Introduction

Earth’s lower mantle is thought to be composed primarily of bridgmanite (Mg1−xFex)SiO3 and ferropericlase (Mg1−xFex)O, where x ~0.1–0.2, coexisting with CaSiO31,2,3. These major mineral phases behave as insulating materials up to conditions of the lowermost mantle, with electrical conductivities on the order of 100 to 102 S/m4,5, many orders of magnitude lower than proposed conductivities of the metallic iron-dominant core (~106 S/m) (e.g., refs. 6,7). Instead of a homogeneous lower mantle, seismic observations over the last several decades have robustly identified multi-scale structures across Earth’s core-mantle boundary8,9. These structures have been grouped into two main categories: (1) two continent-scale “large low-seismic velocity provinces" (LLSVPs), considered to be piles of heterogeneous material or bundles of thermochemically distinct mantle plumes10,11, and (2) numerous mountain-scale “ultralow velocity zones", basal structures discovered within and around the edges of LLSVPs, including at the roots of major mantle plumes like those that source volcanism at Hawai’i, Iceland, and the Gálapagos12,13,14,15,16,17,18,19.

Studies generally agree that the interpretation of these observed structures requires strong compositional contrasts from the surrounding average lower mantle and possibly the presence of partial melt14,20,21. Recent interdisciplinary work on ultralow velocity zones has demonstrated that solid FeO-rich mineral assemblages, consisting of iron-rich (Mg1−xFex)O (x ~0.8–0.95) coexisting with (Mg,Fe)SiO3 and CaSiO3, can produce structures that satisfy the velocity reductions and topographies constrained by seismic observations and geodynamic simulations22,23,24,25,26. Such strong iron enrichment, arising from crystallization of the primordial magma ocean or chemical interactions with the iron core, leads to several unique physical properties observed for the very iron-rich (Mg,Fe)O phase, including high seismic anisotropy27, remarkably low viscosity28, and experimental reports of moderately elevated electrical conductivity (105 to 106 S/m)29,30, orders of magnitude higher than insulators (like typical mantle rocks) but lower than a metal (like the liquid iron-rich core).

The accuracy and origin of these intermediate conductivity values, and the electronic phase diagrams of FeO and iron-rich (Mg,Fe)O more broadly, represent a poorly understood and controversial topic in high-pressure physics and deep Earth science. An insulator-metal transition has been proposed for FeO from measurements of relatively high conductivity (~105 S/m) with weak P–T dependence above ~60 GPa29. In contrast, similarly high conductivity was reported for (Mg0.2Fe0.8)O and (Mg0.05Fe0.95)O but interpreted as insulating behavior up to ~130 GPa30. Meanwhile, standard electronic-structure theory methods focus at T = 0 K, and are not able to properly capture thermal effects, which often dominate in the vicinity of the insulator-metal transition31,32. Such extreme fragility of electronic states is especially pronounced in “strongly correlated"33 electronic systems34,35,36, often featuring tightly-bound d or f orbitals37. Here the Coulomb repulsion between pairs of electrons confined to the same orbital takes center stage, typically resulting in very strong electron-electron scattering and poor conduction at elevated temperature38. Given these complications, several fundamental open questions arise regarding the insulator-metal transition (IMT) in B1-FeO at high pressures: (1) Is there a sharp IMT at high temperature, in the regime characteristic of Earth’s deep mantle? (2) What is the mechanism of electronic transport (i.e., the dominant form of scattering) in this regime? (3) How do orbital selectivity39 and the associated spin-crossover affect the transition region? (4) What are the consequences of these phenomena for the magnitude and P–T dependence of electrical conductivity across deep Earth conditions?

Knowledge of electronic processes in FeO at extreme conditions and consequences for transport properties is essential for understanding phenomena at Earth’s core-mantle boundary, including electromagnetic coupling of the core and mantle and heat flow through this region. To that end, we employ a state-of-the-art “embedded DMFT" (eDMFT) ab initio approach40 that combines dynamical mean field theory (DMFT) methods41,42 and standard density functional theory (DFT) with full charge self-consistency. While some valuable steps in this direction have been taken in previous work29,43,44,45, sufficiently detailed and systematic study of the transition region has not been performed, preventing a clear understanding of the important open questions at hand. Using this approach, we systematically survey the electronic structure of cubic B1-FeO, the crystal structure relevant to Earth’s lower mantle conditions46. An expansive data set featuring calculations at more than 350 temperature-volume conditions (see Supplementary Materials) finely samples the phase diagram up to conditions of Earth’s inner core (300 GPa, 5000 K). This detailed information allows us to accurately determine and physically interpret the boundaries of different transport regimes across the phase diagram.

Results and discussion

Three distinct electronic phases of B1-FeO

Our theoretical calculations reveal three distinct electronic phases in the high-P–T phase diagram of B1-FeO (Fig. 1). At ambient conditions and low degrees of compression, FeO behaves as a Mott insulator, in which both the t2g and eg orbitals exhibit large band gaps at the Fermi energy on the order of several eV and electrons remain bound to their respective nuclei29,45. In contrast, at large degrees of compression, FeO exists as a strongly correlated metal, where one or both the d orbital band gaps are closed, producing a characteristic “quasiparticle" density of states (DOS) peak at the Fermi energy (see also Fig. 2, rightmost panels)29,45.

Fig. 1: Theoretical phase diagram of B1-FeO, as a function of reduced volume Δv and temperature T.
figure 1

Δv = (vo − v)/vo, where vo is the volume of FeO at ambient conditions. The color-coded value of the electronic density of states (DOS) at the Fermi energy is used to distinguish the (gapped) Mott insulator from the metal and the intermediate “quantum critical" regime34,49. The “Brinkman-Rice" crossover line82 marks the thermal destruction of coherent quasiparticles in the metal with increasing temperature83.

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Fig. 2: Evolution of the electronic density of states (DOS) with compression.
figure 2

a–d Metallization is sharp at low temperature (T = 300 K), where closure of the Mott gap in the t2g band leads to immediate emergence of coherent quasiparticle states at the Fermi energy. e–h A broad intermediate “quantum critical" phase arises at higher temperatures (T = 2000 K), with a spectral pseudogap (reduced but finite DOS) and no quasiparticle states, characteristic of an incoherent conductor. The quasiparticle peak appears only at further compression with the closure of the eg gap and onset of the spin crossover phenomenon.

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At intermediate degrees of compression and sufficiently high temperatures, FeO exists in a “quantum critical" (QC) state, which is notably different from either an insulator or a metal. Here, the t2g gap has closed to form a conducting band, but unlike in a conventional metal, the density of states at the Fermi energy is significantly reduced, with a marked absence of quasiparticles (Fig. 2, bottom row). Instead of traveling as coherent waves with minimal scattering as in a metal, electrons in the QC state exhibit incoherent diffusion marked by strong electron-electron scattering with a short mean-free path at the scale of atomic spacing. In this regime, the eg gap remains open and FeO remains in the high-spin state, with four d electrons in the t2g orbital (Fig. 3). We stress that the QC phase arises only at finite temperatures above the insulator-metal phase coexistence region, terminating at the critical end-point Tc ~370 K; the insulator-metal transition assumes first-order character at T < Tc.

Fig. 3: Spin crossover behavior of B1-FeO at high P–T conditions.
figure 3

a Instantaneous local magnetic moment M results show that Fe remains in the high spin state throughout the insulator-metal transition region, with the onset of spin crossover only upon further compression with the closure of the eg gap. b The spin crossover reflects partial charge transfer from the eg to the t2g orbital, with remarkably weak T-dependence.

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Temperature-dependent forms of the IMT

The physical nature of the insulator-metal transition in FeO and the range of pressures spanning the QC region depend strongly on the range of temperatures considered. At low temperatures (T ≤ Tc), FeO transitions directly from a Mott insulator to an “orbitally selective" metal around Δv ~20% (corresponding to P ~58 GPa46). Here the closure of the t2g gap leads to the immediate formation of a quasiparticle peak at the Fermi energy in the t2g orbital (see Fig. 2, top row), while the eg gap remains open. These quasiparticle states are remarkably fragile to thermal excitations, and are suppressed around the “Brinkman-Rice" temperature TBR (see Fig. 1), marking the crossover to the QC phase. As TBR increases with compression, the insulator-metal transition is “smeared out", producing an increasingly wider QC “fan" at Tc < T ≲ 2000 K. The left boundary of the QC region corresponds to a temperature scale where the Mott gap is smeared through thermally activated processes (see Supplementary Materials for precise definition of the corresponding crossover lines shown in Fig. 1).

This behavior becomes qualitatively different at very high temperatures. At T ≳ 2000 K, the quasiparticles are unable to form in the t2g orbital before compression causes the closure of the eg gap, around Δv ~34%. Further compression leads to the onset of spin crossover phenomena and simultaneous formation of a correlated metal, with robust quasiparticles forming in both sectors. The spin crossover extends over a wide compression range with weak temperature dependence (Fig. 3) as previously observed47, and is marked by a partial charge transfer from the eg to the t2g orbital, with one electron remaining in the eg orbital and a drop in the magnetic moment from 4 to ~1.5 Bohr magneton. Unlike T ≲ 2000 K, where the QC region gradually broadens with increasing temperature, here the transition to a quasiparticle metal occurs immediately after the eg gap closure and spin crossover onset, leading to a Brinkman-Rice line with weak temperature dependence and an abridged pressure extent for the QC “fan" at high temperatures. Orbital selectivity and the associated spin crossover phenomena thus dramatically affect the form of the insulator-metal transition behavior at these very high temperatures, producing markedly weak temperature dependence of all physical quantities within the QC region.

We relate our findings to existing knowledge on the experimental phase diagram of FeO by presenting our results as a function of pressure, where pressure is calculated at each volume-temperature condition using the experimentally determined equation of state for B1-FeO46, as shown in Fig. 4. Here we include experimentally estimated phase boundaries for different crystal structures46, as well as the melting curve48. We note that the phase coexistence region, where both insulating and metallic phases are present at T < Tc ~370 K (omitted in Fig. 4, see Fig. 1), is predicted to lie at the center of the experimentally estimated stability field for rhombohedrally distorted rB1-FeO. In addition, we observe that the Brinkman-Rice line below ~2000 K, marking the onset of an orbitally selective metal, traces the experimentally reported B1-B8 transition boundary. These observations raise further questions regarding the relationship between insulator-metal transitions and crystal structures in strongly correlated systems, which merit further investigation but are beyond the scope of this study.

Fig. 4: Conductivities of B1-FeO.
figure 4

a Theoretical phase diagram for B1-FeO, as a function of pressure calculated from volumes using the experimental thermal equation of state46. Color-coded are calculated values for electrical conductivity σ, which span only about one order of magnitude within the entire QC region at magnitudes comparable to the MIR limit (~105 S/m) in other Mott oxides38. Solid and dashed white lines show the geotherm84,85 and experimentally estimated phase boundaries46,48, respectively. b Characteristic “fan-like" evolution of temperature-dependent conductivity curves is shown for 45 GPa ≤ P ≤ 75 GPa, as expected for Mott quantum criticality34,36. Note the markedly weak pressure and temperature dependence of the resistivity at T > 2000 K.

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Consequences for transport properties

The three electronic phases identified for FeO in this study exhibit highly distinct transport properties (Fig. 4). Conductivity in the insulating state is relatively low (~100–103 S/m) and increases with temperature, as expected for thermal activation. In the correlated metallic state, conductivity is large (~106–108 S/m) and decreases with increasing temperature. In contrast, conductivity in the QC state lies at intermediate levels (~104–105 S/m) and displays remarkably weak dependence on both pressure and temperature. As discussed above, transport in the QC state is a consequence of a (poorly) conducting t2g band that lacks the presence of coherent quasiparticles. Unlike in a quasiparticle metal, where the mean-free path for electron-electron scattering is generally much longer than the lattice spacing, conductivity in the QC state lies around the Mott-Ioffe-Regel (MIR) limit (~105 S/m) characterized by a short mean-free path comparable to the lattice spacing38. Physically, the electrons exhibit Brownian-style diffusive motion caused by strong and frequent scattering.

Robustness of theoretical results

The theoretical results we have obtained reveal that at temperatures on the scale of thousands of Kelvin, the insulator to metal crossover displays a significant intermediate regime, in close analogy to what is generally expected for quantum criticality34,49,50. Although there are several aspects of our work that may shift the precise location of the crossover, the general topology of the phase diagram would not be affected. Our result was obtained for specific values of the interaction parameters U and J, which we fixed to the values expected under ambient conditions51. We did so to avoid deliberate “fine-tuning" of input parameters, although we do expect that these interactions should display some volume/pressure dependence. Still, these details should not affect the qualitative and even the semi-quantitative aspects of our results. Similarly, the presence of small concentrations of Fe vacancies or small amounts of Mg substitutions could slightly displace the crossover line positions. We emphasize, however, that the characteristic scale of the electrical conductivity set by the Mott-Ioffe-Regel limit in the QC regime (~105 S/m) should be a robust feature of our results. In particular, the modest pressure and temperature dependence of transport in the QC regime suggests that small shifts in the crossover line positions due to the effects discussed above will not affect the key finding that FeO exhibits intermediate values of electrical conductivity (~105 S/m) at lowermost mantle conditions. Furthermore, various other physical mechanisms (such as different forms of magnetic order) that often play out at low to ambient temperatures (T ~101–102 K) are expected to be negligible at the T ~103 K levels that we consider here. In this sense, the single-site DMFT theory we adopt, which deliberately ignores such magnetic correlations, should be regarded as an accurate solution to the electronic many-body problem under conditions relevant to Earth’s interior.

Comparison to previous results

We quantitatively compare general trends and magnitudes of transport obtained from our theoretical calculations to previous experimental measurements. A shock compression study also reported conductivities on the order of 105.5–106 S/m for pressures between 72 and 155 GPa and at elevated but unconstrained temperatures52. Static compression experiments reported weak temperature dependence of electrical resistance when heating up to ~2500 K in the pressure range ~40–80 GPa and when heating above ~2000 K from 80 to 125 GPa29. The QC region determined in our study spans these P–T conditions and provides a physical basis for the observed weak temperature dependence. Our findings suggest that very shallow minima in resistivity-temperature measurements from these experiments should not be interpreted as marking the location of a sharp insulator-metal transition but could stem from secondary effects, such as phonon (lattice) interactions or defect mobility. In addition, the same experiments reported a plateau in conductivity at around 105 S/m along a pressure range of ~60–120 GPa for T = 1850 K29 (Fig. 5). These measurements of a conductivity plateau (weak pressure dependence) and magnitudes around the Mott-Ioffe-Regel limit (~105 S/m) (Fig. 5) are now clearly explained and supported by the global phase diagram determined in this study, and in particular by the existence of the QC region. Overall, the electronic phase diagram and consequent transport properties determined for B1-FeO in this study provide a clear physical explanation for experimental reports on the material’s conductivity. Our theoretical results capture much of the same features as those reported in previous theoretical works performed by using DFT+DMFT methods for B1-FeO29,44,45,53, although our expansive canvassing of the entire phase diagram provides qualitatively updated insight and interpretation. Specifically, we demonstrate that a clearly defined intermediate regime arises between the insulator and the metal, with distinct spectral and transport signatures.

Fig. 5: Electrical conductivity of FeO compared with other Earth and planetary materials.
figure 5

a Solid lines are calculated in this study, while symbols show previous electrical conductivity experiments on Fe0.96O29, (Mg0.05Fe0.95)O and (Mg0.20Fe0.80)O30, and (Mg0.81Fe0.19)O5, all color-coded by temperature. b Conductivities as a function of depth in the Earth along the geotherm. Solid line is FeO (this study); gray lines show geomagnetic constraints on bulk mantle electrical conductivity (dashed58, dotted59, dot-dash57); points show experimental reports for bridgmanite (perovskite)4, post-perovskite54, pyrolite, and MORB56; blue shading shows theoretical reports for liquid iron alloys7,60,61; gray shading shows the conductivity range of a proposed thin layer at the mantle base65.

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Implications for Earth’s interior

We find that the electrical conductivity of FeO at lower mantle conditions exhibits intermediate values (104–105 S/m) relative to the insulating mantle and metallic core. The lower mantle features conductivity magnitudes of 100 to 102 S/m based on experimental and geomagnetic constraints. Experiments on the major lower mantle phases bridgmanite4, ferropericlase5, and post-perovskite54 have reported conductivities from 100 to at most 103 S/m, similar to experiments on the hydrous silicate phase D55, as well as on pyrolite and mid-ocean ridge basalt (MORB) rocks56 that represent the average lower mantle and subducted oceanic crust, respectively (Fig. 5). These values are in good agreement with depth profiles for conductivity, determined from geomagnetic observations57,58,59. For the metallic outer core, theoretical computations have reported conductivities for liquid iron alloys around 106 S/m7,60,61, similar to experimental measurements on solid iron and iron alloys at high P–T conditions6,62,63,64. The intermediate conductivity values for FeO at lowermost mantle conditions (~105 S/m) are robust even for small amounts of Mg substitution (up to 20%), based on experimental results30, suggesting that iron-rich (Mg,Fe)O in the lowermost mantle would exhibit a unique signature of electrical conductivity relative to coexisting materials.

Interestingly, the base of Earth’s mantle has been suggested to exhibit a unique signature of moderate electrical conductivity, higher than the bulk mantle but lower than outermost core fluid (Fig. 5), that affects electromagnetic coupling of the mantle and core and thus Earth’s rotation and magnetic fields. Specifically, variations in the length of day over periods of several decades, as well as nutations of Earth’s rotation axis on the diurnal timescale, are best explained by a mantle basal layer 1 km thick with conductivity 105 S/m65. Further, low temporal variations of Earth’s magnetic field in the Pacific region have recently been attributed to a non-uniform conducting layer at the mantle base with higher conductance levels in the Pacific, estimated at 6–9 × 108 S compared to 108 S for a global average layer66. This elevated conductance could be approximately explained by 20–30 km thick structures with conductivity ~105 S/m covering around one-third of the mantle base on the Pacific, compatible with typical heights and detection locations of ultralow velocity zones14.

Separately, independent seismic observations of the mantle base combined with geodynamic and mineralogical constraints have recently shown that ultralow velocity zones can be quantitatively explained by highly FeO-rich solid material24,25,26. Geodynamic work has further suggested that these mountain-scale structures may form from a thin layer23,67 that could be difficult to detect seismically68. The bulk conductivity of such features would depend on the interconnection of moderately conductive FeO in the assemblage, which is poorly constrained. However, the remarkably low viscosity of the material (1012 Pa-s) at lowermost mantle conditions28 and its relatively high abundance in ultralow velocity zones suggested by recent work (~20–40%)8,25,26 supports the possibility of interconnected networks of iron-rich (Mg,Fe)O and resulting bulk conductivity similar to 105 S/m.

A solid FeO-rich mineralogy could thus provide a unifying explanation for constraints on Earth’s mantle base from both seismic imaging as well as independent observations of temporal variations in Earth’s rotation and magnetic field. FeO-rich structures could further imply heterogeneous thermal conductivity at the core-mantle boundary, instead of homogeneous heat flow out of the core assumed in some models of mantle dynamics69,70. Using the Wiedemann-Franz law and our calculated conductivity of ~2 × 105 S/m, we estimate an electrical contribution to the thermal conductivity of ~17 W/m-K for FeO at the core-mantle boundary. This value is around two to four times larger than the reported thermal conductivity of the average pyrolitic lowermost mantle71. By a direct calculation of the electrical and the thermal conductivity, recent work on Hubbard models suggested72 that the Lorentz number defining the Wiedemann-Franz law should be somewhat lower at high temperatures, as compared to the conventional value. Nevertheless, our general conclusions should remain qualitatively valid. Solid FeO-rich ultralow velocity zones may thus represent regions of high thermal conductivity at Earth’s mantle base, which could promote the generation of long-lived mantle plumes, influence convection dynamics, and affect crystallization processes in the core73,74,75.

Methods

The eDMFT algorithm we use40,41,42 starts with the calculation of the eigen-energies and the eigen-wavefunctions of the crystal by solving the DFT equations. Next, the correlated orbital subset is projected out as “quantum impurities" by a real-space projectors without downfolding, while the uncorrelated orbitals are treated by DFT, and act as a mean-field bath on the quantum impurities, resulting in a hybridization between the two. The hybridization functions are determined self-consistently by solving the DMFT equation. The quantum impurities are solved by the hybridization expansion continuous time quantum Monte Carlo (CTQMC)76,77 method. The modified charge density derived from combined DFT and DMFT equations is then used as the input of the next DFT iteration. The eDMFT algorithm iterates until full convergence of the charge density, the impurity self-energies, and the lattice Green’s function etc are achieved. Finally, the maximum entropy method78 is employed to analytically continue the Green’s function and the self-energy from the Matsubara frequency to the real frequency axis. The linear augmented plane wave method is used as a basis, as implemented in WIEN2K package79, and the local density approximation (LDA)80 to the exchange and correlation functional is employed in the DFT part. We use a double-counting scheme81 which is known to be exact within the LDA exchange and correlation functional. In each DMFT iteration a huge number (~2.8 × 1010) of Monte Carlo updates is used to reduce the statistical error. A Monkhorst-Pack mesh of at least 12 × 12 × 12k − points is used in the calculation. At the ambient pressure the energy window for projection of the correlated states is ± 10 eV around the Fermi energy. At high pressure the energy window is expanded so that the same number of bands are included for projection as done at ambient pressure. Only the Fe-3d electrons are treated as correlated with Coulomb interaction U = 10.0 eV and Hund’s coupling J = 1.0 eV, which is based on previous constrained DMFT calculations of FeO at ambient pressure51. Throughout the paper we fix the Coulomb interaction U and the Hund’s coupling J as volume independent. Although increased pressure should reduce U and J in real FeO material, it will only quantitatively tune the results in the paper, such as the exact position of the insulator-metal transition.