Electronic structure, magnetism, and optical properties of Fe2SiO4 ...

PHYSICAL REVIEW B 69, 155108 共2004兲

Electronic structure, magnetism, and optical properties of Fe2 SiO4 fayalite at ambient and high pressures: A GGA¿U study Xuefan Jiang1,2,* and G. Y. Guo1,3,† 1

Department of Physics, National Taiwan University, Taipei 106, Taiwan 2 Department of Physics, Changshu Institute of Technology, Changshu 215500, China 3 National Synchrotron Radiation Research Center, Hsinchu 300, Taiwan 共Received 25 October 2003; published 15 April 2004兲 The electronic structure, magnetism, and optical properties of Fe2 SiO4 fayalite, the iron-rich end member of the olivine-type silicate, one of the most abundant minerals in Earth’s upper mantle, have been studied by density-functional theory within the generalized gradient approximation 共GGA兲 with the on-site Coulomb energy U⫽4.5 eV taken into account (GGA⫹U). The stable insulating antiferromagnetic solution with an energy gap ⬃1.49 eV and a spin magnetic moment of 3.65␮ B and an orbital magnetic moment of 0.044␮ B per iron atom is obtained. It is found that the gap opening in this fayalite results mainly from the strong on-site Coulomb interaction on the iron atoms. In this band structure, the top of valence bands consists mainly of the 3d orbitals of Fe2 atoms, and the bottom of the conduction bands is mainly composed of the 3d orbitals of Fe1 atoms. Therefore, since the electronic transition from the Fe2 3d to Fe1 3d states is weak, significant electronic transitions would appear only about 1 eV above the absorption edge when Fe-O orbitals are involved in the final states. In addition, our band-structure calculations can explain the observed phenomena including redshift near the absorption edge and the decrease of the electrical resistivity of Fe2 SiO4 upon compression. The calculated Fe p partial density of states agree well with Fe K-edge x-ray absorption spectrum. The calculated lattice constants and atomic coordinates for Fe2 SiO4 fayalite in orthorhombic structure are in good agreement with experiments. DOI: 10.1103/PhysRevB.69.155108

PACS number共s兲: 71.15.Mb, 75.30.Et, 91.60.Ed

I. INTRODUCTION

The properties of olivine are of considerable interest in geophysics and crystal chemistry, because this mineral is thought to be predominant throughout the Earth’s mantle. Fayalite, Fe2 SiO4 , is the iron-rich end member of the olivine-type silicate, one of the most abundant minerals in Earth’s upper mantle. Fe2 SiO4 is a fairly rare constituent of certain acid alkaline volcanos and granites, and of metamorphic iron formations. However, it is important for being the ferrous component of the Mg-Fe-olivine solid solution series. The olivine-type silicate Fe2 SiO4 has an orthorhombic crystal structure 共Fig. 1兲 with the space group Pnma, in which four formula units are contained in the unit cell.1 In the crystal, Si atoms are coordinated with four O atoms to form SiO4 tetrahedra, as in other silicate minerals. The SiO4 tetrahedra are linked together by the Fe atoms lying between them. The Fe atoms are surrounded by six O atoms. There are two crystallographically inequivalent Fe sites, where Fe1 (4a) ions are sites of inversion symmetry and Fe2 (4c) ions are in the plane of mirror symmetry. The magnetic structure of Fe2 SiO4 fayalite determined by Mo¨ssbouer spectroscopy and neutron-diffraction experiments on single-crystal samples is quite complex and it is reported to be a noncoplanar antiferromagnetic order below a Ne´el temperature of ⬃65 K 共Ref. 2兲共see Fig. 2兲. The magnetic moments at Fe2 sites are either parallel or perpendicular to the b axis, satisfying the symmetry requirement at this site. The Fe1 moments, on the other hand, are parallel to none of three principal axes and their directions show a strong temperature 0163-1829/2004/69共15兲/155108共6兲/$22.50

dependence.2 These behaviors could be stemmed from the strong correlations among the crystal structure and the electronic or magnetic properties. The combination of the competing single-ion anisotropy of different iron sites, exchange interactions among a number of neighboring ions, and Dzyaloshinsky-Moriya interactions might be the origin of the observed magnetocrystalline anisotropy.3,4 At room temperature and pressure Fe2 SiO4 fayalite is a Mott-Hubbard type insulator.5–7 In addition, the effect of pressure on Fe2 SiO4 is another interesting subject, and its high-pressure phases are thought to be predominant throughout the Earth’s mantle.7 Mao and Bell6 found that the optical and electrical properties of fayality undergo marked changes upon compression at

FIG. 1. The structure of Fe2 SiO4 fayalite. The frame represents the orthorhombic unit cell. Large dark balls are Fe, small dark balls are O, light balls are Si.

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XUEFAN JIANG AND G. Y. GUO

FIG. 2. Projection of the magnetic structure on both the Fe1 and Fe2 sites.

temperature of 300 K. A major absorption edge shifts from ultraviolet energies into the near infrared at pressures above 15–18 GPa.7 The electrical resistivity decreases by over five orders of magnitude between 0 and 30 GPa.7 The structural and physical properties of fayalite are of fundamental interest in geophysics and crystal chemistry. Recently, the electronic structure has been studied by experimental investigations for better understanding of the properties.7 The application of ab initio techniques to the study of systems of geophysical interest has expanded considerably and even detailed studies of the thermodynamical properties of some systems are available.8 –11 Quite recently, Cococcioni et al.12 determined the structural, electronic, and magnetic properties of Fe2 SiO4 fayalite by using the densityfunctional theory 共DFT兲 in a plane-wave pseudopotential framework comparing results obtained with local-density approximation 共LDA兲 and generalized gradient approximation 共GGA兲.13 However, their electronic structure calculations within the GGA predicted a metallic ground state, contrary to experimental evidence that indicates an insulating, possibly Mott-Hubbard, behavior at ambient pressure and temperature.12 It is noted that in DFT calculations, the electronic structure of a system with strong electron correlation, such as transition-metal oxides, is not well described. The LDA or GGA often fails to describe systems with strongly correlated d and f electrons. In some cases this can be remedied by introducing a strong intra-atomic 共on-site兲 Coulomb interaction in a Hartree-Fock-like manner, as a replacement of the LDA 共or GGA兲 on-site energy. This approach is commonly known as the LDA⫹U or GGA⫹U method.14 These schemes, yielding quite satisfactory results for a few strongly correlated systems, are considered to be useful approaches. The purpose of the present investigation is to provide a more accurate theoretical understanding of the electronic, magnetic, and optical properties of the Fe2 SiO4 fayalite based on the GGA⫹U calculations. II. THEORY AND COMPUTATIONAL DETAILS

The electronic structure, magnetism, and optical properties of Fe2 SiO4 fayalite are calculated by using the fullpotential projected augmented-wave method15 as implemented in the Vienna ab initio simulation package.16,17 The calculations are based on first-principles DFT with GGA. In order to take the on-site Coulomb interactions into account,

the values of U⫽4.5 eV 共Ref. 12兲 and J⫽0.90 eV 共Ref. 14兲 for Fe2⫹ in Fe2 SiO4 are used in the following spin-polarized GGA⫹U calculations. The simplified Dudarev’s scheme 共Ref. 18兲 for the double counting energy is used. The density of state 共DOS兲 and partial DOS were calculated by the linear tetrahedron method 共Ref. 19兲. The plane-wave cutoff energy of 400 eV and the convergence criteria for energy of 10⫺4 eV are selected. The Monkhorst-pack k-point generation scheme is used with a grid of 4⫻6⫻8 points in the irreducible Brillouin zone 共IBZ兲 for an orthorhombic cell, which means that the number of the k points in the IBZ used in self-consistent calculation is 24. The present expression for the internal structural degrees of freedom for orthorhombic Fe2 SiO4 is in terms of the Wyckoff notation.1 For the magnetic structure, Fe2 SiO4 fayalite is known to be an antiferromagnetic compound with slightly noncollinear arrangement of spin on Fe1 iron site, but this noncollinearity will not be addressed in present calculations for simplicity. In the spin configuration the antiferromagnetic interaction between corner-sharing octahedra is selected 共see Fig. 2兲, because this magnetic structure is consistent with an iron-iron magnetic interaction via a superexchange mechanism through oxygen p orbitals and favorable to lowering energy per unit cell to be a stable structure.12 We first performed an optimization of the geometry of the lattice and internal structural parameters within both the GGA and GGA⫹U schemes. Experimental and theoretical lattice constants and atomic coordinates of Fe2 SiO4 in orthorhombic structure are listed in Table I. A detailed comparison between calculation and experiment shows an overall good agreement between theory and experiment, but GGA⫹U results are in closer agreement with experimental results. Therefore, we calculated the self-consistent electronic structure at optimal lattice constants and internal structural parameters from the GGA⫹U calculation in present work below. Optical properties of Fe2 SiO4 fayalite can be calculated from our electronic band structures with independent electron approximation. Within the one-electron picture, the imaginary part of the complex dielectric function, ⑀ 2 ( ␻ ), was calculated from20

⑀ 2共 ␻ 兲 ⫽

4␲2

␻ 2V

兩具 c 兩 eជ •pជ 兩 v 典兩 2 ␦ 共 E c ⫺E v ⫺ ␻ 兲 , 兺kជ W kជ 兺 c, v

共1兲

where V is the cell volume, ␻ the photon energy, eជ the polarization direction of the photon, and pជ the electron momentum operator. The integral over the k space has been replaced by a summation over special kជ points with corresponding weighting factors W kជ . The momentum matrix elements are evaluated at the same special kជ points as used in the calculation of the DOS. The second summation includes the valence-band states ( v ) and conduction-band states (c), and the subscript E is the corresponding band energy. Then the real part of the dielectric function, ⑀ 1 ( ␻ ), is obtained by the Kramers-Kronig relation. From the complex dielectric function, ⑀ ( ␻ )⫽ ⑀ 1 ( ␻ )⫹i ⑀ 2 ( ␻ ), the linear refractive index reads

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ELECTRONIC STRUCTURE, MAGNETISM, AND . . .

TABLE I. Experimental 共Ref. 1兲 and theoretical lattice constants and atomic coordinates of Fe2 SiO4 in orthorhombic structure. * denotes experimental lattice constants at 25.4 GPa pressure 共Ref. 7兲. GGA ambient pressure

GGA⫹U ambient pressure

GGA⫹U 25.4 GPa pressure

10.488 6.095 4.828 0.7800 0.5147 0.5975 0.0708 0.5929 0.7313 0.9530 0.2923 0.1637 0.0384 0.2885

10.337 (⫺1.44%) 5.947 (⫺0.79%) 4.956 共2.65%兲 0.7754 (⫺0.59%) 0.5148 共0.02%兲 0.5964 (⫺0.18%) 0.0675 (⫺4.66%) 0.5963 共0.57%兲 0.7380 共0.92%兲 0.9476 (⫺0.57%) 0.3005 共2.81%兲 0.1661 共1.28%兲 0.0305 (⫺20.57%) 0.2845 (⫺1.39%)

10.354 (⫺1.28%) 6.115 共0.33%兲 4.878 共1.04%兲 0.7785 (⫺0.19%) 0.5162 共0.29%兲 0.5972 (⫺0.05%) 0.0700 (⫺1.13%) 0.5921 (⫺0.13%) 0.7333 共0.27%兲 0.9499 (⫺0.33%) 0.2904 (⫺0.65%) 0.1669 共1.95%兲 0.0339 (⫺9.11%) 0.2876 (⫺0.31%)

9.782* 5.832* 4.685* 0.7759 0.5217 0.5985 0.0740 0.5881 0.7335 0.9444 0.2877 0.1720 0.0301 0.2774

共Å兲共Å兲共Å兲 u v u v u v u v x y z

a b c Fe2 Si O1 O2 O3

n共 ␻ 兲⫽

Expt. ambient pressure

冉

冑⑀ 21 共 ␻ 兲 ⫹ ⑀ 22 共 ␻ 兲 ⫹ ⑀ 1 共 ␻ 兲 2

冊

1/2

,

共2兲

and the linear absorption coefficient is related to ⑀ 2 by ␣ ⫽ ⑀ 2 ␻ /(nc), where c is the velocity of light in the vacuum. The electron energy loss function is given by Im(⫺1/⑀ ). III. RESULTS AND DISCUSSION

As a test, we have performed GGA calculations for Fe2 SiO4 in the orthorhombic structure. Our GGA band structure 共not shown here兲 for the antiferromagnetic 共AF兲 state is essentially the same as the previous result.12 It results in a metallic ground state, contrary to experimental insulating behavior. Therefore, in the rest of this work, only the GGA ⫹U is used because GGA⫹U gives more reasonable results for Fe2 SiO4 . In Fig. 3共a兲 we present the band structure of Fe2 SiO4 obtained by the GGA⫹U calculations. The corresponding DOS and partial DOS are given in Fig. 3共b兲. Contrary to the GGA result12 that predicted a metallic ground state, an indirect gap of ⬃1.49 eV is opened between Z and ⌫ points 关see Fig. 3共a兲兴. This gap is still underestimated in comparison with the experimental value of about 34 000 cm⫺1 共or 4.22 eV兲 from optical absorption spectra.22–24 Further GGA⫹U band-structure calculations using several different values of the on-site Coulomb U were performed21 and the calculated band gap 共3.74 eV兲 is still smaller than the optical band gap22–24 even when a large U of 16.5 eV was used. This suggests that even the GGA⫹U approach might not be fully adequate for describing the optical excitations in Fe2 SiO4 . In the rest of this paper, we only concentrate on the results obtained using U⫽4.5 eV since this value was regarded as the realistic value for Fe2 SiO4 . 12 Note also a very small dispersion of the topmost valence band in the ⌫Z direction parallel to the c axis but rather flat along other directions perpendicular to the c axis in the ⌫ XSY plane. The

experimental band-edge absorption22 is not so distinct, because an absorption threshold exhibits a tail into the optical gap region, it seems that the experimental results support our findings that Fe2 SiO4 is an indirect gap material. The GGA ⫹U results lead us to the conclusion that the electronic correlations of the Hubbard type are important for the proper gap opening in the Fe2 SiO4 fayalite. In Fig. 3共a兲 we identify three groups of bands. Below the band gap we find two narrow valence bands of only ⬃0.12 eV and ⬃0.32 eV widths, respectively, which are separated from the broad lower valence bands by an additional gap of ⬃0.70 eV. From projected DOS, besides that the Si-projected DOS component in the valence-band energy range is very small, the O-atom orbital weight is concentrated mainly in region of energy ⬃1.36 eV below the top of valence bands, so that Fe(3d)-O(2p) hybridization effects should appear mainly in the lower valence-band energy range. This so-called split-off band comprises four single bands, in which two higher-energy and two lower-energy bands originate mainly from the Fe2(3d) and the Fe1(3d) orbitals, respectively. The broad lower-energy valence bands starting from ⬃1.36 eV below the top of valence bands are constituted from the Fe(3d)-O(2p) hybridization. The conduction bands originate mainly from the Fe(3d) orbitals, where the lower energy conduction bands below about 2.5 eV are attributed mainly to the Fe1(t 2g ) orbitals. All other states that are not included in Figs. 3 play only a negligible role in the given energy interval. Besides the marked flatness of both the top four valence and the lowest conduction bands along 关001兴 (⌫Z) and 关100兴 (⌫X,SY ,ZU,R⌫) directions, the feature of the electronic band structure is characterized by having the relative large dispersion along the 关010兴 direction (⌫Y ,XS,ZT,UR). This result is qualitatively consistent with that from the previous GGA calculation.12 It is concluded that the electronic states are rather localized in the a and c directions and mainly

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FIG. 4. Electronic band structure of Fe2 SiO4 at pressure of 25.4 GPa.

FIG. 3. 共a兲 Electronic band structure of Fe2 SiO4 in the AF state. The zero energy denotes the top of valence bands. 共b兲 Total and partial densities of states 共DOS兲 for the AF state.

extend along the zigzag chains in the b direction, implying a poor quasi-one-dimensional character in the Fe2 SiO4 fayalite. The effect of pressure on the electrical resistivity of Fe2 SiO4 is an interesting problem. The experimental measurements indicate that the electrical resistivity of Fe2 SiO4 fayalite exhibits a rapid decrease on compression.6,7 A redshift in the energy of a strong absorption edge upon compression is observed.6,7 At pressures higher than ⬃39 GPaFe2 SiO4 becomes amorphous.7 To examine the band structure of Fe2 SiO4 at high pressure a set of the structural parameters, namely, a⫽9.782 Å, b⫽5.823 Å, and c ⫽4.685 Å at 25.4 GPa from x-ray diffraction measurement by Williams et al.7 are selected in our calculations below. In order to determine the corresponding internal structural degrees of freedom we first carry out the stable structural calculations through relaxing the ions based on the crystal lattice parameters at 25.4 GPa shown above. Figure 4 shows the calculated electronic energy bands at a pressure of 25.4 GPa. Comparing with the results at ambient pressure in Fig. 3共a兲 we find that the gap becomes narrower, being equal to 0.92 eV. This is consistent with the experimental values of ⬃8000 cm⫺1 共0.99 eV兲共Ref. 6兲 and ⬃9000 cm⫺1 共1.12 eV兲,7 and the Fe(3d) bands become broader at high pressure.

These features can well explain the experimental phenomena at high pressures as mentioned above. The decrease of the energy gap results in the redshift in the energy of a strong absorption edge and the decrease of the electrical resistivity of Fe2 SiO4 upon compression. Note that given the fact that the calculated band gap is nearly three times smaller than the measured optical gap at the ambient pressure, as mentioned above, this good quantitative agreement between the theoretical and experimental band gaps under high pressure may be fortuitous. The calculated spin and orbital magnetic moments from the GGA⫹U for Fe2 SiO4 are listed in Table II. Note that our GGA results give the spin magnetic moments of 3.40␮ B /atom and 3.61␮ B /atom for Fe1 and Fe2, respectively, slightly smaller than previous one of 3.8␮ B /atom for iron atom.12 Within the GGA⫹U, the spin moments of 3.650␮ B and 3.653␮ B per iron atom are also found slightly smaller than the experimental values of 4.40␮ B and 4.41␮ B for Fe1 and Fe2 at a temperature of about 10 K,4 respectively. This experimental value is also higher than the previous theoretical spin moment of 3.8␮ B , 12 and the discrepancy was attributed to the residual orbital contribution. However, the present GGA⫹U calculation gives a rather small orbital moment of 0.044␮ B and 0.024␮ B per Fe1 and Fe2 atoms, respectively. In spite of the same sign of the orbital magnetic moment of Fe 3d orbital as the spin moment, indicating that the 3d orbital is over half filled in accordance with Hund’s rule, our calculated values for the magnetic moment are still slightly smaller than the experimental measurements. In fact, for Fe2 SiO4 , in the fourfold Fe1 position the orientations of the moments are canted away from the corresponding crystallographic axes for T→0 K 共Ref. 7兲 as seen in Fig. 2. The TABLE II. Spin and orbital magnetic moments ( ␮ B /atom) of iron atoms from the GGA⫹U calculation. Spin Fe2 SiO4 Ambient pressure 25.4 GPa pressure

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Fe1 3.650 3.595

Orbital Fe2 3.653 3.590

Fe1 0.044 0.039

Fe2 0.024 0.024


ELECTRONIC STRUCTURE, MAGNETISM, AND . . .

FIG. 5. The calculated 共a兲 imaginary part of the dielectric function ( ⑀ 2 ), 共b兲 refractive index (n), 共c兲 absorption coefficient ( ␣ ), and 共d兲 electron energy loss function 关 Im(⫺1/⑀ ) 兴 of Fe2 SiO4 fayalite. The labels a 共solid lines兲, b 共dashed lines兲, and c 共dotted lines兲 represent the polarization directions of the photon along the crystal axes a, b, and c, respectively.

magnetic structure of Fe2 SiO4 is complex, so that for the magnitudes and directions of the magnetic moments the difference between calculation and experiment might result from the fact that the simple collinear magnetic structures were considered in the present and previous calculations. For Si 3s and O 2p orbitals, the orbital moments are negligibly small. In addition, it is found that the magnetic moment decreases with pressure as seen in Table II. The theoretical calculation of optical spectra is also an interesting problem, which provides us fruitful information on the electronic structure. The calculated spectrum of the imaginary part of the dielectric function is shown in Fig. 5共a兲. Figures 5共b兲–5共d兲 show the calculated spectra of refractive index (n), absorption coefficient ( ␣ ), and electron energy loss function 关 Im(⫺1/⑀ ) 兴 , respectively, of Fe2 SiO4 in the region of 1.49–30.0 eV for the polarization parallel to the a axis 共solid lines兲, b axis 共dashed lines兲, and c axis 共dotted lines兲. The spectra of ␣ in the expanded scale of 1.0– 6.0 eV are inserted in Fig. 5共c兲. We can see that the optical spectra of Fe2 SiO4 exhibit a remarkable anisotropy. The calculated absorption spectra consist of a shoulder in the energy range from about 7 to 11 eV, and a set of strong peaks after about 12 eV. From the partial DOS as shown in Fig. 3共b兲, it turns out that the top of valence bands are built up of Fe2(3d) states and the bottom of conduction bands mainly of Fe1(3d) states. Thus, the lowest electronic transition is essentially of dipole forbidden between the absorption edge or energy gap at 1.49 eV and around 1.9 eV. However, the transitions of 关occupied Fe(3d)]→关unoccupied Fe(3d)] states would be partially allowed through hybridization leading to the weak absorption band structures in the region of 2.0– 4.0 eV. The transitions of O(2p)→Fe(3d) partially take place above 4 eV to result in the intense charge-transfer absorption band in which the oscillator strength for the transitions from the valence band is not completely exhausted around 25 eV. The remainder of the oscillator strength would

FIG. 6. Experimental XANES spectrum of Fe K edge at ambient pressure 共squares兲共Ref. 24兲, compared with the Fe p-DOS at ambient 共circles兲共AP兲 and high 共triangles兲 pressures 共HP兲.

reside in the transition into higher conduction state, i.e., the 3d orbital of cations. The spectra of Im(⫺1/⑀ ) exhibit the intense peak around 23 eV, which is due to the bulk plasmon excitation. The early optical absorption spectra6,23,24 of Fe2 SiO4 were usually limited to the region of low energy below the intense charge-transfer absorption edge at about 34 000 cm⫺1 . 23,24 Note that the absorption bands within this energy region are most likely attributed to excitonic transitions. Unfortunately, current first-principle methods cannot calculate the excitation of excitons. Thus, no corresponding experimental optical spectra can be straightforward compared with present calculated results, for which further optical measurements will be needed. X-ray absorption near-edge spectra 共XANES兲 from a solid provide valuable information on the unoccupied part of the electronic states. For the Fe K edge in Fe2 SiO4 fayalite, XANES essentially probe the Fe p-DOS.25 Therefore, in Fig. 6, we plot the experimental 共squares, labeled Exp兲共Ref. 26兲 XANES spectrum of Fe K edge at ambient pressure and theoretical Fe p-DOS’s at ambient 共circles, labeled AP兲 and high 共triangles; labeled HP兲 pressures. The Fe p-DOS’s have been Gaussian broadened with a 1.0 eV broadening width. From Fig. 6, the Fe p-DOS spectra produce a sharp peak A and two broad peaks B and C, while the high-pressure spectrum shows two better defined peaks B and C in the higher-energy region. Clearly, the maxima of the XANES spectrum in Fig. 6 coincide with the peaks in the Fe p-DOS curves. Overall, we have a reasonable good agreement between theory and experiment. All major features of the absorption peaks and intensities are well reproduced. This suggests that our calculated electronic band structures are reasonable. IV. CONCLUSIONS

In summary, the results of the GGA⫹U band-structure calculations for the Fe2 SiO4 fayalite predict antiferromagnetic indirect insulating band characteristics with an energy

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gap of ⬃1.49 eV and a spin magnetic moment of 3.65␮ B and a small orbital magnetic moment of 0.044␮ B per iron atom. Both the top four valence bands and the lowest conduction bands are mainly constituted by iron atom’s 3d orbitals, so that the strong iron on-site Coulomb interaction is the most important factor for gap opening in this mineral. The present optical spectra calculations indicate that the lowest electronic transition from the Fe2 3d to Fe1 3d states is quite weak. The small transition bands appear gradually only after 1.5 eV and this is followed by a set of intense chargetransfer absorption bands above 4 eV. The good agreement between Fe K edge XANES and the calculated Fe p-DOS

lends further support to the present GGA⫹U band structure for Fe2 SiO4 fayalite.

*Email address: [email protected]

14

†

Email address: [email protected] 1 R.W.G. Wyckoff, Crystal Structures, 2nd ed. 共Krieger, Florida, 1981兲, Vol. 3, Chap. VIII. 2 W. Lottermoser and H. Fuess, Phys. Status Solidi A 109, 589 共1988兲. 3 H. Kato, E. Untersteller, S. Hosoya, G. Kido, and W. Trentmann, J. Magn. Magn. Mater. 140-144, 1535 共1995兲. 4 H. Fuess, O. Ballet, and W. Lottermoser, in Structural and Magnetic Phase Transitions in Minerals, edited by S. Ghose, J.M.D. Coey, and E. Salje 共Springer-Verlag, Berlin, 1988兲. 5 I.G. Austin and N.F. Mott, Science 168, 71 共1970兲. 6 H.K. Mao and P.M. Bell, Science 176, 403 共1972兲. 7 Q. Williams, E. Knittle, R. Reichlin, S. Martin, and R. Jeanloz, J. Geophys. Res. 95, 21 549 共1990兲. 8 B.B. Karki, R.M. Wentzcovitch, S. de Gironcoli, and S. Baroni, Science 286, 1705 共1999兲. 9 D. Alfe`, M.J. Gillan, and G.D. Price, Nature 共London兲 401, 462 共1999兲; 405, 172 共2000兲. 10 A. Laio, S. Bernard, G.L. Chiarotti, S. Scandolo, and E. Tosatti, Science 287, 1027 共2000兲. 11 G. Steinle-Neumann, L. Stixrude, R.E. Cohen, and O. Gulseren, Nature 共London兲 413, 57 共2001兲. 12 M. Cococcioni, A. Dal Corso, and S. de Gironcoli, Phys. Rev. B 67, 094106 共2003兲. 13 J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 共1996兲.

ACKNOWLEDGMENTS

The authors are grateful to the support from the National Science Council of Taiwan 共NSC 91-2816-M-002-0009-6, NSC 92-2816-M-002-0006-6, NSC 92-2112-M-002-037兲. X.F.J. was supported in part by the Natural Science Foundation of Jiangsu Department of Education, China. X.F.J. also profited from discussions with K.C. Chu.

V.I. Anisimov, J. Zaanen, and O.K. Andersen, Phys. Rev. B 44, 943 共1991兲. 15 P.E. Blo¨chl, Phys. Rev. B 50, 17 953 共1994兲. 16 G. Kresse and J. Hafner, Phys. Rev. B 47, 558 共1993兲; 49, 14 251 共1994兲; G. Kresse and J. Furthmu¨ller, Comput. Mater. Sci. 6, 15 共1996兲. 17 G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 共1999兲. 18 S.L. Dudarev, G.A. Botton, S.Y. Savrasov, C.J. Humphreys, and A.P. Sutton, Phys. Rev. B 57, 1505 共1998兲. 19 P.E. Blochl, O. Jepsen, and O.K. Andersen, Phys. Rev. B 49, 16 223 共1994兲. 20 See, e.g., J. Li, C.G. Duan, Z.Q. Gu, and D.S. Wang, Phys. Rev. B 57, 6925 共1998兲. 21 The GGA⫹U band-structure calculations were also performed using many different values of the on-site Coulomb interaction U. The calculated band gaps 共eV兲 are 1.49,2.10,2.70,3.18, 3.58,3.66,3.74 eV, respectively, corresponding to the U values 共eV兲 of 4.5,6.5,8.5,10.5,12.5,14.5,16.5. 22 G. Calas, A. Mancean, J. Petiau, in Synchrotron Radiation Applications in Mineralogy and Petrology, edited by S. S. Augustithis 共Thephrastus, Athens, 1988兲, pp. 77–95. 23 H.G. Smith and K. Langer, Am. Mineral. 67, 343 共1982兲. 24 R.G. Burns, Am. Mineral. 55, 1608 共1970兲; W.A. Runciman, D. Sengupta, and J.T. Gourley, ibid. 58, 451 共1973兲. 25 See, e.g., G.Y. Guo, J. Phys.: Condens. Matter 8, L747 共1996兲; Phys. Rev. B 57, 10 295 共1998兲. 26 M. Wilke, F. Francois, D.E. Petit, G.E. Brown, Jr., and F. Martin, Am. Mineral. 86, 714 共2001兲.

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