Electrical conductivity of SiO₂ at extreme conditions and planetary dynamos

Significance

We find that Earth’s earliest magnetic field may have been produced in a deep magma ocean, and that silicate dynamos may exist in super-Earth exoplanets as well. Ab initio molecular dynamics simulations show that silicate liquids are semimetallic at the extreme pressure and temperature conditions characteristic of planetary interiors. The electrical conductivity shows a remarkable nonmonotonic dependence on pressure that reveals connections to the underlying atomic structure, and highlights broken charge ordering as a novel compression mechanism.

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.

Planetary magnetic fields are produced by a dynamo process via fluid motions in a large rotating body of electrically conducting fluid within the planet’s interior. In present-day Earth, the liquid portion of the iron-rich core produces the magnetic field. Early in Earth’s history and before the inner core began to grow, the core may not have been able to cool sufficiently rapidly to sustain a dynamo (1). However, the rock record contains evidence for an ancient field of similar intensity to today’s field within the first few 100 Ma of Earth’s history (2). What caused this early magnetic field is still unknown.

Earth is thought to have been largely or completely molten early in its history (3). While the shallow portions of the magma ocean cooled quickly (4), a basal magma ocean, separated from the surface by a crystallizing layer, may have survived for a billion years or more (5). Could the basal magma ocean have produced a magnetic field? While a variety of different materials produce planetary magnetic fields, including iron, hydrogen, and ice, silicate dynamos are so far unknown (6).

A key uncertainty is the electrical conductivity $\sigma$ of silicate liquid at the pressure–temperature conditions of the basal magma ocean (100 GPa, 5,000 K). The conductivity must be sufficiently high for the dynamo process to operate. Recent models indicate that $\sigma >{\mathrm{\hspace{0.17em}10}}^3\text{S}\cdot {\text{m}}^{-1}$ to ${10}^4$ S$\cdot$${\text{m}}^{-1}$ is required (7). The possibility of silicate dynamos is not only relevant to early Earth. Silicates appear to be the primary constituents of many super-Earth exoplanets. These planets may have hotter interiors that cool more slowly than Earth and may contain larger and longer-lived basal magma oceans, so that Super-Earths may also have silicate dynamos. The conditions at the base of a 10-Earth-mass Super-Earth mantle are expected to be 1,000 GPa and 13,000 K (8).

Near ambient pressure, the electrical conductivity of silicate liquids is far too small to support dynamo activity, and the dominant charge carriers are ions (9). However, experimental evidence suggests that the electrical conductivity of silicate liquids may be much greater at high pressure and temperature and that the dominant charge carriers may be electrons. Oxide liquids are found to become optically reflective along the Hugoniot at pressures of several hundred gigapascals (10–12). The electrical conductivity is not measured in shock wave studies, but is inferred from the optical reflectivity via a Drude model. Previous theoretical calculations have relied on approximations to the exchange–correlation functional which are known to underestimate the electronic band gap (13, 14), leaving open the question of whether these simulations substantially overestimated $\sigma$.

Existing studies leave unclear the mechanism by which silicate liquids become reflective at high pressure. It is known from experiment and theory that the pressure-induced change of the structure of silicate liquids can be characterized by an increase in the Si–O coordination number from 4 near ambient pressure toward 9 at high pressure (15–17). A recent theoretical study argued that increased coordination was responsible for enhanced electrical conductivity at high pressure (14). However, the connection between coordination change and conductivity is not apparent. It has been suggested that the changes in silicate liquid structure on compression can be characterized by dissociation which, by analogy with that seen in hydrogen and oxygen, might explain closure of the electronic band gap (10, 13). However, hydrogen and oxygen are molecular rather than ionic systems, and it is not clear the extent to which the analogy is suitable.

Here, we find that the electronic conductivity of liquid SiO₂ is sufficiently large to support a silicate dynamo, based on a more accurate approximation to the exchange–correlation functional (HSE06) (18) than used in previous studies. Our results further show a remarkable nonmonotonic dependence of the electrical conductivity on compression. This behavior is crucial for revealing the connection between structural change and electronic and thermodynamic properties at high pressure. We show that a new mode of high-pressure structural change, broken charge ordering, is responsible for the variation of electrical conductivity with density and temperature.

Our first-principles molecular dynamics simulations show that, at high temperature ($T\ge \mathrm{20,000}$ K), the electrical conductivity increases monotonically with compression, reaching half the minimum metallic value at the highest pressure–temperature conditions explored (Fig. 1). For example, at 50,000 K, the conductivity increases by a factor of 3 from $\rho$ = 2.2 g$\cdot$cm⁻³ to $\rho$ = 7.33 g$\cdot$cm⁻³ where $\sigma$ = $4\times {\mathrm{\hspace{0.17em}10}}^5$ S$\cdot$m⁻¹. The electrical conductivity increases with increasing temperature at all densities, for example, by a factor of 30 between 4,000 K and 50,000 K at $\rho$ = 3.67 g$\cdot$cm⁻³.

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).

At lower temperature ($T\le \mathrm{10,000}$ K), the conductivity displays a maximum value at intermediate compression (Fig. 1). The conductivity increases with compression from $\rho$ = 2.2 g$\cdot$cm⁻³ to $\rho$ = 3.67 g$\cdot$cm⁻³ and then decreases on further compression, by a factor of 10 from $\rho$ = 3.67 g$\cdot$cm⁻³ to $\rho =\mathrm{\hspace{0.17em}7.33}$ g$\cdot$cm⁻³ at 6,000 K. The tendency for the conductivity to decrease with compression at very high pressure was also noted in a previous study (14), although that study did not explore the pressure–temperature range necessary to reveal the extremum that we find. The electrical conductivity computed with the HSE06 functional is lower than that computed with PBEsol, as expected, although the difference is not large [$<25$% (Fig. 1)].

To understand our results, we have examined the electronic structure (Fig. 2 and Fig. S1). At high temperature, the electronic density of states (DOS) is nearly free-electron-like. There is no gap, and the DOS at the Fermi level is within 5% of the free-electron value. As the valence electrons are only weakly bound, we expect the Ziman theory (20) to yield a reasonable approximation to the conductivity. As expected, the Ziman formula reproduces our first-principles results for the electrical conductivity to within 30% at $T\ge$20,000 K (Fig. S6). It fails, however, at lower temperature: The Ziman theory cannot capture the local maximum in $\sigma$ that appears near $\rho$ = 3.67 g$\cdot$cm⁻³ for $T\le$10,000 K.

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).

The Ziman formula fails at lower temperature because the electronic structure is no longer free-electron-like (Fig. 2). A prominent pseudogap appears in the electronic DOS, which deepens on compression. The appearance of a pseudogap on cooling, which is in excellent agreement with experimental X-ray absorption near-edge structure (XANES) spectra (21), has a profound effect on the electrical conductivity. To see this, we recall that the conductivity diminishes with the DOS at the Fermi level. In the Mott–Ziman theory (22, 23),

$${\sigma }_{MZ}=g^2{\sigma }_Z,$$ [1]

where ${\sigma }_Z$ is the Ziman result and

$$g=\frac{\overline{N}(\mu )}{N_{Free}(E_F)}$$ [2]

is the ratio of the temperature-smoothed DOS at the chemical potential $\overline{N}(\mu )$ to that of the free-electron system $N_{Free}$. At 10,000 K, the value of $g^2$ diminishes from 0.64 at $\rho =3.67$ g$\cdot$cm⁻³ to 0.22 at $\rho =7.33$ g$\cdot$cm⁻³, accounting for the drop in conductivity over this density range. The good agreement between our results and the Mott–Ziman theory (to within 30% at all conditions; Fig. 1) shows that the pressure-induced decrease in the conductivity at high pressure and low temperature is caused by the decrease in $g$ as the pseudogap deepens on compression.

It remains to relate the appearance of the pseudogap to the structure of the liquid. For liquids, in the nearly free electron limit, the pseudogap appears because of scattering of electronic states from density fluctuations in the liquid as measured by the structure factor, $S(q)$ (24, 25). The amplitude of the first peak in the structure factor varies considerably on compression and cooling (Fig. 3 and Fig. S3). The depth of the pseudogap $\delta$ increases linearly with the amplitude of the first peak in the structure factor $S(q_P)$ as $\delta \approx 2[S(q_P)-1]|w(q_P)|$, where $w(q)$ is the effective electron–ion interaction (25). For example, at 10,000 K, $S(q_P)-1$ increases fourfold from $\rho$ = 3.67 g$\cdot$cm⁻³ to $\rho$ = 7.33 g$\cdot$cm⁻³, in reasonable accord with the threefold decrease in the DOS at the Fermi level over the same range of compression. We see a similar pattern on cooling: As $S(q_P)$ grows (Fig. 3), the pseudogap deepens and the electrical conductivity diminishes.

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 ($2k_F$).

Our simulations highlight a novel mode of compression in silicate liquids that accounts for the growth of the first peak in the structure factor on compression (Fig. 4 and Figs. S4 and S5). At low pressure, it is well known that silicate liquid structure is dominated by charge ordering as is typical of ionically bonded systems: Cations are surrounded by a first-neighbor shell of anions and vice versa, producing well-separated peaks for Si–O and O–O correlations. For example, in SiO₂ liquid at low pressure, the O–O distance (2.6 Å) is much larger than the Si–O distance (1.6 Å) (27). At very high pressure, we find that charge ordering breaks down. Like charges begin to appear in the first coordination shell, Si–O and O–O peaks overlap, and the liquid structure approaches that of isoelectronic neon (Fig. 4). This peak overlap accounts for the large increase in the amplitude of the first peak in the structure factor as density increases from $\rho$ = 3.67 g$\cdot$cm⁻³ to $\rho$ = 7.33 g$\cdot$cm⁻³ (Fig. 3).

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.

Broken charge ordering is also important in high-pressure silicate crystal structures. The novel high-pressure structure of ref. 28 shows remarkably high 10-fold Si–O coordination. Using the reported atomic coordinates and lattice parameters, we have analyzed the predicted structure and found another remarkable feature: The smallest O–O distance (1.754 Å) is less than the smallest Si–O distance (1.790 Å). The crowding of O atoms at high pressure leads to more-efficient packing as shorter-ranged repulsive forces begin to dominate over longer-ranged Coulombic forces. This behavior recalls predictions in simple ionic systems, such as CsI, where the structure approaches that of isoelectronic Xe at high pressure (29). In SiO₂, much higher pressures are required to break the charge ordering compared with CsI, because the ionic bonding in SiO₂ is much stronger. Crowding and increased interaction among O atoms also influences the electronic structure by producing increased hybridization of the O valence states with conduction bands, dominating the electronic DOS in the vicinity of the pseudogap (Fig. 2).

Previous studies have interpreted the variation of the heat capacity along the Hugoniot to indicate dissociation of SiO₂ at large compressions, a pattern of structural change very different than what we have described. To examine this difference more closely, we have computed the heat capacity along the Hugoniot from our simulations. The ab initio heat capacity shows variations with temperature along the Hugoniot very similar to that found experimentally (Fig. 5). We interpret these variations as nondissociative changes in atomic and electronic structure that are common to a wide variety of liquids.

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).

At the highest temperatures ($T>$30,000 K), the heat capacity increases on heating, reflecting the increase with temperature of the electronic contribution to the heat capacity, $C_V^{el}$. The increase of $C_V^{el}$ on heating reflects the growing DOS at the Fermi level with increasing temperature (Fig. 2), behavior that is seen in many silicate liquids (30). At intermediate temperature (10,000 K$<T<$30,000 K), the heat capacity decreases with increasing temperature. At these conditions, the nonelectronic contribution to the heat capacity $C_V^{ion}$ dominates. This behavior is seen in a wide variety of liquids, including simple, nonbonded liquids interacting through hard-sphere or soft-sphere potentials. Fundamental measure theory (31) predicts $C_V\propto T^{-2/5}$ in approximate accord with our results for silica liquid. We have previously found that a variety of silicate liquids closely follow this relation over a wide range of temperatures (30). The heat capacity is everywhere much larger than the Dulong–Petit value, reflecting structural degrees of freedom in the liquid that are unavailable to solids. Indeed, previous studies have shown that the structure of silicate liquids changes substantially on isochoric heating. For example, in SiO₂ liquid at low pressure, the liquid structure, characterized by almost perfect fourfold Si–O coordination at the lowest temperatures, shows increasing numbers of 5- and 3-coordinated Si on isochoric heating (27). The rate of structural change initially increases on heating, and then saturates, producing a local maximum in the heat capacity (32).

Our first-principles predictions of the optical reflectivity agree well with experimental measurements, although experimental uncertainties are significant (Fig. 5). We find that HSE06 predicts smaller values of the reflectivity in better agreement with experiment compared with PBEsol. The smaller value of the HSE06 reflectivity is expected: HSE06 predicts band gaps in crystalline polymorphs of SiO₂ that are in excellent agreement with experiment and much larger than those predicted by PBEsol ($\sim$50%), apparently overcoming the well-known tendency of many functionals to seriously underpredict the band gap (33). In this context, the difference between our HSE06 and PBEsol results is perhaps surprisingly small, amounting to less than 0.02. However, it is important to remember that, while different functionals are often compared in terms of the predicted band gap, our system has no gap, and arguments based on the behavior of functionals in wide-gap systems may not apply to the liquid. We believe that the absence of a gap in our system mutes the effects of exact exchange, reducing the difference between HSE06 and PBEsol. We have not considered explicit temperature dependence of the exchange–correlation functional in our calculations, as a recent investigation shows these effects to be small, amounting to 10% in the value of $\sigma$ (34).

Our results indicate that the magnetic field in early Earth could have been generated by a silicate dynamo. From our results (Fig. 1), we find an interpolated value of the electrical conductivity $\sigma =\mathrm{5,700}$ S$\cdot$m⁻¹ at 100 GPa and 5,000 K. For a 10-Earth-mass super-Earth, we find an even higher value: $\sigma =\mathrm{125,500}$ S$\cdot$m⁻¹ at 1,000 GPa and 13,000 K. While these values of $\sigma$ are submetallic, they exceed the minimum value required in magma ocean dynamo models (7) and those of planetary dynamos in our solar system: The conductivity of the dynamo generating regions of Neptune and Uranus is thought to be 2,000 S$\cdot$m⁻¹ (35). Our predicted values for pure silica may underestimate the conductivity of the magma ocean: Additional components, particularly FeO and CaO, may increase the conductivity. Further, it is known that MgO and MgSiO₃ become optically reflective along the Hugoniot at conditions similar to those at the base of a super-Earth magma ocean (11, 36). We look forward to a broader class of magnetohydrodynamic simulations of the ancient field to include the possibility of an early silicate dynamo. The analysis of our results in terms of the Mott–Ziman theory provides a framework for understanding optical, electrical, and thermodynamic properties in these systems, as well as other liquids that may be accessible to experimental investigation at extreme conditions.

Materials and Methods

Our molecular dynamics simulations are based on density functional theory in the PBEsol (37) approximation, using the projector augmented wave method (38), as implemented in the VASP code (39). Born–Oppenheimer simulations are performed in the canonical ensemble using the Nosé–Hoover thermostat with 96 atoms and run for 10 ps to 15 ps with a 1-fs time step. We assume thermal equilibrium between ions and electrons via the Mermin functional (40). Sampling the Brillouin zone at the Gamma point and a basis-set energy cutoff of 500 eV were found to be sufficient to converge energy and pressure to within 2 meV/atom and 0.2 GPa, respectively. For comparison, we also performed ab initio simulations of neon with the same settings. In addition to standard thermodynamic quantities, we also compute the isochoric heat capacity

$$C_V(V\text{,}T)=C_V^{\mathrm{ion}}(V\text{,}T)+C_V^{\mathrm{el}}(V\text{,}T)$$ [3]

with the ionic contribution given by the fluctuation formula (41)

$$C_V^{\mathrm{ion}}=\frac{{⟨\mathrm{\Delta }E⟩}^2}{k_B{T}^2}$$ [4]

where ${⟨\mathrm{\Delta }E⟩}^2$ is the mean squared fluctuation in the internal energy, and the electronic contribution $C_V^{\mathrm{el}}$ by an accurate fit to the electronic entropy versus temperature using a generalization of the functional form used in our previous studies (30) (Fig. S8 and Supporting Information).

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).

We compute the electrical conductivity via the Kubo–Greenwood formula (42) and the dielectric constant using the Random Phase Approximation (RPA) (43) as implemented in VASP from a series of at least 10 uncorrelated snapshots at each volume–temperature condition. We perform dielectric constant calculations in both PBEsol and HSE06 (18). We found that a 3 × 3 × 3 k-point mesh and 2,000 electronic bands were sufficient to yield converged results. We also compared our findings with the results of a larger system (144 atoms) and found no significant differences in electronic DOS, electrical conductivity, or dielectric constant. We compute the electronic DOS by averaging over at least 20 snapshots well separated in time from the molecular dynamic (MD) trajectories. We sample the Brillouin zone on a $3\times \mathrm{\hspace{0.17em}3}\times \mathrm{\hspace{0.17em}3}$ mesh, and use the Fermi–Dirac distribution with temperature equal to the ionic temperature to smooth the DOS.

Electronic DOS

We compute the electronic DOS by averaging over at least 20 uncorrelated snapshots from our MD trajectories (Fig. S1). We sample the Brillouin zone with k-point meshes up to $5\times 5\times 5$ k points, although convergence is usually achieved for a $3\times 3\times 3$ mesh. We use the Fermi–Dirac distribution with temperature equal to the ionic temperature to smooth the DOS.

Ab Initio Electrical Conductivity

For PBEsol calculations, we compute the static (DC) electrical conductivity from ab initio wave functions using the Kubo–Greenwood formalism as implemented in VASP (42) (Fig. S2). Assuming isotropy, the elements of the conductivity tensor satisfy the relations ${\sigma }_{\alpha \beta }=\sigma {\delta }_{\alpha \beta }$ where ${\delta }_{\alpha \beta }$ is the Kroenecker delta. The frequency-dependent conductivity is

$$\sigma (\omega )=\frac{2\pi e^2{\mathrm{\hslash }}^2}{3m^2\omega \mathrm{\Omega }}\sum _{\overrightarrow{k}}\sum _{ij}[f({ϵ}_i,\overrightarrow{k})-f({ϵ}_j,\overrightarrow{k})]\times {|⟨{\psi }_{i,\overrightarrow{k}}|\nabla |{\psi }_{j,\overrightarrow{k}}⟩|}^2\delta ({ϵ}_i-{ϵ}_j-\mathrm{\hslash }\omega ),$$ [S1]

where the sums are over the Brillouin zone and pairs of states, $f$ is the Fermi occupation, $\psi$ is the wave function, $ϵ$ is the eigenvalue, $\omega$ is the frequency, and $\mathrm{\Omega }$ is the volume of the simulation cell.

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).

In our computations, the $\delta$ function is replaced by a Gaussian of width $\mathrm{\Delta }$ given by the average spacing between eigenvalues weighted by the corresponding change in the Fermi function (44). As the behavior of Eq. S1 becomes unphysical for $\mathrm{\hslash }\omega <\mathrm{\Delta }$, we find the DC conductivity by linearly extrapolating to zero frequency.

Ab Initio Optical Properties

We compute the complex dielectric constant $ϵ$ via RPA as implemented in VASP (43). The reflectivity is (45)

$$r(\omega )=\frac{{[n_0(\omega )-n(\omega )]}^2+{[k_0(\omega )-k(\omega )]}^2}{{[n_0(\omega )+n(\omega )]}^2+{[k_0(\omega )+k(\omega )]}^2},$$ [S2]

where $n=\mathrm{Re}(\kappa )$, $k=\mathrm{Im}(\kappa )$, $\kappa =\sqrt{ϵ}$, and subscript 0 indicates the values of the unshocked material. We use established values (46) $n_0({\omega }_x)=\mathrm{\hspace{0.17em}1.461}$ for fused silica and $n_0({\omega }_x)=\mathrm{\hspace{0.17em}1.546}$ for quartz at ambient conditions, and the frequency of the experimental probe ${\omega }_x$ = 532 nm (2.33 eV); $k_0({\omega }_x)=\mathrm{\hspace{0.17em}0}$ for both phases.

For HSE calculations, we compute the electrical conductivity from the dielectric constant (Fig. S2)

$$\sigma (\omega )=\omega \mathrm{Im}(ϵ).$$ [S3]

We have found that the frequency dependence of the conductivity can be represented by a Drude form plus a Lorentz peak,

$$\sigma (\omega )=A{\omega }^2/({(B-{\omega }^2)}^2-C{\omega }^2)+\sigma (0)/(1+{\omega }^2{\tau }^2),$$ [S4]

which then yields the DC conductivity $\sigma (0)$ as a parameter of the fit.

Structure Factor

Fig. S3 shows our results for the total structure factor. To evaluate the suitability of the nearly free electron picture, we compare the form of the structure factor to that of the electronic DOS. In this limit, the gap appears at the energy $W\approx {\mathrm{\hslash }}^2/(2m){(q_P/2)}^2$ above the bottom of the valence band (24), where $q_P$ is the position of the first peak in the structure factor. For example, at 7.33 g$\cdot$cm⁻³ and 10,000 K, the first peak in the structure factor is located at $q_P=\mathrm{\hspace{0.17em}4.3}{\text{Å}}^{-1}$ (Fig. S3), yielding $W=\mathrm{\hspace{0.17em}18}$ eV, in reasonable agreement with the position of the pseudogap in our simulations (Fig. S1).

We further illustrate the structure of the liquid via the partial radial distribution functions (Fig. S4) and a snapshot from the simulation at 7.33 g$\cdot$cm⁻³ and 50,000 K (Fig. S5).

Mott–Ziman Formalism

The electrical conductivity in the Mott–Ziman theory is given by

$${\sigma }_{MZ}^{-1}=\frac{a_{0}h}{g^{2}2e^2}\frac{k_{TF}^2}{64\pi Z{E}_F^2}\underset{0}{\overset{2k_F}{\int }}{q}^{3}S(q)u{(q)}^{2}dq,$$ [S5]

where $S(q)$ is the structure factor from our simulations, $k_F$ and $k_{TF}$ are the Fermi and Thomas–Fermi wave vectors, respectively, $E_F$ is the Fermi energy, the conductivity quantum $2e^2/h{a}_0=\mathrm{\hspace{0.17em}1.464}\times {10}^6$ S$\cdot$m⁻¹, and we have assumed an unscreened pseudopotential $u(q)=-4\pi Z{e}^2/q^2$ with $Z=4$ for Si and for O. We note that, in general, for a binary system, the total structure factor must be replaced by (47)

$$S(q)u{(q)}^2\to S_{22}(q)u_2{(q)}^2+(1-x)S_{11}(q)u_1{(q)}^2+2{[x(1-x)]}^{1/2}{S}_{12}(q)u_1(q)u_2(q),$$ [S6]

where $S_{ab}$ are the Ahscroft–Langreth partial structure factors, $u_a$ is the pseudopotential of component $a$, and $x$ is the concentration of species 2. In our case, the effective valences of both species are taken to be equal ($Z$ = 4), corresponding to Be and Ne cores for O and Si, respectively. Therefore, the unscreened pseudopotentials $u_1=u_2=u$, and these terms can be factored out: The combined contribution of the partial structure factors is just the total structure factor, and the right-hand side of Eq. S6 reduces to $S(q)u^2(q)$.

Our Mott–Ziman results are shown as dashed lines in Fig. 1. For comparison, we show our Ziman results ($g=\mathrm{\hspace{0.17em}1}$) in Fig. S6.

Hugoniot

We compute the ab initio Hugoniot (Fig. S7) from

$$E(V,T)-E_0=\frac{1}{2}[P(V,T)+P_0](V_0-V),$$ [S7]

where $E_0,P_0,\text{and}{V}_0$ are the energy, pressure, and volume of the initial unshocked state. We compute, with VASP, $E_0$ and $V_0$ at $P_0=\mathrm{\hspace{0.17em}0}$ for initial unshocked states: quartz and fused silica. The latter is obtained by quenching our MD simulations at $\rho =$2.20 g$\cdot$cm⁻³. Solutions to Eq. S7 are found via interpolation of our MD results (48).

Fig. S7.

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

Heat Capacity

Under the assumption of thermal equilibrium between ions and electrons, the total heat capacity is

$$C_V(V,T)=C_V^{\mathrm{ion}}(V,T)+C_V^{\mathrm{el}}(V,T).$$ [S8]

We compute the ionic contribution via the fluctuation formula (41)

$$C_V^{\mathrm{ion}}=\frac{{⟨\mathrm{\Delta }E⟩}^2}{k_B{T}^2},$$ [S9]

where ${⟨\mathrm{\Delta }E⟩}^2$ is the mean squared fluctuation in the internal energy.

Our results for the electronic contribution to the heat capacity at all volumes and temperatures are summarized in Fig. S8. We obtain the electronic contribution from the electronic entropy, which we have found is well represented by

$$S^{\mathrm{el}}(V,T)=\lambda (V){[\ln (T/T_0)]}^2,$$ [S10]

yielding

$$C_V^{\mathrm{el}}=\lambda (V)\ln (T/T_0).$$ [S11]

In Fig. S8, we show an example of the fit obtained. The functional form is a generalization of that used in our previous work over a more limited range of temperatures (30).

In Fig. S9, we show the temperature dependence of the total heat capacity $C_V$ calculated at different volumes for liquid SiO₂. We find a local maximum at temperatures 6,000 K$<T<$ 10,000 K for all volumes. At higher temperatures, the electronic contribution dominates and the heat capacity increases monotonically on heating.

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).

The method we have used for computing the heat capacity has the advantage that variations of $C_V$ with temperature are more robustly recovered. The heat capacity may also be computed via finite difference

$$\overline{C_V}(\overline{T})=\frac{E(T_2)-E(T_1)}{T_2-T_1},$$ [S12]

where the overbars indicate mean values. We demonstrate that our values of the heat capacity are consistent with values computed via Eq. S12 in Fig. S9.