Journal of Nanomaterials

Volume 2017 (2017), Article ID 5038978, 11 pages

https://doi.org/10.1155/2017/5038978

## Thermal Transport and Rectification Properties of Bamboo-Like SiC Polytypes Nanowires

School of Mechanical and Electrical Engineering, Henan University of Technology, Zhengzhou 450007, China

Correspondence should be addressed to Zan Wang; moc.361@zweurt

Received 2 February 2017; Revised 3 May 2017; Accepted 10 May 2017; Published 12 June 2017

Academic Editor: David Cornu

Copyright © 2017 Zan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Bamboo-like SiC nanowires (NWs) have specific geometric shapes, which have the potential to suppress thermal conductivity by phonon boundary scattering. In this work, phonon transport behaviors in the 3C-SiC, 4H-SiC, and 6H-SiC crystal lattices are studied by the Monte Carlo (MC) method, including impurity scattering, boundary scattering, and Umklapp scattering. Phonon relaxation times for Umklapp (U) scattering for the above three SiC polytypes are calculated from the respective phonon spectra, which have not been reported in the literature. Diffuse boundary scattering and thermal rectification with different aspect ratios are also studied at different temperatures. It is found that the thermal conductivities of the bamboo-like SiC polytypes can be lowered by two orders of magnitude compared with the bulk values by contributions from boundary scattering. Compared with bamboo-like 4H-SiC and 6H-SiC NWs, 3C-SiC has the largest U scattering relaxation rate and boundary scattering rate, which leads to its lowest thermal conductivities. The thermal conductivity in the positive direction is larger than that in the negative direction because of its lower boundary scattering relaxation rate.

#### 1. Introduction

SiC is one of the most important third-generation semiconductors in the fields of electronic devices and high-temperature components. Particularly in harsh conditions, SiC devices are much better than Si and GaAs devices [1]. Though SiC has hundreds of stable polytypes, the most commonly used ones are 3C-SiC, 4H-SiC, and 6H-SiC [2]. In recent years, there have been many experimental and theoretical efforts invested in the development of SiC nanostructures [3–5]. Researchers [6–8] have proposed that thermal rectification effects are dependent on the size of the device and the asymmetric geometry. If phonons can be managed to transmit information as electrons and photons, thermal devices will also have more practical applications in phononic devices and the field of thermoelectric conversion.

To determine the boundary and interfacial effects of nanoscale structures, some groups have studied core-shell NWs [9], variable cross-section NWs [10, 11], and rough NWs [12, 13] in recent years. In thermal studies of SiC, Ni et al. [14] and Chantrenne and Termentzidis calculated thermal conductivities and frequency density of states of (DOS) 3C-SiC and 6H-SiC NWs with constant cross-sections by the nonequilibrium molecular dynamics (NEMD) method [15]. Bamboo-like SiC NWs were synthesized by a vapor-liquid-solid mechanism and chemical vapor deposition [16–23]. Termentzidis et al. predicated the thermal conductivities of 3C-SiC and 2H-SiC NWs with constant and variable cross-sections via the NEMD method [24], in which the physical model was similar to that of bamboo-like NWs. Zianni appraised the impact of diameter modulation on the thermal conductivity reduction in core-shell silicon nanowires [25]. By a kinetic theory model, Zianni et al. [26–28] investigated transport properties of phonon and electron in diameter-modulated nanowires and pointed out the amount of disorder which suppresses thermal conductance deeply according to phonon transmission coefficients.

Approaches to investigate nanoscale heat conduction generally fall into two main categories: analytic and numerical methods. Analytic methods incorporate the Boltzmann transport equation (BTE), Green’s theorem, and so on. The most widely used numerical methods are the NEMD and MC methods. In the MD method, macroscopic properties can be obtained by calculating the transmission function of every atom. However, if the system contains a large number of atoms, the computations will be immensely expensive. Furthermore, when the simulated temperature is lower than the Debye temperature, the BTE will be invalid for the exchange between classical statistics and quantum statistics, and the results should be revised by quantum corrections. In contrast to the MD method, the MC method treats atomic thermal vibrations as quantum particles and regards atomic thermal interactions as phonon-phonon scattering. Since the calculation of the transmission function is not required in MC simulations, the computational complexity is reduced to a moderate extent. The transmission parameters and distributions of phonons can be gained statistically without quantum corrections at low temperatures.

Based on the Debye model, Peterson first applied the MC method to solve thermal transport in a one-dimensional model by simplifying the phonon group velocities and polarization [29]. Most later MC methods were derived from Peterson’s work and extended its computational accuracy and geometric complexity [30–37]. Using realistic phonon spectra, the accuracy of the MC method can be improved significantly [38–43]. In particular, sophisticated nanostructures have attracted more interest for studying the effects of phonon ballistic transport [44, 45] and interfacial scattering [46]. Through MC simulations, researchers proposed that the surface roughness is the main factor leading to ultralow and amorphous-limit thermal conductivity [47, 48]. Bong and Wong studied the impact of anisotropy scattering using the MC method and suggested that forward scattering has a more obvious effect on the thermal conductivity than backward scattering [49]. Rickman et al. studied thermal transport across the grain boundary and suggested that one can engineer thermodynamic and transport properties in materials by inducing interfacial layering transitions via changes in temperature or pressure [50].

#### 2. Monte Carlo Method of Phonon Transport

In the presented work, the entire MC simulation process is composed of four steps: initialization, drift, scattering, and statistics. For the initialization, the number of phonons with angular frequency at a given temperature can be calculated by the Bose-Einstein distribution:where denotes the Planck constant and represents the Boltzmann constant.where is the total number of phonons in volume . The phonon spectra are divided into 1000 equal-sized intervals. The first term on the right-hand side of (2) is the total number of transverse acoustic (TA) phonons, and the second term represents the number of longitudinal acoustic (LA) phonons. and denote the DOS of the LA phonons and TA phonons, respectively. is the th frequency interval. The phonon group velocity can be obtained by The temperatures of each cell can be counted by their energies, and this energy can be obtained bywhere is the material volumic energy and represents the phonon polarization.

After initialization, every phonon has a drifting time step . should be less than the average relaxation time for a phonon to avoid missing phonon scattering events. In this work, is set at 1.0 ps. Based on Matthiessen’s formula, the average relaxation time can be calculated by where is the phonon impurity scattering relaxation time [31]. denotes the normal scattering relaxation time, and is the Umklapp scattering relaxation time. is the boundary scattering relaxation time. Given the value of , the average scattering probability can be calculated from Whether a scattering event occurs should be judged by comparing the value of and a generated random number. In this work, is not included in (6). The conventional approach to solve phonon boundary scattering is composed of an indirect method and a direct method. Introduced in Ziman’s work [51] and developed by scholars [52, 53], the indirect method has a simplified , which assumes that the lateral surfaces of the simulated structure are flat. In the direct method, boundary scattering only depends on whether its position exceeds the boundary. In our study, the direct method is adopted. If a boundary scattering event occurs, the phonon will undergo diffuse reflection. The thermal flux and thermal gradient can be gathered by the synchronous ensemble method at each time in Figure 1. The vertical solid lines denote the time point . The interval time between two adjacent dashed lines is time step . The distance between two adjacent solid lines is , where is set to 10^{2}–10^{3} in the simulations. To simplify the explanations in Figure 1, is set to 4. The transverse lines represent the time course of phonon 1 and phonon 2. Solid circles denote impurity scattering or U scattering, and hollow circles denote boundary scattering.