# 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Lattice Dynamics Study on the Thermal Conduction Properties of Single Atomic Layer Films (I)—Formulas for Phonon Linewidth and Thermal Conductivity

DOI: 10.12677/MP.2020.106016, PDF, HTML, XML, 下载: 83  浏览: 184

Abstract: The formulas for lattice vibration frequency, atomic displacement and momentum, lattice vibration energy, an harmonic potential energy and energy flux of lattice vibration in single atomic layer film are derived in this paper on the basis of the lattice dynamics theory, and then the formulas for phonon line width and thermal conductivity are derived with the aid of Green function theory and Green-Kubo formula. The result shows that the thermal conductivity of the film is the sum of contribution from every single phonon which is closely related to phonon’s velocity, energy and lifetime or free path.

1. 引言

2. 单原子层薄膜的晶格动力学理论

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

$H=\underset{k\sigma }{\sum }\left({a}_{k\sigma }^{+}{a}_{k\sigma }+\frac{1}{2}\right)\hslash {\omega }_{k}$ (2)

${u}_{\sigma }\left(l\right)=\frac{1}{N}\underset{k\sigma }{\sum }\sqrt{\frac{\hslash }{2m{\omega }_{k}}}{A}_{k\sigma }{\text{e}}^{ia\stackrel{\to }{k}\cdot \stackrel{\to }{l}}$ (3)

${p}_{\sigma }\left(l\right)=-\frac{i}{N}\underset{k\sigma }{\sum }\sqrt{\frac{\hslash m{\omega }_{k}}{2}}{B}_{k\sigma }{\text{e}}^{ia\stackrel{\to }{k}\cdot \stackrel{\to }{l}}$ (4)

${{H}^{\prime }}_{\sigma }=\frac{\delta }{6}\underset{{l}_{x},{l}_{y}}{\sum }\left\{{\left[{u}_{\sigma }\left({l}_{x},{l}_{y}\right)-{u}_{\sigma }\left({l}_{x}-1,{l}_{y}\right)\right]}^{3}+{\left[{u}_{\sigma }\left({l}_{x},{l}_{y}+1\right)-{u}_{\sigma }\left({l}_{x},{l}_{y}\right)\right]}^{3}\right\}$ (5)

${{H}^{\prime }}_{\sigma }=\underset{k{k}^{\prime }{k}^{″}}{\sum }\text{ }\text{ }V\left(k,{k}^{\prime },{k}^{″}\right){A}_{\sigma }\left(k\right){A}_{\sigma }\left({k}^{\prime }\right){A}_{\sigma }\left({k}^{″}\right)$ (6)

$V\left(k,{k}^{\prime },{k}^{″}\right)=\frac{2{\delta }^{2}{\hslash }^{3}}{9{N}^{2}{m}^{3}}\frac{\Delta \left(k+{k}^{\prime }+{k}^{″}\right)}{\omega \left(k\right)\omega \left({k}^{\prime }\right)\omega \left({k}^{″}\right)}{|\underset{\eta =x,y}{\sum }\mathrm{sin}\frac{{k}_{\eta }a}{2}\mathrm{sin}\frac{{{k}^{\prime }}_{\eta }a}{2}\mathrm{sin}\frac{{{k}^{″}}_{\eta }a}{2}{\text{e}}^{-i\left({k}_{\eta }+{{k}^{\prime }}_{\eta }+{{k}^{″}}_{\eta }\right)a/2}|}^{2}$ (7)

3. 声子Green函数

${G}_{k\sigma ,{k}^{\prime }{\sigma }^{\prime }}^{AA}\left(\omega \right)=\frac{{\omega }_{k}{\delta }_{k,-{k}^{\prime }}{\delta }_{\sigma ,{\sigma }^{\prime }}}{\pi \left[{\omega }^{2}-{\omega }_{k}^{2}-2{\omega }_{k}{M}_{k\sigma }\left(\omega \right)\right]}$ (8)

${M}_{k\sigma }\left(\omega \right)=\frac{18\pi }{{\hslash }^{2}}\underset{{k}_{1}{k}_{2}{q}_{1}{q}_{2}}{\sum }{V}_{3}\left({k}_{1},{k}_{2},k\right){V}_{3}\left({q}_{1},{q}_{2},-k\right)〈{A}_{{k}_{1}\sigma }\left(t\right){A}_{{k}_{2}\sigma }\left(t\right),{A}_{{q}_{1}\sigma }\left({t}^{\prime }\right){A}_{{q}_{2}\sigma }{\left({t}^{\prime }\right)}_{\omega }〉$ (9)

$〈{a}_{k\sigma }^{+}\left(t\right){a}_{{k}^{\prime }{\sigma }^{\prime }}\left(0\right)〉={\delta }_{k{k}^{\prime }}{\delta }_{\sigma {\sigma }^{\prime }}{n}_{k}{\text{e}}^{i{\omega }_{k}t-{\Gamma }_{k\sigma }|t|}$ (10)

$〈{a}_{k\sigma }\left(t\right){a}_{{k}^{\prime }{\sigma }^{\prime }}^{+}\left(0\right)〉={\delta }_{k{k}^{\prime }}{\delta }_{\sigma {\sigma }^{\prime }}\left({n}_{k}+1\right){\text{e}}^{i{\omega }_{k}t-{\Gamma }_{k\sigma }|t|}$ (11)

${M}_{k\sigma }\left(\omega \right)=\frac{36}{{\hslash }^{2}}\underset{{k}_{1}{k}_{2}}{\sum }\underset{±}{\sum }\left[\left({\stackrel{¯}{n}}_{{k}_{2}}+\frac{1}{2}\right)±\left({\stackrel{¯}{n}}_{{k}_{1}}+\frac{1}{2}\right)\right]\frac{{|V\left({k}_{1};{k}_{2};-k\right)|}^{2}\left({\omega }_{{k}_{1}}±{\omega }_{{k}_{2}}\right)}{{\left[\omega +i\left({\Gamma }_{{k}_{1}\sigma }+{\Gamma }_{{k}_{2}\sigma }\right)\right]}^{2}-{\left({\omega }_{{k}_{1}}±{\omega }_{{k}_{2}}\right)}^{2}}$ (12)

${\Gamma }_{k\sigma }=\frac{72}{{\hslash }^{2}}\underset{{k}_{1}{k}_{2}}{\sum }\underset{±}{\sum }\left[\left({\stackrel{¯}{n}}_{{k}_{2}}+\frac{1}{2}\right)±\left({\stackrel{¯}{n}}_{{k}_{1}}+\frac{1}{2}\right)\right]\frac{\omega \left({\omega }_{{k}_{1}}±{\omega }_{{k}_{2}}\right)\left({\Gamma }_{{k}_{1}\sigma }+{\Gamma }_{{k}_{2}\sigma }\right){|V\left({k}_{1};{k}_{2};-k\right)|}^{2}}{{\left[{\omega }^{2}-\left({\omega }_{{k}_{1}}±{\omega }_{{k}_{2}}\right)\right]}^{2}+{\left[2\omega \left({\Gamma }_{{k}_{1}\sigma }+{\Gamma }_{{k}_{2}\sigma }\right)\right]}^{2}}$ (13)

4. 单原子层薄膜的能量通量公式

${\kappa }_{x}=\frac{{k}_{B}{\beta }^{2}}{V}{\int }_{-\infty }^{\infty }\text{d}t〈{S}_{x}\left({t}^{\prime }\right){S}_{x}\left(t\right)〉$ (14)

$S=\frac{1}{2i\hslash m}\underset{i,j}{\sum }\left[R\left(i\right)-R\left(j\right)\right]\underset{\sigma }{\sum }\left\{{p}_{\sigma }\left(i\right)\left[{p}_{\sigma }\left(i\right),{V}_{j}\right]+\left[{p}_{\sigma }\left(i\right),{V}_{j}\right]{p}_{\sigma }\left(i\right)\right\}$ (15)

${S}_{x}=\frac{a}{2m}\underset{\sigma }{\sum }\left\{\underset{j{i}_{1}}{\sum }\left[{p}_{\sigma }\left({i}_{1}\right)\frac{\partial {V}_{j}}{\partial {u}_{\sigma }\left({i}_{1}\right)}+\frac{\partial {V}_{j}}{\partial {u}_{\sigma }\left({i}_{1}\right)}{p}_{\sigma }\left({i}_{1}\right)\right]-\underset{j{i}_{2}}{\sum }\left[{p}_{\sigma }\left({i}_{2}\right)\frac{\partial {V}_{j}}{\partial {u}_{\sigma }\left({i}_{2}\right)}+\frac{\partial {V}_{j}}{\partial {u}_{\sigma }\left({i}_{2}\right)}{p}_{\sigma }\left({i}_{2}\right)\right]\right\}$ (16)

${V}_{j}^{\sigma }=\frac{1}{2}\underset{\sigma }{\sum }\frac{1}{2}k\left\{{\left[{u}_{\sigma }\left({i}_{1}\right)-{u}_{\sigma }\left(j\right)\right]}^{2}+{\left[{u}_{\sigma }\left({i}_{2}\right)-{u}_{\sigma }\left(j\right)\right]}^{2}\right\}$ (17)

${S}_{x}=\underset{k\sigma }{\sum }{n}_{k\sigma }\hslash {\omega }_{k}{v}_{k}^{x}$ (18)

${v}_{k}^{x}=\frac{ka}{m{\omega }_{k}}\mathrm{sin}{k}_{x}a$ (19)

5. 单原子层薄膜的热传导系数公式

${\kappa }_{x}=\frac{{k}_{B}{\hslash }^{2}{\beta }^{2}}{V}\underset{k\sigma {k}^{\prime }{\sigma }^{\prime }}{\sum }{\omega }_{k}{\omega }_{{k}^{\prime }}{v}_{k}^{x}{v}_{{k}^{\prime }}^{x}{\int }_{-\infty }^{\infty }\text{d}t〈{n}_{k\sigma }\left(t\right){n}_{{k}^{\prime }{\sigma }^{\prime }}\left({t}^{\prime }\right)〉$ (20)

${\int }_{-\infty }^{\infty }\text{d}t〈{n}_{k\sigma }\left(t\right){n}_{{k}^{\prime }{\sigma }^{\prime }}\left({t}^{\prime }\right)〉={\int }_{-\infty }^{\infty }\text{d}t〈{a}_{k\sigma }^{+}\left(t\right){a}_{{k}^{\prime }{\sigma }^{\prime }}\left(0\right)〉〈{a}_{k\sigma }\left(t\right){a}_{{k}^{\prime }{\sigma }^{\prime }}^{+}\left(0\right)〉$ (21)

${\kappa }_{x}=\frac{{k}_{B}{\hslash }^{2}{\beta }^{2}}{V}\underset{k\sigma }{\sum }\frac{{\omega }_{k}^{2}{\left({v}_{k}^{x}\right)}^{2}{\stackrel{¯}{n}}_{k\sigma }\left({\stackrel{¯}{n}}_{k\sigma }+1\right)}{{\Gamma }_{k}}$ (22)

${\kappa }_{x}=\frac{3{k}_{B}{\beta }^{2}}{V}\underset{k}{\sum }\text{ }\text{ }{\hslash }^{2}{\omega }_{k}^{2}{L}_{k}^{x}{v}_{k}^{x}{\stackrel{¯}{n}}_{k}\left({\stackrel{¯}{n}}_{k}+1\right)$ (23)

6. 结论

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