PHYSICAL REVIEW B 69, 165107 共2004兲
First-principles calculation of the structure and magnetic phases of hematite
G. Rollmann,1 A. Rohrbach,2 P. Entel,1 and J. Hafner2
1
Institut für Theoretische Physik, Universität Duisburg-Essen, Lotharstrasse 1, 47048 Duisburg, Germany
Institut für Materialphysik and Center for Computational Materials Science, Universität Wien, Sensengasse 8/12,
A-1090 Wien, Austria
共Received 20 June 2003; published 6 April 2004兲
2
Rhombohedral ␣ -Fe2 O3 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 3.4␮ B , which
is about 1.5␮ B below the experimentally observed value. The insulating nature of ␣ -Fe2 O3 is 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 2p states in ␣ -Fe2 O3 . 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 Fe2 O3 we 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 ␣ -Fe2 O3 共hematite兲 is the most common on earth. At ambient conditions, it
crystallizes in the rhombohedral corundum structure 共space
group R3̄c), see Ref. 1. Below the Néel temperature, T N
⫽955 K, ␣ -Fe2 O3 is an antiferromagnetic 共AF兲 insulator
showing weak ferromagnetism above the Morin temperature,
T M⫽260 K, due to a slight canting of the two sublattice
magnetizations.2–5 Below T M , 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.22,23 Especially the nature of the high-pressure
共HP兲 transition of ␣ -Fe2 O3 has recently been discussed in
detail.6 –9,13,19–21 At 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 transition 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 ␣ -Fe2 O3 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
0163-1829/2004/69共16兲/165107共12兲/$22.50
and Rozenberg et al.20 assigned the HP structure to a distorted corundum-type (Rh2 O3 -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 Fe2 O3 are strongly correlated, so methods beyond ordinary density-functional theory 共DFT兲 in the
local spin-density approximation 共LSDA兲 are needed to correctly describe the system in terms of electronic and magnetic properties. As the rhombohedral primitive cell of hematite contains ten atoms 共compared to only two atoms in
simple transition-metal 共TM兲 oxides with the rocksalt structure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
␣ -Fe2 O3 . The Hartree-Fock 共HF兲 study of Catti et al.31 already achieves the correct order of magnetic states at experimental volume, yielding an energy difference between antiferromagnetic and ferromagnetic 共FM兲 alignments of the
spins of 37 meV/共Fe atom兲, although the obtained mixing of
oxygen 2p 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.32 studied ␣ -Fe2 O3
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
69 165107-1
©2004 The American Physical Society
G. ROLLMANN, A. ROHRBACH, P. ENTEL, AND J. HAFNER
PHYSICAL REVIEW B 69, 165107 共2004兲
Coulomb potential improves the prediction for the energy
gap without, however, achieving satisfactory agreement for
other physical properties. In the latter two studies no relaxation of the atoms inside the supercell and also no variation
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 introduced 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.
TABLE I. Positions of the atoms xᠬ ⫽ ␣ a⫹ ␤ b⫹ ␥ c inside the
primitive unit cell of ␣ -Fe2 O3 with respect to the basis vectors
a,b,c of the primitive cell in terms of the two internal degrees of
freedom z Fe and x O . Experimental values are taken from Ref. 6.
Numbers in parentheses represent the uncertainty of the last digit.
II. COMPUTATIONAL MODEL
The calculations have been performed on the basis of
spin-polarized DFT34,35 with the Vienna ab initio simulation
package VASP.36 – 40 For the exchange-correlation functional,
E xc , 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 KohnSham equations were solved via iterative matrix diagonalization based on the minimization of the norm of the residual
vector to each eigenstate and optimized charge- and spinmixing routines.43– 45
A number of eight valence electrons for each Fe atom
(3d 7 4s 1 ) and six valence electrons for each O atom
(2s 2 2p 4 ) were taken into account. The remaining 共core兲
electrons together with the nuclei were described by pseudopotentials in the framework of the PAW method proposed by
Blöchl46 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 integration over the Brillouin zone was performed with a mesh of
8⫻8⫻8 k points generated by following the scheme of
Monkhorst and Pack.47 The 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 Blöchl 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
Atom
Wyckoff
symbol
␣
␤
␥
Fe
共c兲
z
1/2⫺z
1/2⫹z
1⫺z
z
1/2⫺z
1/2⫹z
1⫺z
O
共e兲
x
1⫺x
0
1/2⫺x
1/2⫹x
1/2
z
1/2⫺z
1/2⫹z
1⫺z
z⫽0.10534(6)
1⫺x
0
x
1/2⫹x
1/2
1/2⫺x
x⫽0.3056(9)
0
x
1⫺x
1/2
1/2⫺x
1/2⫹x
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 z Fe and x O , by which the
positions of the atoms inside the primitive cell can be described 共see Table I兲. We have relaxed the positions of the
FIG. 1. 共Color online兲 Hexagonal unit cell of ␣ -Fe2 O3 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.
165107-2
PHYSICAL REVIEW B 69, 165107 共2004兲
FIRST-PRINCIPLES CALCULATION OF THE . . .
TABLE II. Experimentally derived structural parameters for
␣ -Fe2 O3 taken from Ref. 1 共top兲 and Ref. 7 共bottom兲. Distances are
given in Å.
c
13.730
13.747
a
c/a
Fe-Fe 共A兲
Fe-Fe 共B兲
5.029
5.035
2.730
2.730
2.883
2.896
3.982
3.977
␳ ␴⫺ ␳ ␴␳ ␴⫽
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 pressure by Pauling and Hendricks1 and Finger and Hazen7 are
given in Table II. For DFT⫹U calculations, the version described 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
Ĥ⫽
U
2
兺 n̂ m ␴ n̂ m ⬘⫺ ␴ ⫹
m,m , ␴
⬘
U⫺J
2
兺
m⫽m ⬘ , ␴
n̂ m ␴ n̂ m ⬘ ␴ , 共1兲
where n̂ m ␴ 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 required for adding an extra d electron to an Fe atom, U
⫽E(d n⫹1 )⫹E(d n⫺1 )⫺2E(d n ), and a parameter J representing 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
E DFT⫹U ⫽E DFT⫹
U⫺J
2
2
共 n m ␴ ⫺n m
兺
␴兲.
m␴
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 matrix ␳ ␴i j of the d electrons, Lichtenstein et al.51 obtain a rotationally invariant energy functional,
which in the present case is of the form
E DFT⫹U ⫽E DFT⫹
U⫺J
2
兺␴ Tr关 ␳ ␴ ⫺ ␳ ␴ ␳ ␴ 兴 .
pancy matrix ( ␳ ␴ ) 2 ⫽ ␳ ␴ imposed by the second term in Eq.
共3兲. For U⬎J, the term is positive definite, since the eigenvalues ⑀ i of the on-site occupancy matrix are either 0 or 1,
共3兲
This yields exactly the same energy as the DFT functional
E DFT⫹U ⫽E DFT in the limit of an idempotent on-site occu-
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 penalty function driving the on-site occupancy matrices towards
idempotency, whose strength is parametrized by a single parameter (U⫺J). A larger (U⫺J) forces a stricter observance
of the on-site idempotency, achieved by lowering the oneelectron 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,
V ␴i j ⫽
␦ E DFT⫹U
␦ ␳ ␴i j
⫽
␦ E DFT
␦ ␳ ␴i j
⫹ 共 U⫺J 兲
冋
册
1
␦ ⫺ ␳ ␴i j .
2 ij
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 energies 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 solutions 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 rhombohedral 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, energetic, and magnetic properties of the different magnetic
solutions for a volume of 10.06 Å3 are 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 corresponding value in the same publication, but considerably
larger than the value found by Catti et al.31 within their HF
study. The nonmagnetic 共NM兲 solution is highest in energy.
165107-3
PHYSICAL REVIEW B 69, 165107 共2004兲
G. ROLLMANN, A. ROHRBACH, P. ENTEL, AND J. HAFNER
FIG. 2. Total energies 共lower panel兲 and absolute values of magnetic 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.
The AF solutions slightly vary in magnetic moments, with
all moments being ⬇3.4␮ B in magnitude. This is about
1.5␮ B lower than the experimental moments.53 In contrast to
the results of Sandratskii et al.,32 who obtain almost the same
moments for AF and FM ordering, the moments of the FM
solution are about 1.0␮ B lower in our calculation. This reflects 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 distances 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 Å3 per 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 relaxation. 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 0.8␮ B . The small difference in energy between FM and
AF states obtained by Catti et al.29 has already been
explained32 to 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 lowspin 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
AF ⫹⫹⫺⫺
AF ⫹⫺⫹⫺
AF ⫹⫺⫺⫹
FM ⫹⫹⫹⫹
NM
Functional
Reference
Energy
Magnetic moment
c/a ratio
Fe-Fe 共A兲
Fe-Fe 共B兲
GGA
LSDA
HF
GGA
LSDA
GGA
LSDA
GGA
LSDA
HF
GGA
This paper
32
31
This paper
32
This paper
32
This paper
32
31
This paper
0
0
0
211
190
224
204
388
489
37
680
3.44
3.72
4.74
3.38
3.80
3.45
3.74
2.60
3.73
2.77
2.941
4.006
2.70
2.76
2.877
2.959
4.033
3.958
2.69
2.803
4.000
2.85
2.689
4.380
2.83
2.761
4.280
165107-4
PHYSICAL REVIEW B 69, 165107 共2004兲
FIRST-PRINCIPLES CALCULATION OF THE . . .
TABLE IV. Ground-state parameters for the different magnetic states. Volumes V 0 are in Å3 /atom, magnetic moments are in ␮ B /Fe
atom, distances are in Å, and bulk moduli B 0 are in GPa.
Magnetic state
AF ⫹⫹⫺⫺
Expt. 共Ref. 6兲
AF ⫹⫺⫹⫺ 共HS兲
AF ⫹⫺⫹⫺ 共LS兲
AF ⫹⫺⫺⫹ 共HS兲
AF ⫹⫺⫺⫹ 共LS兲
FM ⫹⫹⫹⫹
NM
a
V0
Moment
z Fe
xO
c/a ratio
Fe-Fe 共A兲
Fe-Fe 共B兲
B0
10.00
10.06
10.13
8.85
10.13
8.88
8.79
8.73
3.43
0.1057
0.1053
0.1073
0.0979
0.1031
0.0962
0.0960
0.0966
0.3096
0.3056
0.3122
0.3110
0.3047
0.3120
0.3131
0.3150
2.772
2.731
2.757
2.775
2.690
2.714
2.795
2.811
2.929
2.896
2.976
2.610
2.812
2.526
2.564
2.582
3.998
3.977
3.959
4.052
4.010
4.038
4.111
4.102
173
178, 225 共231,a 258b兲
134
206
169
212
249
271
3.42
1.00
3.48
1.17
Reference 6.
Reference 20.
b
⫹⫺⫹⫺, ⫹⫺⫺⫹, 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 decrease linearly with volume for the AF states, but stay almost
constant around 1 ␮ B for 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 related to the strong superexchange interaction mediated by the
O atoms. But we also note that at low volumes the FM solution 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 ␣ -Fe2 O3 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 formalism 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 BirchMurnaghan 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 overestimation of hybridization of Fe 3d and O 2p states in conventional 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 B 0 ⬃178 GPa closer to our
calculated value. The most recent result of Rozenberg et al.20
is obtained by arbitrarily setting the pressure derivative B 0⬘
⫽⌬B 0 /⌬p in the Murnaghan fit to B 0⬘ ⫽4.
The calculated bulk modulus for the FM state is considerably larger, which is due to the fact that it has been evaluated for a lower volume and coincides well with the experimental result that with increasing pressure hematite is
becoming harder. This can be observed directly when comparing 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 z Fe on the volume. For volumes of
more than 10 Å3 /atom the moments are larger than 3 ␮ B /Fe
atom and z Fe is greater than 0.103. This results in a difference 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 1.5␮ B /Fe atom and z Fe is less than 0.098, leading 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 x O and the c/a ratio.
As not only the volume, but also the magnetic order determine the structural parameters, we now stick to the experimentally observed AF ⫹⫹⫺⫺ state to examine further the
dependence of crystalline structure of ␣ -Fe2 O3 on pressure.
Figure 3 illustrates the dependence of the c/a ratio, the internal degrees of freedom z Fe and x O , 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 p c , a rapid
change, especially in c/a ratio, is observed, followed by
nearly perfect linear behavior above p c . The HS-LS transi-
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G. ROLLMANN, A. ROHRBACH, P. ENTEL, AND J. HAFNER
FIG. 3. Variation of the c/a ratio 共upper panel兲, the structural
parameters x O and z Fe , plotted as z Fe⫹0.25 共middle兲, and Fe-Fe
distances 共lower panel兲 with pressure, together with experimental
data taken from Ref. 20.
tion is obviously accompanied by transitions in these structural 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.6,7,20 The agreement of all parameters and their pressure dependence with experiment is reasonable for pressures below p c . Due to the transition at p c
this agreement becomes poor for pressures above 14 GPa.
B. Electronic structure
In order to analyze electronic properties of ␣ -Fe2 O3 we
calculated the DOS for all energy minima of the curves. In
Fig. 4 the partial DOS of Fe 3d (t 2g and e g ) and O 2p states
for the AF ⫹⫹⫺⫺ solution at experimental volume are
shown. A band gap of ⬇0.3 eV is clearly visible. As is usually 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,33 who applied GGA in combination with the linear muffin-tin orbital 共LMTO兲 method in atomic-sphere approximation 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. 4. Orbital projected DOS 共PDOS兲 for the AF ⫹⫹⫺⫺
solution of Fe2 O3 , calculated within the GGA. The left 共right兲 panel
shows spin-up 共spin-down兲 Fe e g , Fe t 2g , and O 2p projections.
The orbitals are projected onto Wigner-Seitz spheres.
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FIRST-PRINCIPLES CALCULATION OF THE . . .
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 dramatic overestimation of the gap calculated within HF is due
to the complete neglect of electronic correlations. The disagreement with experiment may be traced back to some extent to the amount of mixing of Fe 3d and O 2p 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 2p DOS, whereas it is experimentally confirmed that the valence-band edge in ␣ -Fe2 O3 is dominated
by oxygen 2p states, making hematite a charge-transfer insulator. 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 insulatormetal 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 shortcomings 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 ␣ -Fe2 O3 to be unstable
under even modest compression against a HS-LS transition
coupled to an isostructural transition. Furthermore, the energetic order of the possible magnetic phases changes upon the
transition, with the FM phase being favored at pressures beyond ⬃ 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 correlation 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 ␣ -Fe2 O3 . 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
Fe2 O3 , calculated within the GGA. The left 共right兲 panel shows
spin-up 共spin-down兲 Fe e g , Fe t 2g , and O 2p projections. The
orbitals are projected onto Wigner-Seitz spheres.
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G. ROLLMANN, A. ROHRBACH, P. ENTEL, AND J. HAFNER
FIG. 6. Variation of equilibrium volume, Fe-Fe distance, magnetic 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.
experiment shows O 2p domination and therefore a chargetransfer type of band gap. Furthermore, the Fe 3d band energies from GGA are not in agreement with experimental
valence-band spectra and conduction-band spectra. GGA
also predicts ␣ -Fe2 O3 to 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 properties of hematite obtained with GGA are attributable to the
improper description of the on-site Coulomb repulsion between 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 moments, and band gap on U. The parameter J was kept constant 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 magnetic 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 –58 of Punkkinen 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 increasing U, it increases first until at U⫽5 eV a maximum value
of ⬃10.3 Å3 is 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
FIG. 7. Volume dependence of the free energy and the magnetic 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.
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 equilibrium 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 compared to experiment6,7,20 even 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 favorable. 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 compared to the U⫽4 eV case. Figure 8 shows the pressure
dependence of the c/a ratio, the internal parameters x, z, and
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FIRST-PRINCIPLES CALCULATION OF THE . . .
⬃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 nonmagnetic orthorhombic Rh2 O3 共II兲-type phase at about 50 GPa.
This, however, is beyond the scope of the present study.
B. Electronic structure
FIG. 8. Variation of the c/a ratio 共upper panel兲, the internal
parameters x O and z Fe 共middle兲, and the Fe-Fe distances 共lower
panel兲 with pressure for GGA and GGA⫹U with U⫽4 eV, together with experimental results taken from Ref. 20.
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 ␣ -Fe2 O3 under pressure. Experimentally, Rozenberg et al.20 have shown that ␣ -Fe2 O3
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 parameters 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
The spin-polarized and orbital projected DOS, calculated
in the GGA⫹U with U⫽4 eV is shown in Fig. 9. The onsite 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 2p 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 t 2g 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 2p –Fe 3d gap of nearly 2 eV, in
agreement with experiment.
Figure 10 shows the calculated intensities of the photoemission 共PES兲 and inverse photoemission 共IPES兲 spectra,
compared with the experimental results of Fujimori et al.10
and Ciccacci et al.12 In 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,59 folded 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 correct 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 conduction 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 dominated by Fe 3d states and at E⬃⫺3 eV dominated by O 2p
states is more pronounced than in experiment. This is a consequence 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 optimum from their DOS interpretation. They remarked that already for U⫽3.5 eV, the occupied Fe 3d states are shifted
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G. ROLLMANN, A. ROHRBACH, P. ENTEL, AND J. HAFNER
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-2p
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.
too much towards lower energies. The reason for the disagreement 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 different antiferromagnetic configuration cannot be excluded.
VII. CONCLUSIONS
FIG. 9. Orbital projected DOS 共PDOS兲 for the AF ⫹⫹⫺⫺
solution of Fe2 O3 , calculated within the GGA⫹U with U⫽4 eV.
The left 共right兲 panel shows spin-up 共spin-down兲 Fe e g , Fe t 2g , and
O 2p projections.
We have calculated structural, magnetic, and electronic
properties of rhombohedral ␣ -Fe2 O3 for different volumes of
the primitive unit cell in the framework of DFT using the
GGA. It could be shown that although ground-state properties 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 parameters 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 conventional 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 improved 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
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PHYSICAL REVIEW B 69, 165107 共2004兲
FIRST-PRINCIPLES CALCULATION OF THE . . .
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 2p –Fe 3d to almost pure O 2p
character. As the frontier orbitals determine the chemical reactivity of the surface of a material, we expect a nonnegligible 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 adsorption energy by more than a factor of 2 and a qualitative
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ACKNOWLEDGMENTS
Work at the University of Duisburg has been supported by
the SFB 445 ‘‘Nanoparticles from the Gas phase: Formation,
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supported by the Austrian Science Funds through the Science
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