Study in a Strained Carbon Nanotube and Graphene Junction Using Phonon Wavepacket Analysis. *C* **2023**, *9*, 21.
https://doi.org/10.3390/c9010021

**AMA Style**

Park J.
Thermal Transport Study in a Strained Carbon Nanotube and Graphene Junction Using Phonon Wavepacket Analysis. *C*. 2023; 9(1):21.
https://doi.org/10.3390/c9010021

**Chicago/Turabian Style**

Park, Jungkyu.
2023. "Thermal Transport Study in a Strained Carbon Nanotube and Graphene Junction Using Phonon Wavepacket Analysis" *C* 9, no. 1: 21.
https://doi.org/10.3390/c9010021