Georg K. H. Madsen 1,2 and Pavel Novak´ 2
1 Dept. of Chemistry, University of Aarhus, DK-8000 ˚Arhus C, Denmark.georg@chem.au.dk
2 Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnick´a 10, 162 53 Praha 6, Czech Republic
Abstract. –
The electronic structure of the monoclinic structure of Fe3O4 is studied using both the local density approximation (LDA) and the LDA+U. The LDA gives only a small charge disproportionation, thus excluding that the structural distortion should be sufficient to give a charge order. The LDA+U results in a charge disproportion along the c-axis in good agreement with the experiment. We also show how the effective U can be calculated within the augmented plane wave methods.
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Introduction. –
Magnetite has received a lot of attention both for fundamental and technological reasons. Above the Verwey transition it is a half-metal and has the highest known Tc of 860 K. The crystal structure is the inverse spinel structure with the formal chemical composition Fe3+ A [Fe2+Fe3+]BO 2− 4 . The two octahedrally coordinated B positions are symmetry equivalent and order antiferromagnetically with the tetrahedrally coordinated A site in the cubic Fd¯3m spacegroup. When cooled to the Verwey transition temperature, which lies around 122-125 K depending on sample, the conductivity of magnetite drops abruptly by two orders of magnitude. Originally the structure below the Verwey transition transition was thought to be have iron cations at the B sites order as Fe2+ and Fe3+ along the (001) planes. This turned out to be too simple a model and single crystal diffraction studies showed that the low temperature structure is monoclinic with a √ 2a × √ 2a × 2a supercell and a Cc space group symmetry. A recent NMR study resolved 8 tetrahedral and 16 octahedral environments thereby confirming the Cc space group. [1] However, the diffracted supercell peaks are extremely weak and even a recent synchrotron diffraction study could only resolve three supercell peaks. [2] The structure has therefore only been refined in a a/√ 2×a/√ 2×2a monoclinic subcell with an additional orthorhombic Pmca pseudo symmetry constraint (see refs. [2, 3] for a detailed description of the structure).
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Theoretically the problem of charge order in Fe3O4 has been difficult to attack. First of all the true structure is unknown and even the simplified a/√ 2 × a/√ 2 × 2a subcell is highly complex for a theoretical calculation. Furthermore, the local density approximation (LDA) is not generally applicable to highly correlated transition metal oxides, due to the spurious self interaction. F.inst. if the calculations are done in the originally proposed Verwey model without structural distortions, an LDA calculation converges to a metallic state with the octahedral iron sites in an Fe2.5+ oxidation state. [6] In the LDA+U method an orbital dependent field is introduced which can be shown to give a correction for the self interaction. [7] Consequently two studies have shown that the LDA+U method leads to a charge ordering. [6,8] However, both these studies used the original Verwey model of the structure thereby omitting both the structural distortions and the additional [00 1 2 ] charge modulation. [2] Till date only one calculation has been done on the a/√ 2 × a/√ 2 × 2a structure. [9] Both a pure LDA and three self interaction corrected (SIC)-LDA calculations were performed and the authors reached two surprising conclusions. [9] First of all it was found that the scenario where five d-electrons move in the SIC-LDA potential on both the A and all B sites (which can be interpreted as all the octahedral iron sites being in the Fe3+ oxidation state) was the most stable. [9] Secondly the lowest energy state shows only a small charge disproportion of 0.1 electron. This is of similar magnitude to the charge disproportion found using pure LDA, leading to the conclusion that the charge disproportion is structural of origin. [9] One possible objection to these results could be that the minimal basis set and atomic sphere approximation used for the potential [9] could make the total energies unreliable. A second objection could be that in the SIC-LDA method one must choose a set of localized states for which the SIC is applied. The actual ground state will thus only be found if it is among those tested.
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The meaning of the U parameter was discussed by Anisimov and Gunnarsson, [12] who defined it as the cost in Coulomb energy by placing two electrons on the same site. In an atom the U corresponds to F 0 of the unscreened Slater integrals. [12] F 0 should thus both increase with increased ionicity and as the d-wave function is contracted across the 3d transition series. This is illustrated in Fig. 1 where we show the calculated atomic Slater integrals for chemically relevant 3d ions.
Due to screening the effective U in solids is much smaller than F 0 for atoms. To calculate the effective U Anisimov and Gunnarsson, [12] constructed a supercell and set the hopping integrals connecting the 3d orbital of one atom with all other orbitals to zero. The number of electrons in this non-hybridizing d-shell was varied and F 0 ef f was then calculated from
F 0 ef f =ε3d↑((n + 1)/2, n/2) − ε3d↑((n + 1)/2, n/2 − 1) − εF ((n + 1)/2, n/2) + εF ((n + 1)/2, n/2 − 1) (4)
where ε3d↑ is the spin-up 3d eigenvalue. Using the method of Anisimov and Gunnarsson, U-values have been calculated for the di- and trivalent configurations of the 3d elements in La perovskites. [13] As expected the trends were the same as observed for atoms, Fig. 1, but in smaller magnitude. The trends being, i), an almost linear relation between atomic number and calculated U and ,ii), a constant shift between the M2+ ions, ranging from approximately 6.5 eV (Titanium) to 8.5 eV (Copper), and the M3+ ions, ranging from 8 eV to 10 eV. [13]
The original LDA+U method [10] was based on the linearized muffin tin orbitals basis set, where the individual orbital and hopping terms can be identified. This is not possible within the LAPW method, so the method of Anisimov and Gunnarsson [12] can not be directly applied. Instead the hybridization can be removed putting the d-states into the core or by performing a two-window calculation. To check this procedure we have performed calculations on the well characterized NiO. [11] Two calculations were performed on 2×2×2 supercells each with one impurity site with the d-configuration forced to be as in Eq. (4). The d-character of the APWs at the impurity sites was eliminated by placing the d-linearization energy at a very high value. Using Eq. (4) we hereby got a value of F 0 ef f=5.96 eV. The question is how this value should be used in an LDA+U calculation? When using the spherically averaged form of Eorb , Eq. (1), J simply rescales the orbital term, Eq. (2) and Eq. (3). Furthermore Eq. (3) shows that an occupied and an unoccupied orbital will be split by U −J. Following the interpretation of Anisimov and Gunnarsson [12], this suggest that U − J = F 0 ef f . As the screening of F 2 and F 4 in solids appears to be small, J can to a good approximation be calculated from the atomic values. [12] Using a J = 1.36 eV, Fig. 1, we arrive at U = 7.32 eV in good agreement with earlier values. [11] Furthermore an effective onsite term of F 0 ef f=5.96 eV, leading to a corresponding splitting of occupied and unoccupied bands, is in excellent agreement with experiment [11].
We then applied the method to FeO and Fe2O3 and obtained F 0 ef f = 5.73 eV and 7.33 eV for Fe2+ and Fe3+ respectively. Finally calculations were performed on magnetite using the 2 × 2 × 2 supercell of the cubic high temperature structure. We tried placing both Fe2+ A (corresponding to one calculation with 3.5 spin up and 3 spin down electrons at the impurity site and one calculation with 3.5 spin up and 2 spin down) and Fe 3+ A at the tetrahedrally coordinated A site. Thereby values of F 0 ef f =4.79 eV and 6.33 eV were obtained. For the octahedrally coordinated B site values of F 0 ef f (Fe2+ B )=5.03 eV, F 0 ef f (Fe2.5+ B )=6.21 eV and F 0 ef f (Fe3+ B )=7.38 eV were calculated, showing good internal consistency with the FeO and Fe2O3 values. Using the J parameters derived from the atomic values, Fig. 1, we then obtain U(Fe3+ A )=7.69 eV, U(Fe2+ B )=6.2 eV and U(Fe3+ B )=8.73 eV for magnetite. Our calculated F 0 ef f seem somewhat larger than the U and J values that were earlier calculated for magnetite. [6] However it is not clear what oxidation state was used to calculate these values [6] and furthermore it should be noted that the values for the octahedrally coordinated B site are in very good agreement with what was obtained for the octahedrally coordinated Fe ions in the perovskites. [13]
Computational details. –
With the U values thus fixed we performed calculations on the low-symmetry structure [2] using the L/APW+lo method [17], as implemented in the WIEN2k code. [18] Sphere sizes of 2.0 a.u. and 1.5 a.u. were used for the Fe and O sites respectively. The plane-wave (PW) cut-off was defined by min(RMT )max(kn)=5.7 corresponding to approximately 3700 PW. The Brillouin-zone (BZ) was sampled on a tetrahedral mesh with 100 k points (18 in the irreducible BZ). On the tetrahedrally coordinated A iron sites the calculated F 0 ef f = U − J = 6.33 eV was used. The calculated F 0 ef f = 6.21 eV for Fe2.5+ B was used on all octahedrally coordinated B iron sites so that no charge order was forced.
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