# Influence of oxygen on electronic correlation and transport in iron in the
outer Earth’s core

G. G. Blesio  Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
<EMAIL_ADDRESS>L. V. Pourovskii  CPHT, CNRS, Ecole polytechnique,
Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau, France.
Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France.  M.
Aichhorn  Institute of Theoretical and Computational Physics, Graz University
of Technology, NAWI Graz, Petersgasse 16, Graz, 8010, Austria  M. Pozzo
Department of Earth Sciences and London Centre for Nanotechnology, University
College London, Gower Street, London WC1E 6BT, UK  D. Alfè  Department of
Earth Sciences and London Centre for Nanotechnology, University College
London, Gower Street, London WC1E 6BT, UK  Dipartimento di Fisica Ettore
Pancini, Università di Napoli Federico II, Monte S. Angelo, I-80126 Napoli,
Italy.  J. Mravlje  Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana,
Slovenia

###### Abstract

Knowing the transport properties of iron under realistic conditions present in
the Earth’s core is essential for the geophysical modeling of Earth’s magnetic
field generation. Besides by extreme pressures and temperatures, transport may
be influenced importantly also by the presence of light elements. Using a
combination of molecular dynamics, density functional theory, and dynamical
mean-field theory methods we investigate how oxygen impurities influence the
electronic correlations and transport in the liquid outer Earth’s core. We
consider a case with an oxygen content of $\sim 10\mathrm{atomic}\%$, a value
that is believed to be close to the composition of the core. We find that the
electronic correlations are enhanced but their effect on conductivities is
moderate (compared to pure Fe, electrical conductivity drops by 10% and
thermal conductivity by 18%). The effect of electron-electron scattering
alone, whereas not large, is comparable to effects of the compositional
disorder. We reveal the mechanism behind the larger suppression of the thermal
conductivity and associated reduction of the Lorenz ratio and discuss its
geophysical significance.

## Introduction

The Earth’s magnetic field is generated by a self-excited dynamo that is
driven by convection in the outer core. The thermal conductivity of iron,
which determines the amount of heat flow available for convection, and the
value of electrical conductivity, which determines the dissipation of the
current, are crucial inputs for geophysical models of this geodynamo
mechanism. Direct measurements of transport under extreme pressures and
temperatures that are relevant to Earth’s core are challenging [1, 2] as one
must ensure homogeneous temperature and carefully control the geometry of the
samples [3].

One can access transport properties also from first principle calculations
based on the molecular dynamics (MD)–density functional theory (DFT) method
[4, 5, 6]. These calculations have shown that the electrical and thermal
conductivities have values that are significantly (2-3 times) higher [5, 6]
than earlier established estimates [7, 8]. These earlier estimates were based
on extrapolations that neglected the effects of resistivity saturation [9, 10,
11]. The higher values of thermal conductivity lead to a different geophysical
picture, with an inner core that is younger ($<1$ billion years, whereas
magnetism is known to exist for at least $3.4$ billion years [12, 13] ), and
less thermal convective energy to drive the geodynamo, which is known as “the
new core paradox” [14]. Namely, less thermal energy implies that convection
must be helped by the chemical convection driven by the exsolution of lighter
elements but this was less active before the formation of the inner core.

There has been an ongoing discussion on whether electronic correlations, which
have been shown to be significant in iron under Earth’s core conditions [15,
16], can cause a breakdown in the Mott-Ioffe-Regel resistivity saturation [9,
17]. This debate revolves around whether these correlations are strong enough
to reduce conductivity values to previously established levels [18, 19, 20],
as suggested in a pioneering work (later retracted)[18]. Recent findings
indicate that electron-electron scattering (EES) plays only a moderate role
and represents a small fraction compared to thermal-disorder (electron-phonon)
scattering [21, 22].

One important question remains to be addressed: does alloying with lighter
elements significantly enhance correlations, and if so, to what extent are
conductivities suppressed? The Earth’s core contains a sizable contribution of
lighter elements, primarily silicon and oxygen, with recent research
indicating a high oxygen concentration [23]. The influence of these
substitutions has been broadly investigated using MD-DFT [5, 6, 24, 25], but
without accounting for correlation effects. Oxygen might enhance electronic
correlations by reducing iron 3d shell occupancy toward half-filling [26].
Indeed, recent theoretical work [27] investigated several ordered FeO
structures and found a very strong enhancement of electron-electron scattering
(EES). However, the impact of thermal disorder on EES was neglected in that
study, and the final effect on thermal conductivity, taking into account both
EES and electron-phonon scattering, was not evaluated. Furthermore, the impact
of EES on conductivity in liquid iron, which is most relevant for the dynamo
mechanism, was not clarified.

In this study, we investigate the transport properties of Fe and FeO alloys in
their liquid state at the inner-core boundary (ICB) and core-mantle boundary
(CMB) conditions. We describe the liquid state using MD-DFT and account for
electron-electron scattering (EES) using dynamical mean-field theory (DMFT)
[28, 29, 20, 21]. Specifically, we focus on the Fe0.91O0.09 composition and
observe that its EES rate increases by $\sim 25\%$ compared to pure iron (Fe
), but the conductivities are affected to a lesser extent with only an
$\approx 10\%$ drop in electrical conductivity $\sigma$ and an $\approx 18\%$
drop in thermal conductivity $\kappa$. This change is mainly due to band-
structure effects rather than a direct increase in EES.

In order to quantify the effects of electronic-correlation on transport we
compare the calculated DMFT conductivities with those from the MD-DFT. We find
that the inclusion of EES leads to only a moderate reduction of electrical and
thermal conductivities by roughly 10% and 20%, respectively. This finding is
crucial considering a large body of existing theoretical work that neglects
EES [5, 6, 11, 24]. A related study of Fe-Si alloys has shown similar behavior
[22]. We discuss why the thermal conductivities are suppressed more than the
electrical ones and highlight the geophysical implications of our results.

## Methods

We performed the molecular dynamics calculations using the VASP code [30],
using the projected augmented wave method [31, 32] to describe the
interactions between the electrons and the ions and expanded the single-
particle orbitals as linear combinations of plane waves (PW), including PW
with maximum energies of 400 eV. The molecular dynamics simulations were
performed by sampling the Brillouin zone using the $\Gamma$ point only, and a
time step was 1 fs. The temperature was controlled using the Nosé [33]
thermostat. To compute the DFT electrical and thermal conductivity we used the
modified version of VASP by Dejarlais [34].

The DFT+DMFT self-consistent calculations were performed with a local density
approximation (LDA) approach in the Wien2K code [35, 36], using the TRIQS
library [37, 38, 39, 40] for the DMFT and transport calculations. We used a
local density-density interaction vertex with interaction parameters $U=5.0$
eV, $J_{H}=0.93$ eV, in agreement with the previous studies on pure iron [19,
21], and solved the impurity problem using the continuous-time hybridization-
expansion segment solver [41, 42]. Each calculation was first converged by 25
fully self-consistent DFT+DMFT iterations, where each Monte Carlo run employed
$2\times 10^{10}$ Monte Carlo moves and 200 moves/measurement. Using the
converged Kohn–Sham Hamiltonian, 10 additional DMFT cycles were performed with
the number of Monte Carlo moves increased to $10^{11}$. To obtain clean data
for analytical continuation that we performed using Maximum Entropy method, 20
additional runs (with $2\times 10^{11}$ moves per run) were carried out
starting from the same converged value of the DMFT bath Green’s function and
resetting the random sequence.

We calculated the conductivities within the Kubo linear-response neglecting
the vertex corrections. The electrical and thermal conductivity read [40, 43]

$\sigma_{\alpha\alpha^{\prime}}=\frac{e^{2}}{k_{B}T}K^{0}_{\alpha\alpha^{\prime}}\,~{},~{}\kappa_{\alpha\alpha^{\prime}}=k_{B}\left[K^{2}_{\alpha\alpha^{\prime}}-\frac{\left(K^{1}_{\alpha\alpha^{\prime}}\right)^{2}}{K^{0}_{\alpha\alpha^{\prime}}}\right],$
(1)

where $\alpha$ is the direction ($x$, $y$ or $z$) and $k_{B}$ the Boltzmann
constant. The kinetic coefficients $K^{n}_{\alpha\alpha^{\prime}}$ are

$K^{n}_{\alpha\alpha^{\prime}}=2\pi\hbar\int
d\omega(\beta\omega)^{n}f(\omega)f(-\omega)\Gamma^{\alpha\alpha^{\prime}}(\omega,\omega),$
(2)

where 2 is the spin factor, $f(\omega)$ is the Fermi function, and the
$\Gamma^{\alpha\alpha^{\prime}}$ is given by

$\Gamma^{\alpha\alpha^{\prime}}(\omega,\omega^{\prime})=\frac{1}{V}\sum\limits_{\mathbf{k}}\mathrm{Tr}\left(v^{\alpha}_{\mathbf{k}}A_{\mathbf{k}}(\omega)v^{\alpha^{\prime}}_{\mathbf{k}}A_{\mathbf{k}}(\omega^{\prime})\right)$
(3)

where $V$ is the unit-cell volume, $A_{\mathbf{k}}(\omega)$ is the DMFT
spectral function at momentum $\mathbf{k}$, and $v^{\alpha}_{\mathbf{k}}$ is
the corresponding band velocity in the direction $\alpha$. We also define the
transport distribution

$\Gamma(\omega)=\sum_{\alpha}\Gamma^{\alpha\alpha}(\omega,\omega).$ (4)

We also calculated the response at finite frequency $\Omega$ which yields the
optical electrical and thermal conductivity. These are evaluated by Eq.1 using
the kinetic coefficients evaluated at a finite frequency $\Omega$

$\displaystyle K^{n}_{\alpha\alpha^{\prime}}(\Omega)=2\pi\hbar$
$\displaystyle\int
d\omega\Gamma^{\alpha\alpha^{\prime}}(\omega+\Omega/2,\omega-\Omega/2)(\omega+\Omega/2)^{n}$
$\displaystyle\beta^{n-1}\frac{f(\omega-\Omega/2)-f(\omega+\Omega/2)}{\Omega}.$
(5)

In the momentum sums we retained 14 momentum points. The electron-phonon-only
values were calculated using the Kubo-Greeenwood approximation as implemented
in VASP [34] using 10 momentum points. We checked that upon increasing the
number of momentum points further the results vary by less than 1%.

## Results

We calculate the liquid phase for 67 atoms: all iron (Fe ) or with oxygen
(Fe0.91O0.09 ). We study the liquid at CMB conditions for a temperature
$T=4400$ K, volume $8.64\,$Å3/atom (which corresponds to pressure 132 GPa),
and ICB conditions $T=6350$ K with volume $7.16\,$Å3/atom (pressure 330 GPa).

The calculations were performed for three snapshots separated by 5 ps time
each and the calculations of transport properties was performed by averaging
over the snapshots and spatial directions.

Electronic correlations. Figure 1 displays the imaginary part of self-energies
for a single snapshot in the energy range of $[-1,1]$ eV. Each curve
represents a different site/orbital/spin index, and the thick lines indicate
an average over all of them. Since the electron-electron scattering rate is
$1/\tau=-2\mathrm{Im}\Sigma(\omega\rightarrow 0)$, more negative values of
$\mathrm{Im}\Sigma$ indicate stronger electronic correlations. Our results
show that the scattering increases significantly in the oxygen-rich case for
both ICB and CMB. The distribution of the curves reveals that not only the Fe
sites closest to oxygen are affected, but also the spread of the data is wider
in the oxygen-rich case, and even the self-energies with the smallest
magnitudes are enhanced by oxygen. Overall, the average EES, as given by the
average $\mathrm{Im}\Sigma$, increases by 25%.

Figure 1: Imaginary part of the calculated self-energies on the real axis. In
the top and middle panels, we show the individual self-energies (blue lines)
and their average $\langle\mathrm{Im}\Sigma\rangle$ (black thick). In the
bottom panel, the four average self-energies are shown, at the ICB (full) and
CMB (dashed) conditions.

Conductivities. The calculated optical conductivities are shown in Fig. 2 for
the case of electrical and thermal current on the top and bottom panels,
respectively. The $\omega\rightarrow 0$ values indicate the dc-transport
values. One sees that quantitatively the effect of oxygen on transport is
somewhat weaker than on the EES. Also shown in that figure are the results of
a simplified calculation where one uses the $\langle\Sigma\rangle$ instead of
the individual self-energies. One sees a “self-averaging” effect: the results
of such a calculation are almost indistinguishable from the full calculation.
This also tells that statistical uncertainties of the individual self-energies
will not affect the calculated conductivities.

Figure 2: Optical electrical (top) and thermal (bottom) conductivity for pure
Fe and Fe0.91O0.09 . The ICB and CMB cases are shown on the left and right,
respectively. The dashed lines (that overlap closely with the filled ones)
indicate a simplified calculation using fully (site, orbital, and spin)
averaged self-energies.  Figure 3: Thermal conductivity for pure Fe and
Fe0.91O0.09 for the ICB (top) and CMB cases (bottom). We show the results
obtained with orbitally and site-resolved self-energies (dots) as well as
those calculated using the average self-energy (full line). The latter was
first averaged over all sites and orbitals on the Matsubara grid and then
analytically continued. The differences between the two are very small. With
dash-dotted line, we show results calculated by exchanging the average self-
energy between the Fe and Fe0.91O0.09.

To what extent is the suppression of conductivities in the oxygen-rich case
due to the increase of EES documented in Fig. 1? It turns out that, as
suggested in the earlier work [21] iron under Earth’s core conditions is in a
thermal-disorder dominated case where the changes of EES impact transport
weakly. In Fig. 3 we demonstrate this by additional calculations where we
compute the conductivity of the Fe0.91O0.09 case by using the scattering
information from $\langle\Sigma\rangle$ corresponding to pure Fe calculation
and vice versa for the other case. Quite strikingly, these “exchanged”
calculations are at small frequencies almost indistinguishable from the “non-
exchanged” ones. This tells that the oxygen affects the results through a
structurally induced change in the band dispersions and that the changes in
the EES play an insignificant role. This is further demonstrated in Appendix B
where the scattering is artificially increased and only a weak effect on
transport is seen.

Fig. 4 shows the calculated values of resistivity (top) and thermal
conductivity (bottom) along with data from the literature. Thermal disorder
dominates, and perfect crystalline lattices have much higher conductivities.
The additional influence of compositional disorder is moderate, and the
magnitude of the change is similar to that of including EES on top of the
thermal disorder for a given composition.

Figure 4: Resistivity (top) and thermal conductivity (bottom) for Fe and
Fe0.91O0.09 for the ICB and CMB cases calculated using DMFT (including both
electron-electron and e.-ph scattering) and DFT (e.-ph. only). Previous
calculations (extracted from Ref. [21]) for supercell and perfect lattice
bcc/hcp are shown for the solid case.

Suppression of Lorenz ratio.

Figure 5: (left) Lorenz number for Fe (bullets) and Fe0.91O0.09 (crosses) and
for the ICB (full line) and CMB cases (dashed line). We use the parameter
$\alpha$ to artificially change the magnitude of the EES
$\Sigma\rightarrow\Sigma_{\alpha}=\mathrm{Re}\Sigma+\alpha
i\mathrm{Im}\Sigma$. One sees that the Lorenz number is reduced by the
presence of oxygen. (right) Transport distribution $\Gamma(\omega)$. Solid
curves are the obtained form the full DMFT self energy, dashed lines from a
constant scattering rate approximation.

Interestingly, EES suppresses the thermal conductivities more than the
electrical ones. Figure 5 shows the evolution of the Lorenz number
$L=\kappa/(\sigma T)$ with respect to the strength of EES, which is scaled by
the factor $\alpha$, as described in Appendix B. For pure Fe, the value of $L$
due to electron-phonon scattering is almost identical to the standard value of
2.44$\cdot 10^{-8}$ (W$\Omega$/K2), whereas it is considerably reduced when
EES is taken into account. This reduction occurs because inelastic EES affects
$\kappa$ more strongly than $\sigma$. Specifically, $\kappa$ is determined by
integrating $\Gamma(\omega)\omega^{2}(-df/d\omega)$, where $\Gamma(\omega)$ is
the transport distribution function, $f$ is the Fermi function, and $\omega$
is the frequency. On the other hand, $\sigma$ is calculated by integrating
$\Gamma(\omega)(-df/d\omega)$, which only involves the derivative of the Fermi
function. The conductivity is mostly given by states around $\omega=0$, while
the dominant contribution to thermal conductivity occurs at finite energies
$1.5T\lesssim|\omega|\lesssim 4T$.

The right panel of Figure 5 presents the transport distribution
$\Gamma(\omega)$ for the CMB case, which are evaluated for the actual EES
(full) and compared to a calculation where the energy dependence of scattering
is suppressed and the self-energy Im$\Sigma\rightarrow$const is taken
(dashed), corresponding to a DFT transport distribution. It is evident that
the increase of EES with energy suppresses $\Gamma$ that becomes smaller at
larger energies compared to the DFT transport distribution case. Additionally,
a comparison between Fe and Fe-O is interesting, which is evident from the DFT
transport distributions. The inclusion of oxygen leads to significant
suppression of transport distribution at $\omega=2.5$eV, caused by O-2p
hybridization with the 4s iron states. This explains the smaller Lorenz number
in the oxygen-rich case.

## Discussion

| CMB | ICB
---|---|---
|  $T=$4400 K |  $T=$6350 K
|  $V=$8.64 Å3/at |  $V=$7.16 Å3/at
Case | Fe | Fe0.91O0.09 | Fe | Fe0.91O0.09
$-\mathrm{Im}\langle\Sigma(\omega=0)\rangle\,$(eV) | 0.11 | 0.14 | 0.09 | 0.11
$\sigma_{\mathrm{DFT}}$ $~{}(10^{4}\Omega^{-1}\,$cm${}^{-1})$ | 1.35 | 1.24 | 1.54 | 1.37
$\sigma_{\mathrm{DMFT}}$ $~{}(10^{4}\Omega^{-1}$ cm${}^{-1})$ | 1.22 | 1.10 | 1.41 | 1.28
$1-\sigma_{\mathrm{DMFT}}/\sigma_{\mathrm{DFT}}~{}(\%)$ | 10 | 11 | 9 | 7
$\kappa_{\mathrm{DFT}}\,($W m-1K${}^{-1})$ | 145 | 120 | 243 | 190
$\kappa_{\mathrm{DMFT}}\,($W m-1K${}^{-1})$ | 119 | 99 | 193 | 158
$1-\kappa_{\mathrm{DMFT}}/\kappa_{\mathrm{DFT}}~{}(\%)$ | 18 | 17 | 21 | 17
$L_{\mathrm{DFT}}\,(10^{-8}$ W$\Omega\,$K${}^{-2})$ | 2.43 | 2.20 | 2.48 | 2.18
$L_{\mathrm{DMFT}}\,(10^{-8}$ W$\Omega\,$K${}^{-2})$ | 2.22 | 2.05 | 2.16 | 1.95
Table 1: Summary of results for CMB and ICB conditions, both for Fe and
Fe0.91O0.09.

In summary, our study focused on the impact of oxygen on electronic transport
in liquid iron corresponding to the outer Earth’s core conditions. Because
oxygen diminishes the Fe 3$d$ electronic occupation towards half-filling, it
can be expected to strongly enhance electronic correlations. We indeed find
that the EES is moderately increased. The numerical values of the
conductivities and the Lorenz ratio are given in Table 1. The oxygen
substitution at $\sim 10\%$ level, which is a plausible content for the outer
core, suppresses electrical (thermal) conductivities by about 10(17)$\%$ only.
In both Fe and Fe-O including electronic correlations diminishes electrical
conductivities by less than 10% and thermal conductivities by about 20$\%$.
This reduction is consistent with similar studies [21, 22], which suggests
that it can be used as a rule of thumb when direct calculations are not
feasible.

What are the geophysical implications of electronic correlations? The first
observation is that their effect is moderate. The drastic reduction of
conductivity, which was predicted on the basis of EES enhancement in ordered
Fe-O structures [22], is not observed when thermal disorder effects are
simultaneously included. But it is also clear from our study that neither can
one neglect electronic correlations, since the reduction of conductivities due
to EES is comparable to the reduction of electron-phonon scattering due to
light elements. Importantly, some models that assume a higher heat flow from
the core find a thermally only driven geodynamo for $\kappa$ below a limiting
value that is of order 100W/mK [44, 45]. EES whereas small compared to the
thermal disorder, might in the end provide just the necessary additional
scattering next to the compositional disorder to power the geodynamo
sufficiently. At the very least, whenever one argues the compositional
disorder is important, EES must not be neglected either. Another important
finding is the universal suppression of Lorenz number: thermal conductivities
are affected by EES more than electrical ones [19, 20]. This is seen to be
also an effect of compositional disorder in the Fe-O case but EES enhances
this further, by supressing the contribution of states away from the Fermi
energy. This is important both to properly interpret the high-pressure
measurements that mostly probe $\sigma$ and for the geodynamo, because of the
distinct influence of the two quantities there.

In future studies, it would be interesting to investigate also alloying with
sulfur and silicon [24]. Both elements affect the transport properties
strongly at the DFT level because unlike oxygen that alloys interstitially
[24], they alloy substitutionally and therefore more strongly affect the bond
disorder with perhaps different implications for electronic correlations. The
silicon case was recently investigated [22], and the results for CMB seem
compatible with what we find at comparable concentrations of oxygen, but
sulfur that would act also in an oxidizing way could potentially have a bigger
effect. Whereas the sulfur concentrations are believed to be negligible in the
Earth [46], they are expected to be sizable in extraterrestial planets [47].
Finaly, both thermal disorder and compositional disorder are also important
for Fe oxides that are relevant for the properties of the lower mantle, where,
for example, FeO is predicted to be [48, 49, 50, 51] in a state where
electronic correlations are very strong and the influence of thermal and
compositional disorder might lead to large effects there.

## Acknowledgments

GGB and JM are supported by Slovenian Research Agency (ARRS) under Grant no.
P1-0044 and J1-2458. JM acknowledges discussions with A. Georges. DA and MP
are supported by the U.K. Natural Environment Research Council (NERC) under
Grant no. NE/T000228/1 and NE/R000425/1. Computations were performed on the
supercomputer Vega at the Institute of Information Science (IZUM) in Maribor,
Slovenia and in UK on the UK national service Archer2.

## Author contributions

G.B.B. carried out the DFT + DMFT electronic structure and transport ‘
calculations. D.A. and M.P. carried out the DFT molecular dynamics and
transport calculations. G.G.B., J.M, L.V.P, M.A, and D.A discussed the results
and wrote the paper.

## Competing Interests

The authors declare no competing interests.

##  Appendix A Matsubara self-energies

Figure 6 depicts the imaginary part of the self-energy as a function of
Matsubara frequency at ICB (top) and CMB (bottom). The site, spin, and orbital
average is performed and the corresponding self-energy is indicated by a full
line, whereas the individual self-energies smoothly span the full range,
indicated by shading. One sees the enhancement of EES for the oxygen-rich case
in terms of larger magnitude (i.e. more negative values) of $\Sigma(i\omega)$.

Figure 6: Imaginary part of the Matsubara self-energy for the Fe (blue) and
Fe0.91O0.09 (orange) configurations and ICB (top) and CMB cases (bottom). The
lines represent the average over all sites, orbitals, and spins, while the
shadow area shows the range of values. For both temperatures, the
configuration with oxygen is below the pure iron one.

##  Appendix B Effects of increased electron-electron scattering

In order to investigate how results depend on the strength of EES we performed
trial conductivity calculation where we artificially set
$\Sigma\rightarrow\Sigma_{\alpha}=\mathrm{Re}\Sigma+\alpha
i\mathrm{Im}\Sigma$, thus increasing the scattering by a scale factor
$\alpha$. The calculated optical conductivities and thermal optical
conductivities are shown in Fig. 7. One sees a weak dependence on $\alpha$
around $\alpha=1$. As discussed in Ref. [21], the dependence of conductivities
on the EES is weaker than what one would expect from Matthiessen’s rule. The
breakdown of Matthiessen’s rule was also documented as a function of the
substitutional disorder [11].

Figure 7: The optical conductivity and thermal optical conductivity of Fe and
Fe0.91O0.09 configuration with a scale factor $\alpha$ applied to the
imaginary part of calculated average self-energy.

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