# 单原子层薄膜热传导性质的晶格动力学研究(II)——数值计算与结果分析Lattice Dynamics Study on the Thermal Conduction Properties of Single Atomic Layer Films (II)—Numerical Calculations and Analysis on the Results

DOI: 10.12677/MP.2020.106017, PDF, HTML, XML, 下载: 84  浏览: 202

Abstract: Based on the formulas derived previously by the authors, the variation of the phonon line width, phonon free path, the contribution of phonon to thermal conductivity with its wave vector are calculated. The results show that the phonon with a shorter wave vector, or with a lower frequency, has a longer free path and a greater contribution to the thermal conductivity. When the size of the film increases, the more phonon free path distribution is concentrated in the low wave vector region, the phonon free path in the low wave vector region is larger, and the contribution of the short wave vector phonon, that is the low frequency phonon, to the thermal conductivity of the film is bigger. The larger the size of the film, the larger the thermal conductivity, and when the size N exceeds 40 lattice constants, there is a linear relationship between the thermal conductivity and logN. When the size tends to infinity, the thermal conductivity tends to diverge, so single atomic layer film with large size has excellent heat dissipation performance.

1. 引言

${\omega }_{k}^{2}=\frac{2k}{m}\left(2-\mathrm{cos}{k}_{x}a-\mathrm{cos}{k}_{y}a\right)$ (1)

${v}_{k}^{x}=\frac{ka}{m{\omega }_{k}}\mathrm{sin}{k}_{x}a$ (2)

${\Gamma }_{k}=\frac{72}{{\hslash }^{2}}\underset{{k}_{1}{k}_{2}}{\sum }\underset{±}{\sum }\left[\left({\stackrel{¯}{n}}_{{k}_{2}}+\frac{1}{2}\right)±\left({\stackrel{¯}{n}}_{{k}_{1}}+\frac{1}{2}\right)\right]\frac{\omega \left({\omega }_{{k}_{1}}±{\omega }_{{k}_{2}}\right)\left({\Gamma }_{{k}_{1}}+{\Gamma }_{{k}_{2}}\right){|V\left({k}_{1};{k}_{2};-k\right)|}^{2}}{{\left[{\omega }^{2}-\left({\omega }_{{k}_{1}}±{\omega }_{{k}_{2}}\right)\right]}^{2}+{\left[2\omega \left({\Gamma }_{{k}_{1}}+{\Gamma }_{{k}_{2}}\right)\right]}^{2}}$ (3)

${\kappa }_{x}=\frac{3{k}_{B}{\beta }^{2}}{V}\underset{k}{\sum }{\hslash }^{2}{\omega }_{k}^{2}{L}_{k}^{x}{v}_{k}^{x}{\stackrel{¯}{n}}_{k}\left({\stackrel{¯}{n}}_{k}+1\right)$ (4)

2. 声子色散关系与群速度

(a) N = 40(b) N = 80

Figure 1. The phonon’s frequency vs. its wave vector

(a) N = 40(b) N = 80

Figure 2. The phonon’s group velocity vs. its wave vector

3. 声子寿命与平均自由程

(a) N = 40(b) N = 80

Figure 3. The phonon’s linewidth vs. its wave vector at T = 100 K

(a) N = 40(b) N = 80

Figure 4. The phonon’s freepath vs. its wave vector at T = 100 K

(a) N = 40(b) N = 80

Figure 5. The Convergence of phonon’s linewidth at T = 100 K

4. 单原子层薄膜热传导及尺寸效应

(a) N = 40(b) N = 80

Figure 6. The phonon’s thermal conductivity vs. its wave vector at T = 100 K

(a) N = 40(b) N = 80

Figure 7. The Convergence of phonon’s thermal conductivity at T = 100 K

Figure 8. The thermal conductivity vs. size of single atomic layer film at T = 100 K

5. 总结与讨论

 [1] Zhu, L. and Li, B. (2014) Low Thermal Conductivity in Ultrathin Carbon Nanotube. Scientific Reports, 4, Article ID: 4917. https://doi.org/10.1038/srep04917 [2] Tang, Q. (2004) A Molecular Dynamics Simulation: The Effect of Finite Size on the Thermal Conductivity in a Single Crystal Silicon. Molecular Physics, 102, 1959-1964. https://doi.org/10.1080/00268970412331292777 [3] Cruz, A., Termentzidis, K., Chantrenne, P., Kleber, X., et al. (2011) Molecular Dynamics Simulations for the Prediction of Thermal Conductivity of Bulk Silicon and Silicon Nan-owires: Influence of Interatomic Potentials and Boundary Conditions. Journal of Applied Physics, 110, Article ID: 034309. https://doi.org/10.1063/1.3615826 [4] Xu, X., Pereira, L.F.C., Wang, Y., et al. (2014) Length-Dependent Thermal Conductivity in Suspended Single-Layer Graphene. Nature Communications, 5, Article ID: 3689. https://doi.org/10.1038/ncomms4689 [5] 黄建平, 贺业鹏. 单原子层薄膜热传导性质的晶格动力学研究(I)-声子线宽和热传导系数公式[J]. 现代物理, 2020, 10(6): 140-145. [6] 黄建平, 唐婧. 分子机器中链状分子热传导性质的理论研究[J]. 现代物理, 2017, 7(6): 227-234. https://doi.org/10.12677/mp.2017.76026