Modeling Heat Conduction in Nanoporous Silicon with Geometry Distributions

Formation of nanoporous structures is an effective approach to manipulate heat conduction and has been experimentally demonstrated to greatly reduce thermal conductivity. Thermal conductivity reduction of nanoporous materials depends on structural parameters such as size, shape, and position of the pores and their distributions, which are hard to explore in experiments. In this work, by systematically performing ray-tracing Monte Carlo simulations of nanoporous silicon crystal, we evaluate the impacts of the structural parameters on the thermal conductivity reduction. As a result, we find that the thermal conductivity reduction with spherical pores is insensitive to the spatial configuration and size distribution of the pores, even in the regime of quasi-ballistic phonon transport. Although the sensitivity does increase as phonon scattering by the pores becomes directional (as for rectangular pores) and porosity increases, the overall results deliver aspects that are useful in practice: the thermal conductivity of nanoporous structures can be well described by a phonon scattering model with a single length scale.

Figure 1

A schematic of phonon transport in nanoporous $\mathrm{Si}$ with different types of pores (a) spherical pores and (b) rectangular pores, respectively. Phonon transmittance calculation by ray-tracing Monte Carlo (RTMC) simulation is illustrated in (a). Phonon is incident on the x-y plane at z = 0 with azimuthal angle φ and polar angle θ. Transmittance τ is calculated by counting the number of phonons reaching the x-y plane at z = L. Phonon-pore scattering is treated as diffusive, and a perfect specular boundary condition is applied along x and y directions. d and a denote a diameter of a spherical pore and the side length of a rectangular pore, respectively.

Figure 2

Volume fraction dependence of effective lattice thermal conductivity (κ_eff) of nanoporous $\mathrm{Si}$ at 300 K for spherical pores with different diameters. κ_bulk denotes the thermal conductivity of bulk $\mathrm{Si}$. Dashed and dotted lines are results of Eucken model [11] and Bruggeman model [12, 13]. Solid lines denote geometrical scattering (GS) model [Eq. (3)] [16, 19]. Filled circles are experimental results previously reported [2].

Figure 3

Impact of diameter distribution of randomly located spherical pores on thermal conductivity reduction of nanoporous $\mathrm{Si}$ at 300 K. The standard deviation σ of Gaussian for diameter distribution is varied as 0 (black open circle), 0.25 (red open square), 0.5 (green open triangle), and 0.75 (blue open diamond). Mean diameter is set to 100 nm.

Figure 4

Impact of alignment of spherical pores on thermal conductivity reduction of nanoporous $\mathrm{Si}$ at 300 K. Three different configurations (random, cubic, and hexagonal) considered in this calculation are illustrated in the inset figure. In this calculation, the diameter of the spherical void is fixed to 100 nm.

Figure 5

Influence of the bimodal distribution on thermal conductivity of nanoporous $\mathrm{Si}$ at 300 K for different volume fractions. The open circles and solid lines are the results from RTMC and GS model calculations, respectively.

Figure 6

(a) Side-length dependent κ_eff of nanoporous $\mathrm{Si}$ at 300 K for randomly placed rectangular pores with different volume fractions (red: 10%, black: 30%). Squares and solid lines are results of RTMC simulation and GS model, respectively. (b) A schematic of phonon-pore scattering with spherical void. (c) A schematic of phonon-pore scattering with rectangular void.

Figure 7

Impact of alignment of rectangular pore on thermal conductivity reduction calculated by RTMC simulation. Squares and diamonds are results of RTMC simulation with cubic alignment and hexagonal alignment, respectively. Solid lines are results of RTMC simulation and GS model, respectively.

Figure 8

Impact of pore width of rectangular pores on thermal conductivity reduction calculated by RTMC simulation. Squares and diamonds are results of RTMC simulation with cubic alignment and hexagonal alignment, respectively. Solid lines are results of RTMC simulation and GS model, respectively. Circles are a ratio of calculated thermal conductivity of hexagonal and cubic alignment.

Figure 9

Impact of tortuosity on thermal conductivity reduction calculated by RTMC simulation. Four different configurations are considered (straight, random tortuosity, tortuosity with x-y plane, and tortuosity with z-x plane) as illustrated in the inset. Solid line and dashed line are results of GS model and modified GS model in consideration of actual scattering cross-section on tortuosity structure, respectively. In this calculation, the side length of a rectangular void is fixed to 100 nm.

Figure 10

Normalized cumulative thermal conductivity of bulk $\mathrm{Si}$ at 300 K calculated from first-principles-based anharmonic lattice dynamics [29].