Lattice thermal conductivities of two ${\mathrm{SiO}}_2$ polymorphs by first-principles calculations and the phonon Boltzmann transport equation

Lattice thermal conductivities of two ${\mathrm{SiO}}_2$ polymorphs, i.e., $\alpha$ quartz (low) and $\alpha$ cristobalite (low), were studied using first-principles anharmonic phonon calculation and linearized phonon Boltzmann transport equation. Although $\alpha$ quartz and $\alpha$ cristobalite have similar phonon densities of states, phonon frequency dependencies of phonon group velocities and lifetimes are dissimilar, which results in largely different anisotropies of the lattice thermal conductivities. For $\alpha$ quartz and $\alpha$ cristobalite, distributions of the phonon lifetimes effective to determine the lattice thermal conductivities are well described by energy and momentum conservations of three phonon scatterings weighted by phonon occupation numbers and one parameter that represents the phonon-phonon interaction strengths.

Figure 1

Crystal structures of $\alpha$ quartz (left) and $\alpha$ cristobalite (right). The space-group types are $P{3}_{2}21$ and $P{4}_1{2}_{1}2$, respectively.

Figure 2

Lattice thermal conductivities of $\alpha$ quartz and $\alpha$ cristobalite calculated at 300 K with different $q$-point sampling meshes using the PBEsol exchange correlation potential. Experimental lattice parameters were employed for these calculations. The lattice thermal conductivities are plotted as a function of a number of sampled phonon modes, i.e., product of a number of sampled $q$ points and $3n_{\text{a}}$, where $n_{\text{a}}=9$ for $\alpha$ quartz and $n_{\text{a}}=12$ for $\alpha$ cristobalite.

Figure 3

Phonon band structures and DOS of (a) $\alpha$ quartz and (b) $\alpha$ cristobalite. In the DOS plots on the right hand sides of the band structures, the solid and dotted curves depict the partial DOS of Si and O, respectively, and the curves under shadow show the total DOS. The special point symbols of wave vectors follow the convention provided in the Bilbao crystallographic server [52].

Figure 4

(a) Mode contributions of lattice thermal conductivities $\kappa \left(\omega \right)$ at 300 K and (b) distributions of direct-vector-products of group velocities $\mathrm{w}\left(\omega \right)$ [see Eq. (9)] calculated for $\alpha$ quartz and $\alpha$ cristobalite with respect to phonon frequency. Both in (a) and (b), dotted and solid curves depict their $xx$ and $zz$ components, respectively.

Figure 5

Ratios between $zz$ and $xx$ elements of cumulative lattice thermal conductivities, ${\kappa }_{zz}^{\text{c}}\left(\omega \right)/{\kappa }_{xx}^{\text{c}}\left(\omega \right)$, in $\alpha$ quartz (solid curve) and $\alpha$ cristobalite (dashed-dotted curve) at 300 K.

Figure 6

Phonon lifetimes of $\alpha$ quartz and $\alpha$ cristobalite at 300 K with respect to phonon frequency. Each dot corresponds to one phonon mode. The points are sampled on the $19\times 19\times 19$ mesh for $\alpha$ quartz and $19\times 19\times 14$ mesh for $\alpha$ cristobalite in the respective Brillouin zones.

Figure 7

$\tilde{\kappa }\left(\omega \right)$, mode contributions of lattice thermal conductivities of $\alpha$ quartz and $\alpha$ cristobalite calculated with $\tilde{P}={\tilde{P}}_{\text{av}}$ at 300 K as a function of phonon frequency. Dotted and solid curves depict ${\tilde{\kappa }}_{xx}$ and ${\tilde{\kappa }}_{zz}$, respectively.

Figure 8

${\left(3n_{\mathrm{a}}\right)}^2{P}_{\lambda }$ of $\alpha$ quartz and $\alpha$ cristobalite with respect to phonon frequency. Here ${\left(3n_{\mathrm{a}}\right)}^2$ is multiplied with $P_{\lambda }$ to align the scale between $\alpha$ quartz and $\alpha$ cristobalite. Each dot corresponds to one phonon mode. The points are sampled on the $19\times 19\times 19$ mesh for $\alpha$ quartz and $19\times 19\times 14$ mesh for $\alpha$ cristobalite in the respective Brillouin zones.

Figure 9

${\tilde{\tau }}_{\lambda }$, phonon lifetimes of $\alpha$ quartz and $\alpha$ cristobalite calculated with $\tilde{P}={\tilde{P}}_{\text{av}}$ at 300 K as a function of phonon frequency (black dots). To compare, ${\tau }_{\lambda }\times {10}^{-1}$ (Fig. 6) are shown as the gray dots behind the black dots. The points are sampled on the $19\times 19\times 19$ mesh for $\alpha$ quartz and $19\times 19\times 14$ mesh for $\alpha$ cristobalite in the respective Brillouin zones.

Figure 10

Lattice thermal conductivities at 300 K with respect to cutoff distances of atomic pairs used to compute third-order force constants employing (a) $\alpha$-quartz $3\times 3\times 2$ supercell (162 atoms), (b) $\alpha$-quartz $2\times 2\times 2$ supercell (72 atoms), (c) $\alpha$-quartz unit cell (9 atoms), (d) $\alpha$-cristobalite $3\times 3\times 2$ supercell (216 atoms), (e) $\alpha$-cristobalite $2\times 2\times 2$ supercell (96 atoms), and (f) $\alpha$-cristobalite unit cell (12 atoms). The selected cutoff distances are those closest to but below (a) $2,...,11\AA$, (b) $2,...,8\AA$, (c) 2, 3, 3.5, and 4 Å, (d) $2,...,12\AA$, (e) $2,...,10\AA$, and (f) $2,...,5\AA$, respectively. The filled circles depict ${\kappa }_{zz}/2$ for $\alpha$ quartz and ${\kappa }_{zz}$ for $\alpha$ cristobalite, and the open circles show ${\kappa }_{xx}$. The cross symbols present the numbers of supercells with displacements that were used to compute the third-order force constants with the respective cutoff distances. The rightmost points correspond to the results obtained without using the cutoff distances. Lines are eye guides.