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1 IRD, 209 rue La Fayette, 75480 Paris Cedex 10, France
2 Institut de Minéralogie et de Physique des Milieux Condensés (IMPMC), UMR CNRS 7590, Universités Paris VI et VII, IPGP, Campus Boucicaut, 140 rue de Lourmel, 75015 Paris, France
Correspondence: * E-mail: marc.blanchard{at}impmc.jussieu.fr
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ABSTRACT |
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-Fe2O3) was computed using ab initio quantum mechanical calculations. Frequencies of the normal vibrational modes and Born effective charges were computed using the density functional theory (DFT) with and without the addition of a Hubbard U correction. The infrared reflection spectra of a single crystal of hematite were calculated as well as the infrared powder absorption spectrum using an electrostatic model that takes into account the shape of hematite particles. The theoretical behavior of the absorption bands is in agreement with experimental observations and provides a firm basis for the interpretation of the bands in term of vibrational modes. Overall, results suggest that the use of DFT + U, which is necessary to describe correctly the electronic and magnetic properties of hematite, does not improve noticeably the prediction of vibrational properties.
Key Words: Iron oxide • density functional theory • infrared spectroscopy
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INTRODUCTION |
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Unfortunately, limitations related to the density functional theory occur when considering Fe oxides and Fe-bearing compounds, which are major phases of many geological environments. These minerals are indeed deeply involved in redox and bio-geochemical processes and magnetic oxides are able to record the fluctuations of the Earth’s magnetic field over geologic times (e.g., Théveniaut and Freyssinet 1999). The spectacular red to yellow colors of tropical soils can also be ascribed to the occurrence of finely divided Fe oxide particles, hematite (-Fe2O3) and goethite (FeOOH), respectively (Fritsch et al. 2005).
Hematite, the most common of all Fe oxides, is an insulator with a corundum-type structure. The width of its band gap (2 eV; Mochizuki 1977) is the result of the strong on-site Coulomb repulsion that occurs between the Fe 3d electrons. This feature is a challenge for theory and recent computational work has demonstrated that methods beyond the regular density functional theory (DFT) are needed to describe correctly the structural, electronic, and magnetic properties of hematite (e.g., Punkkinen et al. 1999; Bandyopadhyay et al. 2004; Rollmann et al. 2004; Velev et al. 2005). All these studies took into account the electron correlations by adding a Hubbard U correction (DFT + U). However, while the effect of this U parameter on the electronic and magnetic properties of bulk hematite is well documented, no first-principles study of the vibrational properties has been done so far. To our knowledge, the only phonon calculation available is reported by Chamritski and Burns (2005), who employed an atomistic simulation method with interatomic potentials. Experimentally, the frequencies of the infrared normal modes and the optical constants have been determined from reflection and thermal emission spectra using dispersion theory (Onari et al. 1977; Glotch et al. 2006). Several works have also focused on the effect of particle size, shape, and orientation on hematite infrared spectra (e.g., Rendon and Serna 1981; Serna et al. 1987; Wang et al. 1998).
In this paper, we present the first theoretical infrared spectrum of an Fe oxide computed using first-principles methods. This work compares the results obtained from DFT and DFT + U calculations where the magnitude of the Hubbard U correction is determined with a self-consistent approach based also on a first-principles method. The infrared reflection and powder absorption spectra are calculated and compared with experimental spectra. The effect of particle shape is also taken into account. Thus, we assess whether it is necessary to consider the on-site Coulomb interaction U for the description of the vibrational properties of hematite.
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COMPUTATIONAL AND EXPERIMENTAL METHODS |
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where I are the Lagrange multipliers defining the strength of the potential shifts and PI, the projection operator. Then, the interaction parameter U associated with site I can be written as
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where the first and second terms correspond to the non-interacting and interacting response functions respectively. In the procedure described above, the value of U is calculated from the GGA ground state (Cococcioni and de Gironcoli 2005). However, Kulik et al. (2006) have shown that U should be consistently obtained from the GGA + U ground state itself. Therefore U was computed from several input values, Uin and the final value of the Hubbard U was extrapolated from the linear dependence between Uin and Uout.
Vibrational and dielectric properties
The infrared spectrum can be obtained knowing the analytical part of the dynamical matrix and dielectric quantities such as the Born effective charges and the electronic dielectric tensor. For DFT calculations, displacements and frequencies of the normal vibrational modes at the Brillouin zone center were calculated using linear response theory (Baroni et al. 2001). They were derived from the second-order derivative of the total energy with respect to atomic displacements, using the PHONON code (Baroni et al. 2001; http://www.pwscf.org). For DFT + U calculations, the phonon frequencies were computed employing the frozen-phonon approach. In this case, the analytical part of the dynamical matrix was calculated from the first-order derivative of the atomic forces with respect to atomic displacements. The amplitudes of the displacements from the relaxed structure were typically 0.1% of the lattice parameter. We have verified, in the DFT case, that both methods give the same vibrational frequencies within 1%.
For both DFT and DFT + U calculations, the Born effective charges were obtained from finite differences of the bulk polarization induced by small atomic displacements away from the relaxed structure. The Born effective charge (Z*) can be defined as
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where 
is the volume of the unit cell, P is the macroscopic polarization, and u is the set of atomic coordinates (i, j, and 
are the Cartesian and atomic indexes, respectively). Resta et al. (1993) have shown that the macroscopic polarization is linear in atomic displacement to a good approximation. The polarization was computed using the Berry-phase approach (King-Smith and Vanderbilt 1993; Resta 1994). For the k-space integrations in the Berry-phase calculations, the number of k-points parallel to the direction of polarization has been increased to 14.
Finally, the calculated LO-TO splittings have been fitted to the experimental ones to determine the electronic dielectric tensors since its calculation is not implemented for DFT + U in the PHONON code.
IR spectrum modeling
The infrared reflection spectrum of a single crystal of hematite has been computed from the real and imaginary parts of the dielectric constant (
1 and 2, respectively), by applying the two following relations:
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where R is the reflectivity at normal incidence and the two parameters n and k are, respectively, the real and imaginary parts of the refractive index.
In solids, the vibrational frequencies (which determine the peak positions in infrared spectra) can be influenced by the coupling of the atomic vibrations with a macroscopic electric field. This field is related to the longitudinal polarization of optical modes in large crystals or to the polarization charges occurring at the surfaces of small dielectric particles. In powder absorption spectra, the vibrational frequencies therefore depend on the macroscopic shape of the measured particles. To describe the frequency dependence from the particle shape, we have determined the powder infrared spectrum of hematite using a model similar to the one developed by Balan et al. (2001). This model, developed for thin disks, can be easily extended to small particles of arbitrary ellipsoidal shape because in both cases the electric and polarization fields are homogeneous inside the particle. The case considered assumes particles with a size smaller than the light wavelength.
Experimental IR spectroscopy
A commercial hematite sample has been used. Approximately 0.8 mg of dried sample were gently ground with 300 mg of KBr. The mixture was pressed to produce a KBr pellet and the transmission infrared spectrum was recorded at room temperature using a FTIR spectrometer (Nicolet Magna 560) in the 250–4000 cm–1 range with a resolution of 2 cm–1.
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RESULTS AND DISCUSSION |
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c). It is characterized by hexagonal close-packed (001) layers of O atoms, with Fe atoms filling 2/3 of the octahedral sites. Calculations have been carried out on the rhombohedral primitive cell, which contains ten atoms. All Fe and O atoms are crystallographically equivalent, in the (4c) and (6e) Wyckoff positions, respectively. Therefore, only two internal degrees of freedom (i.e., xFe and xO) are needed to describe the position of the atoms inside the primitive cell (Table 1
). The structure has been fully relaxed with the optimization of the lattice parameters as well as the atomic coordinates. The final lattice parameters and bond lengths are reported in Table 2, while the structure is shown in Figure 1. The relation between the hexagonal and rhombohedral cells can be found in Rollmann et al. (2004). In DFT calculations, under the GGA approximation, the unit-cell volume is only 0.6% larger than the experimental one (Finger and Hazen 1980). This good agreement is also apparent for Fe-Fe and Fe-O bond lengths. Iron atoms align along the hexagonal c-axis and are distributed alternatively with a short and large Fe-Fe distance. Looking in more detail, the shortest Fe-Fe bonds are longer and FeO6 octahedra are more distorted than experimentally observed.
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= – 0.1 to + 0.1 eV was applied to the Fe d electrons. Results show that the convergence of U with the size of the supercell is already achieved for the primitive unit cell. These calculations have been done for Uin = 0, 1, and 2 eV. We obtain a linear dependence, which indicates that the U derived from the GGA ground state is valid, in the hematite case. The final result for the Hubbard U is 3.3 ± 0.1 eV and is used in the rest of the paper. In Table 2
, we report the relaxation done with GGA + U. The use of an on-site Coulomb interaction parameter affects significantly the crystal structure. Despite the fact that the unit cell expands significantly with a volume 4.0% larger than the experimental one, we note that a better agreement is achieved for the c/a ratio of the hexagonal unit cell as well as the two internal degrees of freedom (xFe and xO), improving the description of the crystal structure. The bulk modulus has been calculated to test the description of hematite compressibility. Five negative and five positive increments of deformation, keeping the c/a ratio constant, were applied to the relaxed geometry. The energy curve as a function of the applied deformation was fitted using a polynomial whose second derivative at null deformation constrains the bulk modulus, according to the following relation:
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V0 is the equilibrium volume. The GGA value for the bulk modulus is found to be 177 GPa, and it increases to 192 GPa with GGA + U. These values show a certain improvement using the GGA + U method even if they are lower than the experimental data. The most recent results of Rozenberg et al. (2002) indicate a value of 258 GPa, but the literature shows that the hematite bulk modulus is subject to considerable uncertainty. This underestimation by GGA and GGA + U has already been reported by Rollmann et al. (2004). It is noteworthy that the homothetic deformation applied here may introduce some uncertainties since the c/a ratio decreases with pressure (Rozenberg et al. 2002).
Hematite is antiferromagnetic below the Morin temperature (260 K) with the Fe spins aligning along the [111] axis of the rhombohedral unit cell, equivalent to the hexagonal c-axis (Morin 1950). Above the Morin transition and below the Néel temperature, 948 K, hematite shows a weak ferromagnetism due to a slight canting of the spins lying in the basal plane, i.e., the plane perpendicular to the hexagonal c-axis (Searle and Dean 1970; Levinson 1971). Beyond 948 K, hematite becomes paramagnetic. All our calculations have been set up with the antiferromagnetic (+ – – +) configuration (according to our representation of the primitive cell) since it is known to be the most stable (Rollmann et al. 2004). This magnetic ordering means that, along the hexagonal c-axis, Fe atoms separated by a short distance have opposite magnetic moments while the more distant Fe atoms have equal magnetic moments. After relaxation, the magnetic moment on each Fe atom is 3.87 µB and increases to 4.19 µB for U = 3.3 eV. These values are consistent with the theoretical results of Rollmann et al. (2004) who found for the same antiferromagnetic state, 3.43 and ~4.15 µB/Fe atom, for U = 0 and 4 eV, respectively (according to our definition of U). For comparison, the experimental value is 4.6–4.9 µB/Fe atom (Kren et al. 1965; Coey and Sawatzky 1971).
We have also investigated the electronic structure of hematite by calculating the total and partial densities of states (DOS) as illustrated in Figures 2 and 3. DOS have been calculated with a 16 x 16 x 16 k-point grid and a smearing width of 0.005 Ry. Our results are in agreement with the findings of the previous theoretical studies (e.g., Punkkinen et al. 1999; Bandyopadhyay et al. 2004; Rollmann et al. 2004; Velev et al. 2005). Hematite is properly described as an insulator by DFT. The band gap is underestimated but an important improvement is obtained by taking into account the on-site Coulomb repulsion. In fact, the energy gap increases from 0.7 eV in DFT to 1.5 eV with U = 3.3 eV. The latter value compares better with the experimental value, i.e., 2.0 eV (Mochizuki 1977). In DFT calculations, the top of the valence band is marked by a strong hybridization of the O p states and Fe d states, i.e., Fe states and O states are in equivalent amount (Fig. 2). We expect however, from experimental observations, to see a domination of O 2p states at the valence-band edge. This is what we observe from DFT + U calculations. Comparing Figures 2 and 3, the relative intensity of the DOS between Fe 3d and O 2p in the range –2.0/0.0 eV is smaller in GGA + U. Such a system corresponds to a charge-transfer insulator, in agreement with experiments (Lad and Henrich 1989; Dräger et al. 1992).
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We can note 30 normal modes at the center of the Brillouin zone (
). Among these modes, 12 are Raman active (2 one-dimensional A1g and 5 two-dimensional Eg) and 10 modes are infrared active vibrations (2 one-dimensional A2u and 4 two-dimensional Eu). Phonon frequencies of the normal modes of hematite were calculated at the center of the Brillouin zone using the relaxed structures and compared with experimental values (Table 3). The analysis of eigenvectors has allowed us to describe the symmetry of each vibrational mode. Thus, regarding infrared active vibrations, the two A2u modes correspond to an in-phase vibration of Fe atoms in the direction of the hexagonal c-axis. At the same time, the group of three O atoms (i.e., O atoms having the same zO in Fig. 1) performs a rotation around the c-axis associated with a vertical motion along the same axis. No relative motion between Fe atoms or between O atoms is observed. On the other hand, the four two-dimensional Eu modes correspond to motions of Fe atoms in the plane perpendicular to the c-axis. In these vibrational modes, the motion of the O atoms is more complex and cannot be described in a simple way. All these observations made from DFT and DFT + U results, are in full agreement with the symmetry analysis done by Cowley (1969). Although the overall agreement between theoretical and experimental frequencies is satisfactory, noticeable differences are present and it is difficult from Table 3 to assess an actual improvement of the use of GGA + U over GGA. In particular, besides a change in the ratio of different frequencies, the frequencies present an overall change due to the overestimation of the cell volume. The effect of the cell volume results in an overall shift toward lower frequencies, as can be seen in Figure 4 where we compare the experimental and the theoretical frequencies. This effect can be quantified by comparing the linear regression of the theoretical data with the ideal 1:1 ratio in Figure 4. Beyond this effect, we remark that, over the whole frequency range, GGA + U frequencies are less dispersed around the theoretical regression than the GGA ones, thus we can say that GGA + U slightly improves the theoretical vibrational modes of hematite.
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c space group (Table 4). For convenience, Born effective charge tensors are expressed relative to Cartesian axes (Table 5). The magnitude of the zz component of the effective charge tensor (Z*zz) is smaller than in the (x,y) plane. Onari et al. (1977) calculated the Born effective charges of hematite from the measured phonon frequencies. They found that the effective charges of the Fe atoms are 3.75e and 4.05e for an electric field parallel and perpendicular to the hexagonal c-axis, respectively. Effective charges obtained from DFT + U calculations are much closer to the experimental values with only a slightly stronger anisotropy between the (x,y) plane and the perpendicular direction. This anisotropy is still stronger for the values given by regular DFT. The same comments are applicable to O atoms.
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(
), were determined by fitting the LO-TO splitting to the experimental data (Table 6). As the crystal lattice of hematite belongs to the hexagonal system, any macroscopic second-rank tensor properties such as the dielectric tensor will have two independent parts [i.e., the (x,y) plane and zz components]. Therefore 
() was determined by minimizing the residual differences between the theoretical and experimental LO-TO splittings of the two A2u modes while 
() [or 
()] was obtained in the same manner from Eu modes.
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(
) was calculated as a function of the light frequency using the theoretical phonon frequencies, normal vibrational modes and Born effective charge tensors. An ad-hoc damping parameter of 6 cm–1 was used. The static dielectric tensor, (0), shown in Table 6, presents the same features as the electronic one, (). The anisotropy predicted by the calculations is larger than the observed one and the use of DFT + U improves significantly the agreement with the experimental data. The theoretical infrared reflection spectra for ordinary and extraordinary rays were subsequently derived and compared to the experimental spectra (Figs. 5 and 6). For the ordinary ray spectrum, the electric vector of the incident light is perpendicular to the hexagonal c-axis. Therefore this spectrum corresponds to the Eu modes that are polarized in this basal plane. In the same way, the extraordinary ray spectrum corresponds to the A2u modes that are polarized along the c axis because the electric vector of the incident light is parallel to the same axis. The hematite ordinary ray spectrum has three broad absorption bands centered at 319, 456, and 560 cm–1 and one narrow band at 227 cm–1. Over the whole frequency range, the reflectivity of theoretical spectra is higher than that of the experimental spectra of Glotch et al. (2006). This is not too worrying since the overall reflectivity of the experimental spectra from Onari et al. (1977) was also higher. For both DFT and DFT + U calculations, the absorption bands are shifted toward low frequencies because of the underestimation of the transverse optical frequencies. The frequencies of the two lowest bands are better reproduced by DFT + U but the intensity of the band at 227 cm–1 is underestimated. The experimental extraordinary ray spectrum has two broad bands centered at 314 and 558 cm–1. The agreement with the DFT spectrum is good; we only note a small shift (~30 cm–1) of the band at 558 cm–1. This shift increases with the use of DFT + U, providing a less satisfactory agreement. Concluding, from a comparison of the shape of the computed and measured infrared spectra, it is not possible to directly detect an improvement of the GGA + U approach.
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and 8): platy particles perpendicular to the c-axis, spherical particles, and ellipsoidal particles with four c/a ratios. The theoretical spectra were computed using ext = 1.00, i.e., dielectric constant of the vacuum. They are compared with the resonances of the trace of the imaginary part of the dielectric tensor, Im[ ()]. These resonances correspond to the transverse optical (TO) modes of an infinite crystal. Therefore, this comparison enables us to quantitatively determine the influence of the particle shape on the infrared spectrum.
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and 3 of Serna et al. 1987; Fig. 5 of Wang et al. 1998).
Finally the theoretical spectra are compared with the experimental infrared absorption spectrum that we collected (Fig. 9). For this modeling, the dielectric constant of the external medium was fixed to the KBr value, i.e., ext = 2.25. Considering both the relative position and intensity of absorption bands, the best agreement between experiment and theory is achieved for oblate ellipsoids with a c/a ratio ranging from about 0.4 to 1 (i.e., spherical). An unambiguous assignment of the absorption bands is then possible. The high-frequency absorption approximately centered on 570 cm–1 is due to the absorption of both a Eu and a A2u mode; the mode polarized parallel to the c-axis (A2u) being at higher wavenumber. The bands at 479 and 337 cm–1 are assigned to Eu modes while the less intense band at 382 cm–1 corresponds to the other A2u mode. The magnitude of the theoretical peak shift as a function of particle shape shows that the experimental peak broadening is accounted for by a distribution of the depolarization field values. This distribution is likely related to the occurrence of particles with various and non-trivial shapes in the experimental sample.
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–9 highlight the differences between DFT and DFT + U results that have already been described in the previous sections. Infrared active modes are slightly shifted in frequency and the main difference is related to the relative intensities of the first two Eu modes. It is however very likely that the experimental intensity of the first Eu mode is intermediate between the intensities given by both models, as it was suggested by the reflection spectra (Figs. 5 and 6).
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ACKNOWLEDGMENTS |
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TopAbstractIntroductionComputational and experimental...Results and discussionAcknowledgmentsReferences cited |
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Footnotes |
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MANUSCRIPT RECEIVED October 4, 2007; MANUSCRIPT ACCEPTED January 7, 2008
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and // represent the vibrational modes polarized perpendicular and parallel to the hexagonal c-axis respectively.
