Unexpected 3+ valence of iron in FeO2, a geologically important material lying “in between” oxides and peroxides

Recent discovery of the pyrite FeO2, which can be an important ingredient of the Earth’s lower mantle and which in particular may serve as an extra source of water in the Earth’s interior, opens new perspectives for geophysics and geochemistry, but this is also an extremely interesting material from physical point of view. We found that in contrast to naive expectations Fe is nearly 3+ in this material, which strongly affects its magnetic properties and makes it qualitatively different from well known sulfide analogue - FeS2. Doping, which is most likely to occur in the Earth’s mantle, makes FeO2 much more magnetic. In addition we show that unique electronic structure places FeO2 “in between” the usual dioxides and peroxides making this system interesting both for physics and solid state chemistry.


Details of the calculations
For ab initio calculations we used full-potential Wien2k 1 and pseudopotential Quantum ESPRESSO (QE) 2 codes. The cutoff energy for wave-functions in QE was chosen to be 45 Ry. The parameter of the plane-wave expansion in Wien2k calculations was set to be R MT K max = 7, where R MT is the smallest atomic sphere radii (R O MT = 1.6 a.u., R Fe MT = 1.8 a.u.) and K max -plane wave cut-off. The crystal structures for the calculations were taken from Ref. 3,4 . The on-site Hubbard U = 6 eV and Hund's intra-atomic exchange J H = 0.9 eV were estimated in QE 5 on the same Wannier functions, which were used in construction of a small noninteracting Hamiltonian used in subsequent GGA+DMFT calculations 6,7 . Note, that these values agree well with results of previous calculations of U for other Fe oxides at high pressure 8 . It is shown in the main part of paper that Hubbard correlations have minor influence on the electronic properties of FeO 2 . Therefore we have not investigated how they change with U.
We used segment CT-QMC method to solve an impurity problem 9 . The Hamiltonian includes Fe 3d and O 2p states and was constructed on a fine-grid 16 × 16 × 16 mesh in k−space. The double counting correction was set to E dc = U(n DMFT − 1 2 ) 6 , with n DMFT the total number of 3d electrons self-consistently obtained within DMFT.
For the calculation we used the crystal structure presented in 4 (in the extended data tables 1 and 2). We chose the one corresponding to the powder data.

GGA results
In Fig. S1 results of the nonmagnetic GGA calculations for FeO 2 and FeS 2 are presented. One may see that FeS 2 is an insulator, while FeO 2 is metallic. We note in passing that the transition to insulating state found in Ref. 10 at larger volumes obviously corresponds to smaller pressures, below critical value P c = 76 GPa, at which FeO 2 can be synthetized 4 .
In the GGA calculations for defect and doped FeO 2 we allowed a relaxation of the crystal structure. Using total energy calculations we estimated the nearest neighbor exchange coupling, J, in the case of 25% Fe vacancies (J = 20 K) and simulated the uniform magnetic susceptibility in the classical Heisenberg model using QMC calculations as implemented in ALPS code 11 (see Fig. S4).

Dynamical electronic correlations (DFT+DMFT results) and magnetic properties of FeO 2
We used wannier function projection procedure 12 as realized in the Quantum ESPRESSO (QE) 2 code to extract noninteracting GGA hamiltonian H DFT , which included both Fe 3d and O 2p states. Full many-body Hamiltonian to be solved by the GGA+DMFT is written in the form:   Here U σ σ αβ is the Coulomb interaction matrix,n d iασ is the occupation number operator for the d electrons with orbitals α or β and spin indexes σ or σ on the i-th site. The termĤ dc stands for the d-d interaction already accounted for in the DFT, so called double-counting correction, which was chosen to beĤ dc =Ū(n dmft − 1 2 )Î. Here n dmft is the self-consistent total number of d electrons obtained within the DFT+DMFT,Ū is the average Coulomb parameter for the d shell.
The elements of U σ σ αβ matrix are parameterized by U and J H according to procedure described in Ref. 13 . The effective impurity problem for the DMFT was solved by the hybridization expansion Continuous-Time Quantum Monte-Carlo method (CT-QMC) 9 . Spectral functions on real energies were calculated by Maximum Entropy Method (MEM) 14 . The values of Coulomb repulsion parameter U and Hund's exchange parameter J H were found to be U = 6.0 eV and J H =0.9 eV using constrained LDA calculations 5 . The calculations were performed with the AMULET code 7 .
Spectral function for undoped FeO 2 calculated for inverse temperature β = 1/T =20eV −1 is shown in Fig. S2. One may see that the main effect of the electronic correlations is a renormalization of the spectra in the vicinity of the Fermi level, m * /m ∼ 1.2-1.6 (depending on the orbital).
We which, e.g., for β = 15 eV −1 was found to be 2.32 µ 2 B . In DMFT it is calculated as where n m are the occupancies of corresponding orbitals. Thus, there will be three contributions to m 2 z , which come from the t 2g orbitals, the e g orbitals, and mixed t 2g /e g terms: m 2 z t 2g = 1.08µ 2 B , m 2 z e g = 1.04µ 2 B and the rest, which is due to the mixed t 2g /e g contribution. The fact that m 2 z t 2g ∼ 1µ 2 B is due to the low-spin d 5 configuration (t 3 2g ↑ t 2 2g ↓) of Fe 3+ ions. It is more interesting to discuss how so large contribution from the e g orbitals appears.
The answer is that this moment is not due to Fe electrons, but actually it comes from oxygen holes. Due to a rather large oxidation state of Fe some of the electrons move from oxygen to iron (and create holes on oxygen sites; for the review see 15,16 ). In a conventional DFT calculation this is a well known effect, which results in decrease of the local magnetic moment (the so-called decrease of the moment due to hybridization or covalency). But in paramagnetic DMFT we calculate not an ordered, but fluctuating moment, which is then squared and averaged over all QMC sweeps. Each time an electron comes from an oxygen it can have the same spin as Fe electrons (which are in d 5 low-spin configuration), or can have an opposite spin. Intra-atomic Hund's rule dictates the same spin configuration. On each QMC step in the impurity problem we have contributions both from the t 2g and e g orbitals to the total m 2 z . The total magnetic moment is zero, since the spin moment in paramagnetic DMFT is fluctuating, but m 2 z is not. Due to intra-atomic Hund's exchange in paramagnetic DMFT oxygen electrons will increase m 2 z (not the total moment!) in a strong contrast to the DFT, where hybridization leads to decrease of the local moment. Moreover, since t ↑ 2g states are completely filled in the case of Fe 3+ and pdσ bonding is much stronger than pdπ this contribution will largely affect m 2 z eg . The uniform magnetic susceptibility, χ u (T ), was calculated as a response to small external field (corresponding to Zeeman splitting of 0.01 eV) in DFT+DMFT and presented in Fig. 3 of the main text. In order to identify the origin of its nonmonotonic temperature behavior we performed an analysis of the spin susceptibility χ p−h (T ) employing the particle-hole bubble approximation (see e,g. Ref. 17 ): HereĜ(k, iω n ) = [(iω n + µ)Î −Ĥ DFT (k) +Σ(iω n )] −1 is the lattice Green's function, iω n are the fermionic Matsubara frequencies, µ is the chemical potential,Î is the identity operator andΣ(iω n ) is the local self-energy. The operatorĤ DFT (k) denotes the effective Hamiltonian computed by projection onto a set of Wannier functions with symmetry of p and d states. Spin susceptibility without the account of electronic correlations i.e. corresponding to DFT is obtained by settingΣ(iω n ) =0.
Temperature dependence of χ p−h (T ) corresponding to DFT and DMFT solutions is presented in Fig. S3 (left panel). First of all, one may see that χ DMFT p−h (T ) computed with DFT+DMFT single-particle Green functions basically preserves all features of the temperature evolution of χ u (T ) obtained in direct DFT+DMFT calculation. Namely, it demonstrates a decrease in the low-temperature region followed by a minimum and a quasi-linear increase in the region of higher temperatures. Secondly, we found that the particle-hole bubble χ DFT p−h (T ) computed within non-interacting Green functions (Σ(iω n ) =0) is strongly influenced by small variations of the chemical potential due to temperature broadening (Fig. S3 (right panel)). Specifically, temperature dependence of χ DFT p−h (T ) obtained using the fixed chemical potential corresponding to T = 0 K does not reproduce the low-temperature behavior of χ DMFT p−h (T ). At the same time χ DFT p−h (T ) computed with temperature-dependent µ is qualitatively similar to χ DMFT p−h (T ). These observations along with the fact that electronic correlations do not cause a serious transformation of the energy spectrum allow to suggest that single-particle properties, in particular peculiarities of the single-particle density of states in the vicinity of the Fermi play a dominant role in the formation of the non-monotonic behavior of the uniform susceptibility.
This situation is thoroughly discussed in literature. In particular, it was mentioned that there is indeed an important contribution to the magnetic susceptibility in metals related to the temperature dependence of chemical potential 18 . Also, it was shown that thermal excitations of the low-energy states of the energy spectrum forming sharp peaks below the Fermi level can be responsible for similar anomalies of the uniform susceptibility in iron-based superconductors. Therefore, we conclude that anomalies of the temperature behavior of magnetic susceptibility the FeO 2 are most likely to be defined by its low-energy band structure. The quasi-linear growth of χ u (T ) for T > T * ∼ 750 K is then due to a strong peak in the DFT density of states centered at ∼ −0.2 eV.