HIGH-PRESSURE CHARACTERISTICS OF α Fe 2 O 3 USING DFT+U

Transcription

1 HIGH-PRESSURE CHARACTERISTICS OF α Fe 2 O 3 USING DFT+U G. ROLLMANN and P. ENTEL Institute of Physics, University of Duisburg-Essen, Duisburg, Germany A. ROHRBACH and J. HAFNER Center for Computational Materials Science, University of Vienna, Sensengasse 8/12, Vienna, Austria (Received...) Abstract We have calculated structural, magnetic and electronic properties of corundum-type α-fe 2 O 3 from first principles using densityfunctional theory (DFT) and the DFT+U method to account for correlation effects in this material. Although the correct magnetic ground state is obtained by pure DFT, the magnetic moments and the band gap are too small, and a predicted structural phase transition coupled with a transition from the insulating high-spin to a metallic low-spin phase at a pressure of 14 GPa is not seen in experiment. We find that considering the Coulomb interaction directly by including a Hubbardlike term U into the density functional greatly improves the results with respect to band gap and magnetic moments. The phase transition is shifted to higher pressures with increasing values of U and disappears for U > 3 ev. Best overall agreement of structural, magnetic and electronic properties with experimental data is obtained for U = 4 ev. Keywords: Correlation effects, Density functional theory, DFT+U, Hematite, High pressure study, Phase transitions Corresponding author: Tel.: , fax: , georg@thp.uni-duisburg.de 1

2 1. INTRODUCTION The study of transition metal (TM) oxides by first principles computational methods is very appealing from a theoretical point of view for several reasons. First, they often exhibit unusual, sometimes unique physical and chemical properties. Second, their accurate description poses a challenge to the underlying theoretical methods, because of, e.g., strongly correlated d-electrons as in the case of α-fe 2 O 3. Therefore these systems can act as benchmark systems to test and compare different theoretical models. Third, the corresponding unit cells often contain many atoms, so the simulation of their properties by computationally costly ab initio methods can in many cases only be performed using the most up-to-date computational hardware and software. Of the Fe oxides, hematite (α-fe 2 O 3 ) is the most stable and the most common on earth. It has the same stoichiometry but a different crystal structure compared to the technologically very important γ-fe 2 O 3 (maghemite) used in magnetic applications, e.g., for data storage. At ambient conditions, α- Fe 2 O 3 crystallizes in the rhombohedral corundum structure. The hexagonal unit cell is displayed in Fig. 1. Between the Morin temperature of 260 K and the Néel temperature of 955 K, hematite is an antiferromagnetic (AF) insulator with a band gap of 2 ev. At lower temperatures, one observes weak ferromagnetism due to a slight canting of the sublattice magnetizations (Searle and Dean, 1970; Levinson, 1971; Chow and Keffer, 1974; Sandratskii and Kübler, 1996). The positions of the atoms inside the primitive unit cell can be described in terms of two internal degrees of freedom, commonly named x O and z Fe. In recent years, experimental work on bulk α-fe 2 O 3 has been focused on the high-pressure (HP) characteristics of hematite (Sato and Akimoto, 1979; Finger and Hazen, 1980; Olsen, Cousins, Gerward, Jhans et al., 1991; Pasternak, Rozenberg, Machavariani, Naaman et al., 1999; Rozenberg, Dubrovinsky, Pasternak, Naaman et al., 2002; Badro, Fiquet, Struzhkin, Somayazulu et al., 2002). It has been observed that at pressures around 50 GPa the material undergoes a structural phase transition with a volume reduction of about 10% followed by an electronic transition from an insulating high-spin (HS) to a metallic low-spin (LS) phase (Badro, Fiquet, Struzhkin, Somayazulu et al., 2002). The HP crystal structure has been confirmed to be of distorted corundum type (Rh 2 O 3 -II) (Pasternak, Rozenberg, Machavariani, Naaman et al., 1999; Rozenberg, Dubrovinsky, Pasternak, Naaman et al., 2002) with a single Fe 3+ cation. There has also been a number of experimental investigations regarding the electronic structure of hematite (Fujimori, Saeki, Kimizuka, Taniguchi et al., 1989; Lad and Henrich, 1989; Ciccacci, Braicovich, Puppin and Vescovo, 1991; Uozumi, Okada, Kotani, Zimmermann et al., 1997). On the basis of X-ray and ultraviolet photoemission spectroscopic (PES) data from these experiments, hematite is generally considered a charge-transfer 2

3 rather than a Mott-Hubbard insulator according to the definition of Zaanen et al. (Zaanen, Sawatzky and Allen, 1985), because the upper edge of the valence band is clearly dominated by oxygen 2p states. In contrast to that, there are only a few theoretical works on α-fe 2 O 3 based on first-principles calculations. This may be due to the fact that the primitive unit cell contains many atoms, which makes the calculations computationally demanding, in contrast to simulations of TM oxides with the simpler rocksalt structure. Already in 1995, Catti et al. calculated structural and electronic properties of hematite using the Hartree-Fock (HF) method (Catti, Valerio and Dovesi, 1995). But because of the complete neglect of correlation effects in this approach, their calculated band gap of 11 ev is much too high, and the bands are too narrow. Using density functional theory (DFT) in the local spin density approximation (LSDA), Sandratskii et al. calculated the order of magnetic states at experimental volume (Sandratskii, Uhl and Kübler, 1996) and obtained the correct AF ground state. In 1996, Punkkinen et al. used DFT in the generalized gradient approximation (GGA) and accounted for the on-site Coulomb repulsion at the Fe sites by a Hubbard like term U to calculate electronic properties of hematite (Punkkinen, Kokko, Hergert and Väyrynen, 1999). They find that the inclusion of even a modest value of U considerably improves the results compared to conventional GGA. In both these studies, to our knowledge, neither relaxation of the atoms nor a change of the cell shape was taken into account. In the following sections we present our results from ab initio calculations regarding the structure and stability of α-fe 2 O 3. Details of the simulations are given in the following section, results of both DFT and DFT+U are presented and discussed in Sec. 3 followed by a conclusion in Sec COMPUTATIONAL DETAILS We have performed total energy calculations based on DFT (Hohenberg and Kohn, 1964; Kohn and Sham, 1965) in the GGA for different volumes of the unit cell allowing for full relaxation of the cell shape and the positions of the atoms inside the unit cell in terms of the two internal degrees of freedom x O and z Fe. For the exchange correlation functional we chose a form proposed by Perdew and Wang in 1991 (Perdew and Wang, 1992). A number of eight (six) valence electrons was taken into account for each Fe (O) atom, the remaining core electrons together with the nuclei were described by pseudopotentials following the projector augmented wave (PAW) method as implemented in the Vienna ab initio simulation package VASP (Blöchl, 1994; Kresse and Furthmüller, 1996; Kresse and Joubert, 1999). The energy cutoff for the plane wave basis set was kept fixed at a constant value of 550 ev throughout the calculations. The integration over the irreducible part of the Brillouin zone was done using the linear tetrahedron method with Blöchl corrections 3

4 on an k-point mesh. To account for the electronic correlation present in α-fe 2 O 3, we also performed DFT+U (Solovyev, Dederichs and Anisimov, 1994; Liechtenstein, Anisimov and Zaanen, 1995) calculations by incorporating a Hubbard-like term into the density functional following the formalism proposed by Dudarev et al. (Dudarev, Botton, Savrasov, Humphreys et al., 1998). The Coulomb repulsion is therefore considered explicitly; its strength can be adjusted by a single parameter U J, where J is an approximation to the Stoner exchange parameter and was held at a constant value of 1 ev in our calculations. The relaxation of the atoms was performed using the conjugate gradient method. Resulting energy vs. volume curves were fitted to a Birch-Murnaghan equation of state (Murnaghan, 1944) to obtain the equilibrium volume, the bulk modulus and the volume dependence of pressure. Local magnetic moments were calculated by projecting the spin density onto spheres around the Fe atoms of radius 1.2 Å. 3. RESULTS AND DISCUSSION Figure 2 shows the total energy and the magnetic moments of the Fe atoms as a function of the volume per atom for the nonmagnetic (NM) state as well as for the magnetic states calculated from density functional theory. The global energy minimum corresponds to an AF solution with an equilibrium volume of Å 3 per atom, which is in good agreement with the experimental value of Å 3 per atom (Finger and Hazen, 1980). The same holds true for other ground state structural quantities like unit cell dimensions and internal degrees of freedom. We compare our results with corresponding experimentally derived values from Finger et al. in Table I. While the deviation for the internal degree of freedom z Fe is very small, x O is overestimated by about 1.3 %. The errors in lattice constants c and a, which compensate each other in the calculation of the volume, lead to a difference of 1.5 % in the c/a ratio. Similar deviations are observed for the interatomic distances. Despite the fact that the results for these structural parameters are quite close to experiment, other quantities, especially those related to the electronic structure of hematite, are in serious disagreement with experimental data. The band gap is calculated to 0.3 ev, which is an order of magnitude smaller than the measured value. The obtained mixing of Fe 3d with O 2p states is too strong, resulting in ground state magnetic moments of ±3.4 µ B per Fe atom (see Fig. 2), which is about 1.5 µ B lower than the experimental values. For a detailed analysis see Rollmann et al. (Rollmann, Rohrbach, Entel and Hafner, 2004). At experimental volume the other two states, namely FM and NM, are separated from the AF state by large energy differences of 155 mev and 4

5 272 mev per atom. For lower volumes, the situation is different. The curves for the AF and the FM solutions cross at a volume of 8.58 Å 3 per atom, and stay in principle parallel to each other for lower volumes. The reason for this is twofold. First, the equilibrium volume of the FM solution is with 8.79 Å 3 per atom much lower than that of the AF state. Second, due to a HS LS transition at a volume of 8.86 Å 3 per atom, the energy curve of the AF state shows a distinct kink resulting in the observed two-minima shape. By placing a common tangent to the curves, the pressure at which this transition occurs is obtained to be 14 GPa. This is not in agreement with experiment where such a transition is not observed. The inaccurate description of the HP characteristics and also some of the ground state properties of hematite related to the electronic structure in the framework of conventional DFT can be attributed to the improper description of the on-site Coulomb repulsion between the localized Fe 3d electrons. The DFT+U method takes into account this Coulomb repulsion directly by including a Hubbard-like term U into the density functional. Depending on the magnitude of this parameter, it is possible to obtain very accurate results for magnetic moments, band gap, structural parameters and electronic properties of hematite (Rollmann, Rohrbach, Entel and Hafner, 2004). In Fig. 3 the volume dependence of the magnetic moments and the energy per atom for DFT+U with U = 2 ev and U = 4 ev is shown. Note that U = 1 ev corresponds to the DFT limit, as in this case the difference U J vanishes, because we kept J at a constant value of 1 ev. Zero energy corresponds to the global energy minima of the AF solutions. The curves for U = 4 ev have been shifted upwards in energy by 100 mev for better readability of the diagram. We note that the collapse of the magnetic moments is shifted towards lower volumes with increasing U. For a value of U = 4 ev, it has disappeared from the volume range considered. The FM state has its energy always above the AF state, so we do not observe a phase transition as in the pure DFT calculations. This becomes even more obvious when we take a look at the dependence of an important structural parameter, the c/a ratio, on pressure. From Fig. 4 we see that the existence of the transition due to the collapse of the magnetic moments in ordinary DFT is indeed related to a structural phase transition. While the experimental behavior is recovered in DFT+U calculations for lower pressures, the calculated c/a values deviate considerably from experiment at higher pressures. But already for a value of U = 2 ev, as we saw before, the transition is shifted to lower volumes and higher pressures. For U = 4 ev, the transition has vanished, and nearly perfect agreement with the experimental dependence of c/a on pressure is achieved. Not only at high pressures does DFT+U lead to an improvement but also we note that at zero pressure the calculated c/a ratio is closer to experiment 5

6 than the value obtained by using pure DFT. This can be confirmed by taking a look at Table I, where we have compared structural ground state quantities with experimental values. The already quite good agreement of DFT values becomes even better in the framework of DFT+U. Only the equilibrium volume of Å 3 per atom is slightly large, but this is only because the error cancellation is not so pronounced here. The cell dimensions themselves are much closer to experiment than in pure DFT. The same holds true for the interatomic distances, where the deviations are smaller than 0.6 %. Regarding the electronic and magnetic properties, we observe that for U = 4 ev the magnetic moments of the Fe atoms are 4.0 µ B in magnitude, only about 0.9 µ B lower than the experimental result. No considerable improvement can be obtained by increasing the value of U, which instead would lead to the disagreements of the calculated band structure properties with available experimental PES data, as was shown by Rollmann et al. (Rollmann, Rohrbach, Entel and Hafner, 2004). 4. CONCLUSION In this work we have presented results from first-principles calculations regarding structural, magnetic and electronic properties of corundum-type α- Fe 2 O 3. Using pure DFT in the GGA, we were able to reproduce ground state structural parameters as well as some basic features related to electronic and magnetic structure, like insulating, AF ground state, qualitatively. But corresponding magnetic moments and band gap turn out to be too small. Furthermore, a structural phase transition coupled to a HS LS insulator metal transition at 14 GPa is predicted by pure DFT which is not seen in experiment. This can be related to an improper description of the on-site Coulomb repulsion in conventional DFT. It is observed that an introduction of a Hubbard like term U into the calculations changes the high-pressure behavior of magnetic moments and c/a ratio. The phase transition in this case is shifted to lower volumes with increasing U, and disappears for U = 4 ev. At this level, nearly perfect agreement with experiment concerning the geometric structure of α-fe 2 O 3 is reached. In addition to that, the accuracy of ground state structural parameters of this material, which was already reasonable in the framework of pure DFT, can be further improved using DFT+U, resulting in deviations well below 1 %. We therefore consider DFT+U with a value of U = 4 ev as the appropriated method for calculating properties of hematite from first principles, because this method predicts and excellent agreement of the electronic structure with experimental observation. Acknowledgements This work has been supported by the German Science Foundation through 6

7 the SFB 445 Nano-Particles from the Gasphase: Formation, Structure, Properties. The calculations have been carried out at the Regional Computer Center of the University of Cologne (RRZK). References Badro, J., G. Fiquet, V. V. Struzhkin, M. Somayazulu et al. (2002). Nature of the high-pressure transition in Fe 2 O 3 hematite. Phys. Rev. Lett. 89, Blöchl, P. E. (1994). Projector augmented-wave method. Phys. Rev. B 50, Catti, M., G. Valerio and R. Dovesi (1995). Theoretical study of electronic, magnetic, and structural properties of α-fe 2 O 3 (hematite). Phys. Rev. B 51, Chow, H. and F. Keffer (1974). Soft surface magnons and the first-order magnetic phase transition in antiferromagnetic hematite. Phys. Rev. B 10, 243. Ciccacci, F., L. Braicovich, E. Puppin and E. Vescovo (1991). Empty electron states in Fe 2 O 3 by ultraviolet inverse-photoemission spectroscopy. Phys. Rev. B 44, Dudarev, S. L., G. A. Botton, S. Y. Savrasov, C. J. Humphreys et al. (1998). Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study. Phys. Rev. B 57, Finger, L. W. and R. M. Hazen (1980). Crystal structure and isothermal compression of Fe 2 O 3, Cr 2 O 3, and V 2 O 3 to 50 kbars. J. Appl. Phys. 51, Fujimori, A., M. Saeki, N. Kimizuka, M. Taniguchi et al. (1989). Photoemission satellites and electronic structure of Fe 2 O 3. Phys. Rev. B 34, Hohenberg, P. and W. Kohn (1964). Inhomogeneous electron gas. Phys. Rev. 136, B864. Kohn, W. and L. J. Sham (1965). Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133. Kresse, G. and J. Furthmüller (1996). Efficient iterative schemes for ab initio total-energy calculations using a plane wave basis set. Phys. Rev. B 54, Kresse, G. and D. Joubert (1999). From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, Lad, R. J. and V. E. Henrich (1989). Photoemission study of the valence-band electronic structure in Fe x O, Fe 3 O 4, and α-fe 2 O 3 single crystals. Phys. Rev. B 39, Levinson, L. M. (1971). Temperature dependence of the weak ferromagnetic moment of hematite. Phys. Rev. B 3,

8 Liechtenstein, A. I., V. I. Anisimov and J. Zaanen (1995). Density-functional theory and strong interactions: Orbital ordering in mott-hubbard insulators. Phys. Rev. B 52, R5467. Murnaghan, F. D. (1944). The compressibility of media under extreme pressures. Proc. Natl. Acad. Sci. U.S.A. 30, 244. Olsen, J. S., C. S. G. Cousins, L. Gerward, H. Jhans et al. (1991). A study of the crystal structure of Fe 2 O 3 in the pressure range up to 65 GPa using synchrotron radiation. Phys. Scr. 43, 327. Pasternak, M. P., G. K. Rozenberg, G. Y. Machavariani, O. Naaman et al. (1999). Breakdown of the mott-hubbard state in Fe 2 O 3 : A first-order insulator-metal transition with collapse of magnetism at 50 GPa. Phys. Rev. Lett 82, Perdew, J. P. and Y. Wang (1992). Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 45, Punkkinen, M. P. J., K. Kokko, W. Hergert and I. J. Väyrynen (1999). Fe 2 O 3 within the LSDA + U approach. J. Phys.: Condens. Matter 11, Rollmann, G., A. Rohrbach, P. Entel and J. Hafner (2004). First-principles calculation of the structure and magnetic phases of hematite. Phys. Rev. B 69, Rozenberg, G. K., L. S. Dubrovinsky, M. P. Pasternak, O. Naaman et al. (2002). High-pressure structural studies of hematite Fe 2 O 3. Phys. Rev. B 65, Sandratskii, L. M. and J. Kübler (1996). First-principles LSDF study of weak ferromagnetism in Fe 2 O 3. Europhys. Lett. 33, 447. Sandratskii, L. M., M. Uhl and J. Kübler (1996). Band theory for electronic and magnetic properties of α-fe 2 O 3. J. Phys.: Condens. Matter 8, 983. Sato, Y. and S. Akimoto (1979). Hydrostatic compression of four corundum-type compounds: α-al 2 O 3, V 2 O 3, Cr 2 O 3 and α-fe 2 O 3. J. Appl. Phys. 50, Searle, C. W. and G. W. Dean (1970). Temperature and field dependence of the weak ferromagnetic moment of hematite. Phys. Rev. B 1, Solovyev, I. V., P. H. Dederichs and V. I. Anisimov (1994). Corrected atomic limit in the local-density approximation and the electronic structure of d impurities in Rb. Phys. Rev. B 50, Uozumi, T., K. Okada, A. Kotani, R. Zimmermann et al. (1997). Theoretical and experimental studies on the electronic structure of M 2 O 3 (M = Ti, V, Cr, Mn, Fe) compounds by systematic analysis of high-energy spectroscopy. J. Electron Spec. and Rel. Phen. 83, 9. Zaanen, J., G. A. Sawatzky and J. W. Allen (1985). Band gaps and electronic structure of transition-metal compounds. Phys. Rev. Lett. 55,

9 TABLES Table I: Ground state structural parameters for α Fe 2 O 3 calculated in the framework of pure DFT and DFT+U with U = 4 ev in comparison with experimental values obtained by Finger et al. (Finger and Hazen, 1980). x O and z Fe denote the two internal degrees of freedom for the positions of the atoms inside the primitive unit cell, c and a are the dimensions of the hexagonal unit cell. Interatomic distances are given for nearest neighbor Fe Fe and Fe O pairs. 9

10 FIGURE CAPTIONS Figure 1: Hexagonal unit cell of rhombohedral α-fe 2 O 3. The Fe atoms (blue) are arranged in a bilayer structure and coordinated octahedrally by six O atoms (red) which form close packed basal planes. Figure 2: Total energy and magnetic moments of the Fe atoms for the different magnetic states. The dashed line indicates the experimental volume. The solid line corresponds to the transition at a pressure of 14 GPa. Figure 3: Energy (relative to some arbitrary energy zero) and magnetic moments for the AF and the FM states for DFT+U with U = 2 ev and U = 4 ev. The energy curves for U = 2 ev are shifted towards higher energies by 100 mev for better readability. Figure 4: Dependence of the c/a ratio on pressure for pure DFT and DFT+U with U = 2 ev and U = 4 ev. 10

11 Table 1: G. Rollmann et al. z Fe x O c a Fe Fe Fe O c/a Ref Exp DFT DFT+U 11

12 Figure 1: G. Rollmann et al. 12

13 Moment (µ Β / Fe atom) Energy (mev / atom) NM AF FM Volume / atom (Å 3 ) Figure 2: G. Rollmann et al. 13

14 Moment (µ Β / Fe atom) AF U = 4 ev FM U = 4 ev FM U = 2 ev AF U = 2 ev Energy (mev / atom) Volume / atom (Å 3 ) Figure 3: G. Rollmann et al. 14

15 2.9 c/a ratio pure GGA U = 4 ev U = 2 ev Exp. (Ref. 3) Pressure (GPa) Figure 4: G. Rollmann et al. 15