Electrical conductivity of SiO2 at extreme conditions and planetary dynamos

Abstract

Ab intio molecular dynamics simulations show that the electrical conductivity of liquid SiO₂ is semimetallic at the conditions of the deep molten mantle of early Earth and super-Earths, raising the possibility of silicate dynamos in these bodies. Whereas the electrical conductivity increases uniformly with increasing temperature, it depends nonmonotonically on compression. At very high pressure, the electrical conductivity decreases on compression, opposite to the behavior of many materials. We show that this behavior is caused by a novel compression mechanism: the development of broken charge ordering, and its influence on the electronic band gap.

Keywords: Earth’s mantle; density functional theory; electrical conductivity; high pressure; silicate liquids.

Fig. 1.

Ab initio electrical conductivity computed with PBEsol (full lines and closed symbols) and with HSE06 (open symbols). We also compare with the Mott–Ziman theory (dashed lines; see text) and the minimum metallic conductivity of Mott (19) (gray band).

Fig. 2.

Electronic DOS at (Top) 50,000 K and (Bottom) 10,000 K over the density range explored compared with the free-electron DOS at $\rho$ = 3.67 g$\cdot$cm⁻³ (black dashed line), and the contribution of the oxygen p band at the same density (dashed green line).

Fig. S1.

Electronic DOS for different densities and temperatures. The continuous blue line is the free-electron DOS.

Fig. S6.

Ab initio electrical conductivity computed with PBEsol (full lines and closed symbols) and with HSE06 (open symbols) compared with the Ziman formula ($g=1$, dashed lines).

Fig. 3.

Structure factors at (Top) $\rho$ = 7.33 g$\cdot$cm⁻³ and (Bottom) $T$ = 10,000 K.

Fig. S3.

Total structure factor for SiO₂ at different densities and temperatures. The dashed black lines indicate the upper limit of the Ziman integral ($2k_F$).

Fig. 4.

Radial distribution function of SiO₂ (green) compared with that of isoelectronic Ne (black) at 10,000 K at lesser (Top) and greater (Bottom) density. (Inset) The comparison at $\rho$ = 7.33 g$\cdot$cm⁻³ and 50,000 K (SiO₂: red, Ne: black). Schematics illustrate (Top) charge-ordering characteristic of silicate liquids, consisting of alternating neighbor shells of cations (blue) and anions (red), and (Bottom) the broken charge-ordered structure characteristic of extreme pressure in which the nearest-neighbor shell contains like as well as unlike charged ions.

Fig. S4.

Total radial distribution function of SiO₂ (green) and the partial radial distribution functions O–Si (solid red), O–O (long-dashed red), and Si–Si (short-dashed red) at 10,000 K and the density indicated and (Inset) at $\rho$ = 7.33 g$\cdot$cm⁻³ and 50,000 K.

Fig. S5.

Snapshot from the simulation at $\rho$ = 7.33 g$\cdot$cm⁻³ and 50,000 K showing (blue) Si and (red) O atoms and distances nearer than 2.0 Å shown as bonds. We note that the choice of distance cutoff is somewhat arbitrary, as the first coordination shell is not well defined at these conditions, as is evident from the radial distribution function (Fig. S4). Our choice emphasizes the similar number of O–O to O–Si nearest nieghbors and thus the breakdown of charge ordering.

Fig. 5.

(Top) Isochoric heat capacity and (Bottom) reflectivity at 532 nm for SiO₂ along the fused silica (red) and quartz (blue) Hugoniots. Lines are our results, while experimental data are shown as open circles (10) and squares (26). The black circles indicate the heat capacity computed via finite difference between the two Hugoniots. In Bottom, the colored bands indicate the sensitivity to the exchange–correlation functional comparing PBEsol results (upper lines) with those of HSE06 (lower lines).

Fig. S8.

Electronic heat capacity as a function of temperature at various densities from our simulations. (Inset) Electronic entropy from our simulations at $\rho$ = 7.33 g$\cdot$cm⁻³ (symbols) and the fit to these results (Eq. S10).

Fig. S2.

Comparison of electrical conductivity computed from the Kubo–Greenwood method at $\rho$ = 7.33 g$\cdot$cm⁻³ and $T$ = 10,000 K (green), $\rho$ = 7.33 g$\cdot$cm⁻³ and $T$ = 20,000 K (orange), and $\rho$ = 3.67 g$\cdot$cm⁻³ and $T$ = 10,000 K (black). Also shown is the conductivity computed from the dielectric constant at $\rho$ = 7.33 g$\cdot$cm⁻³ and $T$ = 10,000 K in PBEsol (short-dashed green) and HSE06 (long-dashed green).

Fig. S7.

Principle Hugoniots of fused silica (red) and quartz (blue) as computed from our first principles simulations (lines) and measured experimentally (symbols) (10).

Fig. S9.

Total heat capacity as a function of temperature at various densities from our simulations. (Inset) Comparison of the total heat capacity computed with the fluctuation formula (Eq. S9) and the electronic entropy (Eq. S11) at $T$ = 8,000 K (blue) and $T$ = 10,000 K (red) to that computed via finite difference (Eq. S12) with $T_1$ = 8,000 K and $T_2$ = 10,000 K (gray).