Charge disproportionation and site-selective local magnetic moments in the post-perovskite-type Fe$_2$O$_3$ under ultra-high pressures

The archetypal $3d$ Mott insulator hematite, Fe$_2$O$_3$, is one of the basic oxide components playing an important role in mineralogy of Earth's lower mantle. Its high pressure-temperature behavior, such as the electronic properties, equation of state, and phase stability is of fundamental importance for understanding the properties and evolution of the Earth's interior. Here, we study the electronic structure, magnetic state, and lattice stability of Fe$_2$O$_3$ at ultra-high pressures using the density functional plus dynamical mean-field theory (DFT+DMFT) approach. In the vicinity of a Mott transition, Fe$_2$O$_3$ is found to exhibit a series of complex electronic, magnetic, and structural transformations. In particular, it makes a phase transition to a metal with a post-perovskite crystal structure and site-selective local moments upon compression above 75 GPa. We show that the site-selective phase transition is accompanied by a charge disproportionation of Fe ions, with Fe$^{3\pm \delta}$ and $\delta \sim 0.05$-$0.09$, implying a complex interplay between electronic correlations and the lattice. Our results suggest that site-selective local moments in Fe$_2$O$_3$ persist up to ultra-high pressures of $\sim$200-250 GPa, i.e., sufficiently above the core-mantle boundary. The latter can have important consequences for understanding of the velocity and density anomalies in the Earth's lower mantle.


Introduction
Being model objects for studying the Mott transition phenomenon, the iron-bearing oxides play an important role in the mineralogy of Earth's lower mantle and outer core [1][2][3][4][5][6][7][8]. Because of their complex electronic, magnetic, and crystal structure behavior under high pressure-temperature conditions, these compounds have been of considerable recent interest [1,2,[6][7][8][9][10]. It is known that upon compression these materials exhibit a magnetic collapse -a crossover from a high-spin to low-spin state of iron ions, resulting in drastic changes of their physical properties [13][14][15][16]. In fact, the anomalous behavior of their bulk modulus, density, and elastic properties is essential to understanding the seismic observations and dynamic processes in the Earth's lower mantle and outer core [1][2][3][4][5], e.g., for interpretation of the anomalous seismic behavior at the bottom 400 km of Earth's mantle, in the so-called D" region. The high-pressure electronic and structural properties of ferric oxide, hematite (α-Fe 2 O 3 ), the "classical" example of a Mott insulating material, is of particular interest for both fundamental science and technological applications. Its high-pressure properties, such as, rich allotropic behavior, release of oxygen resulting in the appearance of a homologous series of nFeO·mFe 2 O 3 oxides (with wüstite, FeO and Fe 2 O 3 as the end-members), and the unclear role of Fe 3+ in the nature and dynamics of the Earth's lower mantle have attracted much recent attention in geophysics and geochemistry [6-8, 10, 17].
Under ambient conditions, Fe 2 O 3 crystallizes in a rhombohedral corundum (R3c space group) crystal structure with Fe 3+ cations located in distorted oxygen octahedra [18,19]. It is antiferromagnetic below ∼260 K (Morin spin-flop transition temperature) and exhibits weak ferromagnetism (canted antiferromagnetism with a small net ferromagnetic moment of ∼0.002 µ B ) between 260 K and the Néel temperature of ∼956 K [18,19]. Photoemission spectroscopy measurements show that Fe 2 O 3 is a Mott-Hubbard insulator with a large energy gap of about 2.5 eV [20][21][22][23]. Upon compression above ∼50 GPa Fe 2 O 3 undergoes a sharp first-order phase transition to a metallic state (Mott insulator-metal transition) which is accompanied by a remarkable collapse of the lattice volume by about 10% [2,6,10,[24][25][26][27][28][29][30]. The phase transition has been generally assigned to a high-spin (HS) to low-spin (LS) crossover of Fe 3+ ions [1], with a complex coexistence (with equal relative abundance) of the non-magnetic and high-spin components in the Mössbauer spectra at pressures above 50 GPa [2,7]. It was shown that the transition is accompanied by a structural transformation to the high-pressure crystal structure of Fe 2 O 3 . The latter was previously assigned either to an orthorhombic perovskite [24] or a Rh 2 O 3 -II-type crystal structure (P bcn) [10,25].
Only recently, based on single-crystal diffraction, the lattice structure has been resolved to be a distorted double-perovskite GdFeO 3 -type (DPv) structure (space group P 2 1 /n) [2,7,31]. Furthermore, the fine details of this phase transition seems to depend very much on "thermal prehistory" of a sample, showing that the Rh 2 O 3 -II-type structure may appear upon heating to about 1800 K with subsequent quenching to low temperatures [7]. Upon further compression above ∼72 GPa, DPv-type Fe 2 O 3 makes a transition to a new high-pressure polymorph whose crystal structure still remains controversial, with two proposed candidates: either a CaIrO 3 -type post-perovskite (PPv) or orthorhombic Aba2 structures [2,7,31,32].
Whereas the electronic properties of the low-pressure corundum (R3c) phase of Fe 2 O 3 are now well understood from, e.g., the LDA+U method [33,34] (LDA+U : density functional theory calculations within the local density approximation plus Hubbard U approach) or the DFT+DMFT calculations [35][36][37][38][39][40][41] (DMFT: dynamical mean-field theory of correlated electrons), the high-pressure properties of Fe 2 O 3 , e.g., its electronic structure, complex coexistence of the HS and LS states observed in the Mössbauer spectroscopy, a rich variety of structural polymorph and details of the phase diagram in the megabar pressure range still remain enigmatic [6-8, 10, 24-29, 31, 32]. Very recently Greenberg et al. [2] detailed the pressure-induced Mott transition in the DPv-type Fe 2 O 3 at about 50 GPa. In our present work, we extend this study focusing on a long-standing challenge of the electronic and magnetic properties of Fe 2 O 3 under ultra-high pressures. We provide a microscopic theory of the high-pressure electronic structure and magnetic state of Fe 2 O 3 up to compression above the core-mantle boundary conditions. Our results reveal that above 75 GPa Fe 2 O 3 adopts a post-perovskite crystal structure, which is characterized by site-selective local moments, with local moments on half of the Fe sites collapsed into the LS state. The Fe 3d electrons on the rest of the Fe sites remain localized in a high-spin (S=5/2) state up to ultra-high pressures of ∼200-250 GPa, well above the core-mantle boundary. We predict that the site-selective local-moments phase is accompanied by a charge disproportionation of Fe ions, with Fe 3±δ and δ ∼ 0.05-0.09, implying a complex interplay between electronic correlations and the lattice.

Results and Discussion
We employ a state-of-the-art fully self-consistent in charge density DFT+DMFT approach [35][36][37][38][39][40][42][43][44][45][46] to compute the electronic structure, magnetic state, and crystal structure properties of Fe 2 O 3 under pressure. Our results for the calculated enthalpy (with the high-spin R3c phase taken as a reference) are summarized in Fig. 1. The pressure-induced evolution of the instantaneous local moments is shown in Fig. 2. Overall, the calculated electronic and lattice properties of Fe 2 O 3 agree well with available experimental data. In agreement with previous studies [1,2], at ambient pressure, we obtain a Mott insulating solution with a large energy gap of ∼2.5 eV (see Supplementary Fig. 1). The calculated local magnetic moment is ∼4.8 µ B , clearly indicating that at ambient pressure the Fe 3d electrons are strongly localized and form a high-spin S=5/2 state (Fe 3+ ions have a 3d 5 configuration with three electrons in the t 2g and two in the e g orbitals).
Our results for the equilibrium lattice constant a = 5.61 a.u. and bulk modulus K 0 ∼ 187 GPa (with K ≡ dK/dP fixed to 4.1) are in good quantitative agreement with experiment [2,7] Furthermore, we examine the high-pressure electronic structure and phase stability of paramagnetic Fe 2 O 3 in the orthorhombic Aba2 and the CaIrO 3 -type post-perovskite (PPv) crystal structures, i.e., for the two structural candidates for the high-pressure metallic phase proposed from experiment [2,7,31]. Our total-energy calculations within a single-site DFT+DMFT method for the Aba2 phase reveal its remarkable thermodynamic instability (as high as ∼3.71 eV/f.u. above the PPv phase). We note that while this value is large, ∼740 meV/atom, it is not unrealistic. Here, we refer to two new metastable phases of SiO 2 , coesite-IV and coesite-V, which have been synthesized experimentally [47]. The phases are energetically highly unfavorable -at 38 GPa, where coesite-V and coesite-IV are nearly degenerate in enthalpy in theoretical calculations, the calculated enthalpy difference between them and the ground state was found to be ∼390 meV/atom. The values calculated for carbon polymorphs, for example, such as diamond and C 60 are even higher. DFT+DMFT calculations with different computational parameters (U and J) show that the obtained result is robust, suggesting that Aba2 Fe 2 O 3 is metastable at high pressures. Our result agrees well with recent x-ray diffraction studies which reveal that the Aba2 phase is in fact metastable with a stability range limited to low-temperatures [7]. Moreover, the latter is experimentally found to transform into the PPv structure upon annealing to high temperatures [7].
Interestingly, the crystal structure analysis of the Aba2 structure of Fe 2 O 3 reveals the existence of two sufficiently different Fe-Fe bond distances: the Fe-Fe pairs with a short inter-atomic distance of ∼2.43Å and the rest with that of 2.73Å. While the Aba2 phase is non-magnetic (according to the low-temperature Mössbauer spectroscopy [2]), this may suggest dimerization of the Fe-Fe ions in Aba2 Fe 2 O 3 . As a consequence, it implies a possible importance of non-local correlation effects to explain the appearance of the metastable Aba2 phase at low temperatures. Based on our theoretical results, we predict a structural phase transition from the DPv to PPv phase above ∼75 GPa, in quantitative agreement with available experiments [2,7]. The phase transition is accompanied by about 2.6 % collapse of the lattice volume and is associated with formation of a metallic state (both Fe A and B sublattices are metallic). In Fig. 3   We also notice that the sign of δ is opposite to what is expected: the small-volume octahedral Fe B cations, appear to hold more electrons than the prismatic Fe A, located in the largest oxygen cage. This behavior is a consequence of emptying of the antibonding e σ g states of the octahedral Fe B sites at the MI to SSMI phase transition, which leads to a different strength of covalent p-d bonding for the Fe A and B sites [56]. Moreover, upon further compression of the lattice below ∼0.6 V 0 , above ∼190 GPa, the charge disproportion is found to change sign upon transition to a (conventional) metallic state in the DPv phase. In the PPv Fe 2 O 3 it tends to decrease below δ ∼ 0.05, implying a complex interaction between electronic correlations, local magnetism, and the lattice on microscopic level. We note that a similar behavior has been recently suggested to occur in nanocrystalline Fe 2 O 3 under pressure [57]. In addition, we point out a similarity of the electronic and magnetic behavior of Fe 2 O 3 to that observed in the rare-earth nickelates (RNiO 3 with R=Sm, Eu, Y, or Lu) [58][59][60][61][62]. In fact, the latter exhibit a site-selective Mott transition, characterized by a two-sublattice symmetry breaking, with formation of site-selective local moments with localized Ni

Methods
We have employed the DFT+DMFT approach to explore the electronic structure, local magnetic state of Fe 3+ ions, and crystal structure stability of paramagnetic Fe 2 O 3 under pressure using the DFT+DMFT method [42][43][44][45][46] implemented with plane-wave pseudopotentials [63][64][65]. We start by constructing the effective low-energy O 2p -Fe 3d Hamiltonian [Ĥ DFT σ,αβ (k)] using the projection onto Wannier functions to obtain the p-d Hubbard Hamiltonian (in the density-density approximation) structures taken from experiment [2,7], and calculated enthalpy within DFT+DMFT. To compute pressure we fit the calculated total energies using the third-order Birch-Murnaghan equation of states separately for the low-and high-volume regions.

Data availability
The data that support the findings of this study are available from the corresponding authors upon request.

Code availability
The DFT+DMFT code employed in this study is available from the corresponding authors upon request.

Supplementary Information
In our DFT+DMFT calculations we used the average Coulomb interaction U = 6 eV and Hund's exchange J = 0.86 eV for the Fe 3d shell as was estimated previously [1,2].  In Supplementary Fig. 11 we show our results for the orbitally-resolved hybridization function [−Im(∆(ω))] for the Fe A and B sites of the DPv (top) and PPv (bottom) phases of Fe 2 O 3 . Our results indicate sufficiently different hybridization strength for the low-lying (e.g., Fe B t 2g ) and the upper-lying (e.g., Fe B e g ) orbitals of the Fe A and B sites. This hybridization strength difference is larger for the low-spin Fe B sites with respect to that in the high-spin Fe A site. While this difference is sizable near the Fermi level, it becomes even more pronounced near -4 eV− -5 eV below the Fermi level, where the O 2p states are located. This sufficiently different hybridization strength seems to mediate the critical value of the crystal field splitting determined from the Hund's coupling J energy scale (∆ cf = 3J determined from the equality of the energies of the HS and the LS states in the atomic limit).
In our calculations we employ a basis set of atomic-centered symmetry-constrained Wannier functions [3][4][5]. For this purpose the localized atomic orbitals of a given symmetry φ σ µk (Fe 3d and O 2p) are projected onto the subspace of the Bloch functions ψ σ ik (onto the energy bands in a selected energy range near the Fermi level). In this scheme, the Wannier functions are defined as are the matrix elements of the projection operator expressed in the basis of local orbitals φ σ νk : Upon construction of the Wannier basis set we ensure orthogonality of the Wannier orbitals |w σ νk by computing the overlap matrix O σ µν (k) = i P σ * iµ (k)P σ iν (k). This gives us the expression for the orthonormal Wannier functions | w σ The orthonormal projectors P σ iν (k) are used to evaluate the matrix elements of the low-energy DFT HamiltonianĤ σ DFT (k) within the orthonormal Wannier basis set as