Model for thermal conductivity in nanoporous silicon from atomistic simulations

By means of molecular dynamics simulations, we have studied heat transport in nanoporous silicon, finding that the Eucken model, widely adopted in the description of thermal transport in macroporous systems, breaks down when pores are nanometer-sized. Present atomistic results are used to inform an extension to this model, effectively describing the relationship between thermal conductivity and interface density, here identified as the key structural characteristic of a porous sample. Our model, validated against a range of pore sizes and distributions, provides a robust framework for the interpretation of the atomistic results, as well as suggesting how to estimate the average pore size through thermal transport measurements.

Figure 1

Ordered porous Si samples with same porosity $\phi =0.14$, but different pore diameter $d_p$. Pictures show a 1.36-nm-thick longitudinal section. Heat flux is generated along the $z$ direction (see text).

Figure 2

Red triangles: local porosity of a typical random porous Si sample calculated along the $z$ direction. Horizontal solid line: average porosity. Shaded area: standard deviation for porosity.

Figure 3

Random porous Si samples with different porosity $\phi$. Pictures show a 1.36-nm-thick longitudinal section. The zoomed frames provide additional details on the atomic scale structure. We remark that the simulation cell is as thick as 13.6 nm and, therefore, matter is not discontinuous. Heat flux is generated along the $z$ direction (see text).

Figure 4

Inverse thermal conductivity $1/\kappa$ vs inverse cell size $1/L_z$ for ordered porous Si samples with different porosity. Symbols are typically as large as the standard deviation of the calculated thermal conductivity.

Figure 5

Thermal conductivity $\kappa$ as a function of cell size $L_z$ for random porous Si system with different porosity. The horizontal solid lines (shaded area) represent the average values (standard deviation).

Figure 6

Thermal conductivity $\kappa$ as a function of porosity $\phi$ in ordered porous Si samples with fixed number of pores. Black solid line: effective model provided by Eq. (11). Red dashed line: standard Eucken model provided by Eq. (4). Dots are typically as large as the standard deviation on ${\kappa }_{\mathrm{eff}}$. Symbols are typically as large as the standard deviation of the calculated thermal conductivity.

Figure 7

Thermal conductivity $\kappa$ as a function of porosity $\phi$ in ordered porous Si samples with either fixed number of pores $N_p$ (black squares) and fixed pore diameter $d_p=3.3$ nm (red triangles) and $d_p=5.0$ nm (green dots). Solid lines: effective model provided by Eq. (11). Symbols are typically as large as the standard deviation of the calculated thermal conductivity.

Figure 8

Detail of the microscopic structure nearby a typical spherical pore with increasing diameter $d_p$. The yellow circle identifies approximately the shell corresponding to the highly-defected region.

Figure 9

Number of defected atoms (per pore) as a function of the pore diameter in ordered porous Si samples. Red squares: systems with constant number of pores. Green dots (blue triangles): systems with pore diameter fixed at $d_p=3.3$ nm $(d_p=5.0$ nm).

Figure 10

Thermal conductivity $\kappa$ as a function of the number of pores $N_p$ (bottom horizontal scale) and pore diameter $d_p$ (top horizontal scale) in ordered porous Si samples with $\phi =0.28$. Symbols are typically as large as the standard deviation of the calculated thermal conductivity.

Figure 11

Thermal conductivity $\kappa$ as a function of porosity $\phi$ in random (red dots) and ordered (black squares) porous Si samples. Solid lines: effective model provided by Eq. (11). Symbols are typically as large as the standard deviation of the calculated thermal conductivity.

Figure 12

Symbols: Thermal conductivity $\kappa$ as function of the interface density $\Psi$ (top panel) and porosity $\phi$ (bottom panel) for all systems here investigated. Solid lines: effective model provided by Eqs. (12) and (11) for the top and bottom panels, respectively. Shaded area: the deviation of the calculated AEMD data (symbols) from such models.