0
Research Papers: Micro/Nanoscale Heat Transfer

# Thermal Properties for Bulk Silicon Based on the Determination of Relaxation Times Using Molecular Dynamics

[+] Author and Article Information
Javier V. Goicochea

Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213

Pittsburgh Supercomputing Center, Pittsburgh, PA 15213

Cristina Amon

Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, M5S 1A4, Canada

J. Heat Transfer 132(1), 012401 (Oct 22, 2009) (11 pages) doi:10.1115/1.3211853 History: Received May 07, 2008; Revised July 23, 2009; Published October 22, 2009

## Abstract

Molecular dynamics simulations are performed to estimate acoustical and optical phonon relaxation times, dispersion relations, group velocities, and specific heat of silicon needed to solve the Boltzmann transport equation (BTE) at 300 K and 1000 K. The relaxation times are calculated from the temporal decay of the autocorrelation function of the fluctuation of total energy of each normal mode in the $⟨100⟩$ family of directions, where the total energy of each mode is obtained from the normal mode decomposition of the motion of the silicon atoms over a period of time. Additionally, silicon dispersion relations are directly determined from the equipartition theorem obtained from the normal mode decomposition. The impact of the anharmonic nature of the potential energy function on the thermal expansion of the crystal is determined by computing the lattice parameter at the cited temperatures using a NPT (i.e., constant number of atoms, pressure, and temperature) ensemble, and are compared with experimental values reported in the literature and with those computed analytically using the quasiharmonic approximation. The dependence of the relaxation times with respect to the frequency is identified with two functions that follow the functional form of the relaxation time expressions reported in the literature. From these functions a simplified version of relaxation times for each normal mode is extracted. Properties, such as group and phase velocities, thermal conductivity, and mean free path, needed to further develop a methodology for the thermal analysis of electronic devices (i.e., from nano- to macroscales) are determined once the relaxation times and dispersion relations are obtained. The thermal properties are validated by comparing the BTE-based thermal conductivity against the predictions obtained from the Green–Kubo method. It is found that the relaxation times closely resemble the ones obtained from perturbation theory at high temperatures; the contribution to the thermal conductivity of the transverse acoustic, longitudinal acoustic, and longitudinal optical modes being approximately 30%, 60%, and 10%, respectively, and the contribution of the transverse optical mode negligible.

<>

## Figures

Figure 1

Flow chart of calculations performed in this work

Figure 2

Temperature dependence of the lattice parameter for the SW potential

Figure 3

Relaxation rates as a function of the frequency for LA and LO modes at 300 K (a) and 1000 K (b). The vertical line indicates the first Brillouin zone.

Figure 4

Relaxation rates as a function of the frequency for TA and TO modes at 300 K (a) and 1000 K (b)

Figure 5

Relaxation rates for LO and TO modes at 300 K and 1000 K with respect to the reduced wave vector

Figure 6

Fitted relaxation rates at 1000 K. Log-log scale.

Figure 7

Relaxation rates as a function of the wave vector for the LO and TO modes at 1000 K

Figure 8

Dispersion relations obtained at 300 K and 1000 K. The thick solid line represents the analytical dispersion relation obtained at 0 K using the equilibrium lattice parameter. The MD results (thin solid and dashed lines) are fitted using a fourth-order polynomial. The symbols represent domains of different sizes.

Figure 9

Group (solid lines) and phase (dashed lines) velocities at 300 K (bold lines) and 1000 K (thin lines) in the [100] direction

Figure 10

Phonon mean free path at 300 K (a) and 1000 K (b) as a function of the frequency. The arrows and the dashed lines represent the edge of the Brillouin zone.

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections