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Tight-Binding Investigation of Thermal Conductivity of Graphene and Few-Layer Graphene Systems

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Abstract

The thermal conductivities of few-layer graphene per stratum are analytically calculated and compared with the single-layer value within the tight-binding Hamiltonian model and Green’s function formalism. The results show a decrease in the intra-plane thermal conductivity by increasing the number of layers. Moreover, the change in its magnitude varies less as the number of layers exceeds two.

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Authors and Affiliations

  1. Department of Physics, Razi University, Kermanshah, Iran

    Hamze Mousavi & Jabbar Khodadadi

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  1. Hamze Mousavi
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  2. Jabbar Khodadadi
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Correspondence to Hamze Mousavi.

Appendices

Appendix 1

The EVs of the Hamiltonian Eq. 1 are given as follows:

For \(N_\mathrm{p}=2\), they are obtained as

$$\begin{aligned} {\mathcal {E}}^{(1)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(2)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\left( \frac{t_{\bot }}{2}\right) ^{2}}+\frac{t_{\bot }}{2}, \end{aligned}$$
(14)
$$\begin{aligned} {\mathcal {E}}^{(3)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(4)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\left( \frac{t_{\bot }}{2}\right) ^{2}}-\frac{t_{\bot }}{2}, \end{aligned}$$
(15)

for \(N_\mathrm{p}=3\), they are calculated as

$$\begin{aligned} {\mathcal {E}}^{(1)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(2)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\left( \sqrt{2}\frac{t_{\bot }}{2}\right) ^{2}} +\sqrt{2}\frac{t_{\bot }}{2}, \end{aligned}$$
(16)
$$\begin{aligned} {\mathcal {E}}^{(3)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(4)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\left( \sqrt{2}\frac{t_{\bot }}{2}\right) ^{2}} -\sqrt{2}\frac{t_{\bot }}{2}, \end{aligned}$$
(17)
$$\begin{aligned} {\mathcal {E}}^{(5)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(6)}_{0}({\varvec{k}}) =|\epsilon _{{\varvec{k}}}|, \end{aligned}$$
(18)

while for \(N_\mathrm{p}=4\), we have

$$\begin{aligned} {\mathcal {E}}^{(1)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(2)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\frac{3+\sqrt{5}}{2}\left( \frac{t_{\bot }}{2}\right) ^{2}} +\left( \frac{1+\sqrt{5}}{2}\right) \frac{t_{\bot }}{2}, \end{aligned}$$
(19)
$$\begin{aligned} {\mathcal {E}}^{(3)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(4)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\frac{3+\sqrt{5}}{2}\left( \frac{t_{\bot }}{2}\right) ^{2}} -\left( \frac{1+\sqrt{5}}{2}\right) \frac{t_{\bot }}{2}, \end{aligned}$$
(20)
$$\begin{aligned} {\mathcal {E}}^{(5)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(6)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\frac{3-\sqrt{5}}{2}\left( \frac{t_{\bot }}{2}\right) ^{2}} +\left( \frac{1-\sqrt{5}}{2}\right) \frac{t_{\bot }}{2}, \end{aligned}$$
(21)
$$\begin{aligned} {\mathcal {E}}^{(7)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(8)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\frac{3-\sqrt{5}}{2}\left( \frac{t_{\bot }}{2}\right) ^{2}} -\left( \frac{1-\sqrt{5}}{2}\right) \frac{t_{\bot }}{2}, \end{aligned}$$
(22)

and for \(N_\mathrm{p}=5\), the EVs are as follows:

$$\begin{aligned} {\mathcal {E}}^{(1)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(2)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\left( \sqrt{3}\frac{t_{\bot }}{2}\right) ^{2}} +\sqrt{3}\frac{t_{\bot }}{2}, \end{aligned}$$
(23)
$$\begin{aligned} {\mathcal {E}}^{(3)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(4)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\left( \sqrt{3}\frac{t_{\bot }}{2}\right) ^{2}} -\sqrt{3}\frac{t_{\bot }}{2}, \end{aligned}$$
(24)
$$\begin{aligned} {\mathcal {E}}^{(5)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(6)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\left( \frac{t_{\bot }}{2}\right) ^{2}}+\frac{t_{\bot }}{2}, \end{aligned}$$
(25)
$$\begin{aligned} {\mathcal {E}}^{(7)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(8)}_{0}({\varvec{k}}) =\sqrt{|\epsilon _{{\varvec{k}}}|^{2}+\left( \frac{t_{\bot }}{2}\right) ^{2}}-\frac{t_{\bot }}{2}, \end{aligned}$$
(26)
$$\begin{aligned} {\mathcal {E}}^{(9)}_{0}({\varvec{k}})= & {} -{\mathcal {E}}^{(10)}_{0}({\varvec{k}}) =|\epsilon _{{\varvec{k}}}|. \end{aligned}$$
(27)

Appendix 2

\(\xi _{xx}({\mathcal {E}},\,T)\) for \(N_\mathrm{p}=2\) is determined by

$$\begin{aligned} \xi _{xx}({\mathcal {E}},\,T)=\xi _{0}(T)\sum ^\mathrm{FBZ}_{{\varvec{k}}} \left\{ \frac{\sin ^{2}\left( \sqrt{3}k_{x}\frac{a}{2}\right) \cos ^{2}(k_{y} \frac{a}{2})}{|\epsilon _{{\varvec{k}}} |^{2}+\left( \frac{t_{\bot }}{2}\right) ^{2}}\sum _{b=1}^{4} \left[ \mathfrak {I}\left( \frac{1}{E-{\mathcal {E}}^{(b)}_{0}({\varvec{k}})} \right) \right] ^{2}\right\} ,\nonumber \\ \end{aligned}$$
(28)

for \(N_\mathrm{p}=3\), the result is

$$\begin{aligned} \xi _{xx}({\mathcal {E}},\,T)= & {} \xi _{0}(T)\sum ^\mathrm{FBZ}_{{\varvec{k}}} \sin ^{2}\left( \sqrt{3}k_{x}\frac{a}{2}\right) \cos ^{2}\left( k_{y}\frac{a}{2}\right) \left\{ \frac{1}{|\epsilon _{{\varvec{k}}}|^{2}+\left( \sqrt{2}\frac{t_{\bot }}{2}\right) ^{2}} \right. \nonumber \\&\quad \left. \times \,{\sum ^{4}_{b=1}}^{\prime }\left[ \mathfrak {I}\left( \frac{1}{E-{\mathcal {E}}^{(b)}_{0} ({\varvec{k}})} \right) \right] ^{2}+\frac{1}{|\epsilon _{{\varvec{k}}}|^{2}}{\sum ^{6}_{b=5}}^{\prime } \left[ \mathfrak {I}\left( \frac{1}{E-{\mathcal {E}}^{(b)}_{0}({\varvec{k}})} \right) \right] ^{2}\right\} , \end{aligned}$$
(29)

when \(N_\mathrm{p}=4\), it is found that

$$\begin{aligned} \xi _{xx}({\mathcal {E}},\,T)= & {} \xi _{0}(T)\sum ^\mathrm{FBZ}_{{\varvec{k}}} \sin ^{2}\left( \sqrt{3}k_{x}\frac{a}{2}\right) \cos ^{2}\left( k_{y}\frac{a}{2}\right) \nonumber \\&\times \left\{ \frac{1}{|\epsilon _{{\varvec{k}}}|^{2}+\left( \frac{3+\sqrt{5}}{2}\right) \left( \frac{t_{\bot }}{2}\right) ^{2}}{\sum ^{4}_{b=1}}^{\prime }\left[ \mathfrak {I}\left( \frac{1}{E-{\mathcal {E}}^{(b)}_{0}({\varvec{k}})} \right) \right] ^{2} \right. \nonumber \\&\left. +\,\frac{1}{|\epsilon _{{\varvec{k}}}|^{2}+\left( \frac{3-\sqrt{5}}{2}\right) \left( \frac{t_{\bot }}{2}\right) ^{2}}{\sum ^{8}_{b=5}}^{\prime } \left[ \mathfrak {I}\left( \frac{1}{E-{\mathcal {E}}^{(b)}_{0}({\varvec{k}})} \right) \right] ^{2}\right\} , \end{aligned}$$
(30)

and \(N_\mathrm{p}=5\) leads to

$$\begin{aligned} \xi _{xx}({\mathcal {E}},\,T)= & {} \xi _{0}(T)\sum ^\mathrm{FBZ}_{{\varvec{k}}} \sin ^{2}\left( \sqrt{3}k_{x}\frac{a}{2}\right) \cos ^{2}\left( k_{y}\frac{a}{2}\right) \left\{ \frac{1}{|\epsilon _{{\varvec{k}}}|^{2}+\left( \sqrt{3}\frac{t_{\bot }}{2}\right) ^{2}} {\sum ^{4}_{b=1}}^{\prime }\right. \nonumber \\&\left. \times \left[ \mathfrak {I}\left( \frac{1}{E-{\mathcal {E}}^{(b)}_{0} ({\varvec{k}})} \right) \right] ^{2} \right. \nonumber \\&\left. +\,\frac{1}{|\epsilon _{{\varvec{k}}}|^{2}+\left( \frac{t_{\bot }}{2}\right) ^{2}} {\sum ^{8}_{b=5}}^{\prime } \left[ \mathfrak {I}\left( \frac{1}{E-{\mathcal {E}}^{(b)}_{0}({\varvec{k}})} \right) \right] ^{2}+\frac{1}{|\epsilon _{{\varvec{k}}}|^{2}}{\sum ^{10}_{b=9}}^{\prime }\right. \nonumber \\&\left. \times \left[ \mathfrak {I}\left( \frac{1}{E-{\mathcal {E}}^{(b)}_{0}({\varvec{k}})} \right) \right] ^{2}\right\} , \end{aligned}$$
(31)

where \(\sum ^{\prime }\) shows sum over just some bands but not all.

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Mousavi, H., Khodadadi, J. Tight-Binding Investigation of Thermal Conductivity of Graphene and Few-Layer Graphene Systems. Int J Thermophys 36, 2638–2646 (2015). https://doi.org/10.1007/s10765-015-1946-8

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