Effect of substrate on thermal conductivity of single-walled carbon nanotubes

We analyze numerically the thermal conductivity of single-walled carbon nanotubes placed on a flat rigid substrate. We demonstrate that the character of thermal conductivity depends crucially on the interaction between the nanotube and the substrate. In particular, we reveal that unlike the well-established anomalous thermal conductivity of isolated carbon nanotubes, the nanotube placed on a substrate demonstrates normal thermal conductivity due to the appearance of a narrow gap in the frequency spectrum of acoustic phonons. Copyright c © EPLA, 2009 Carbon-based structures display the highest measured thermal conductivity of any known material at moderate temperatures. The discovery of carbon nanotubes (CNTs) in 1991 [1] has led to suggestions that this new material could have a thermal conductivity greater than that of diamond and graphite [2]. Experimental data [3] demonstrated that conductivity of CNTs at room temperature can exceed the value of 3000W/mK which is two orders of magnitude larger than the estimates of earlier experiments that used macroscopic mat samples. Further studies employing different experimental methods confirmed that CNTs demonstrate anomalously high thermal conductivity. However, recent experiments [4] which measured the dependence of thermal conductivity on the length of individual single-walled CNTs placed on a Si substrate, revealed that the coefficient of thermal conductivity saturates for longer nanotubes. This observation contradicts earlier theoretical results based on the molecular-dynamics simulations [5–10]. Therefore, a natural question arises: What is the effect of the substrate on the thermal conductivity of single-walled carbon nanotubes? In this letter, we study numerically a transfer of thermal energy along an isolated single-walled CNT and analyze the effect of a rigid substrate on the character of ther(a)E-mail: [email protected] mal conductivity and its dependence on the nanotube length. We employ two different methods: i) a direct modeling of the heat transfer by means of the moleculardynamics simulations, and ii) the study of the equilibrium multi-particle dynamics based on the Green-Kubo formalism. We demonstrate that fixing a nanotube on a substrate changes dramatically its thermal conductivity. The nanotube placed on a flat rigid surface displays a finite conductivity, in a sharp contrast to an isolated CNT that displays anomalous thermal conductivity. For modeling thermal conductivity of long nanotubes, we consider the nanotubes with the indices (m, 0) (zigzag CNT) and (m,m) (armchair CNT). Figure 1(a) shows schematically a nanotube placed on a flat rigid surface (e.g., a surface of Si crystal [4]). It is convenient to describe the structure of an ideal nanotube as that created by an operation of screw rotation by an elementary unit cell defined between two neighboring carbon atoms located at the surface of the cylindrical tube. We introduce such a operator as S(∆z, δφ), that moves the point with the cylindrical coordinates (z, φ) to the point with the coordinates (z+∆z, φ+ δφ), i.e. a shift along the axis by the distance ∆z and its rotation by the angle δφ. This operation is commutative, i.e. any two operators S1 = S(∆z1, δφ1) and S2 = S(∆z2, δφ2) are linearly independent. The operation S 1 S n 2 means the transformation of