Monte Carlo Simulation of Thermal Conductivities of Silicon Nanowires

One-dimensional (1D) materials such as various kinds of nanowires and nanotubes have attracted considerable attention due to their potential applications in electronic and energy conversion devices. The thermal transport phenomena in these nanowires and nanotubes could be significantly different from that in bulk material due to boundary scattering, phonon dispersion relation change, and quantum confinement. It is very important to understand the thermal transport phenomena in these materials so that we can apply them in the thermal design of microelectronic, photonic, and energy conversion devices. While intensive experimental efforts are being carried out to investigate the thermal transport in nanowires and nanotube, an accurate numerical prediction can help the understanding of phonon scattering mechanisms, which is of fundamental theoretical significance. A Monte Carlo simulation was developed and applied to investigate phonon transport in single crystalline Si nanowires. The Phonon-phonon Normal (N) and Umklapp (U) scattering processes were modeled with a genetic algorithm to satisfy both the energy and the momentum conservation. The scattering rates of N and U scattering processes were given from the first perturbation theory. Ballistic phonon transport was modeled with the code and the numerical results fit the theoretical prediction very well. The thermal conductivity of bulk Si was then simulated and good agreement was achieved with the experimental data. Si nanowire thermal conductivity was then studied and compared with some recent experimental results. In order to study the confinement effects on phonon transport in nanowires, two different phonon dispersions, one based on bulk Si and the other solved from the elastic wave theory for nanowires, were adopted in the simulation. The discrepancy from the simulations based on different phonon dispersions increases as the nanowire diameter decreases, which suggests that the confinement effect is significant when the nanowire diameter goes down to tens nanometer range. It was found that the U scattering probability engaged in Si nanowires was increased from that in bulk Si due to the decrease of the frequency gap between different modes and the reduced phonon group velocity. Simulation results suggest that the dispersion relation for nanowire solved from the elasticity theory should be used to evaluate nanowire thermal conductivity as the nanowire diameter reduced to tens nanometer INTRODUCTION In recent years, low-dimensional structures have attracted much attention for their potential application in thermoelectric devices. The performance of thermoelectric devices depends on the figure of merit ZT, given by ) / ( 2 T K T ZT ρ α = (1) where T K T , , , ρ α are the Seebeck coefficient, absolute temperature, electrical resistivity and total thermal conductivity, respectively. For a material to have a high ZT, one requires a high thermoelectric power α (Seebeck coefficient), a low electrical resistvity and a low thermal conductivity. A material with a figure of merit of around 1.0 was first reported over four decades ago, but since then, little progress has been made in finding new materials with enhanced 1 Copyright © 2005 by ASME ZT values at room temperature [1]. A big challenge to enhance the ZT value is to decrease the thermal conductivity and at the same time not to produce a deterioration of electronic transport. Nanostructures such as nanowires and quantum wells provide a promising method for the enhancement of ZT through controlling the phonon and electron transport [2]. Phonon transport at the nanoscale differs from that at the macroand microscale for several fundamental reasons. One is that size confinement changes the phonon dispersion relation. This change causes the phonon group velocity to differ from that in bulk material. A more complicated problem is that the phonon scattering rate also differs from that in bulk material [3]. This affects the phonon mean free path, which in turn changes the lattice thermal conductivity of the nanostructure. In addition, as the size of the nanostructure decreases below the phonon mean free path and starts to approach the wavelength of the dominant phonon, λ , the validity of phonon particle transport theory becomes questionable and wave theory should be applied to interpret heat transport in nanostructures. So far there are two distinct methods for the analysis of heat conduction in nanowires. Those include the solution of the Boltzmann transport equation (BTE) and the molecular dynamics simulation method. Based on the lifetime assumption and accounting for the modified nanowire acoustic phonon dispersion relation, it is possible to predict nanowire thermal conductivity [4-8] from the solution of the BTE. Because great simplifications must be introduced to produce a closed form solution, the results usually deviate greatly from experimental findings. Certain assumptions lead to erroneous explanations of particular phenomena [9]. The molecular dynamics simulation method uses Newton’s second law to describe the movement of a large number of atoms in the nanowire. Thermal conductivity can be extracted from averaging the positions and velocities of the atoms [10]. However, this method is limited by the knowledge of interatomic potential and computation ability. In this paper, the Monte Carlo simulation method is used to trace the phonon movement in a Si nanowire. Three phonon scattering, boundary scattering and impurity scattering processes are considered. A genetic algorithm is used to guarantee energy and wave-vector conservation conditions for N and U scattering. Bulk silicon and silicon nanowire thermal conductivities are calculated. Simulation results agree well with that from previous investigations and experimental results. II. OVERVIEW OF MONTE CARLO METHOD The Boltzmann equation for phonon transport in the presence of a temperature gradient is written as c g t N dT dN T V ) ( ∂ ∂ = ∇ • r (2) where is the group velocity, , (3) g V r ω ∇ = K g V r r N is the distribution function, K r is the phonon wavevector, T is local temperature, is the phonon frequency, and ω c t N ) ( ∂ ∂ is the rate of change of due to collision. On the left side of equation (2) can be replaced by , the equilibrium Planck distribution. Consequently equation (3) can be read as N N 0 N ∑ Φ − Φ = ∇ • ' )] ( ) , ( ) ( ) , ( [ ' ' ' 0 K g K N K K K N K K dT dN T V r