Main

Paleomagnetic studies indicate that Earth’s magnetic field was present by at least 3.45 billion years ago (Ga; for example, refs. 1,2,3) and possibly by 4.2 Ga (ref. 4). At present, Earth’s dynamo is driven by thermochemical convection in the liquid outer core. The solid inner core is thought to be essential for the dynamo, because the release of heat and light elements upon inner core crystallization facilitates the necessary thermal and compositional convection. However, this mechanism may not have worked for early Earth. The high thermal conductivities of Fe and Fe alloys at the core–mantle boundary (CMB), as inferred from recent experiments (90–250 W m−1 K−1)5,6, indicate that such rapid cooling and, therefore, the solid inner core are young (<700 Myr). These high conductivities are broadly consistent with estimates from density functional theory molecular dynamics (DFT-MD) simulations (75–150 W m−1 K−1)7,8. A young inner core (<1.5 Ga) is supported by paleomagnetic interpretations (for example, refs. 9,10) and thermal evolution models (for example, refs. 11,12). However, some measurements have indicated a possible lower thermal conductivity of iron metal, which would imply an older inner core age13.

Several models have been proposed to explain the presence of an early (<1.5 Ga) geodynamo without the solid inner core, including (1) a thermally driven convection dynamo in the liquid core (for example, ref. 14), (2) precession and tides by the Moon15, and (3) an exsolution-driven dynamo, where the precipitation of light elements releases gravitational energy and drives core convection, and dynamo generation16,17.

An alternative, which is the main focus of this work, is dynamo generation in a basal magma ocean (BMO). At the end of the planet accretion phase, especially after the Moon-forming impact, Earth would have had a deep magma ocean (MO; for example, refs. 18,19). During MO crystallization, melt sinks towards the CMB and forms a BMO if the melt is denser than the ambient solid mantle where the mantle adiabat crosses the liquidus20. Even if the melt were initially less dense than the surrounding solid mantle, the melt probably densifies by FeO enrichment, as FeO behaves as an incompatible element (for example, refs. 21,22). As the BMO cooling rate is controlled by the solid-state mantle convection above the BMO, the BMO lifetime could be several billion years20,23.

With sufficiently high conductivity, an Fe-enriched BMO at the CMB can produce a dynamo for billions of years24,25. The viability of a dynamo in a convection zone is often assessed by the magnetic Reynolds number Rm. The magnetic Reynolds number is an approximate ratio of advection to magnetic diffusivity terms in the induction equation and is defined as

$${R}_\mathrm{m}={\mu }_{0}vL\sigma ,$$
(1)

where μ0 is the permeability of free space, v is the convective flow velocity, L is a scale for convection in the flow and σ is the electrical conductivity. Numerical simulations26 indicate that a self-sustaining dipolar dynamo requires Rm > 40 for a rapidly rotating planetary object (with a small Rossby number).

BMOs can also occur in other terrestrial planets and moons (for example, ref. 27), including present-day Mars28,29 and super-Earths, which are massive Earth analogues ranging from 1 to 10 Earth masses M⊕ with radii typically less than 1.6 R⊕. Super-Earths orbiting close to their host stars or those with heat sources, such as tides, can host surface or deep MOs (‘lava worlds’; for example, ref. 30). By contrast, super-Earths can naturally have long-lived BMOs without external heat sources given that iron enrichment and densification of melt during crystallization is a natural process (for example, ref. 31). Iron generally becomes more incompatible under high pressures, partitioning into and densifying the melt (for example, ref. 31), making it more probable that super-Earths have BMOs32. Another interesting factor is that a BMO would decelerate core-cooling by acting as a heat sink and, thereby, hinder a core convection-driven dynamo33, initially making the BMO a potentially primary location for dynamo generation. Mini-Neptunes, which can be as massive as 20 M⊕ with radii of 1.7–3.9 R⊕, are planets with thick H- and He-rich atmospheres and icy or rocky cores. If mini-Neptunes have MOs34, they may also be candidates to host MO-driven dynamos. Dynamo generation in these exoplanets could potentially be tested by future observations of planetary magnetic fields, although such detections remain challenging (for example, refs. 35,36).

Whether a BMO or MO can produce a dynamo critically depends on its electrical conductivity. For Earth’s BMO to produce a dynamo, the electrical conductivity σ must exceed 104 S m−1 to satisfy Rm > 40 (L = 300 km and v = 1 cm s−1)37. The measured electrical conductivities of silicates and oxides below 140 GPa are lower than 102.5 S m−1 (refs. 38,39). By contrast, their electrical conductivity can reach 104 S m−1 to 105 S m−1 at ~400–1,400 GPa, a pressure range relevant to super-Earths (MgO (refs. 40,41,42,43), SiO2 (ref. 44), MgSiO3 (refs. 42,45), Mg2SiO4 (refs. 42,46) and natural olivine samples47).

One of the main unknowns is the effect of iron; previous DFT-MD calculations indicate that iron-bearing oxides or silicate may have a higher electrical conductivity by orders of magnitude than iron-free counterparts because the 3d states of Fe increase the electron density at the Fermi level37,48,49. DFT-MD calculations by Stixrude et al.37 indicate that the electrical conductivity of (Mg0.75,Fe0.25)O at Earth’s BMO conditions (100–140 GPa and 4,000–6,000 K) can exceed 104 S m−1, which is favourable for an Fe-rich BMO on Earth. This implies that Fe-rich BMOs or MOs in super-Earths could generate dynamos at lower pressures and, thus, in smaller planets than previously thought. However, such a strong effect of Fe on the electrical conductivity at high pressures has not yet been experimentally tested.

The main focus of this work is to investigate the effect of Fe on the electrical conductivity of MO analogue materials and constrain the likelihood of dynamo generation in the BMO and MO of super-Earths. To achieve this, we conduct shock experiments, DFT-MD calculations and one-dimensional thermal evolution calculations (Methods). Our shock experiments were performed at the OMEGA EP laser at the Laboratory for Laser Energetics at the University of Rochester. We use ferropericlase, (Mg,Fe)O as the analogue material for the BMO and MO. The compositions of (Mg,Fe)O that we explore here are MgO, (Mg0.98,Fe0.02)O and (Mg0.95,Fe0.05)O. We focus on relatively Fe-poor samples because Fe-rich samples tend to be opaque and are not compatible with our target design, which was made for semitransparent targets. The details of (Mg,Fe)O target fabrication are described in Supplementary Section 1.

Shock experiments and DFT-MD calculations

Hugoniot relations

We performed nine successful decaying shock experiments (Table 1). In a decaying shock experiment, a shock wave decays in a target and generates continuous Hugoniot data points. We used a 1.25-ns square pulse to generate the decaying shock. The target design for our shock experiments is shown in Fig. 1a. The diagnostic tools—the Velocimetry Interferometric System for Any Reflector (VISAR)50 and the streaked optical pyrometer (SOP)51—observed the target from the side opposite to the laser drive (for example, refs. 44,50). We used polystyrene (CH) as an ablator. The α-quartz layers were transparent, and the (Mg,Fe)O samples were transparent or semitransparent. VISAR detects the shock velocity at the shock front at 532 nm (ref. 50). We used a fast Fourier transform to extract shock velocity information from the raw VISAR images (for example, ref. 44). Figure 1b shows an example VISAR dataset for shot 38693 on (Mg0.95,Fe0.05)O. The shock velocity as a function of time was extracted from the fringe motion. The vacuum velocity, which was not corrected for the index of refraction of each layer, is shown by the orange line. At (1), the shock wave breaks out from the polystyrene (CH) ablator into the α-quartz. The shock wave reaches (Mg,Fe)O at (2) and the α-quartz anvil at (3) and breaks out to vacuum at (4), with a slight increase in the shock velocity. An epoxy layer is visible at (3) in this example. The straight light grey fringes are caused by reflections of the VISAR probe off the quartz–vacuum interfaces. The reflectivity was estimated from the VISAR data, where high counts (dark colours) indicate high reflectivity.

Table 1 Target information and Hugoniot values for (Mg,Fe)O samples
Full size table
Fig. 1: Target design, and example of VISAR and SOP data.
Fig. 1: Target design, and example of VISAR and SOP data.
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a, Schematic view of the target design. The laser is focused on the CH (polystyrene) ablator, and VISAR and SOP observe the passage of the shock through the target from the side opposite to the laser drive. b, Example of VISAR data. The x axis is the time in nanoseconds, and the y axis is the position in micrometres. The velocity in vacuum without correction for the index of refraction is shown as the orange line. c, SOP data with the SOP count shown as the orange line. The data for b and c are from shot ID 38693. (1)–(4) represent the breakout to the pusher quartz, (Mg,Fe)O, anvil quartz and vacuum. α-Qz, α-quartz.

The SOP measures shock-front spectral radiance in the 590–850 nm range. Figure 1c shows raw SOP data with the orange line showing the SOP count for shot 38693. The numbers (1)–(4) correspond to the interfaces described above. The SOP counts in the (Mg,Fe)O region are slightly smaller than for the surrounding α-quartz pusher and anvil, indicating that (Mg,Fe)O has a lower spectral radiance than the α-quartz layers. The methods of estimating the shock velocity and temperature are described in Supplementary Section 2.

The shock Hugoniot relations at the α-quartz pusher and (Mg,Fe)O interface obtained from the shock experiments on MgO, (Mg0.98,Fe0.02)O and (Mg0.95,Fe0.05)O samples are shown in Fig. 2 and listed in Table 1 (see also Supplementary Section 3 for fitting parameters). These relations were determined from the VISAR data and impedance-matching to α-quartz. Figure 2a shows the relation between particle velocity up and shock velocity Us, which is broadly consistent with previous shock experiments40,41,52. Figure 2b shows the density and pressure relation along the Hugoniot, which is also consistent with previous experiments as well as our DFT-MD calculations on MgO. Generally, our shock experiments provide higher particle velocities and densities at a given pressure than the previous shock experiments40,41,52. Shot ID 38694 has low reflectivity due to its low shot energy, which affected data quality. Details of the fitting lines are listed in Supplementary Section 3. Overall, no statistically detectable differences were observed between MgO and (Mg,Fe)O in the Hugoniot relations.

Fig. 2: Hugoniot relations for MgO and (Mg,Fe)O.
Fig. 2: Hugoniot relations for MgO and (Mg,Fe)O.
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a,b, The US versus up (a) and ρ versus P (b) relations based on our shock experiments and impedance-matching method. The horizontal bars indicate ± one standard deviation. The orange–yellow colours indicate MgO, blue colours indicate (Mg0.95,Fe0.05)O and green colours represent (Mg0.98,Fe0.02)O. The tan, grey and dark red plus marks indicate results from previous work40,41,52. The grey lines show the fit to our results together with the previous data. The blue and red cross marks in b represent DFT-MD calculations for B2 and liquid phases, respectively.

Reflectivity and temperature of (Mg,Fe)O

The reflectivities of MgO and (Mg,Fe)O as a function of the shock velocity and the corresponding pressure are shown in Fig. 3a (Supplementary Sections 2 and 3). Although the reflectivity data have considerable scatter, especially at high shock velocity and pressure, the general trend is that the reflectivity increases with increasing pressure, which is consistent with previous shock experiments (for example, refs. 40,44). The reflectivity is nearly at the lowest detectable level (a few per cent) at the shock velocity Us ≲ 18 km s−1 and increases at higher shock velocity. This might indicate a solid–liquid phase transition, but such a signature was not clearly observed in the temperature data, as discussed below. There is no statistical difference in reflectivity between MgO and (Mg,Fe)O, and our measured reflectivity is generally consistent with previous shock experiments on MgO (refs. 40,42,52). Moreover, our experimental data are broadly consistent with the previous data49 and our new DFT-MD calculations, but there are some notable differences. At Us < 22 km s−1, DFT-MD calculations predict higher reflectivity than the experiments, whereas above 22 km s−1, they predict lower reflectivity than the experiments.

Fig. 3: Reflectivity, temperature and d.c. conductivity of MgO and (Mg,Fe)O.
Fig. 3: Reflectivity, temperature and d.c. conductivity of MgO and (Mg,Fe)O.
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a,b,c, Reflectivity (a), temperature (b) and d.c. conductivity (c). The bottom and top x axes represent the shock velocity and pressure, respectively. The colour scheme is the same as in Fig. 2, although the colours are more transparent in this figure. The solid lines represent the mean, and the shading represents ±standard deviation. Reflectivity estimates from ref. 49 are included in a, and these are consistent with the DFT-MD calculations in this work. MgO B1, B2 and liquid phases are included in b43. Note that temperature data for 39882 are not shown as we could not obtain a physically reasonable result. In c, estimates of the electrical conductivity obtained from ref. 48 for MgO, (Mg0.95,Fe0.05)O and (Mg0.75,Fe0.25)O are shown as the brown solid line and the blue and grey dashed lines.

By contrast, the pyrometric temperature measurements and DFT-MD calculations match well (Fig. 3b). Some data deviate from the overall data trend, such as (Mg0.95,Fe0.05)O (shot ID 38694). This dataset has low SOP counts because of the lower energy, which therefore, potentially affects the uncertainties of the temperature estimate. The B1, B2 and liquid phases of MgO are shown in Fig. 3b in yellow, light blue and light red with the phase curves from previous work43. Based on these phase curves, most, if not all, of our data fall in the liquid phase of MgO. It is possible that the (Mg,Fe)O and MgO were solid at low pressures (P ≲ 570 GPa or Us < 18 km s−1), but we did not observe a clear plateau in the SOP data, which could be a sign of melt44. The (Mg,Fe)O targets probably have a lower melting temperature than MgO, which further supports the finding that all of our (Mg,Fe)O were in the melt phase. Combining this with the reflectivity data, it is inconclusive whether we saw post-shock solid (Mg,Fe)O in these experiments.

d.c. conductivity of (Mg,Fe)O

Our calculated d.c. conductivity values are shown in Fig. 3c. As seen in the reflectivity and temperature data, we do not see a statistical difference between MgO and (Mg,Fe)O. The d.c. conductivity at Us < 18 km s−1 is omitted as its reflectivity is at the detection limit and the flat reflectivity profile may indicate bound electron contribution or melting, not the d.c. conductivity. The d.c. conductivity estimates based on our shock experiments broadly agree with our DFT-MD calculations. However, our estimate differs substantially from previous DFT-MD work on MgO, (Mg0.95,Fe0.05)O and (Mg0.75,Fe0.25)O (ref. 48), which are shown as the beige solid line and the blue and grey dashed lines in Fig. 3c. The lines for MgO and (Mg0.95,Fe0.05)O seem to overlap in this figure, but their values differ by approximately six times (for example, their electrical conductivities are 13 S m−1 and 84 S m−1 at 488 GPa, respectively). These DFT-MD calculations underestimate the d.c. conductivity compared with our shock experiments for MgO and (Mg0.95,Fe0.05)O by orders of magnitude. There are several reasons for this discrepancy. First, the DFT-MD calculations conducted by Holmstrom et al.48 focus on the pressure ranges at 0–300 GPa and 2,000–10,000 K, which are lower than our experimental P and T conditions, and therefore these calculations may not be directly comparable with our experiments. Second, Holmstrom et al.48 state that their analytical model (Supplementary Section 3) may not be very accurate over a broad pressure range, because the d.c. conductivity depends on the pressure nonlinearly due to several effects, including the iron high-spin to low-spin transition.

The observation of a negligible iron effect on the electrical conductivity can be explained because MgO itself can be metallic under high pressures due to its bandgap closure40 and, therefore, the iron contribution becomes a secondary effect. Another potential explanation is that the Fe content in our sample was small (up to 0.05), and a higher Fe content may be required to test this Fe effect. Further DFT-MD simulations and experiments with different target designs over a broader range (0–500 GPa) would be needed to address this question.

Implications for dynamos in super-Earths

Electrical conductivity scaling relation

Our shock experiments robustly show the Hugoniot relations of (Mg,Fe)O. However, temperatures along the Hugoniot are much higher than those in planetary interiors. Here we suggest an Arrhenius-like electrical conductivity fit (see a similar fit in ref. 48) for our MgO, (Mg0.98,Fe0.02)O and (Mg0.95,Fe0.05)O based on our experiments:

$$\sigma \approx{\sigma }_{0}\exp \left(-\frac{\Delta E+P\Delta V}{RT}\right).$$
(2)

We assume that the reference electrical conductivity σ0, the activation energy ΔE and the activation volume ΔV are constant, even though they may depend on the temperature T and pressure P. The best parameter values based on our experiments are σ0 = (3.2897 ± 0.5595) × 105 S m−1, ΔE = 345.919 ± 19.277 kJ mol−1 and ΔV = (−10.765 ± 5.859) × 10−8 m3 mol−1. The P in equation (2) is in pascals. This simple model can reasonably reproduce the Hugoniot data in the pressure range 467–1,400 GPa (Supplementary Fig. 3), which corresponds to CMB pressures of 3.5–9 M⊕ planets, but the model is less certain between 467 and 570 GPa because the reflectivity data with Us < 18 km s−1 were not taken into account. Here we assume that the electrical conductivity is approximately described by the d.c. conductivity (zero frequency). This is reasonable because currents associated with the macroscopic planetary magnetic field vary over much longer timescales than microscopic electron scattering timescales, and therefore, assuming a near-zero frequency for electromagnetic fields is appropriate here.

Magnetic field strength for Earth and super-Earths

Here we discuss the implications of our experiments for Earth and super-Earth interiors. The solid lines in Fig. 4a show the minimum electrical conductivity required for dynamo generation in the BMO as a function of the planetary mass. The black, dark grey and light grey lines represent core mass fractions (CMF) of 0.20 (Mars-like), 0.32 (Earth-like) and 0.68 (Mercury-like), respectively. The reason why the required minimum electrical conductivity decreases as the planet mass increases is closely related to the magnetic Reynolds number (equation (1)), which needs to be larger than 40 to drive a dynamo. Generally speaking, a larger planet has a larger convection length L as well as a convective flow velocity v, which increases Rm, and therefore, dynamo generation becomes easier for larger planets. Moreover, a smaller core (a smaller CMF) leads to a larger mantle and, therefore, also leads to a large value of L. This explains why planets with a smaller CMF require a smaller electrical conductivity to generate a dynamo.

Fig. 4: Electrical conductivity for a BMO-driven dynamo and time evolution of the dynamo.
Fig. 4: Electrical conductivity for a BMO-driven dynamo and time evolution of the dynamo.
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a, The solid lines represent the minimum electrical conductivity required for dynamo generation in BMOs in super-Earths. The x axis is the planetary mass Mp normalized by the Earth mass M⊕. The black, grey and light grey lines correspond to CMFs of 0.20, 0.32 and 0.68. The dashed lines and dashed-dotted lines represent the electrical conductivity based on equation (2) using the P and T condition at the CMB and the average P and T in the BMO. b, Time evolution of the magnetic field strength at the planetary surface as a function of time for a 6 M⊕ planet. The orange lines represent the BMO-driven dynamo, and the grey lines represent the core-driven dynamo. Details of the best fit, MLT, CIA and MAC are discussed in the main text.

The dashed lines represent the electrical conductivity of MgO and (Mg0.95,Fe0.05)O estimated from equation (2) using the P and T conditions at the CMB. The required minimum electrical conductivity is achieved when the planet mass is larger than ~2.9 M⊕ at CMF = 0.20–0.32. This indicates that super-Earths that are more massive than 3 M⊕ can generate dynamos in their BMOs at these CMF values. If the planetary CMF is Mercury-like (CMF = 0.68), the required conductivity can be met at the planetary mass Mp ≈ 5.4 M⊕. The dash-dotted lines show the electrical conductivities at the average pressures and temperatures of the BMOs, indicating that large BMOs can host large conductivity gradients. Additionally, Fig. 4a may indicate that dynamo generation in Earth’s BMO condition may not be possible, but we remain agnostic whether this is the case, as our electrical conductivity model is experimentally constrained to the planetary mass range 3.5–9 M⊕, which may not be applicable to ~1 M⊕.

Figure 4b shows the predicted magnetic field strength at the planetary surface Bs as a function of time for a planet of 6 M⊕ with CMF = 0.32. The orange lines represent the BMO-driven dynamo, and grey lines represent the core-driven dynamo. Here we show four different models for BMO-driven and core-driven dynamos. Each model represents a different exponent of a scaling law that connects the Rossby number and convective power. The details can be found in Lherm et al.25, but we summarize each model here. The solid line represents the best-fitting model, which is a scaling law developed by Lherm et al.25 based on previous magnetohydrodynamics simulation data53. The dash, dashed-dotted and dotted lines represent models based on mixing length theory (MLT), the Coriolis-inertia-Archimedean (CIA) model and the magnetic-Archimedean-Coriolis (MAC) model. In the BMO, the MAC model does not predict a magnetic field because the magnetic Reynolds number is below 40. At 2.76 Gyr, core-driven dynamo ceases because the core fully crystallizes. Subsequently, the BMO-driven dynamo ends at 2.84 Gyr for the same reason. Even though the four scaling laws provide various Bs values, the general trend is that the BMO-driven dynamo is much stronger than the core-driven dynamo throughout the time. This is partly because the presence of a BMO decreases the heat flow from the core, which also weakens the core dynamo33. Additionally, the BMO is physically closer to the surface than the core, and the magnetic field strength drops sharply as the distance from the source increases (equation (17)). Thus, a BMO-driven dynamo can be the dominant source of the surface magnetic field for billions of years, which crucially includes the formative stages of planetary evolution.

Our thermal and magnetic evolution model does not directly address a dynamo produced in a surface MO, but we briefly discuss its likelihood here. Assuming a set of very conservative values for a planet of 6 M⊕, σ = 3,000 S m−1, L = 2,000 km and v = 1 cm s−1, then Rm = 75, which is larger than the critical value of 40. Thus, dynamo generation in such a surface MO is probable, but its lifetime could be short-lived if external heating sources such as tidal heating and stellar irradiation are absent. Thus, the BMO-driven dynamo may provide a more stable, long-lasting source of magnetic field generation for super-Earths.

In this work, we focus on (Mg,Fe)O, as it is a principal component of a BMO, but silicate, such as (Mg,Fe)SiO3, is another main BMO component. Nevertheless, the general behaviour of (Mg,Fe)O and other elements, such as SiO2 (ref. 44) and MgSiO3 (ref. 42), are similar (they are insulators at low pressures and become conductors at high pressures). Silicate contribution may modestly increase the electrical conductivity (at most by less than a factor of two), but the effects would be limited.

In conclusion, our experimental results for (Mg,Fe)O are broadly consistent with our DFT-MD results, previous shock experiments and previous DFT-MD simulations of MgO. Based on this consistency, we infer that MgO and (Mg,Fe)O also have similar d.c. conductivities. This conclusion differs from previous predictions that Fe could increase the electrical conductivity by orders of magnitude. Our model indicates that BMO-driven dynamos are probably stronger than the core-driven dynamos of super-Earths that are larger than 3–6 M⊕. The BMO-driven dynamo can last for a few billion years, which may be detectable with future observations of super-Earths. These experiments and models contribute to an emerging picture of exoplanet habitability and provide support for the presence of a dynamo, even if the core itself cannot produce a dynamo.

Methods

Experimental configuration and analysis

We chose (Mg,Fe)O as an MO analogue material for several reasons. First, (Mg,Fe)O is a principal planetary building block at high pressures54, and understanding its behaviour is essential for characterizing planetary interiors. Additionally, MgO–FeO solid solutions have been well characterized at lower pressures, thus providing a calibration when robustly comparing how the material behaves at lower and higher pressures. Moreover, existing DFT-MD calculations for (Mg,Fe)O at Earth’s CMB condition37 can be directly compared with our experiments.

In our shock experiments, we used a 1,100-μm energy spot size (distributed phase plates). We used a 1.25-ns square pulse shape, and the energy range was 365–828 J (Table 1). This generated a decaying shock wave in the target material, with several Hugoniot relations in one shot55. The target design is shown in Fig. 1a. Our target consists of a 3 mm × 3 mm × 20 μm polystyrene (CH) ablator, a 3 mm × 3 mm × 50 μm α-quartz pusher, a 1.5 mm × 1 mm × 58–80 μm (Mg,Fe)O sample and a 1.5 mm × 1 mm × 150 μm α-quartz anvil with a 2 mm × 1 mm × 200 μm α-quartz witness. These targets were held together with 1–3-μm-thick epoxy. The laser drive was directed at the ablator (CH), causing it to ionize into plasma and expand in the direction of the laser drive. This expansion generated a shock wave that propagated through the rest of the target while conserving momentum.

Based on the shock velocity obtained from the VISAR data, we calculated the corresponding particle velocity and pressure at the interface between the α-quartz pusher and (Mg,Fe)O using the impedance-matching method. The pre- and post-shock conditions are related by the Rankine–Hugoniot equations:

$${\rho }_{0}{U}_\mathrm{s}=\rho ({U}_\mathrm{s}-{u}_\mathrm{p}),$$
(3)
$$P={P}_{0}+{\rho }_{0}{U}_\mathrm{s}{u}_\mathrm{p},$$
(4)
$$E-{E}_{0}=\frac{1}{2}(P+{P}_{0})\left(\frac{1}{{\rho }_{0}}-\frac{1}{\rho }\right),$$
(5)

where ρ is the density, P is the pressure and E is the internal energy. The subscript 0 indicates the pre-shock parameters. Us is the shock velocity, and up is the particle velocity. Here, as is typical, the approximation P0 = 0 is valid because P0 ≪ P. The initial density ρ0 and the index of refraction n for MgO are 3.58 g cm−3 and 1.743 (refs. 43,52) and those of α-quartz are 2.648 g cm−3 and 1.547 (ref. 44).

Drude-type d.c. conductivity model

Using the reflectivity, pressure, temperature and density from the shock experiments, we calculated the band-pass conductivity of (Mg,Fe)O with a Drude-type model. This method has been widely used (for example, ref. 44), and we briefly summarize the approach here. The d.c. conductivity is given by

$${\sigma }_{0}=\frac{{n}_\mathrm{e}{e}^{2}\tau }{{m}_\mathrm{e}},$$
(6)

where ne is the electron density, e is the charge on an electron, me is the electron mass and τ is the electron scattering time, which is given by

$$\tau =\frac{{R}_{0}}{{v}_\mathrm{e}}=\frac{2}{{v}_\mathrm{e}}{\left(\frac{3}{4{\uppi }{n}_\mathrm{i}}\right)}^{1/3},$$
(7)

where R0 is the interatomic distance (\({R}_{0}=2{(\frac{3}{4\uppi {n}_\mathrm{i}})}^{1/3}\)). ve is the electron velocity, which is the larger of the Fermi velocity (\(\frac{\hslash }{{m}_{\mathrm{e}}}{(3{\pi }^{2}{n}_{\mathrm{e}})}^{1/3}\)) and the thermal velocity (\(\sqrt{2{k}_\mathrm{B}T/{m}_\mathrm{e}}\), where kB is the Boltzmann constant and T is the temperature). The electron number density ne = Zni for conductors, where ni = ρNA/M and where ni is the number density of ions, NA is Avogadro’s number, M is the average atomic molar mass and Z is the ionization factor. The connection between σ0 and the reflectivity is that we use the latter to determine Z, which allows the determination of ne. The reflectivity R of a material is

$$R={\left\vert \frac{n-{n}_{0}}{n+{n}_{0}}\right\vert }^{2},$$
(8)

where n and n0 are the indices of refraction of the compressed and uncompressed states, respectively. These indices can be complex numbers. In the Drude–Sommerfeld model:

$$n=\sqrt{{n}_\mathrm{b}^{2}-\frac{{\omega }_\mathrm{p}^{2}}{{\omega }^{2}(1+{\mathrm{i}/\omega \tau })}},$$
(9)

where nb is the bound electron contribution, ω is the VISAR frequency at 532 nm and ωp is the plasma frequency. The complex denominator in the second term under the square root accounts for electron motion from collisions as well as electromagnetic radiation. Typically, τω ≈ 1 in our experimental range. The plasma frequency is

$${\omega }_\mathrm{p}=\sqrt{\frac{{n}_\mathrm{e}{e}^{2}}{{\epsilon }_{0}{m}_\mathrm{e}}},$$
(10)

where ϵ0 is the permittivity of free space, and me is the mass of an electron. We assume nb = 1.598 (ref. 40), as nb being a function of density does not significantly affect the result. Given n0, nb, ni, ω and τ, we then parameterize Z until it matches the measured reflectivity. Then ne can be derived based on equations (8) and (9) and the d.c. conductivity from equation (6).

DFT-MD calculations

We calculated the material properties, including the ab initio Hugoniot, electrical conductivities and reflectivities of MgO, using first-principles molecular dynamics simulations, closely following previous work on the SiO2 system56. Our simulations are based on density functional theory with the PBEsol approximation57 using the projector augmented plane wave method as implemented in VASP58,59. We used projector augmented wave pseudopotentials where the frozen core radius (in Bohr radii) and the numbers of electrons are Mg (2.0, 8) and O (1.52, 6). Born–Oppenheimer simulations were performed with the NVT ensemble using the Nosé–Hoover thermostat. These ran for 8 ps with a 0.5-fs time step. Thermal equilibrium between the ions and electrons was assumed via the Mermin functional60. We sampled the Brillouin zone at the gamma point and used a basis-set energy cutoff of 600 eV. The simulation supercells contained 128 atoms for both the liquid and B2 phase.

For each density ρ examined, we performed simulations at various temperatures T. Both quantities determine the internal energy E and pressure P of the system. We then interpolated these results to find the values of T and P that satisfy equation (5). This process was then repeated for other densities to obtain the ab initio Hugoniot. To calculate the conductivity, we first performed molecular dynamics simulations at ρ and T conditions along the computed Hugoniot, and then we extracted ten uncorrelated snapshots from the simulation. These snapshots were used to compute the electronic contribution to the frequency-dependent electrical conductivity via the Kubo–Greenwood formula:

$${\sigma }_\mathrm{el}(\omega )=\frac{2\uppi {e}^{2}{\hslash }^{2}}{3{m}_\mathrm{e}^{2}\omega \varOmega }\mathop{\sum}\limits_{\bf{k}}\mathop{\sum}\limits_{i,\;j}(\;{f}_{i}-{f}_{j})\,\left\vert \langle{\psi }_{i}\left\vert \nabla \right\vert {\psi }_{j} \rangle\right\vert^{2}\,\delta ({\epsilon }_{i}-{\epsilon }_{j}-\hslash \omega ).$$
(11)

Here the first summation is over the Brillouin zone and the second summation is over pairs of states i and j, where fi,j is the Fermi occupation, k is the wave vector, ψi,j is the wavefunction, ϵi,j is the corresponding single-electron eigenvalue, Ω is the simulation volume, ω is the frequency, ℏ is the reduced Planck constant and me is the electron mass. In practice, the δ function is replaced by a Gaussian with a width given by the average spacing between eigenvalues weighted by the corresponding change in the Fermi function61. We found that σel converged well with a 2 × 2 × 2 k-point mesh, 4,000 electronic bands, 128 atoms and our choice of pseudopotentials.

Equation (11) provides the real component of the electrical conductivity σ1(ω), and the imaginary part σ2(ω) is obtained via the Kramers–Kronig transformation:

$${\sigma }_{2}(\omega )=\frac{2}{\uppi }{\mathcal{P}}\int_{0}^{\infty }\frac{{\sigma }_{1}(\nu )\omega }{{\omega }^{2}-{\nu }^{2}}\,\mathrm{d}\nu,$$
(12)

where \({\mathcal{P}}\) denotes the Cauchy principal value and ν is the integration variable. The refractive index n(ω) and extinction coefficient k(ω) are calculated by

$$n(\omega )=\frac{1}{\sqrt{2}}\sqrt{\left\vert \epsilon (\omega )\right\vert +{\epsilon }_{1}(\omega )},$$
(13)
$$k(\omega )=\frac{1}{\sqrt{2}}\sqrt{\left\vert \epsilon (\omega )\right\vert -{\epsilon }_{1}(\omega )},$$
(14)

where ϵ(ω) = ϵ1(ω) + iϵ2(ω) is the complex dielectric function related to the complex conductivity by ϵ1(ω) = 1 − σ2(ω)/(ωϵ0) and ϵ2(ω) = σ1(ω)/(ωϵ0). The reflectivity is then calculated as

$$R({\omega }_{x})=\frac{{\left[n({\omega }_{x})-{n}_{0}({\omega }_{x})\right]}^{2}+{\left[k({\omega }_{x})-{k}_{0}({\omega }_{x})\right]}^{2}}{{\left[n({\omega }_{x})+{n}_{0}({\omega }_{x})\right]}^{2}+{\left[k({\omega }_{x})+{k}_{0}({\omega }_{x})\right]}^{2}},$$
(15)

where ωx = 532 nm, which is the frequency of the experimental probe, and n0(ωx) represents the refraction index of the unshocked material. The extinction coefficient at ambient conditions k0(ωx) was set to zero. Finally, the d.c. conductivity was obtained by extrapolating the frequency-dependent electronic contribution from equation (11) to ω = 0 using a linear fit at small ω.

Evolution model for super-Earths

We modelled the structural, thermal and magnetic evolution of super-Earths with masses in the range MP = 1–10 M⊕, where M⊕ = 5.972 × 1024 kg is the mass of the Earth, and with CMFs of {0.2, 0.32, 0.68}, corresponding to the CMF values of Mars, Earth and Mercury, respectively. We used the model in Lherm et al.25.

First, we defined a reference internal structure model over a wide range of CMB temperatures TCMB, with a solid MgSiO3 mantle and an Fe core62. We estimated the possible formation of an iron-enriched BMO using the intersection between an adiabatic temperature profile based on a recent equation of state63,64,65 and melting curves65,66. We redefined the temperature profile of the solid mantle using a thermal boundary layer model67, where the reference viscosity was parameterized as \({M}_\mathrm{P}^{-2(\theta -4\psi )/3(4+\theta )}\), with θ = 30 and ψ = 0 (ref. 68).

Next, we obtained the thermal evolution of the planet by computing the evolution of the temperature at the CMB with coupled energy budgets of the core, BMO and solid mantle (for example, refs. 69,70,20,71,72):

$$\frac{{\rm{d}}{T}_\mathrm{CMB}}{{\rm{d}}t}=\frac{{Q}_\mathrm{P}-{Q}_\mathrm{R,m}-{Q}_\mathrm{R,b}-{Q}_\mathrm{R,c}}{{\tilde{Q}}_{S,m}+{\tilde{Q}}_\mathrm{S,b}+{\tilde{Q}}_\mathrm{S,c}+{\tilde{Q}}_\mathrm{G,b}+{\tilde{Q}}_\mathrm{G,c}+{\tilde{Q}}_\mathrm{L,b}+{\tilde{Q}}_\mathrm{L,c}},$$
(16)

where heat flow at the surface (QP) and the radiogenic heating terms (QR,x) in the mantle (x = m), the BMO (x = b) and the core (x = c) can be written explicitly. The secular cooling (QS,x), gravitational energy (QG,x), and latent heat (QL,x) terms are expressed as functions \({\tilde{Q}}_{X}(r)\) that depend on only r.

Finally, we determined the magnetic evolution of the planet. In an adiabatic and well-mixed layer involving a rapidly rotating, convective and electrically conductive fluid, dynamo operation requires that (1) convection supplies enough power to balance ohmic dissipation73,74 and that (2) magnetic induction is larger than magnetic diffusion26. The convective power required to sustain a dynamo is determined using entropy budgets (for example, refs. 69,70,71,72) and must at least satisfy the condition of a positive entropy of dissipation. In addition, the magnetic Reynolds number must exceed a critical value, here 40 (ref. 26), for induction to overcome diffusion (for example, refs. 24,37). If these criteria are met and a dynamo is active in the BMO or the core, we can estimate the resulting internal magnetic field intensities B with magnetic scaling laws26,53. We then obtain the magnetic field at the surface BS with

$${B}_\mathrm{S}=\frac{1}{7}{\left(\frac{{R}_\mathrm{D}}{{R}_\mathrm{P}}\right)}^{3}B,$$
(17)

where RP is the radius of the planet and RD is the upper radius of the dynamo region, that is the top of the BMO or the CMB. We used equation (2) as the electrical conductivity of BMO, assuming that the difference in the electrical conductivities of (Mg,Fe)O and (Mg,Fe)2SiO4 is small. The interior thermal profile and the effect of radiogenic heating are discussed in Supplementary Sections 4 and 5.