First-principles calculation of the structure and magnetic phases of hematite
G. Rollmann,1A. Rohrbach,2 P. Entel,1 and J. Hafner21
Institut fu¨r Theoretische Physik, Universita¨t Duisburg-Essen, Lotharstrasse 1, 47048 Duisburg, Germany 2Institut fu¨r Materialphysik and Center for Computational Materials Science, Universita¨t Wien, Sensengasse 8/12,
A-1090 Wien, Austria
共Received 20 June 2003; published 6 April 2004兲
Rhombohedral ␣-Fe2O3 has been studied by using density-functional theory 共DFT兲 and the generalized
gradient approximation共GGA兲. For the chosen supercell all possible magnetic configurations have been taken into account. We find an antiferromagnetic ground state at the experimental volume. This state is 388 meV/共Fe atom兲below the ferromagnetic solution. For the magnetic moments of the iron atoms we obtain 3.4B, which
is about 1.5Bbelow the experimentally observed value. The insulating nature of␣-Fe2O3is reproduced, with
a band gap of 0.32 eV, compared to an experimental value of about 2.0 eV. Analysis of the density of states confirms the strong hybridization between Fe 3d and O 2 p states in ␣-Fe2O3. When we consider lower
volumes, we observe a transition to a metallic, ferromagnetic low-spin phase, together with a structural transition at a pressure of 14 GPa, which is not seen in experiment. In order to take into account the strong on-site Coulomb interaction U present in Fe2O3we also performed DFT⫹U calculations. We find that with
increasing U the size of the band gap and the magnetic moments increase, while other quantities such as equilibrium volume and Fe-Fe distances do not show a monotonic behavior. The transition observed in the GGA calculations is shifted to higher pressures and eventually vanishes for high values of U. Best overall agreement, also with respect to experimental photoemission and inverse photoemission spectra of hematite, is achieved for U⫽4 eV. The strength of the on-site interactions is sufficient to change the character of the gap from d-d to O-p-Fe-d.
DOI: 10.1103/PhysRevB.69.165107 PACS number共s兲: 71.20.Be, 61.50.Ks, 71.15.Mb, 71.30.⫹h
I. INTRODUCTION
Among the iron oxides, corundum-type␣-Fe2O3 共 hema-tite兲is the most common on earth. At ambient conditions, it crystallizes in the rhombohedral corundum structure 共space group R3¯ c), see Ref. 1. Below the Ne´el temperature, TN ⫽955 K, ␣-Fe2O3 is an antiferromagnetic 共AF兲 insulator showing weak ferromagnetism above the Morin temperature,
TM⫽260 K, due to a slight canting of the two sublattice magnetizations.2–5Below TM, the direction of the magnetic moments is parallel to the 关111兴 axis of the hexagonal unit cell. The localization of the iron 3d electrons due to the strong on-site Coulomb repulsion results in a large splitting of the d bands and a band gap of 2 eV. As the upper edge of the valence band is dominated by oxygen p states, hematite is generally considered to be a charge-transfer rather than a Mott-Hubbard insulator.
There has been a considerable amount of experimental work on hematite, regarding its structural, magnetic, and electronic properties in bulk1–3,6 –21 as well as in nanoparticles.22,23Especially the nature of the high-pressure
共HP兲 transition of ␣-Fe2O3 has recently been discussed in detail.6 –9,13,19–21At pressures around 50 GPa a shrinkage of the crystal volume of about 10% together with a collapse of the magnetic moments and a concurrent insulator-metal tran-sition due to the breakdown of the d-d correlation has been observed. This transition was found to be isostructural, but subsequent to a change in crystalline structure.21 For quite a long time the HP phase of␣-Fe2O3 was considered to be of the orthorhombic perovskite type, containing Fe2⫹and Fe4⫹ sites in the same amount.7,8,9,13 Recently, Pasternak et al.19
and Rozenberg et al.20 assigned the HP structure to a dis-torted corundum-type (Rh2O3-II) phase with a single Fe3⫹ cation.
From a theoretical point of view, the study of hematite is very attractive because of the challenges it poses to theory. The Fe d electrons in Fe2O3are strongly correlated, so meth-ods beyond ordinary density-functional theory 共DFT兲in the local spin-density approximation共LSDA兲are needed to cor-rectly describe the system in terms of electronic and mag-netic properties. As the rhombohedral primitive cell of hema-tite contains ten atoms 共compared to only two atoms in simple transition-metal共TM兲oxides with the rocksalt struc-ture兲, considerable computational effort is required to obtain reasonable results concerning the crystalline, electronic, and magnetic structures of hematite.
This might be responsible for the fact that, in contrast to studies of NiO, FeO, and similar TM monoxides, which have been studied extensively using ab initio methods,24 –30 there are only a few theoretical works4,5,16,31,32,33 concerning ␣-Fe2O3. The Hartree-Fock共HF兲 study of Catti et al.31 al-ready achieves the correct order of magnetic states at experi-mental volume, yielding an energy difference between anti-ferromagnetic and anti-ferromagnetic 共FM兲 alignments of the spins of 37 meV/共Fe atom兲, although the obtained mixing of oxygen 2 p and iron 3d electrons is very weak. In addition, the energy gap is grossly overestimated while the predicted bandwidth is too small. Sandratskii et al.32studied␣-Fe2O3 using conventional LSDA. They find an AF bilayer sequence for the Fe moments to be lowest in energy: 489 meV/共Fe atom兲 lower than the FM state. In a recent LDA⫹U study,
Punkkinen et al.33 concentrate on the density of states
共DOS兲. They find that even a modest value of the on-site
Coulomb potential improves the prediction for the energy gap without, however, achieving satisfactory agreement for other physical properties. In the latter two studies no relax-ation of the atoms inside the supercell and also no varirelax-ation of the volume of the cell was taken into account.
In our study, we present results of a complete relaxation process of cell size and atomic positions, in the framework of DFT, based on the generalized gradient approximation
共GGA兲 and the projector augmented wave 共PAW兲 method. We also performed GGA⫹U calculations to account for the
strong on-site Coulomb repulsion of the Fe d electrons. This paper is organized as follows. In Sec. II we describe the computational details, after that we briefly discuss in Sec. III how the on-site Coulomb interaction parameter U is intro-duced in the framework of DFT. We present results of the GGA calculations in Sec. IV, while results concerning GGA⫹U will be given in Sec. V. Concluding remarks and
open key questions can be found in Sec. VI.
II. COMPUTATIONAL MODEL
The calculations have been performed on the basis of spin-polarized DFT34,35with the Vienna ab initio simulation package VASP.36 – 40 For the exchange-correlation functional,
Exc, we chose a semi-local form proposed by Perdew and Wang41 in 1991 using the GGA denoted hereby PW91. The spin interpolation of Vosko et al.42 was used. The Kohn-Sham equations were solved via iterative matrix diagonaliza-tion based on the minimizadiagonaliza-tion of the norm of the residual vector to each eigenstate and optimized charge- and spin-mixing routines.43– 45
A number of eight valence electrons for each Fe atom (3d74s1) and six valence electrons for each O atom (2s22 p4) were taken into account. The remaining 共core兲 electrons together with the nuclei were described by pseudo-potentials in the framework of the PAW method proposed by Blo¨chl46 and adapted by Kresse and Joubert.36
The one-electron Kohn-Sham wavefunctions as well as the charge density were expanded in a plane-wave basis set. The results reported in this paper correspond to an energy cutoff of 550 eV. It was checked that increasing the cutoff to 800 eV did not change total energies by more than 3 meV/ atom. Energy differences were affected by less than 0.3 meV/atom. For the calculation of total energies, the integra-tion over the Brillouin zone was performed with a mesh of 8⫻8⫻8 k points generated by following the scheme of Monkhorst and Pack.47The DOS have been calculated with a 16⫻16⫻16 k point mesh.
An increase to 20⫻20⫻20 k points did not lead to any visible change in the total DOS. The integration over the irreducible part of the Brillouin zone was carried out using the linear tetrahedron method with Blo¨chl corrections. The relaxation of the crystalline structure was performed using a Gaussian-smearing approach with a width of 0.2 eV to speed up convergence. Pressure and bulk modulus were derived from the calculated pressure-volume relation by fitting the data to a Murnaghan equation of state.48
All calculations were carried out under rhombohedral symmetry constraints. The complete hexagonal unit cell of
hematite is shown in Fig. 1, together with the rhombohedral primitive cell used in our calculation. All iron atoms have an equivalent octahedral environment. Therefore, electronic and magnetic properties will be the same at each iron site. The octahedra built by oxygen atoms and centered by iron atoms are slightly rotated against each other. We note that there are two types of pairs of Fe atoms, which are characterized by a short Fe-Fe distance 共type A兲 and by a larger distance共type B兲 along the hexagonal axis. There are two internal degrees of freedom, commonly named zFe and xO, by which the positions of the atoms inside the primitive cell can be de-scribed 共see Table I兲. We have relaxed the positions of the
FIG. 1. 共Color online兲 Hexagonal unit cell of ␣-Fe2O3 共left兲
together with the rhombohedral primitive cell共right兲. Blue spheres represent Fe atoms, red spheres display the O atoms. The bilayer structure for the Fe atoms is clearly visible. Along the关111兴 axis there are two sorts of pairs of Fe atoms, one共denoted as type A兲 with a short and one 共type B兲with a larger Fe-Fe distance. The O atoms form close-packed basal planes, each Fe atom is coordinated octahedrally by six O atoms.
TABLE I. Positions of the atoms xᠬ⫽␣a⫹b⫹␥c inside the
primitive unit cell of ␣-Fe2O3 with respect to the basis vectors a,b,c of the primitive cell in terms of the two internal degrees of
freedom zFe and xO. Experimental values are taken from Ref. 6.
Numbers in parentheses represent the uncertainty of the last digit.
Atom Wyckoff symbol ␣  ␥ Fe 共c兲 z z z 1/2⫺z 1/2⫺z 1/2⫺z 1/2⫹z 1/2⫹z 1/2⫹z 1⫺z 1⫺z 1⫺z z⫽0.10534(6) O 共e兲 x 1⫺x 0 1⫺x 0 x 0 x 1⫺x 1/2⫺x 1/2⫹x 1/2 1/2⫹x 1/2 1/2⫺x 1/2 1/2⫺x 1/2⫹x x⫽0.3056(9)
atoms in terms of these parameters, as well as the c/a ratio of the supercell, by using the conjugate gradient method until the forces on each atom were below 5 meV/Å. Structural parameters of hematite derived experimentally at zero pres-sure by Pauling and Hendricks1 and Finger and Hazen7 are given in Table II. For DFT⫹U calculations, the version
de-scribed in the following section was used. III. DFT¿U METHOD
The on-site Coulomb repulsion amongst the localized TM 3d electrons is not described very well in a spin-polarized DFT treatment. A conceptually better method 共DFT⫹U)
consists of combining the DFT with a Hubbard-Hamiltonian, thereby considering this Coulomb repulsion explicitly. In the present calculations we use a simple DFT⫹U version. It is
based on a model Hamiltonian with the form49
Hˆ⫽U
2 m,m
兺
⬘, nˆmnˆm⬘⫺⫹U⫺J
2 m⫽
兺
m⬘, nˆmnˆm⬘, 共1兲 where nˆm is the operator yielding the number of electrons occupying an orbital with magnetic quantum number m and spin at a particular site.The Coulomb repulsion is characterized by a spherically averaged Hubbard parameter U describing the energy re-quired for adding an extra d electron to an Fe atom, U ⫽E(dn⫹1)⫹E(dn⫺1)⫺2E(dn), and a parameter J repre-senting the screened exchange energy. While U depends on the spatial extension of the wave functions and on screening,
J is an approximation of the Stoner exchange parameter and
almost constant J⬃1 eV 共see Ref. 50兲. The spin-polarized DFT⫹U energy functional is obtained by subtracting the
Mott-Hubbard contributions already existing in the DFT functional 共‘‘double-counting’’ contributions兲50 and after some straightforward algebra one obtains
EDFT⫹U⫽EDFT⫹ U⫺J 2
兺
m 共 nm⫺nm 2 兲 . 共2兲This form is not invariant under a unitary transformation of the orbitals. However, when replacing the number operator by the on-site density matrixi j of the d electrons, Lichten-stein et al.51obtain a rotationally invariant energy functional, which in the present case is of the form
EDFT⫹U⫽EDFT⫹
U⫺J
2
兺
Tr关⫺兴. 共3兲
This yields exactly the same energy as the DFT functional
EDFT⫹U⫽EDFT in the limit of an idempotent on-site
occu-pancy matrix ()2⫽ imposed by the second term in Eq.
共3兲. For U⬎J, the term is positive definite, since the
eigen-values ⑀iof the on-site occupancy matrix are either 0 or 1,
⫺⫽
兺
i ⑀i
⫺共⑀
i
兲2⬎0,
where the sum on the right-hand side is over all eigenvalues ⑀i of the on-site occupancy matrix . Hence the second term in Eq. 共3兲can be interpreted as a positive definite pen-alty function driving the on-site occupancy matrices towards idempotency, whose strength is parametrized by a single pa-rameter (U⫺J). A larger (U⫺J) forces a stricter observance
of the on-site idempotency, achieved by lowering the one-electron potential locally for a particular metal d orbital and in turn reducing the hybridization with, e.g., O atoms. The one-electron potential is given by the functional derivative of the total energy with respect to the electron density, i.e., in a matrix representation, Vi j⫽ ␦EDFT⫹U ␦i j ⫽␦EDFT ␦i j ⫹共U⫺J兲
冋
1 2␦i j⫺i j 册
. 共4兲It is recognized that filled d orbitals, which are localized on one particular site, are moved to lower energies, by ⫺(U ⫺J)/2, whereas empty d orbitals are raised to higher
ener-gies by (U⫺J)/2. For the implementation of the DFT⫹U
approach within VASP, see Ref. 52.
IV. RESULTS AND DISCUSSION OF GGA CALCULATIONS A. Crystal structure and magnetization
The total energies per atom and magnetic moments of the Fe atoms as a function of the volume per atom are displayed in Fig. 2. The curves belonging to different magnetic solu-tions are indexed according to the orientation of the local spin density around the Fe atoms. As zero energy the global energy minimum was taken, corresponding to an AF ⫹⫹ ⫺⫺ state, which means that Fe atoms belonging to type A pairs 共short distance, see Sec. II兲 have opposite magnetic moment, while Fe atoms belonging to type B pairs 共larger distance兲have equal magnetic moments. Each point on every curve represents the result obtained after full relaxation of the cell shape (c/a ratio兲, atomic positions共under rhombo-hedral symmetry constraints兲, and magnetic moments for the corresponding volume.
For a discussion we first draw our attention to the order of states at experimental volume. Corresponding structural, en-ergetic, and magnetic properties of the different magnetic solutions for a volume of 10.06 Å3are presented in Table III. The energy differences between the AF ⫹⫹⫺⫺ and the other two AF states agree well with results of the LSDA calculation reported by Sandratskii and co-workers.32 The total energy of the FM solution is comparable to the corre-sponding value in the same publication, but considerably larger than the value found by Catti et al.31within their HF study. The nonmagnetic 共NM兲solution is highest in energy.
TABLE II. Experimentally derived structural parameters for
␣-Fe2O3taken from Ref. 1共top兲and Ref. 7共bottom兲. Distances are
given in Å.
c a c/a Fe-Fe共A兲 Fe-Fe共B兲
13.730 5.029 2.730 2.883 3.982
The AF solutions slightly vary in magnetic moments, with all moments being ⬇3.4B in magnitude. This is about 1.5Blower than the experimental moments.53In contrast to the results of Sandratskii et al.,32who obtain almost the same moments for AF and FM ordering, the moments of the FM solution are about 1.0B lower in our calculation. This re-flects the frustration of the magnetic exchange interactions in the FM configuration.
In an ideal corundum-type structure the c/a ratio is 2.8333. The unit cell of hematite is slightly shortened in c direction with respect to this ideal cell, the experimentally measured c/a ratio at zero pressure is 2.73共Ref. 7兲. We find that the calculated values for the c/a ratios for all three AF states agree well with this number, while the c/a ratios for the FM and the NM state are somewhat larger and closer to the ideal value. The internal structural parameters共and hence the Fe-Fe distances兲also show a pronounced dependence on the magnetic configuration. For the stable AF ⫹⫹⫺⫺ phase, we note good agreement with experiment, in the other magnetic configurations substantial differences are observed. These effects are largest in the FM configuration where smaller values of z and larger x parameters lead to a strongly increased difference of the bond lengths in Fe-Fe 共A兲 and Fe-Fe 共B兲 pairs. The contraction of the shortest Fe-Fe dis-tances contributes to the quenching of the magnetic moments in the FM phase.
When comparing our results to other theoretical work we first note that we get the same order of states as Sandratskii
et al. But while the energy differences between the AF
phases are similar, we get the FM solution closer to the AF states by 110 meV per Fe atom. From a calculation of the total energy for the FM state at a volume of 10.06 Å3per Fe atom and a fixed c/a ratio of 2.73, without relaxing the atomic positions, we obtained a value which is 135 meV above the corresponding energy obtained with full relax-ation. Therefore we ascribe the discrepancy to the results of Sandratskii and co-workers to the fact that we have allowed for full relaxation of the cell shape (c/a ratio兲and the atomic positions inside the supercell. Sandratskii et al. present their results for a fixed geometry. This might also be the reason for their FM magnetic moments being about as large as the AF ones, while in our calculations the moments differ by more than 0.8B. The small difference in energy between FM and AF states obtained by Catti et al.29 has already been explained32to be due to the complete neglect of correlation in HF theory.
When we consider lower volumes we observe a collapse of the magnetic moments, from a high-spin 共HS兲 to a low-spin共LS兲state, which occurs at 8.86 Å3 for the AF⫹⫹⫺⫺ solution, and around 9.1, 9.3 and 9.2 Å3 for the
TABLE III. Order of the magnetic states at experimental volume, 10.06 Å3/atom. Energies are in meV/共Fe atom兲, magnetic moments are in (B/Fe atom兲, and distances are in Å共where NM stands for nonmagnetic兲.
Magnetic state Functional Reference Energy Magnetic moment c/a ratio Fe-Fe共A兲 Fe-Fe共B兲
AF⫹⫹⫺⫺ GGA This paper 0 3.44 2.77 2.941 4.006
LSDA 32 0 3.72
HF 31 0 4.74 2.70 2.877 4.033
AF⫹⫺⫹⫺ GGA This paper 211 3.38 2.76 2.959 3.958
LSDA 32 190 3.80
AF⫹⫺⫺⫹ GGA This paper 224 3.45 2.69 2.803 4.000
LSDA 32 204 3.74
FM⫹⫹⫹⫹ GGA This paper 388 2.60 2.85 2.689 4.380
LSDA 32 489 3.73
HF 31 37
NM GGA This paper 680 2.83 2.761 4.280
FIG. 2. Total energies共lower panel兲and absolute values of mag-netic moments of the Fe atoms 共upper panel兲 as a function of the volume per atom. Dotted vertical lines mark the experimental value
共Ref. 7兲(10.06 Å3/atom) for the volume per atom of hematite at
ambient conditions and are drawn to guide the eye. As zero energy the global energy minimum is taken, corresponding to the AF ⫹
⫹⫺⫺state, at a volume of 10.00 Å3/atom. The straight solid line is tangent to the AF ⫹⫹⫺⫺ and the FM⫹⫹⫹⫹ curves and marks a structural transition at a pressure of 14 GPa.
⫹⫺⫹⫺, ⫹⫺⫺⫹, and ⫹⫹⫹⫹ solutions, respectively. Related to this are kinks in the energy vs volume curves. These HS-LS transitions are more pronounced for the AF states compared to the FM solution, resulting in two clearly visible minima at least in the AF ⫹⫺⫹⫺ and ⫹⫺⫺⫹ energy curves. In the LS phases, the magnetic moments de-crease linearly with volume for the AF states, but stay almost constant around 1Bfor the FM state. For very low volumes we did not succeed in stabilizing a LS AF ⫹⫺⫹⫺ phase. Below 7.6 Å3 we observe the collapse of the AF ⫹⫺⫹⫺ solution and obtain the NM state as the most stable solution. We note that at low pressure the AF solutions are much more stable than the FM and the NM solutions. This is re-lated to the strong superexchange interaction mediated by the O atoms. But we also note that at low volumes the FM so-lution is slightly lower in energy than the AF⫹⫹⫺⫺phase
共although the energy difference is rather small, e.g., 2.5 meV/atom for a volume of 8.46 Å3/atom, but this is not of importance for the further discussion兲, corresponding to a FM ground state at high pressures. By placing a common tangent to the FM and the AF⫹⫹⫺⫺ curves, we obtain a value for the pressure, for which a HS-LS AF-FM transition will occur, of 14 GPa. The corresponding volume reduction is 10.5%. As the rhombohedral corundum-type structure has been reported to be the stable phase of␣-Fe2O3 in the 0–50 GPa regime, we conclude that this transition is an artifact of the simulation method, namely, the DFT in combination with the GGA. It will be shown that when the GGA⫹U
formal-ism is applied, the observed transition is shifted to higher pressures with increasing U and eventually disappears for large values of U.
We have calculated structural parameters for each state by fitting the data in the volume regions corresponding to local minima of the energy curves to a third-order Birch-Murnaghan equation of state共Ref. 54兲. The results are given in Table IV. For two of the AF curves, structural parameters were calculated for both HS and LS phases. The equilibrium volume for the magnetic ground state agrees well with the experimental value,7 although the corresponding magnetic moment is too low. This is probably due to the overestima-tion of hybridizaoverestima-tion of Fe 3d and O 2 p states in conven-tional GGA. We also obtain a value for the bulk modulus which is lower than the values found experimentally.6,7,20
However, it must be emphasized that the experimental value of the bulk modulus is subject to considerable uncertainty. Sato and Akimoto’s value of 231 GPa is derived by a fit over a very small pressure range of p⬍3 GPa, whereas a fit to pressures up to 10 GPa leads to B0⬃178 GPa closer to our calculated value. The most recent result of Rozenberg et al.20 is obtained by arbitrarily setting the pressure derivative B0
⬘
⫽⌬B0/⌬p in the Murnaghan fit to B0⬘
⫽4.The calculated bulk modulus for the FM state is consid-erably larger, which is due to the fact that it has been evalu-ated for a lower volume and coincides well with the experi-mental result that with increasing pressure hematite is becoming harder. This can be observed directly when com-paring the bulk moduli for the HS and LS solutions of the AF ⫹⫺⫹⫺ and the AF⫹⫺⫺⫹ phases.
Common to all magnetic states 共and the experimental finding兲is the dependence of the magnetic moment of the Fe atoms and the parameter zFeon the volume. For volumes of more than 10 Å3/atom the moments are larger than 3B/Fe atom and zFe is greater than 0.103. This results in a differ-ence between Fe-Fe 共A兲 and Fe-Fe 共B兲 distances of about 1.0–1.1 Å. For volumes below 9 Å3/atom the moments are smaller than 1.5B/Fe atom and zFeis less than 0.098, lead-ing to differences in Fe-Fe distances of more than 1.5 Å. We conclude that the volume reduction seems to result in an increase of Fe-Fe 共B兲 distances and a strong decrease of Fe-Fe 共A兲 distances. In contrast to that a clear trend is not visible for the parameter xOand the c/a ratio.
As not only the volume, but also the magnetic order de-termine the structural parameters, we now stick to the experi-mentally observed AF⫹⫹⫺⫺ state to examine further the dependence of crystalline structure of ␣-Fe2O3 on pressure. Figure 3 illustrates the dependence of the c/a ratio, the in-ternal degrees of freedom zFe and xO, and the Fe-Fe and Fe-O distances on pressure for the AF ⫹⫹⫺⫺ phase. For comparison, experimental data from Ref. 19 is included. The straight solid lines mark the pressure of 14 GPa at which the AF-FM HS-LS transition takes place in our simulation. Up to that value, the structural parameters from our simulation change almost linearly with pressure. Around pc, a rapid change, especially in c/a ratio, is observed, followed by nearly perfect linear behavior above pc. The HS-LS
transi-TABLE IV. Ground-state parameters for the different magnetic states. Volumes V0 are in Å 3
/atom, magnetic moments are inB/Fe
atom, distances are in Å, and bulk moduli B0are in GPa.
Magnetic state V0 Moment zFe xO c/a ratio Fe-Fe共A兲 Fe-Fe共B兲 B0
AF⫹⫹⫺⫺ 10.00 3.43 0.1057 0.3096 2.772 2.929 3.998 173 Expt.共Ref. 6兲 10.06 0.1053 0.3056 2.731 2.896 3.977 178, 225共231,a258b兲 AF⫹⫺⫹⫺ 共HS兲 10.13 3.42 0.1073 0.3122 2.757 2.976 3.959 134 AF⫹⫺⫹⫺ 共LS兲 8.85 1.00 0.0979 0.3110 2.775 2.610 4.052 206 AF⫹⫺⫺⫹ 共HS兲 10.13 3.48 0.1031 0.3047 2.690 2.812 4.010 169 AF⫹⫺⫺⫹ 共LS兲 8.88 1.17 0.0962 0.3120 2.714 2.526 4.038 212 FM⫹⫹⫹⫹ 8.79 0.0960 0.3131 2.795 2.564 4.111 249 NM 8.73 0.0966 0.3150 2.811 2.582 4.102 271 aReference 6. bReference 20.
tion is obviously accompanied by transitions in these struc-tural parameters, whereas we do not see a transition in the experimental values.
The c/a values from our calculation are larger than the ones found in experiment.6,7,20The agreement of all param-eters and their pressure dependence with experiment is rea-sonable for pressures below pc. Due to the transition at pc this agreement becomes poor for pressures above 14 GPa.
B. Electronic structure
In order to analyze electronic properties of ␣-Fe2O3 we calculated the DOS for all energy minima of the curves. In Fig. 4 the partial DOS of Fe 3d (t2gand eg) and O 2 p states for the AF ⫹⫹⫺⫺ solution at experimental volume are shown. A band gap of⬇0.3 eV is clearly visible. As is usu-ally the case for calculations based on DFT, this is much smaller共by about a factor of 6兲than the experimental value of 2.0 eV 共Ref. 55兲. For comparison, Punkkinen and co-workers,33who applied GGA in combination with the lin-ear muffin-tin orbital 共LMTO兲method in atomic-sphere ap-proximation共ASA兲, obtained a value of 0.51 eV for the band gap, Sandratskii et al.32 got a value of 0.75 eV by using
FIG. 3. Variation of the c/a ratio共upper panel兲, the structural parameters xO and zFe, plotted as zFe⫹0.25共middle兲, and Fe-Fe
distances 共lower panel兲 with pressure, together with experimental data taken from Ref. 20.
FIG. 4. Orbital projected DOS 共PDOS兲 for the AF ⫹⫹⫺⫺ solution of Fe2O3, calculated within the GGA. The left共right兲panel
shows spin-up 共spin-down兲 Fe eg, Fe t2g, and O 2 p projections.
LSDA and the augmented-spherical wave method 共ASW兲, and Catti et al. calculated about 11 eV within HF. The small discrepancies between our and previous DFT calculations can be ascribed to the use of the ASA in the LMTO and ASW work 共whereas we use a full-potential technique兲, the dra-matic overestimation of the gap calculated within HF is due to the complete neglect of electronic correlations. The dis-agreement with experiment may be traced back to some ex-tent to the amount of mixing of Fe 3d and O 2 p states, which plays an important role in the characterization of the insulating nature of hematite. Whereas the mixing is nearly absent in the HF study and clearly underestimated in the LSDA calculations, a strong mixing across the entire range of the valence band is present in our calculations. At the upper edge of the valence band we observe Fe states and O states in comparable amounts. The Fe 3d DOS is even a bit larger than the O 2 p DOS, whereas it is experimentally con-firmed that the valence-band edge in ␣-Fe2O3 is dominated by oxygen 2 p states, making hematite a charge-transfer in-sulator. This deficiency in our calculations originates from the underestimation of the on-site Coulomb interaction of the Fe d electrons and can be compensated by introducing an additional parameter in the density functional, as will be shown in Sec. VI.
Figure 5 shows the partial DOS’s for the FM solution at the equilibrium volume of the FM phase of 8.79 Å3/atom. We find half-metallic hematite under these conditions, the valence-band edge being clearly dominated by Fe 3d states. The HS-LS transition therefore turns out to be an insulator-metal transition.
V. CONCLUSIONS ON DFT CALCULATIONS Using gradient corrections in the exchange-correlation functional we obtain a quite reasonable description of the structural properties at low pressures and the magnetic ground state of hematite. But still several evident shortcom-ings of the DFT remain. The magnetic moments of the Fe atoms and the band gap are too small, and the character of the gap contradicts the accepted charge-transfer character. Most importantly, however, we find␣-Fe2O3 to be unstable under even modest compression against a HS-LS transition coupled to an isostructural transition. Furthermore, the ener-getic order of the possible magnetic phases changes upon the transition, with the FM phase being favored at pressures be-yond ⬃ 14 GPa. This contradicts the observed stability of the corundum-type AF⫹⫹⫺⫺ phase up to about 50 GPa and demonstrates the need to account for electronic correla-tion effects.
VI. GGA¿U CALCULATIONS A. Crystal structure and magnetization
As discussed before, conventional DFT gives the correct stable antiferromagnetic ground state but underestimates the band gap and magnetic moments of␣-Fe2O3. It also predicts the valence-band edge to be Fe 3d dominated, whereas the
FIG. 5. Orbital projected DOS 共PDOS兲for the FM solution of Fe2O3, calculated within the GGA. The left 共right兲 panel shows
spin-up 共spin-down兲 Fe eg, Fe t2g, and O 2 p projections. The
experiment shows O 2 p domination and therefore a charge-transfer type of band gap. Furthermore, the Fe 3d band en-ergies from GGA are not in agreement with experimental valence-band spectra and conduction-band spectra. GGA also predicts␣-Fe2O3to undergo a magnetic phase transition at⬃ 14 GPa, which is not found in experiment. On the other hand, the GGA prediction of the equilibrium volume agrees quite well with experimental data.
The inaccurate results for some of the ground-state prop-erties of hematite obtained with GGA are attributable to the improper description of the on-site Coulomb repulsion be-tween the localized Fe 3d electrons. The DFT⫹U method by
explicitly taking into account the on-site Coulomb repulsion improves the predictions for the band gap, band energies, and magnetic moments. Figure 6 illustrates the dependence of equilibrium volume, Fe-Fe 共A兲 distance, magnetic mo-ments, and band gap on U. The parameter J was kept con-stant at 1 eV for all calculations.
GGA⫹U stabilizes the high-spin AF ⫹⫹⫺⫺ ground state as in the GGA limit, U⫽1 eV. The experimental mag-netic moment is reached only for U⫽8 –9 eV, whereas for the band gap full agreement is obtained for moderate values of U⬃5 eV. The results for magnetic moment and band gap are in good agreement with former GGA⫹U calculations
using the tight-binding LMTO-ASA method56 –58of Punkki-nen et al.,33 using the experimental structure. The choice of an appropriate value for U is the same in both cases. The volume shows only a weak dependence on U. With increas-ing U, it increases first until at U⫽5 eV a maximum value of⬃10.3 Å3is reached and then decreases slightly for larger
U values. The agreement with experiment is reasonable
throughout the entire U range. The Fe-Fe 共A兲 distance reaches its maximum value of ⬃2.95 Å already for U ⫽2 eV and decreases slightly when further increasing U. The agreement with the experimental value of ⬃2.896 Å
共Ref. 7兲is best for U⫽5 eV. The Fe-Fe distance reflects the
behavior of the volume with varying U. The c/a ratio slightly decreases from c/a⬃2.77 for GGA at an equilibrium volume of 10.00 Å3/atom to ⬃2.73 for GGA⫹U with U
⫽5 eV at an equilibrium volume of 10.30 Å3/atom and to
⬃2.72 for GGA⫹U with U⫽7 eV at an equilibrium volume of 10.24 Å3/atom. Agreement with the experimental value of
c/a⫽2.73 is again best for U⬃5 eV. Concerning the bulk modulus, we can observe an increase from the GGA value of 173 GPa at an equilibrium volume of 10.00 Å3/atom to the GGA⫹U value for U⫽4 eV with 177.4 GPa at an equilib-rium volume of 10.30 Å3/atom. For U⫽7 eV, the bulk modulus increases to 178.8 GPa at an equilibrium volume of 10.24 Å3/atom. The bulk modulus remains too small com-pared to experiment6,7,20even for large U parameters.
Figure 7 shows the volume dependence of the total energy and the magnetic moment for U⫽1, 4, and 6 eV.
For the GGA limit, U⫽1 eV, the transitions between the LS and HS AF ⫹⫹⫺⫺ phases are clearly visible, but the HS solution is the stable one. Near a volume of
⬃9.3 Å3/atom, the LS FM phase becomes energetically fa-vorable. The observed pressure at the phase transition is
⬃14 GPa. The LS FM phase at this pressure is half metallic, whereas the HS AF⫹⫹⫺⫺ phase is weakly insulating with a band gap of⬃0.5 eV. At U⫽4 eV, the AF ⫹⫹⫺⫺ HS to LS phase transition no longer occurs, the FM HS to LS phase transition happens at ⬃8.5 Å3/atom.
The U⫽7 eV case does not show any significant changes concerning phase transition and energetical stability com-pared to the U⫽4 eV case. Figure 8 shows the pressure dependence of the c/a ratio, the internal parameters x, z, and
FIG. 6. Variation of equilibrium volume, Fe-Fe distance, mag-netic moment, and band gap with U for the GGA⫹U calculations.
The exchange parameter J was kept constant at 1 eV. So U
⫽1 eV corresponds to the GGA limit.
FIG. 7. Volume dependence of the free energy and the mag-netic moment for GGA⫹U with U⫽1 共GGA兲, 4, and 7 eV. The tangent on the GGA curves mark the phase transition from the AF⫹⫹⫺⫺ to the ferromagnetic phase for U⫽1 eV.
the Fe-Fe distances for GGA and GGA⫹U with U⫽4 eV, together with experimental results taken from Rozenberg
et al.20
It can be seen that experimentally there is no HS to LS phase transition for the stable AF⫹⫹⫺⫺ phase. Therefore, only the calculations with U⭓4 eV give a proper qualitative description for the behavior of ␣-Fe2O3 under pressure. Ex-perimentally, Rozenberg et al.20 have shown that ␣-Fe2O3 undergoes a structural phase transition at a pressure of
⬃50 GPa, but they did not see a phase transition from the AF ⫹⫹⫺⫺ phase to the FM phase for pressures up to 50 GPa. In this pressure range the calculated structural param-eters show very good agreement with experiment.
A symmetry analysis of the relaxed structures of the AF ⫹⫹⫺⫺ order at low volumes has shown that for the GGA case, the structure of the bulk undergoes a slight distortion into a structure with monoclinic symmetry at a volume of
⬃6.6 Å3/atom, corresponding to a pressure of ⭓100 GPa. For U⫽4 eV, the structural distortion happens at a volume of ⬃7.3 Å3/atom, corresponding to a pressure of
⬃80–90 GPa. The densities of states in this volume range show that the structural phase transition is followed by an insulator to metal transition. Especially in the GGA⫹U case,
the band gap closes abruptly. Whereas at a volume of
⬃7.4 Å3/atom, the band gap is⬃1.2 eV, it drops to zero at
⬃6.9 Å3/atom. However, as shown by Rozenberg et al., these transitions are preceded by a transition to a nonmag-netic orthorhombic Rh2O3共II兲-type phase at about 50 GPa. This, however, is beyond the scope of the present study.
B. Electronic structure
The spin-polarized and orbital projected DOS, calculated in the GGA⫹U with U⫽4 eV is shown in Fig. 9. The on-site Coulomb repulsion is found to have a profound influence on the electronic spectrum. The occupied Fe 3d states are shifted to higher binding energies, breaking the strong Fe 3d –O 2 p hybridization characteristic for the electronic structure calculated in the GGA. The occupied states close to the Fermi energy are now entirely oxygen dominated, with only a small admixture of Fe t2g minority states. The empty Fe 3d minority states are up shifted, increasing the width of the gap. In contrast to the GGA predicting hematite to be an insulator with a very narrow gap created by the exchange splitting in the Fe 3d band, GGA⫹U with U⫽4 eV leads to a charge-transfer type O 2 p –Fe 3d gap of nearly 2 eV, in agreement with experiment.
Figure 10 shows the calculated intensities of the photo-emission 共PES兲 and inverse photoemission 共IPES兲 spectra, compared with the experimental results of Fujimori et al.10 and Ciccacci et al.12In spite of the fact that the one-particle DFT eigenvalues have no direct physical meaning, it is well established that calculated DOS and experimental PES can be compared on a qualitative basis, especially when the bands are very broad. In this sense the theoretical spectra are approximated by the sum over the partial DOS, weighted with the partial photoionization cross section of Yeh and Lindau,59folded with a Gaussian to simulate the effect of a finite instrumental resolution. The analysis of the PES and IPES spectra confirms the conclusion that at least moderately strong on-site interactions must be included to achieve a cor-rect description of the electronic structure of hematite. In the GGA limit and even with U⫽2 eV, the narrow gap at the Fermi level is completely covered by the finite resolution of the spectra, the lower edge of the valence band is found at too low binding energies, and the dominant peak in the con-duction band is too close to the Fermi level. U⫽4 eV leads to reasonable agreement with experiment. The structure in the valence band with separated peaks at E⬃⫺7 eV domi-nated by Fe 3d states and at E⬃⫺3 eV dominated by O 2 p states is more pronounced than in experiment. This is a con-sequence of the fact that the on-site Coulomb potential acts only on the Fe 3d states, leading to a reduction of the Fe-O hybridization which is too strong. However, GGA⫹U with U⫽4 eV leads undoubtedly to a marked improvement of the calculated electronic structure.
Punkkinen et al.33 found a value of U⫽2 eV as an opti-mum from their DOS interpretation. They remarked that al-ready for U⫽3.5 eV, the occupied Fe 3d states are shifted
FIG. 8. Variation of the c/a ratio 共upper panel兲, the internal parameters xO and zFe 共middle兲, and the Fe-Fe distances 共lower
panel兲 with pressure for GGA and GGA⫹U with U⫽4 eV, to-gether with experimental results taken from Ref. 20.
too much towards lower energies. The reason for the dis-agreement with our conclusions is twofold.
共i兲Punkkinen et al. used a different version of the DFT⫹
U Hamiltonian which is not rotationally invariant.
共ii兲There are significant differences already at the GGA level. As no information on the magnetic structure is given in the paper, the possibility that the calculation refer to a differ-ent antiferromagnetic configuration cannot be excluded.
VII. CONCLUSIONS
We have calculated structural, magnetic, and electronic properties of rhombohedral␣-Fe2O3for different volumes of the primitive unit cell in the framework of DFT using the GGA. It could be shown that although ground-state proper-ties such as c/a ratio and atomic positions agree well with experimental data, electronic properties such as band gap, but also magnetic moments, are in substantial disagreement with experiment. Furthermore we observe a HS-LS AF-FM insulator-metal transition at a pressure of 14 GPa which is not seen in experiment. Above that pressure structural pa-rameters differ strongly from the corresponding experimental values. This transition turns out to be due to the improper description of the on-site Coulomb interaction in conven-tional DFT in combination with the GGA. When introducing a Hubbard like term in the density functional共DFT⫹U), the
transition is shifted to lower volumes with increasing U and eventually disappears for U⭓4 eV, resulting in strongly im-proved values for magnetic moments, internal degrees of freedom, and band gap as well as better agreement of the density of states with experimental PES and IPS spectra. A further increase of U does not lead to better agreement, the band gap gets too large and occupied Fe 3d states are shifted to too low energies. We consider U⫽4 to be the value for
FIG. 9. Orbital projected DOS 共PDOS兲 for the AF ⫹⫹⫺⫺ solution of Fe2O3, calculated within the GGA⫹U with U⫽4 eV.
The left共right兲panel shows spin-up共spin-down兲Fe eg, Fe t2g, and
O 2 p projections.
FIG. 10. Valence-band 共UPS兲 共Ref. 10兲 and conduction-band
共IPS兲spectra共Ref. 12兲for Fe-3d states together with an estimate of the photoemission spectra calculated from the Fe-3d and the O-2 p DOS using photoionization cross sections of Yeh and Landau共Ref. 59兲for different values of U. A smearing parameter of⫽0.8 was used to broaden the DOS.
which best overall agreement is achieved.
An important result of this study is the profound change in the semiconducting gap from a d-d exchange gap to an O 2 p –Fe 3d charge-transfer gap, paralleled by the change of the frontier orbitals 共the highest occupied valence states兲 from strongly hybridized O 2 p –Fe 3d to almost pure O 2 p character. As the frontier orbitals determine the chemical re-activity of the surface of a material, we expect a non-negligible influence of the on-site Coulomb repulsions on the stable surface termination and on the adsorption of atoms and molecules on hematite surfaces. Recent work by two of us60 on the adsorption of small molecules such as CO on NiO共100兲has demonstrated that the strong correlation effects present in these materials lead to a reduction of the adsorp-tion energy by more than a factor of 2 and a qualitative
change in the adsorption geometry共tilted vs perpendicular兲, markedly improving agreement with experiment. Although the on-site Coulomb repulsion is stronger in NiO than in Fe2O3 (U⫽6.3 eV vs U⫽4 eV), we expect similar effects in hematite as well. Due to the much more complex crystal structure, even the surface termination could be influenced.
ACKNOWLEDGMENTS
Work at the University of Duisburg has been supported by the SFB 445 ‘‘Nanoparticles from the Gas phase: Formation, Structure, Properties’’; work at the University of Vienna was supported by the Austrian Science Funds through the Science College ‘‘Computational Materials Science.’’
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