Thermal conductivity and its relation to atomic structure for symmetrical tilt grain boundaries in silicon

We perform a systematic study of thermal resistance and conductance of tilt grain boundaries (GBs) in Si using classical molecular dynamics. The GBs studied are naturally divided into three groups according to the structural units forming the GB core. We find that, within each group, the GB thermal conductivity strongly correlates with the excess GB energy. All three groups predict nearly the same GB conductivity extrapolated to the high-energy limit. This limiting value is close to the thermal conductivity of amorphous Si, suggesting similar heat transport mechanisms. While the lattice thermal conductivity decreases with temperature, the GB conductivity slightly increases. However, at high temperatures it turns over and starts decreasing if the GB structure undergoes a premelting transformation. Analysis of vibrational spectra of GBs resolved along different directions sheds light on the mechanisms of their thermal resistance. The existence of alternating tensile and compressive atomic environments in the GB core gives rise to localized vibrational modes, frequency gaps creating acoustic mismatch with lattice phonons, and anharmonic vibrations of loosely bound atoms residing in open atomic environments.

Figure 1

(a) Typical temperature profile for the (210) GB with the mean temperature of 500 K. The hot and cold thermostats are located at $y=0$ and $y=\pm 50$ nm, respectively. GB-1 and GB-2 are two GBs existing in the simulation block due to periodic boundary conditions. (b) Enlargement of the encircled region showing the procedure for computing the temperature jump $\mathrm{\Delta }T$. The black line is the fit by Eq. (4). The blue lines are the linear portions of the fitted function given by Eqs. (6) and (7).

Figure 2

The HCACF computed for c-Si using four interatomic potentials at 500 K (a) and 1000 K (b), for l-Si at 2000 K (c), and for a-Si at 300 K (d). Note the difference in the timescales between panels (a) and (b) and panels (c) and (d). The potentials tested: SW [32], T89 [75], MOD1 [76], and MOD2 [33].

Figure 3

Thermal conductivity computed using the GK method with the T89 potential [75] plotted as a function of the upper integration limit $\tau$. The solid line is a fit with the function $\kappa \left(\tau \right)={\kappa }_{\infty }tanh(\tau -{\tau }_0)$ discussed in the text. The dashed blue line indicates the converged value.

Figure 4

Groups of atoms selected for the calculation of local vibrational DOS of crystalline lattice (green) and GB atoms (red) in the (a) (210) GB and (b) (650) GB. The GB plane is horizontal and the tilt axis is normal to the page.

Figure 5

GB energy $\gamma$ as a function of tilt angle $\theta$. The inset figures represent typical GB structures with fundamental structural units A, B, and C shown on the right. The structures are projected along the [001] tilt axis normal to the page. The blue atoms have a diamond-like local environment whereas the orange atoms have local environments deviated from diamond. The square symbols represent the GBs used for the systematic study of Kapitza conductance at the fixed temperature of 750 K. The green circles represent GBs selected for a less systematic study at different temperatures. The color of the squares reflects the different structural categories discussed in the text. The solid red and blue lines were obtained by Read-Shockley fits [83]. The dashed black lines serve as a guide for the eye.

Figure 6

Energy of selected GBs as a function of the number $\lambda$ of atoms removed from the GB (fraction of atomic plane). (a) Tilt angle $\theta \le 43.6^{\circ }$. (b) Tilt angle $\theta \ge 53.1^{\circ }$. The lines are drawn to guide the eye.

Figure 7

Atomic structure of the (650) GB at different temperatures. The atoms with diamond-like local environment are shown in blue, while atoms with distorted environments are shown in orange. Note the high level of disordering in the dislocation cores forming the boundary at the highest temperature (1600 K).

Figure 8

The GB order parameter as a function of temperature. (a) (310) and (210) GBs show a linear decrease in the order parameter due to atomic vibrations. (b) GBs show an accelerated decrease of order at high temperatures approaching the melting point (1687 K) caused by premelting. The order parameters in c-Si and l-Si (at 1690 K) are shown for comparison.

Figure 9

(a) Temperature profiles for low-angle (relative to ${90}^{\circ }$) GBs at 750 K. The profiles were fitted by Eq. (4) and shifted to zero by subtracting the average temperature of the simulation block. (b) Kapitza conductance of all GBs studied in this work as a function of GB energy at 750 K. Note that the results naturally break into three groups as discussed in the text. The lines represent linear fits intended to highlight the trends. The inlays show the plots of the GB conductance ${\sigma }_K$ versus the tilt angle.

Figure 10

Thermal width $w$ (a) and GB conductivity ${\kappa }_K$ (b) as functions GB energy at 750 K. The lines are drawn to highlight the trends. The inlays show the plots of $w$ and ${\kappa }_K$ versus the tilt angle.

Figure 11

GB conductance of Si GBs at 750 K plotted against the GB excess volume per unit GB area. The color of the points represents the structural groups as in Figs. 9 and 10.

Figure 12

The thermal conductance of GBs plotted as a function of temperature. (a) GBs remaining ordered up to the melting point. (b) GBs undergoing structural disordering at high temperatures. The lines are included as a guide to the eye.

Figure 13

Temperature dependence of thermal conductivity of c-Si (computed by two methods), a-Si, and two representative GBs. The line is a $1/T$ fit of the c-Si data.

Figure 14

Vibrational density of states for diamond silicon at several temperatures.

Figure 15

Atomic vibrational spectra of (210) and (650) GBs in Si in comparison with DOS in c-Si, l-Si, and a-Si at (a) 300 K and (b) 1600 K.

Figure 16

(a) Selected atomic sites (shown in green and labeled A through F) in the (210) GB at 300 K. The sites are ranked according to the distance from the GB center and their energy is plotted as a function of this distance. The lines are drawn to highlight the trends. Panels (b), (c), and (d) show the vibrational density of states of the selected GB sites in the $x$, $y$, and $z$ directions, respectively. The projection on the DOS-frequency plane shows the overall DOS of the GB core in the respective direction in comparison with that of diamond cubic (DC) Si.

Figure 17

(a) Selected atomic sites (shown in green and labeled A through F) in the (650) GB at 300 K. The sites are ranked according to the distance from the GB center and their energy is plotted as a function of this distance. The lines are drawn to highlight the trends. Panels (b), (c), and (d) show the vibrational density of states of the selected GB sites in the $x$, $y$, and $z$ directions, respectively. The projection on the DOS-frequency plane shows the overall DOS of the GB core in the respective direction in comparison with that of diamond cubic (DC) Si.