Thermal conductivity accumulation in amorphous silica and amorphous silicon

We predict the properties of the propagating and nonpropagating vibrational modes in amorphous silica ($a$-SiO${}_2$) and amorphous silicon ($a$-Si) and, from them, thermal conductivity accumulation functions. The calculations are performed using molecular dynamics simulations, lattice dynamics calculations, and theoretical models. For $a$-SiO${}_2$, the propagating modes contribute negligibly to thermal conductivity (6$\%$), in agreement with the thermal conductivity accumulation measured by Regner et al. [Nat. Commun. 4, 1640 (2013)]. For $a$-Si, propagating modes with mean-free paths up to 1 $\mu$m contribute 40$\%$ of the total thermal conductivity. The predicted contribution to thermal conductivity from nonpropagating modes and the total thermal conductivity for $a$-Si are in agreement with the measurements of Regner et al. The accumulation in the measurements, however, takes place over a narrower band of mean-free paths (100 nm–1 $\mu$m) than that predicted (10 nm–1 $\mu$m).

Figure 1

(a) Radial distribution function $g\left(r\right)$ for the 4608-atom $a$-SiO${}_2$ structure created from a melt-quench technique. The radial distribution function compares well with the experimental measurement from Ref. [54]. Inset: Small sample of the $a$-SiO${}_2$ structure showing the Si-O tetrahedral bond network. Bond lengths range between 1.6 and 1.8 Å. (b) Radial distribution function for the 4096-atom $a$-Si structure created by the modified WWW algorithm. The radial distribution function compares well with the experimental measurement from Ref. [57]. Inset: Small sample of the $a$-Si structure. Bond lengths range between 2.3 and 2.7 Å.

Figure 2

Vibrational DOS of $a$-SiO${}_2$ and $a$-Si plotted on a log-log scale. Both models show an ${\omega }^2$ scaling at low frequency. The dashed lines indicate an extrapolation of the DOS based on this scaling. The DOS for $a$-Si has two peaks similar to the DOS of the crystalline phase [62]. The DOS for $a$-SiO${}_2$ is flat over most of the spectrum, with a high-frequency gap that separates the modes involving Si-O interactions [47].

Figure 3

Longitudinal (left panel) and transverse (right panel) structure factors [Eq. (10)] for (a) $a$-SiO${}_2$ and (b) $a$-Si. The wave vectors are normalized by ${\kappa }_{\max }=2\pi /a$, where $a$ is 4.8 Å ($a$-SiO${}_2$) and 5.43 Å ($a$-Si), based on the lattice constants of the crystalline phases [48, 56].

Figure 4

Vibrational mode lifetimes predicted by NMD [Eq. (20)] and the time scales extracted from the structure factors [Eq. (13)] for (a) $a$-SiO${}_2$ and (b) $a$-Si. For $a$-Si, a clear ${\omega }^{-2}$ scaling is observed at low frequencies, while the lifetimes plateau at higher frequencies and over a wider range of frequencies than for $a$-SiO${}_2$.

Figure 5

Vibrational mode diffusivities predicted from NMD [using Eqs. (5) and (20) with the DOS sound speed from Table 1] and the AF theory [Eq. (8)]. Also shown are extrapolations based on an ${\omega }^{-2}$ scaling with Eqs. (5) and (6) for $a$-SiO${}_2$ and $a$-Si, and an additional ${\omega }^{-4}$ scaling for $a$-Si. For both systems, the diffusivities are larger than the high-scatter limit [Eq. (21)] except at high frequencies, where the modes are localized.

Figure 6

Thermal conductivities of $a$-SiO${}_2$ and $a$-Si predicted using the GK method and Eq. (1). For $a$-SiO${}_2$, the GK-predicted thermal conductivity is size independent, indicating that there is not an important contribution from propagating modes. For $a$-Si, there is a clear size dependence, indicating the importance of propagating modes.

Figure 7

(a) Predicted thermal conductivity accumulation function [Eq. (24)] for $a$-SiO${}_2$ compared with experimental broadband frequency domain reflectance measurements by Regner et al. [10] and thin-film measurements from Refs. [16, 17]. The predicted thermal conductivity accumulation demonstrates that the propagating contribution is negligible in our model, which is in accord with the experimental measurements. (b) Predicted thermal conductivity accumulation function for $a$-Si compared with experimental measurements by Regner et al. and thin films fabricated by sputtering (Experiment A) [5, 31, 32] and chemical vapor deposition (Experiment B) [7, 8, 30, 33]. The predicted thermal conductivity accumulation demonstrates that the propagating contribution is significant for $a$-Si. We note that thermal conductivities as high as 6 W/m-K (not plotted) have been measured for $a$-Si thin films deposited using hot-wire chemical vapor deposition [8].