Physics Based Analytical Thermal Conductivity Model For Metallic Single Walled Carbon Nanotube

Abstract

Single-Walled Carbon Nanotube (SWCNT) based Very Large Scale Integrated circuit (VLSI) interconnect is one of the emerging technologies, and has the potential to overcome the thermal issues persisting even with the advanced copper based interconnect. This is because of it’s promising electrical and thermal transport properties. It can be stated that thermal energy transport in SWCNTs is highly anisotropic due to the quasi one dimensionality, and like in other allotropes of carbon, phonons are the dominant energy carriers of heat conduction. In case of conventional interconnect materials, copper and aluminium, although their thermal conductivity varies over orders of magnitude at temperatures below100 K, near room temperature and above they have almost constant value. On the other hand, the reported experimental studies on suspended metallic SWCNTs illustrate a wide variation of the longitudinal lattice thermal conductivity (κl) with respect to the temperature(T)and the tube length(L)at low, room and high temperatures. Physics based analytical formulation of κl of metallic SWCNT as a function of L and T is essential to efficiently quantify this emerging technology’s impact on the rising thermal management issues of Integrated Circuits. In this work, a physics based diameter independent analytical model for κl of metallic SWCNT is addressed as a function of Lover a wide range of T. Heat conduction in metallic SWCNTs is governed by three resistive phonon scattering processes; second order three phonon Umklapp scattering, mass difference scattering and boundary scattering. For this study, all the above processes are considered, and the effective mode dependent relaxation time is determined by the Matthiessen’s rule. Phonon Boltzmann transport equation under the single mode relaxation time approximation is employed to derive the non-equilibrium distribution function. The heat flux as a function of temperature gradient is obtained from this non-equilibrium distribution function. Based on the Fourier’s definition of thermal conductivity, κl of metallic SWCNT is formulated and the Debye approximations are used to arrive at analytical model. The model developed is validated against both the low and high temperature experimental investigations. At low temperatures, thermal resistance of metallic SWCNT is due to phonon-boundary scattering process, while at high temperatures it is governed by three phonon Umklapp scattering process. It is understood that apart from form factor due to mass difference scattering, boundary scattering also plays the key role in determining the peak value. At room temperature, κl of metallic SWCNT is found to be an order of magnitude higher than that of most of metals. The reason can be attributed to the fact that both sound velocity and Debye temperature which have direct effect on the phonon transport in a solid, are reasonably higher in SWCNTs. Though Umk lapp processes reduce the κl steeper than 1/T beyond room-temperature, it’s magnitude is round1000 W/m/K upto 800 K for various tube lengths, which confirms that this novel material is indeed an efficient conductor of heat also, at room-temperature and above.