Quantitatively predicting modal thermal conductivity of nanocrystalline Si by full-band Monte Carlo simulations

Thermal transport in nanocrystalline Si is of great importance for many advanced applications. A better understanding of the modal thermal conductivity of nanocrystalline Si will be expected. In this work, the efficient variance reduced Monte Carlo simulation with full band phonon dispersion is applied to study the modal thermal conductivity of nanocrystalline Si. Importantly, the phonon modal transmissions across the grain boundaries which are modeled by the amorphous Si interface are calculated by the mode-resolved atomistic Green's function method. The predicted ratios of thermal conductivity of nanocrystalline Si to that of bulk Si agree well with that of the experimental measurements in a wide range of grain size. The ratio of thermal conductivity of nanocrystalline Si is decreased from 54% to 3% and the contribution of phonons with mean free path larger than the grain size increases from 30% to 96% as the grain size decreases from 550 to 10 nm. This work demonstrates that the full band Monte Carlo simulation using phonon modal transmission by the mode-resolved atomistic Green's function method can effectively capture the phonon transport picture in complex nanostructures, and therefore can provide guidance for designing Si based devices with better performance.

Figure 1

(a) The schematic of the nanocrystalline Si. The grain boundaries, which are represented by red, blue, and green planes, are perpendicular to the $x, y$, and $z$ direction, respectively. The red dashed circle shows the zoom-in of the grain boundary, which is modeled by the a-Si interface. (b) The unit cell (${\mathrm{UC}}_{\mathrm{mc}}$) used in the calculation of full band phonon dispersion for the MC simulations and the unit cell (${\mathrm{UC}}_{\mathrm{agf}}$) used in the calculation of phonon modal transmission by the mode-resolved AGF method.

Figure 2

(a) The phonon modal transmissions calculated by the mode-resolved AGF method (colored dots). The normalized $k_{\mathrm{y}}=0.167\mathrm{and}{k}_{\mathrm{z}}=0.167$. The value of transmission is according to the color bar. (b) Phonon modal transmissions by interpolation for the MC simulations (colored dots). The corresponding modal transmissions by the mode-resolved AGF method are these in (a). The black solid lines in (a) and (b) are calculated by lattice dynamics (LD) using GULP [48] for leading the eye. (c) Phonon modal transmission by interpolation for all the phonon modes and the modal transmission by the AGF method. The thickness of a-Si interface is 5 CC (2.716 nm) in (a), (b), and (c). (d) Phonon modal transmissions by interpolation for all the phonon modes across the a-Si interface with thickness of 4 (blue dots), 5 (red dots), and 6 (dark grey dots) CC.

Figure 3

(a) The ratio of $\kappa$ of nc-Si (${\kappa }_{\mathrm{nc}\text{−}\mathrm{Si}}$) with grain sizes of 550 nm (red triangle) and 76 nm (blue triangle) to that of bulk $\mathrm{Si}\left({\kappa }_{\mathrm{bulk}}\right)$ versus the thickness of a-Si interfaces 4 CC (2.172 nm), 5 CC (2.716 nm) and 6 CC (3.259 nm). The horizontal red and blue dashed lines represent the experimental measurements [8] for the nc-Si with grain size 550 nm and 76 nm, respectively. (b) The modal $\kappa$ of nc-Si and bulk Si. (c) The phonon MFP of bulk Si and nc-Si. The pink and blue dashed lines are for reference. (d) The ratios of MFPs of nc-Si to that of bulk Si. In (b) to (d), the nc-Si with grain size 550 and 76 nm are represented by the red and blue dots, respectively.

Figure 4

(a) The spectral $\kappa$ of bulk Si (black dotted line) and nc-Si with a-Si interface thickness of 4, 5, and 6 CC versus frequency. (b) Normalized cumulative $\kappa$ of bulk Si and nc-Si versus frequency. The $\kappa$ of nc-Si is normalized by the $\kappa$ of bulk Si. The grain size of nc-Si is 550 nm in (a) and (b). (c) The spectral $\kappa$ of bulk Si (black dotted line) and nc-Si with grain size of 550 nm (red dashed line) and 76 nm (blue dashed line) versus frequency. (d) The normalized cumulative $\kappa$ of bulk Si and nc-Si with grain sizes of 550 nm (red line) and 76 nm (blue line).

Figure 5

(a) The ratio of $\kappa$ of nc-Si to that of bulk Si versus grain size. The thickness of a-Si interface is varied from 4 to 6 CC. The experiment measurements of nc-Si with grain sizes of 550, 144, 76 nm [8] (triangle), 30 nm [49] (star), and 9.7 nm [13] (square) are shown for comparison. The normalized cumulative $\kappa$ versus frequency (b) and phonon mean free path (c) for bulk Si and nc-Si with grain sizes of 550, 144, 76, 30, and 10 nm. The thickness of a-Si interface is 5 CC in (b) and (c).