# Impact of Electronic Correlations on High-Pressure Iron: Insights from Time-
Dependent Density Functional Theory

K. Ramakrishna  Electronic mail<EMAIL_ADDRESS>Center for Advanced
Systems Understanding, D-02826 Görlitz, Germany  Helmholtz-Zentrum Dresden-
Rossendorf, D-01328 Dresden, Germany  M. Lokamani  Helmholtz-Zentrum Dresden-
Rossendorf, D-01328 Dresden, Germany  A. Baczewski  Sandia National
Laboratories, Albuquerque, NM 87185, USA  J. Vorberger  Helmholtz-Zentrum
Dresden-Rossendorf, D-01328 Dresden, Germany  A. Cangi  Electronic mail:
<EMAIL_ADDRESS>Center for Advanced Systems Understanding, D-02826 Görlitz,
Germany  Helmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden, Germany

###### Abstract

We present a comprehensive investigation of the electrical and thermal
conductivity of iron under high pressures at ambient temperature, employing
the real-time formulation of time-dependent density functional theory (RT-
TDDFT). Specifically, we examine the influence of a Hubbard correction (+U) to
account for strong electron correlations. Our calculations based on RT-TDDFT
demonstrate that the evaluated electrical conductivity for both high-pressure
body-centered cubic (BCC) and hexagonal close-packed (HCP) iron phases agrees
well with experimental data. Furthermore, we explore the anisotropy in the
thermal conductivity of HCP iron under high pressure, and our findings are
consistent with experimental observations. Interestingly, we find that the
incorporation of the +U correction significantly impacts the ground state and
linear response properties of iron at pressures below 50 GPa, with its
influence diminishing as pressure increases. This study offers valuable
insights into the influence of electronic correlations on the electronic
transport properties of iron under extreme conditions.

††preprint: AIP/123-QED

##  I Introduction

The properties of elemental iron under high pressure are of great interest due
to its prominence in Earth’s core Anderson (1986); Mao, Bassett, and Takahashi
(1967); Anzellini  _et al._ (2013); Kraus  _et al._ (2022). Two solid phases
of iron have been broadly studied. Under ambient conditions, the stable
structure of iron is a single-atom body-centered cubic (BCC) cell. At room
temperature and pressures relevant to Earth’s core (360 GPa) Kuwayama  _et
al._ (2008); Tateno  _et al._ (2010), a single-atom hexagonal close-packed
(HCP) cell appears to be the stable structure Dewaele  _et al._ (2006); Tateno
_et al._ (2010); Mao  _et al._ (1990). The phase boundary between the two
phases depends on the magnetic ordering in the BCC phase. Ferromagnetic BCC is
the energetically favored ground state compared to paramagnetic BCC Zhang,
Johansson, and Vitos (2011). Electronic correlations are significant in iron
but become less significant under pressure because the electronic kinetic
energy increases more rapidly than the potential energy.

Experiments using x-ray magnetic circular dichroism and x-ray absorption
spectroscopy indicate a BCC-HCP transition Bancroft, Peterson, and Minshall
(1956); Takahashi and Bassett (1964) between 12-18 GPa. The disappearance of
the net magnetic moment indicates the HCP phase is non-magnetic Mathon  _et
al._ (2004). As the pressure increases, 3$d$ electrons delocalize further and
form hybridized 3$d$-4$p$ states. This goes along with a reduction in magnetic
moment Iota  _et al._ (2007).

Material properties are commonly modeled using density functional theory
(DFT)Hohenberg and Kohn (1964); Kohn and Sham (1965). While successful, strict
semi-local approximations typically fail to capture even the qualitative
physics of strongly correlated materials. However, methods that incorporate a
Hubbard correction (DFT+$U$) for the on-site Coulomb interaction have found
great success in computing the structural and electronic properties of these
challenging materials Anisimov, Aryasetiawan, and Lichtenstein (1997);
Cococcioni and De Gironcoli (2005). Dynamical mean field theory (DMFT)+DFT is
an alternative approach for augmenting conventional DFT, which treats
electronic correlations differently Kotliar  _et al._ (2006); Paul and Birol
(2019); Sánchez-Barriga  _et al._ (2009); Sponza  _et al._ (2017).

The standard approach for computing the electrical conductivity uses the Kubo-
Greenwood (KG) formula Kubo (1957); Greenwood (1958). Commonly, it is
evaluated on Kohn-Sham (KS) orbitals, eigenvalues, and occupation numbers.
This method has been widely used to compute the electrical conductivity in
various systems, including strongly coupled Desjarlais, Kress, and Collins
(2002); Witte  _et al._ (2017); Schörner  _et al._ (2022), as well as liquid
and solid metals Pozzo, Desjarlais, and Alfè (2011); de Koker, Steinle-
Neumann, and Vlček (2012); Vlček, De Koker, and Steinle-Neumann (2012); Korell
_et al._ (2019); French and Mattsson (2014); Kleinschmidt  _et al._ (2023).
However, employing KS quantities in the KG formula neglects the interaction
kernel, which is crucial for capturing collective effects that govern
electronic excitations and determine transport properties Pourovskii  _et al._
(2017).

A more accurate alternative that captures electronic excitations is time-
dependent density functional theory (TDDFT). Within TDDFT, the standard
approach is to employ linear-response TDDFT (LR-TDDFT), which calculates the
interacting response function using an interaction kernel that incorporates
electron correlations through the Hartree and exchange-correlation (XC)
kernel. The effectiveness of LR-TDDFT in computing transport properties with
full wavenumber and frequency resolution has been extensively studied for both
solid materials Cazzaniga  _et al._ (2011) and liquid aluminum Ramakrishna
_et al._ (2021). An alternative method is real-time propagation in time-
dependent density functional theory (RT-TDDFT)Yabana and Bertsch (1996);
Baczewski  _et al._ (2016, 2021). RT-TDDFT directly simulates the microscopic
form of Ohm’s law by obtaining the time-dependent physical current from the
system’s linear response to an electric field. According to Ohm’s law, the
electrical conductivity can be directly computed from the induced time-
dependent current Ramakrishna  _et al._ (2023); Dornheim  _et al._ (2023).

While TDDFT captures the transport properties of simple metals like aluminum
quite well, its performance for less simple metals has not yet been
established. DFT+$U$ has been traditionally used in evaluating static
properties Cococcioni and De Gironcoli (2005), but the Hubbard correction has
not yet been included in the TDDFT calculations of dynamical properties such
as the electrical conductivity.

In this work, we investigate the influence of electronic correlations on iron
under pressure. Initially, we calculate the equation of state (EOS) using DFT
and incorporate Hubbard corrections. By investigating the static properties,
we establish a foundation for computing electronic transport properties. To
evaluate the electrical conductivity of BCC and HCP iron under high pressure,
we employ RT-TDDFT. We specifically account for strong electronic correlations
by incorporating a Hubbard correction, allowing us to assess its impact on the
computed conductivity. Additionally, utilizing this methodology, we
investigate the thermal conductivity and its anisotropy of HCP iron under high
pressure, covering pressure conditions relevant to the Earth’s core-mantle
boundary (up to 150 GPa). Moreover, we demonstrate agreement of our RT-TDDFT
results with the experimental anisotropy measurements reported by Ohta et al.
Ohta  _et al._ (2018).

##  II Methodological and Computational Details

Ground-state DFT calculations of the EOS and generation of the initial KS
orbitals are performed using the full-potential linearized augmented plane-
wave code implemented in Elk et al. (2023). A _k_ -point grid of $20\times
20\times 20$ points and $16\times 16\times 16$ points up to $60$ bands per
atom are considered for the BCC and HCP phases, respectively in a unit cell.
The $c/a$ ratio is fixed at 1.60 throughout for HCP Vekilova  _et al._ (2015);
Hausoel  _et al._ (2017) in the experimental range Glazyrin  _et al._ (2013);
Dewaele  _et al._ (2006) as the variation with pressure is weak at higher
pressures Steinle-Neumann, Stixrude, and Cohen (2004); Steinle-Neumann, Cohen,
and Stixrude (2004). The Perdew-Burke-Ernzerhof (PBE) Perdew, Burke, and
Ernzerhof (1996) exchange-correlation (XC) functional is used. The
calculations with the Hubbard correction are also carried out with spin-
polarized PBE around the mean-field (AMF) Petukhov  _et al._ (2003) form with
double counting Czyżyk and Sawatzky (1994); Cococcioni and De Gironcoli (2005)
with an enhanced description of magnetic and structural properties (see Table.
1 in Appendix A). The Hubbard parameters ($U=0.125$ Ha and $J=0.033$ Ha) used
in this work are well-tested for high-pressure iron Glazyrin  _et al._ (2013);
Mohammed  _et al._ (2010); Vekilova  _et al._ (2015) and in the range used in
earlier studies of ambient iron Cococcioni and De Gironcoli (2005).

We utilize RT-TDDFT Yabana and Bertsch (1996) to compute the electrical
conductivity based on the microscopic form of Ohm’s law

$j_{\alpha}(\bm{r},t)=\sum_{\beta}\int
d\bm{r}^{\prime}\int_{-\infty}^{t}dt^{\prime}\
\sigma_{\alpha\beta}(\bm{r},\bm{r}^{\prime},t-t^{\prime})\,E_{\beta}(\bm{r}^{\prime},t^{\prime})\,,$
(1)

where an external electric field
$\bm{E}(\bm{r}^{\prime},t)=-\partial\bm{A}(\bm{r}^{\prime},t)/\partial t$
induces a current density $\bm{j}(\bm{r},t)$, with $\bm{A}(\bm{r}^{\prime},t)$
denoting the associated external vector potential. To this end, we solve the
time-dependent KS equations

$\hat{H}_{\mathrm{{\scriptscriptstyle
S}}}\psi_{n,k}(\bm{r},t)=i\frac{\partial}{\partial t}\psi_{n,k}(\bm{r},t)$ (2)

for the KS orbitals $\psi_{n,k}(\bm{r},t)$ with the effective Hamiltonian

$\hat{H}_{\mathrm{{\scriptscriptstyle
S}}}=\frac{1}{2}\left[-i\nabla+\bm{A}_{\mathrm{{\scriptscriptstyle
S}}}(\bm{r},t)\right]^{2}+v_{\mathrm{{\scriptscriptstyle S}}}(\bm{r},t)\,,$
(3)

where $v_{\mathrm{{\scriptscriptstyle
S}}}(\bm{r},t)=v_{\mathrm{{\scriptscriptstyle
ext}}}(\bm{r},t)+v_{\mathrm{{\scriptscriptstyle
H}}}(\bm{r},t)+v_{\mathrm{{\scriptscriptstyle XC}}}(\bm{r},t)$ is the KS
potential which is a sum of the external, the Hartree, and XC potentials,
while the effective vector potential $\bm{A}_{\mathrm{{\scriptscriptstyle
S}}}(\bm{r},t)=\bm{A}(\bm{r},t)+\bm{A}_{\mathrm{{\scriptscriptstyle
XC}}}(\bm{r},t)$ comprises the sum of the external vector potential and the XC
contribution Vignale and Kohn (1996); Vignale, Ullrich, and Conti (1997).
Also, note that we use Hartree atomic units throughout. Solving the time-
dependent KS equations yields the induced current density Vignale and Kohn
(1996) as

$\bm{j}(\bm{r},t)=\Im\left[\sum_{i}^{N}\psi_{n,k}^{*}(\bm{r},t)\nabla\psi_{n,k}(\bm{r},t)\right]+n(\bm{r},t)\bm{A}_{\mathrm{{\scriptscriptstyle
S}}}(\bm{r},t)\ .$ (4)

Assuming the electric field is constant in space and its time-dependence is
treated within the dipole approximation, we obtain the dynamical electrical
conductivity as

$\sigma_{\alpha\beta}(\omega)=\frac{\tilde{j}_{\alpha}(\omega)}{\tilde{E}_{\beta}(\omega)}\,,$
(5)

where we integrated Eq. (1) over the spatial coordinate and performed Fourier
transforms over the time variable with $\tilde{j}_{\alpha}(\omega)=\int dt\int
d\bm{r}\ j_{\alpha}(\bm{r},t)e^{i\omega t}$ and
$\tilde{E}_{\beta}(\omega)=\int dt\ E_{\beta}(t)e^{i\omega t}$. Note that both
the current and electric field are vectors, while the electrical conductivity
is a tensor. To achieve a sufficient resolution in frequency space and to
guarantee that the electric current eventually reaches a stable state, the
duration of the real-time propagation is crucial. The energy window of the
simulation is inversely related to the full-width half maximum (FWHM) of the
pulse, and the spectral energy resolution is inversely proportional to the
total propagation time.

Within the RT-TDDFT calculations, a sigmoidal pulse of vector amplitude 0.1
a.u. is applied with a peak time of 2 a.u. having an FWHM of 0.5 a.u. for a
total simulation time of up to 2000 a.u (1 a.u.=24.819 attoseconds) providing
an energy resolution up to 0.5 $\mathrm{m}\mathrm{H}\mathrm{a}$ in frequency
space. The pulse mentioned above (blue solid) is applied in the $z$ direction
shown in Fig. 1a) with the induced current density (green solid) along $z$.
The dynamical electrical conductivity (red solid) obtained from Eq. (5) is
shown in Fig. 1b) where the DC conductivity is obtained from a Drude fit
(black dashed) at low energies.

Figure 1: a) Sigmoidal pulse (blue) in the $z$ direction vs. time with the
induced current density (green) along $z$ vs. time, b) dynamical electrical
conductivity component $\sigma_{zz}$. The black dashed line is the Drude fit
that yields the DC conductivity. The inset plot shows the dynamical electrical
conductivity at low energies.

##  III Results

###  III.1 Equation of State

By employing ground-state DFT in conjunction with the Hubbard correction, we
examine, as shown in Fig. 2, the EOS, the electronic density of states (DOS),
and the phonon DOS of iron across a pressure range of approximately 180 GPa.
In doing so, we specifically analyze the impact of electronic correlations on
the EOS. In Fig. 2a), we present our findings (depicted as red and orange
curves) alongside previous experimental measurements Boehler  _et al._ (2008);
Dewaele  _et al._ (2006); Sakai  _et al._ (2014); Fei  _et al._ (2016); Ono
_et al._ (2010) (denoted by black symbols) and theoretical predictions
Pourovskii (2019); Sha and Cohen (2010); Zhuang  _et al._ (2021); Sola,
Brodholt, and Alfè (2009); Steinle-Neumann, Cohen, and Stixrude (2004)
(denoted by green symbols). Our results are obtained by fitting total energies
from DFT calculations using the PBE functional (GGA) Perdew, Burke, and
Ernzerhof (1996) for various volumes to the Vinet EOS Vinet  _et al._ (1987,
1989). The calculations with a Hubbard correction (+$U$) are also carried out
for the spin-polarized case using the PBE functional around the mean-field
form with double counting Czyżyk and Sawatzky (1994); Cococcioni and De
Gironcoli (2005).

We first examine the equation of state (EOS) shown in Fig. 2a). In the BCC
phase, the inclusion of the GGA$+U$ correction yields a relatively modest but
notable improvement over the GGA functional alone. The GGA$+U$ calculations
effectively shift the results towards a better agreement with the experimental
findings reported by Dewaele et al.Dewaele  _et al._ (2006). Moreover, our
GGA$+U$ results are consistent with the findings obtained from LDA
calculations augmented by dynamical mean field theory (DMFT) calculations, as
reported by Pourvoskii et al.Pourovskii (2019). However, our findings suggest
that employing a GGA functional alone is sufficient to reasonably capture the
ground state properties Cococcioni and De Gironcoli (2005) of the
ferromagnetic BCC phase, particularly when considering the significant static
exchange splitting present in the system.

As pressure increases, the antiferromagnetic HCP phase becomes the stable
phase Steinle-Neumann, Stixrude, and Cohen (2004); Cohen and Mukherjee (2004);
Lizárraga  _et al._ (2008); Thakor  _et al._ (2003), starting at approximately
13 GPa (refer to Fig. 7 in Appendix A for the enthalpy curve). In this phase,
electronic correlations play a more significant role as the static exchange
splitting reduces bonding. Previous studies have considered these dynamical
many-body effects using dynamical mean field theory (DMFT) to account for the
impact of the +$U$ correction on the EOS Pourovskii  _et al._ (2014).

In the HCP phase, we observe a similar trend to the BCC phase. At lower
pressures, HCP iron exhibits strong correlation effects. While our results
obtained using GGA agree well with available DFT and quantum Monte Carlo (QMC)
data in the literature Sha and Cohen (2010); Zhuang  _et al._ (2021); Sola,
Brodholt, and Alfè (2009); Steinle-Neumann, Cohen, and Stixrude (2004), they
visibly deviate from experimental measurements. Consequently, our GGA$+U$
calculations provide more accurate results and are in agreement with the
LDA+DMFT results of Pourvoskii et al.Pourovskii (2019) in the pressure range
of 13-30 GPa. However, even with the inclusion of the Hubbard correction to
improve the treatment of strong correlations, the volume change in the EOS
still deviates from experimental dataBoehler  _et al._ (2008); Dewaele  _et
al._ (2006); Sakai  _et al._ (2014); Fei  _et al._ (2016); Ono  _et al._
(2010) by approximately 5% at around 70 a.u.3/atom. The GGA predictions yield
a $c/a$ ratio of less than 1.60, which further decreases with increasing
volume Pourovskii  _et al._ (2014), while DMFT predicts a nearly constant
$c/a$ ratio of approximately 1.60 regardless of volume, consistent with
experimental measurements Glazyrin  _et al._ (2013); Dewaele  _et al._ (2006).
In the pressure range of 40-50 GPa, the correlation effects are slightly
better captured by LDA+DMFT compared to GGA$+U$. At higher pressures (>100
GPa), our GGA results align well with experimental data Boehler  _et al._
(2008); Dewaele  _et al._ (2006); Sakai  _et al._ (2014); Fei  _et al._
(2016); Ono  _et al._ (2010) and previous theoretical efforts Sha and Cohen
(2011); Steinle-Neumann, Cohen, and Stixrude (2004); Zhuang  _et al._ (2021)
as the importance of correlations diminishes.

Fig. 2b) shows the spin-resolved DOS of HCP iron at high pressure computed
using DFT and DFT+$U$. At relatively low pressure (17.7 GPa), DFT+$U$ results
in substantial changes in the occupancy of the $3d$ electrons near the Fermi
energy. The DOS in the HCP phase at ambient pressure computed using DFT+DMFT
Hausoel  _et al._ (2017) is additionally shown for reference. There is a
striking resemblance in the peak feature above the Fermi energy between the
DFT+$U$ and DMFT results, even though the DMFT results are calculated at
ambient pressure.

Fig. 2c) shows the phonon DOS obtained via finite differences force constants
as implemented in phonopy Togo  _et al._ (2023); Togo (2023) at pressures of
71.5 GPa and 138 GPa. They agree with the experimental results obtained using
nuclear resonant inelastic x-ray scattering (NRIXS) at pressures of 71 GPa and
136 GPa respectively Gleason, Mao, and Zhao (2013). The figure clearly
demonstrates that the influence of the Hubbard correction on the phonon
density of states (DOS) is insignificant within the considered pressure
ranges, consistent with the available experimental data.

Figure 2: a) Equation of state (EOS) for BCC and HCP iron. Experimental data
for ambient temperature (T=300 K) stems from Refs. Boehler  _et al._ , 2008;
Dewaele  _et al._ , 2006; Sakai  _et al._ , 2014; Fei  _et al._ , 2016; Ono
_et al._ , 2010. Theoretical data at T=0 K and ambient temperature (T=300 K)
stems from Refs. Pourovskii, 2019; Sha and Cohen, 2010; Zhuang  _et al._ ,
2021; Sola, Brodholt, and Alfè, 2009; Steinle-Neumann, Cohen, and Stixrude,
2004, b) Spin-resolved density of states (DOS) of HCP iron at 17.6 GPa
computed using DFT and DFT+$U$. Contribution from the 3d states is indicated
by the dotted lines. HCP DOS at ambient pressure stems from Ref. Hausoel  _et
al._ , 2017, c) Phonon density of states for HCP iron at 71.5 GPa (bottom) and
138 GPa (top) computed using DFT and DFT+$U$. Experimental data on HCP iron
for pressures 71 GPa and 136 GPa at ambient temperature stems from Ref.
Gleason, Mao, and Zhao, 2013.

###  III.2 Dynamical Conductivity

In the following, we investigate the impact of electronic correlation effects
on the dynamical conductivity of iron in both the BCC and HCP phases.

####  III.2.1 BCC Iron

The frequency-dependent conductivity (Eq. 5) obtained using RT-TDDFT (red
solid) including Hubbard corrections for BCC iron under ambient conditions is
shown in Fig. 3. Due to the symmetry of the BCC lattice ($a=b=c$), it is
sufficient to compute a single diagonal component of the conductivity tensor
($\sigma_{ab}$). In our case, we compute the $\sigma_{zz}$ component and
compare our results with the KG results (green dashed) by Alfè et al. Alfè,
Pozzo, and Desjarlais (2012), at a slightly elevated temperature (T=500 K),
and experimental results by Paquain Paquin (1995), Palik Palik (1998) and
Cahill Cahill  _et al._ (2019) at ambient temperature. Overall, the RT-TDDFT
results are in remarkable agreement with the experimental results by Paquain
Paquin (1995) and Palik Palik (1998) throughout the considered frequency
range, and with the experimental results by Cahill Cahill  _et al._ (2019) at
lower frequencies. In contrast, the KG method, widely regarded as the standard
approach for computing electrical conductivities, falls short of providing an
accurate description across the entire frequency domain. In not displaying the
drop in conductivity up to $50$ eV, the KG method averages over the feature at
$\sim$55 eV which stems from 3p to conduction band transitions.

The DC conductivities obtained from the KG method (9.03
$\mathrm{M}\mathrm{S}\mathrm{m}^{-1}$), extracted through a Drude-like fit
using different _k_ -point sampling and cell sizes Alfè, Pozzo, and Desjarlais
(2012), exhibit values close to those reported in experimental studies
($\sim$10 $\mathrm{M}\mathrm{S}\mathrm{m}^{-1}$) Gomi  _et al._ (2013); Seagle
_et al._ (2013). Similarly, the DC conductivities derived from RT-TDDFT+U
yield a value of 8.83 $\mathrm{M}\mathrm{S}\mathrm{m}^{-1}$, further
confirming their consistency with experimental observations.

It is common to verify simulation data in terms of the f-sum rule Mahan (2013)

$\frac{1}{\pi}\int_{0}^{\infty}d\omega\ \sigma(\omega)=n_{e},$ (6)

where $n_{e}$ is the number of electrons in the simulation cell. Evaluated in
a frequency range up to $\sim$5 Ha (i.e., $\sim$136 eV) and normalized to the
f-sum rule of the experimental data by Paquin Paquin (1995), this yields ratio
of 0.95 for RT-TDDFT compared to 0.97 using the KG formula obtained by Alfè et
al. Alfè, Pozzo, and Desjarlais (2012) evaluated up to $\sim$5.5 Ha (i.e.,
$\sim$150 eV directly from the f-sum rule) providing a measure of the quality
of our calculations.

Figure 3: Dynamical electrical conductivity of BCC iron at ambient conditions
computed using RT-TDDFT with Hubbard corrections. DC values for the RT-TDDFT
(red) and KG (green) results are indicated by the arrows. The inset plot shows
the dynamical electrical conductivity at low energies. Experimental data stems
from Refs. Paquin, 1995; Cahill  _et al._ , 2019; Palik, 1998. Theoretical
data at T=500 K evaluated using the Kubo-Greenwood formula stems from Ref.
Alfè, Pozzo, and Desjarlais, 2012.

####  III.2.2 HCP Iron

In the HCP phase, we evaluate the conductivity tensor similar to the BCC phase
discussed earlier but along various directions and analyze the corresponding
tensor components. This is relevant because the asymmetrical HCP lattice
($a=b\neq c$), as opposed to the BCC lattice, requires the computation of
additional tensor components and could be of great aid for characterizing
materials exhibiting the Hall effect Hall  _et al._ (1879); Elliott,
Shallcross, and Sharma (2022). Fig. 4 shows the frequency-dependent
conductivity of HCP iron at 23 GPa obtained using RT-TDDFT with Hubbard
corrections along the $z$ ($\sigma_{zz}$ \- red solid) and $x$ ($\sigma_{xx}$
\- blue dashed) directions. Notably, no experimental or theoretical data exist
for direct comparison under these conditions, making our study the first
benchmark for future experiments in this domain.

The DC conductivities are represented by arrows, with the $z$ and $x$
directions yielding DC values ranging from 7.42
$\mathrm{M}\mathrm{S}\mathrm{m}^{-1}$ to 1.65
$\mathrm{M}\mathrm{S}\mathrm{m}^{-1}$, respectively. These values align with
experimental resistivity (inverse of conductivity) measurements Seagle  _et
al._ (2013); Gomi  _et al._ (2013). The significant anisotropy in the DC
conductivity holds particular relevance for experiments conducted with diamond
anvil cells (DAC), as crystallographic orientations play a vital role in
determining the observed behavior.

Figure 4: Dynamical electrical conductivity of HCP iron at 23 GPa computed
using RT-TDDFT with Hubbard $U$ corrections along the $z$ (red) and $x$ (blue)
directions. The inset plot shows the dynamical electrical conductivity at low
energies. Anisotropy in the DC conductivity is inferred from the arrows.

###  III.3 DC Conductivity with Pressure

Now we turn to the DC conductivity evaluated using RT-TDDFT at high pressures.
Fig. 5 shows the DC conductivity as a function of pressure in the BCC and HCP
phases at ambient temperature (T=300 K). The results in the BCC phase are
indicated by solid red circles and those in the HCP phase by empty red
circles. In the BCC phase, our predicted DC conductivity is in the range
reported by several experiments Yousuf, Sahu, and Rajan (1986); Gomi  _et al._
(2013); Paquin (1995); Palik (1998). With increasing pressure in the BCC
phase, the DC conductivity drops in agreement with the experiments Gomi  _et
al._ (2013); Seagle  _et al._ (2013). There is a further drop in the
conductivity observed near 13 GPa which marks the transition between the BCC
and the HCP phase. Note that a similar drop in the conductivity during the
phase transition has been observed for several other materials Balchan and
Drickamer (1961); Garg, Vijayakumar, and Godwal (2004). In contrast to the
experimental results Gomi  _et al._ (2013); Seagle  _et al._ (2013) and our
results, ab initio calculations by Sha et al. Sha and Cohen (2011) report an
increase in the conductivity for the BCC phase with pressure although the drop
in the conductivity during the phase transition is captured. The discrepancies
in the theoretical results can be attributed to the use of density functional
perturbation theory (DFPT) by Sha et al. to evaluate the electrical
resistivity, whereas our work employs the more effective approach of RT-TDDFT.
By considering electronic correlations more accurately, as demonstrated
earlier in the analysis of dynamical electrical conductivity, RT-TDDFT proves
to be a superior method for capturing the relevant physics.

Our results in the HCP phase exhibit good agreement with experimental data
from Seagle and Gomi et al.Seagle  _et al._ (2013); Gomi  _et al._ (2013), as
well as with the findings of C. Zhang et al.Zhang  _et al._ (2018) up to 50
GPa. Additionally, our results align well with the measurements on HCP iron by
Y. Zhang et al.Zhang  _et al._ (2020), as indicated by the black dotted line,
particularly for pressures below 50 GPa. Notably, our results differ
significantly from the calculations by Sha et al.Sha and Cohen (2011), as
their estimations overestimate the conductivity in the HCP phase. The black
dashed line represents the model-based pressure and temperature dependence of
the resistivity, derived by Seagle et al.Seagle  _et al._ (2013), which agrees
well with our findings for pressures above 50 GPa.

Figure 5: Pressure versus DC conductivity at ambient temperature (T=300 K).
The results of this work are indicated by solid and empty red circles for BCC
and HCP phases respectively. The drop in the electrical conductivity around
$\sim$13 GPa indicates the BCC$\rightarrow$HCP phase transition. The red
dashed line is the linear fit to our HCP data. The black dashed line is a
pressure and temperature-based model by Seagle et al. Seagle  _et al._ (2013).
The black dotted line is the fitted results to the experimental measurements
by Y. Zhang et al. Zhang  _et al._ (2020). Theoretical data stems from Ref.
Sha and Cohen, 2011. Experimental data stems from Refs. Gomi  _et al._ , 2013;
Seagle  _et al._ , 2013; Zhang  _et al._ , 2018.

###  III.4 Thermal Conductivity in HCP Iron with Pressure

Figure 6: Electronic components of the thermal conductivity of HCP iron along
different directions (top panel - $z$ direction, bottom panel - $x$ direction)
vs pressure. The inclusion of Hubbard corrections ($+U$) is additionally
shown. The dashed and the solid lines are the linear fit to RT-TDDFT and RT-
TDDFT+$U$ results respectively. The red and the yellow curves shown are
computed using the ideal Lorenz number ($L_{0}$) and 1.22$\times L_{0}$
respectively. Experimental anisotropy data (black hollow squares and black
hollow circles) at ambient temperature (T=300 K) stems from Ref. Ohta  _et
al._ , 2018. Estimated data (black hollow triangles) at the core-mantle
boundary (CMB) conditions from Ref. Ohta  _et al._ , 2018.

Finally, we analyze the electronic component of the thermal conductivity of
HCP iron along the $z$ (top panel) and $x$ (bottom panel) directions, as
depicted in Fig. 6. The thermal conductivity is determined from the electrical
conductivity using the Wiedemann-Franz law Franz and Wiedemann (1853). Given
the pressure and temperature conditions in this study, we expect the
Wiedemann-Franz law to be valid, as both energy and charge transport is
predominantly governed by the same free carriers, with inelastic scattering
effects becoming less significant. For our calculations, we employ the ideal
Lorenz number ($L_{0}$=2.44$\times 10^{-8}$ W$\Omega$K-2), as well as an
experimentally determined Lorenz number (1.22$\times L_{0}$) established by
Secco et al.Secco (2017) under ambient conditions, for comparison. It is worth
noting that the conventional Lorenz number used under high-pressure and high-
temperature conditions exhibits substantial variations across previous studies
Pozzo  _et al._ (2012); de Koker, Steinle-Neumann, and Vlček (2012); Pozzo
_et al._ (2014); Ohta  _et al._ (2016); Konôpková  _et al._ (2016); Xu  _et
al._ (2018).

Our findings exhibit excellent agreement with the experimental data obtained
from laser-heated diamond anvil cell (DAC) measurements conducted by Ohta et
al.Ohta  _et al._ (2018) (depicted as black hollow squares and black hollow
circles for $\kappa_{z}$ and $\kappa_{x}$, respectively) at pressures below 50
GPa. Notably, the Hubbard corrections play a crucial role, particularly at
lower pressures where electronic correlations are more pronounced. Neglecting
the Hubbard corrections leads to a deviation of up to 40$\%$ along the $z$
direction around 13 GPa. We extend our calculations to the pressure regime
encountered at the core-mantle boundary (CMB) in Earth’s interior (P$\sim$135
GPa). Here, we additionally present the estimated thermal conductivity
(depicted as black hollow triangles) by Ohta Ohta  _et al._ (2018). At CMB
conditions, we report a $\kappa_{z}/\kappa_{x}$ ratio of 3.03, which is
comparable to the value of $2.56\pm 0.48$ estimated by Ohta et al.Ohta  _et
al._ (2018). The thermal conductivity of pure iron at CMB conditions is
estimated by Seagle et al.Seagle  _et al._ (2013) to be 145 Wm-1K-1. Our
predictions fall within this range, with a value of $158.4$ Wm-1K-1 ($197.4$
Wm-1K-1 when considering 1.22$\times L_{0}$), aligning with previous
theoretical effortsde Koker, Steinle-Neumann, and Vlček (2012); Pozzo  _et
al._ (2012). Taking the trace average of the conductivity tensor
$\sum_{i=(x,y,z)}\sigma_{ii}/3$ into account, the resulting thermal
conductivity near CMB conditions (P$\sim$135GPa) is approximately 87 Wm-1K-1
(approximately 109 Wm-1K-1 when considering 1.22$\times L_{0}$). This analysis
emphasizes the impact of electronic correlations on thermal conductivity at
pressures up to 50 GPa, underscoring the importance of employing the RT-TDDFT
methodology with Hubbard corrections for accurate simulation and modeling.

In recent studies, the total thermal conductivity, accounting, in addition,
for the coupling between phonons and magnons, has been split into various
contributions, as demonstrated by Nikolov et al.Nikolov  _et al._ (2022).
Their work highlights the significant contribution of the electronic component
(approximately 80-90$\%$) at ambient temperatureBäcklund (1961); Wu, Liu, and
Luo (2018). For the purpose of this study, we have thus focused solely on the
electronic contribution to thermal conductivity. Experimental measurements of
ferromagnetic BCC iron at ambient conditions show a total thermal conductivity
of 77.3 Wm-1K-1, with the electronic contribution estimated at 69.4 Wm-1K-1
according to the Wiedemann-Franz law Fulkerson, Moore, and McElroy (1966). In
comparison, our calculations yield a thermal conductivity of 64.6 Wm-1K-1
(78.6 Wm-1K-1 using 1.22$\times L_{0}$) according to the Wiedemann-Franz law.

##  IV Conclusions

In this study, we comprehensively investigated the influence of electronic
correlations on various properties of solid iron at high pressures. These
properties include ground state characteristics such as the EOS, the
electronic DOS, and the phonon DOS, as well as electronic transport properties
including electrical and thermal conductivity. To perform our analysis, we
employed RT-TDDFT with Hubbard corrections and compared the results with those
obtained using the conventional KG method for transport properties.

Our findings clearly demonstrate that electronic correlation effects have a
significant impact on the investigated properties up to pressures of 50 GPa
Pourovskii  _et al._ (2014). Moreover, our work highlights the advantages of
utilizing RT-TDDFT (including Hubbard corrections) as a more reliable method
for computing electronic transport properties compared to the current state-
of-the-art approach, such as the KG method.

We envision promising applications of this methodology in studying structural
phase transitions and spin response Kaa  _et al._ (2022) in materials under
extreme conditions, made possible by recent advancements in free-electron
lasers Zastrau  _et al._ (2021); Cerantola  _et al._ (2021). The frequency-
dependent electrical conductivity, along with the analysis of anisotropy in
HCP iron, can serve as a valuable benchmark for future experiments,
particularly in the context of advanced diagnostics.

Furthermore, the role of magnetism in iron-nickel alloys under Earth-core
conditions has been shown to be significant, highlighting the importance of
electronic correlations, which were previously assumed to be negligible
Vekilova  _et al._ (2015); Hausoel  _et al._ (2017). In the future,
incorporating electronic correlations into the ab initio analysis of
electronic transport properties in the liquid phase of iron and iron alloys
relevant to Earth-core conditions would be an important task to pursue.

###### Acknowledgements.

This work was partially supported by the Center for Advanced Systems
Understanding (CASUS) which is financed by Germany’s Federal Ministry of
Education and Research (BMBF) and by the Saxon state government out of the
State budget approved by the Saxon State Parliament. Computations were
performed on a Bull Cluster at the Center for Information Services and High-
Performance Computing (ZIH) at Technische Universität Dresden and on the
cluster Hemera at Helmholtz-Zentrum Dresden-Rossendorf (HZDR). Sandia National
Laboratories is a multimission laboratory managed and operated by National
Technology & Engineering Solutions of Sandia, LLC, a wholly-owned subsidiary
of Honeywell International Inc., for the U.S. Department of Energy’s National
Nuclear Security Administration under contract DE-NA0003525. This paper
describes objective technical results and analysis. Any subjective views or
opinions that might be expressed in the paper do not necessarily represent the
views of the U.S. Department of Energy or the United States Government.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Appendix

A comparison between the calculated lattice constant, magnetic moment, and
bulk modulus within several approximate DFT schemes, ab initio methods, and
experimental results are shown in Table. 1.

Table 1: Comparison between the calculated lattice constant, magnetic moment,
and bulk modulus within several approximate DFT schemes, experimental and
theoretical results for BCC iron.

Method |  $a_{0}$ (a.u.) |  $\mu_{0}$ ($\mu_{b}$) |  $B_{0}$ (GPa)
---|---|---|---
Spin-pol. LDA | 5.21 | 2.02 | 259
Spin-pol. PBEsol | 5.28 | 2.09 | 213
Spin-pol. PBE | 5.37 | 2.19 | 169
Spin-pol. PBE+$U$ | 5.39 | - | 166
Spin-pol. LDAa | 5.20 | 1.95 | -
Spin-pol. LDA+$U^{a}$ | 5.35 | 2.73 | -
Spin-pol. PBEsola | 5.35 | 2.12 | -
Spin-pol. PBEsol+$U^{a}$ | 5.35 | 2.71 | -
Spin-pol. PBEa | 5.20 | 2.19 | -
Spin-pol. PBE+$U^{a}$ | 5.37 | 2.09 | -
Spin-pol. LDAe | 5.22 | 2.33 | 210
LDA+DMFTf | 5.39 | - | 172
Expt. | 5.42b | 2.22c | 173d

11footnotetext: Ref. Tavadze  _et al._ , 2021.22footnotetext: Ref. Chiarotti,
1995.

33footnotetext: Ref. Kikuchi, Suzuki, and Katayama, 1990.44footnotetext: Ref.
Adams  _et al._ , 2006.

55footnotetext: Ref. Cococcioni and De Gironcoli, 2005.66footnotetext: Ref.
Pourovskii  _et al._ , 2014.

Fig. 7 shows the variation of enthalpy with pressure for the BCC and HCP
phases of iron. The inset plot indicates the relative difference in enthalpy
between the phases with the phase transition occurring at $\sim$12.9 GPa in
agreement with the range observed in experiments Bancroft, Peterson, and
Minshall (1956); Takahashi and Bassett (1964).

Figure 7: Enthalpy versus pressure for BCC and HCP phases of iron. The inset
plot shows the relative difference in enthalpy ($\Delta H_{BCC}-\Delta
H_{HCP}$) between the phases versus pressure.

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