Phonon–phonon scattering rates in single walled carbon nanotubes Pierre Gautreau a , Yanbiao Chu a , Tarek Ragab a,b,c , Cemal Basaran a,⇑ a Electronic Packaging Laboratory, State University of New York at Buffalo, 102 Ketter Hall, Buffalo, NY 14260, USA b Nanotechnology Research Laboratory, University of Tabuk, 71491, Saudi Arabia c Faculty of Engineering, Alexandria University, 21526, Egypt article info Article history: Received 16 January 2015 Accepted 23 February 2015 Available online 8 April 2015 Keywords: Phonon–phonon interaction Scattering rate Single walled carbon nanotube abstract A theoretical model for the phonon–phonon scattering rates of carbon nanotubes (CNTs) is developed using the carbon specific Brenner’s potential. This model allows for the calculation of mode specific pho- non–phonon scattering rates, via direct computation of the three-phonon strength of interaction. This direct calculation provides further accuracy to the previously existing model, which relied on continuum mechanics approximations to the strength of interactions. The contributions of each phonon branch to the total phonon–phonon scattering rates are analyzed. The results for longitudinal optical and longitu- dinal acoustic phonons of a (10, 10) CNT suggest very different behavior for each vibrational mode. While the results presented are specific to the (10, 10) metallic CNTs, the method is directly applicable to CNTs of other chiralities. Ó 2015 Elsevier B.V. All rights reserved. 1. Introduction It is very difficult to obtain Carbon Nano Tube (CNT) material properties for specific chiralities experimentally,as a result computational modeling has been extensively used to obtain ther- mal, mechanical, electrical properties [1–13]. In order to model and compute electrical conductivity and Joule heating, it is necessary to calculate electron–phonon scattering rates in a CNT, using models proposed in the literature [2–5,14,15]. For metallic CNTs under electrical loading, the continued scat- tering between electrons and phonons results in a non-equilibrium phonon distribution associated with the creation of hot phonons. The effects of hot phonons on Joule heating [16] and wind forces [17] have been reported. If these hot phonons are not allowed to relax back toward the equilibrium phonon distribution, the cre- ation of phonons in electron–phonon scattering process may result in a net accumulation of hot phonons. To avoid such pile-ups of phonons during the simulation of CNTs, it is important to under- stand and calculate the strength of interactions between all phonons. Several groups have studied phonon–phonon scattering rates either experimentally [18,19] or theoretically [20–24]. Raman scat- tering experimental results presented in Refs. [18,19] report a decay rate of optical phonons at 300 K of 1 ps and 1.1 ps respectively. In Ref. [20] Pennington and his group make use of the phonon Boltzmann transport equation to calculate the optical phonon relaxation times for various CNTs. The results presented in Ref. [20] vary from 0.2 to 10.2 ps at 300 K depending on the chi- rality of the CNT. While experimental and theoretical results seem to agree on the magnitude of the phonon–phonon scattering rates, they may not necessarily reflect the true behavior of each individ- ual phonon–phonon interactions because of the averaging process used. The Raman scattering results presented in Ref. [18] are for very specific frequencies and do not show the full frequency spec- trum that phonons can span. Moreover, most theoretical models for phonon–phonon scattering rates calculate constant scattering rates for all phonons. However, during three phonon processes, there is no physical reason why the scattering rates should be con- stant for all phonon–phonon scattering mechanisms. The use of constant rates is a very rough approximation, which can be improved by analyzing the full phonon bands. According to the ab initio study of thermal transport properties, Broido et al. [23] reported that the three-phonon scattering is the dominant phonon scattering mechanisms around room temperature. In Ref. [21], Hepplestone and Srivastava derived a formalism to calculate pho- non specific scattering rates based on the anharmonic part of a generalized three dimensional lattice potential. The authors of Ref. [21] were able to show that phonon–phonon scattering rates have large variations for different phonon wavevector values. Their approach relied on the calculation of anharmonic scattering rates making use of a continuum mechanics approach to obtain the constant used in the strength of interaction formalism. http://dx.doi.org/10.1016/j.commatsci.2015.02.046 0927-0256/Ó 2015 Elsevier B.V. All rights reserved. ⇑ Corresponding author. Tel.: +1 716 645 3667. E-mail address: cjb@buffalo.edu (C. Basaran). Computational Materials Science 103 (2015) 151–156 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci