under pressure Materials Science, Univesity of Minnesota, Minneapolis, Minnesota 55455, USA NY , USA

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1 Spin transition in (Mg,Fe)SiO 3 perovskite under pressure Koichiro Umemoto a, Renata M. Wentzcovitch a Yonggang G. Yu a Ryan Requist b,1 a Minnesota Supercomputing Institute and Department of Chemical Engineering Materials Science, Univesity of Minnesota, Minneapolis, Minnesota 55455, USA b Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY , USA 1 2 Key words: spin transition, ferrous iron, atomic and magnetic configurations, perovskite, lower mantle, first principles Corresponding author, umemoto@cems.umn.edu, fax: addresses: umemoto@cems.umn.edu (Koichiro Umemoto), wentzcov@cems.umn.edu (Renata M. Wentzcovitch), yonggang@cems.umn.edu (Yonggang G. Yu), Ryan.Requist@physik.uni-erlangen.de (Ryan Requist). 1 Present address: Theoretische Festkörperphysik, Universität Erlangen-Nürnberg, Staudtstrasse 7, Erlangen, Germany. Preprint submitted to Elsevier 29 September 2008

2 3 Abstract We present a density functional study of the pressure-induced spin transition in ferrous iron (Fe 2+ ) at the A site in MgSiO 3 perovskite. We address the influence of iron concentration and configuration (structural and magnetic), as well as technical issues such as the influence of the exchange correlation functional (LDA versus GGA) on the spin transition pressure. Supercells containing up to 160 atoms were adopted to tackle these issues. We show that there are preferred configurations for high-spin and low-spin iron and that the spin transition pressure depends strongly on iron concentration and all the issues above. Across the spin transition, irons move into the middle of distorted octahedra causing drastic changes in the d states configuration and a blueshift in the band gap. Such blueshift should decrease the contribution of ferrous iron to the electrical conductivity and increase its contribution to the radiative conductivity in the lower mantle. Both LDA and GGA results suggest that the spin transition can occur in the pressure range of the lower mantle and of previous experiments. The transition range can encompass the entire lower mantle passing through a mixed spin state caused by cation disorder and magnetic entropy. 2

3 21 1 Introduction The lower mantle of the Earth is believed to consist mainly of iron-bearing magnesium silicate perovskite, (Mg,Fe)SiO 3, with a little Al 2 O 3, and smaller amounts of (Mg,Fe)O and CaSiO 3. The presence of iron in perovskite affects several lower mantle properties: elastic and seismic properties (Kiefer et al., 2002; Li et al., 2005; Tsuchiya & Tsuchiya, 2006; Stackhouse et al., 2007), the post-perovskite transition pressure (Caracas & Cohen, 2005; Mao et al., 2005; Ono & Oganov, 2005; Stackhouse et al., 2006; Tateno et al., 2007) and electrical and thermal conductivities (Burns, 1993; Katsura et al., 1998; Xu et al., 1998; Badro et al., 2004), to mention a few. The spin state of iron in perovskite, which depends on its oxidation state, is a crucial factor in determining these properties. To date, there have been several studies to clarify what type of iron, ferrous (Fe 2+ ) and/or ferric (Fe 3+ ), and which site, A and/or the B, that are responsible for the spin transition at lower mantle pressures. However, there are some discrepancies between these studies An x-ray emission spectroscopy (XES) experiment by Badro et al. (2004) detected two distinct spin transitions, the first at about 70 GPa and the second at about 120 GPa. It was proposed that at 70 GPa the high-spin (HS) state transformed to a mixed-spin (MS) state and at 120 GPa all irons transformed to low-spin (LS). Infrared radiation was no longer absorbed after the transition to the LS state. Another XES experiment by Li et al. (2004) showed a gradual spin transition which was not completed at 100 GPa. It was suggested that in aluminum-free samples up to 100 GPa, all ferric irons acquired LS, while half of ferrous irons in the A site were in the intermediate-spin (IS) state. A gradual spin transition was also reported by a synchrotron Mössbauer spec- 3

4 46 47 troscopy experiment (Jackson et al., 2005) It was suggested that ferric iron was responsible for the spin transition which ended around 70 GPa The spin transitions in ferrous and ferric iron have been intensively investigated also theoretically (Cohen et al., 1997; Li et al., 2005; Hofmeister, 2006; Zhang & Oganov, 2006; Stackhouse et al., 2007; Bengtson et al., 2008). Calculations so far showed that ferric iron undergoes a spin transition at lower mantle pressures. The transition pressure was found to depend strongly on how ferric iron replaces Mg and Si, with two ferric irons or one ferric iron and one aluminum (Li et al., 2005; Zhang & Oganov, 2006; Stackhouse et al., 2007). This may be related to the observed gradual spin transition in ferric iron. The choice of the exchange-correlation functional used in these calculations, the local-density approximation (LDA) or the generalized-gradient approximation (GGA), affects strongly the transition pressure as well (Stackhouse et al., 2007; Bengtson et al., 2008). GGA studies in ferrous iron have insisted that the spin transition should not take place at lower mantle conditions while LDA studies show that the transition could take place within lower mantle pressures but at still rather high values (Hofmeister, 2006; Zhang & Oganov, 2006; Stackhouse et al., 2007; Bengtson et al., 2008). It is still hard to say that all of the properties of the spin transition in ferrous iron whether the transition is sharp or gradual, whether or not the transition occurs in the lower mantle, and what are the corresponding changes in the infrared absorption spectrum have been accounted for In the present first-principles study, we investigate further the spin states (HS, LS, and IS states) in ferrous iron in Mg 1 x Fe x SiO 3 perovskite. For ferrous iron, HS, LS, and IS states have magnetic moments per iron of 4µ B (5 majorityspin and 1 minority-spin electrons, S = 2), 0µ B (3 and 3, S = 0), and 2µ B (4 4

5 and 2, S = 1), respectively. The dependence of the spin transition pressure on iron concentration x, atomic configurations, and magnetic ordering is clarified. Both LDA and GGA results suggest that the spin transition through a MS state occurs within the pressure range of experiments and of the lower mantle. We also address the change of atomic and electronic structures across the spin transition, which should greatly affect thermal and electrical conductivities Computational method Calculations were performed using LDA and GGA (Perdew & Zunger, 1981; Perdew et al., 1996). The pseudopotentials for Fe, Si, and O were generated by Vanderbilt s method (Vanderbilt, 1990). The valence electronic configurations used are 3s 2 3p 6 3d 6.5 4s 1 4p 0, 3s 2 3p 1, and 2s 2 2p 4 for Fe, Si, and O. Core radii for all quantum numbers l are 1.8, 1.6, and 1.4 a.u. for Fe, Si and O. The pseudopotential for Mg was generated by von Barth-Car s method. Five configurations, 3s 2 3p 0, 3s 1 3p 1, 3s 1 3p 0.5 3d 0.5, 3s 1 3p 0.5, and 3s 1 3d 1 with decreasing weights 1.5, 0.6, 0.3, 0.3, and 0.2, respectively, were used. Core radii for all quantum numbers l are 2.5 a.u.. The plane-wave cutoff energy is 40 Ry. As described later, we used several atomic configurations with various supercell sizes and shapes. The k-point meshes used for the Brillouin zone sampling in each cell are fine enough to achieve convergence within 1mRy/iron in the total energy. They are described in the captions of figures showing atomic configurations. We used variable-cell-shape molecular dynamics (Wentzcovitch, 1991; Wentzcovitch et al., 1993) for structural optimization under arbitrary pressures. 5

6 95 3 Results and Discussion Effect of iron concentration First we investigate the effect of iron concentration on the spin transition. For each iron concentration from 6.25% (x = ) to 100% (x = 1), the smallest unit cell is taken (Fig. 1). Hereafter we refer to them as configuration 0 in order to distinguish them from other configurations which will be generated later. Only the FM state is considered for the HS state in configuration 0. Calculated HS-LS transition pressures are shown by blue points in Fig. 2. GGA gives higher transition pressures, by 50 GPa, than LDA for all iron concentrations. The transition pressure decreases with increasing iron concentration, being 13 GPa (62 GPa) by LDA (GGA) at x = 1. Below x = 0.25 the transition pressure is approximately constant. This trend agrees with Bengtson et al. (2008) but disagrees with Stackhouse et al. (2007), where the transition pressure in FeSiO 3 (calculated to be 284 GPa) was higher than in Mg Fe SiO 3. We have not found a spin transition to the IS state; the IS state always has higher enthalpy than the LS state in our calculations The HS-LS spin transition is accompanied by significant change in the atomic structure (Fig. 3). The size of ferrous iron decreases through the transition. This decrease destablizes iron in the high symmetry site (with z Fe = ±0.25) where iron sits in the HS state. Optimization from an initial geometry for the HS iron atom displaced brings it back to the symmetric position. In the HS state, irons are 8-fold coordinated in a bicapped trigonal prism. Across the spin transition, five bonds among eight shorten and the other three lengthen. In particular, one of the two longest Fe-O bonds in the HS state shortens from 6

7 Å to 1.830Å at 120 GPa and x = (LDA). This 16% decrease of the Fe-O bond is much larger than that expected from the volume reduction of just 0.4%. As a result, in the LS state, irons end up being 6-fold coordinated and in strongly distorted octahedra. The displacement of irons is crucial for the spin transition. Without the displacement, the spin transition does not occur at least up to 180 GPa for all iron concentrations. The displacement plays an important role in the electronic structure as discussed later in section Effect of atomic and magnetic ordering Next we study the effects of atomic and magnetic ordering on the spin transition. For x = 0.5, we prepare five atomic configurations (Fig. 4). The first three configurations have unit cells of 20 atoms, while configurations 4 and 5 are generated in the supercell. This is done to take into account special distributions of irons along the (110) plane of the perovskite structure, whose importance will be clarified later. FM configuration 1 is the same as HS configuration 0. The AFM states have an additional degree of freedom, the spin of iron, leading to two or three spin configurations for each atomic configuration. AFM 1 is stable in the atomic configurations 1, 2, and 5, while AFM 2 is stable in the atomic configurations 3 and 4. Enthalpies depend on atomic configurations and spin state and the dependence becomes stronger as pressure increases and iron-iron distances decrease (Fig. 5 (a) and (b)). At higher pressure the AFM states have lower enthalpies than the FM states, while at lower pressures the enthalpy difference between the FM and AFM states is very small. The spin transition pressure depends on the atomic configuration as shown in Fig. 5 (c). Within this set of configurations it varies by 7

8 GPa. Configurations 4 and 5 have the lowest transition pressures, 49 and 52 GPa. Because of its low enthalpy and transition pressure, configuration 4, with irons preferentially on the (110) plane, is important. From this we infer that similar cations prefer to exist in the same column For x = 0.125, which is close to the expected concentration in the lower mantle and those investigated experimentally, we consider fourteen atomic configurations shown in Fig. 6. For configurations 1 to 9, we use supercells with 80 atoms. Configurations 3, 4, 7, 8, and 9 correspond respectively to configurations referred to as SSM2, 3, 4, 1, and 5 in Stackhouse et al. (2007). Configurations 10, 11, and 12 are extensions of configurations 3, 4, and 5 for x = 0.5. In configurations 10 (1 1 4 supercell with 80 atoms) and 11 ( supercell with 160 atoms), irons are placed on (001) and (110) planes, respectively. In addition, we prepared configurations 13 (4 1 1 supercell with 80 atoms) and 14 (1 4 1 supercell with 80 atoms) in which irons are placed on (010) and (100) planes. Again, we can see the enthalpy dependence on atomic configurations for each spin state, the dependence becoming more accentuated with increasing pressure (Fig. 7 (a) and (b)). Configuration 11 has the lowest enthalpy for AFM-HS and LS states, whereas configuration 10 has the lowest for FM-HS at high pressure, but, its enthalpy is very close to that of configuration 11 FM. Hence, irons tend to order in the (110) plane like the case of x = 0.5. These results together indicate that the HS-LS transition pressure depends simultaneously on the cation and magnetic ordering (Fig. 7 (c)). Its dependence on the magnetic ordering is not so large though. For most atomic configurations, transition pressures from or to FM and AFM states differ by 5 GPa. For configurations 2, 7, and 12, the differences are larger but still GPa at most. Variation in the tranition pressures for 8

9 atomic configurations 1 9 is not so large either, 15 GPa, being consistent with the previous calculation on SSM1 5 configurations (Stackhouse et al., 2007). In contrast, configuration 11 AFM, in which irons are placed on the (110) plane as in configuration 4 of x = 0.5, has a LDA transition pressure as low as 56 GPa which is significantly smaller than those of other configurations. Therefore, for the HS-LS transition, the effect of cation ordering, i.e., elastic and chemical interactions, is much stronger than that of the magnetic ordering, i.e., the exchange interaction. We then generated equivalent AFM configurations containing irons in the (110) plane for other iron concentrations as well and calculated the spin transition pressures (red points in Fig. 2). It is clear that ordering of irons in the (110) plane greatly reduces the transition pressure at low iron concentration Electronic structure The electronic structure changes with the spin transition as shown in Fig. 8. Although LDA is known to underestimate the band gap by as much as 50%, it offers trends that are in good agreement with experiments. For configuration 0 in the HS state at x = and 120 GPa, there is a small band gap between the first and second peaks of the minority-spin d states of iron. At 0 GPa, this band gap vanishes and the band structure is metallic. On the other hand, as pressure increases, the band structure becomes nonmetallic and the band gap increases, since Fe-O bondlengths decrease and the crystal field splitting increases. From the spatial distributions of wave functions shown in Fig. 8, we see that iron d states do not simply split into lower e g and higher t 2g states as usually assumed (Li et al., 2004; Hofmeister, 2006). This is because the A site 9

10 does not have cubic symmetry. It is surrounded by eight oxygens in a bicapped trigonal prism configuration. In the LS state, as a result of the displacement of iron to the middle of distorted octahedra, the d states split into the lower occupied t 2g -derived and the higher empty e g -derived states. Then the band gap greatly increases, indicating a blueshift across the spin transition (Badro et al., 2004, 2005). This is opposite to the case of magnesiowüstite in which a redshift was reported (Tsuchiya et al., 2006; Goncharov et al., 2006; Keppler et al., 2007). The LDA band gaps for the HS and the LS states at 120 GPa are 0.3 ev and 1.75 ev, respectively In all other configurations, we observed iron displacement and blueshift in the band gap across the spin transition. Since in configuration 11 the distance between irons are smaller than in configuration 0, the electronic bands originated from iron d states are more dispersive. The band gaps (0.2 ev for the AFM state and 1.45 ev for the LS state) are slightly smaller than those of configuration 0. These values are lower bounds, because the LDA tends to underestimate the band gap. Therefore, in the absence of shallow or deep defect levels in the gap, iron-bearing perovskite in the LS state should not absorb near-infrared radiation and should be transparent to most of the blackbody radiation from the core at 2,500 K whose peak appears at 1.07 ev (1159 nm). It is consistent with the experimental observation that the radiation of the Nd:YAG laser with the wave length of 1064 nm could not heat (Mg 0.9,Fe 0.1 )SiO 3 in the LS state at 120 GPa (Badro et al., 2004). The blueshift across the spin transition should reduce the electrical conductivity due to thermally excited electrons and increase the radiative conductivity in the lower mantle. 10

11 Structural disorder and configurational entropic stabilization We have seen that configurations with irons close to each other in the (110) plane (configuration 4 with x = 0.5 and configuration 11 with x = 0.125) have low enthalpies and spin transition pressures. Irons prefer to order especially at high pressure and in the LS state. The extreme limit is the decomposition into pure phase: MgSiO 3 and FeSiO 3. In fact, static enthalpy calculation for x = indicates that the dissociated phases have lower enthalpy than any configuration generated so far (Fig. 9). But at high temperature the configurational entropy stabilizes Mg Fe SiO 3 against the dissociation products. For reference we may assume Mg Fe SiO 3 is an ideal solid solution of MgSiO 3 and FeSiO 3, the configurational entropy S conf per Mg Fe SiO 3 is given by S conf = k B (x log x+(1 x) log(1 x)) where k B is Boltzmann constant and x = The configurational entropy of the dissociated products is 0 (x = 0 and 1). The difference in Gibbs free energy between Mg Fe SiO 3 in configuration 0 and the dissociated products, G, G = H T S conf = H Mg0.875 Fe SiO 3 (0.875H MgSiO H FeSiO3 ) T S conf. (1) When G < 0, Mg Fe SiO 3 is energetically stable with respect to the dissociation. Fig. 9 shows that G is negative at low pressure and high temperature; at 26 GPa, configuration 0 should be stable with respect to the dissociation over 1,300 K by LDA ( 600 K by GGA). This is consistent with the synthesis of (Mg,Fe)SiO 3 perovskite with 10% iron concentration at 26 GPa and 2,000 K (e.g., Kudoh et al. (1990); Parise et al. (1990); Fei et al. (1994); Jackson et al. (2005)). Although at 300 K the dissociation products are energetically stable with respect to Mg Fe SiO 3 ( G > 0), 11

12 there has not been any experimental report of the dissociation, as far as we know. This could indicate that cation diffusion among the A sites in quenched samples is suppressed or not fast enough to induce the dissociation in experimental time scales at room temperature. Then, it may be assumed that in Mg Fe SiO 3 all atomic configurations appear at least locally at room temperature. According to this assumption, the spin transition should occur in each configuration with different transition pressures, giving rise to a MS state even at 0 K. It should be noted that this MS state is fundamentally different from the IS state; in the MS state, a fraction of the HS irons transforms to the LS state and both states exist simultaneously. Our calculations have not stabilized irons in the IS state. The transition pressure should be continuous between lower and higher bounds. The higher bound of the transition pressure is given by that of configuration 0, i.e., 98 (158) GPa by LDA (GGA). The lower bound should be that of configuration 11 or of FeSiO 3 : 56 or 16 GPa (116 or 76 GPa) by LDA (GGA), respectively. However, these configurations assume (110) plane segments and FeSiO 3 islands of infinite lengths. The transition pressures we calculated correspond to these limit lengths. These lengths are unlikely to occur in practice. Instead, it is appropriate to suppose that there are (110) iron plane segments and 3D FeSiO 3 islands of various lengths. Larger planes and islands should have lower transition pressures. Configuration 0 corresponds to the lower limits for the (110) plane segment and FeSiO 3 island lengths. Variation in plane segment and island sizes should also lead to variation in spin transition pressure. This argument suggests a gradual spin transition in ferrous iron in the A sites between 16 and 98 GPa (76 and 158 GPa) by LDA (GGA) if cation diffusion is not significant. Therefore, both LDA and GGA results indicate that the gradual spin transition in ferrous iron in the A site can occur in the pressure range of previous experiments and relevant 12

13 to the lower mantle. The gradual spin transition observed by Li et al. (2004) may be due to a transition between a MS state, due to random configurations, to the LS state, as opposed to a transition from the IS state, which has higher enthalpy than other states, to the LS state Fig. 9 indicates that G by LDA becomes positive beyond 40 GPa in the lower mantle, i.e., that Mg Fe SiO 3 with completely random iron distribution is no longer stable beyond 40 GPa. On the other hand, GGA indicates that Mg Fe SiO 3 with random cation distribution should be stable up to 120 GPa, i.e., almost in the entire lower mantle, except in the D layer. Reality may exist somewhere between LDA and GGA results. Beyond the dissociation pressure, diffusion could start at the high temperatures of the lower mantle. Although we do not know how fast diffusion occurs, it should not be fast enough to induce the complete dissociation since (Mg,Fe)SiO 3 is experimentally reported to be stable at lower mantle conditions (Serghiou et al., 1998). Cation diffusion should facilitate clustering of irons and increase the number of iron-(110) plane segments and FeSiO 3 islands. Sample annealing may enhance cation diffusion and clustering of irons at high pressure and, consequently, facilitate the spin transition. This could be related to the complete spin transition up to 120 GPa in the experiment with sample annealing by Badro et al. (2004). In the lower mantle, cation diffusion and iron clustering might be further faciliated at high temperature in geological time scale. Hence the spin transition pressure could be lower in the lower mantle than in laboratories, inferring a possible relationship between the spin transition with the blueshift in the band gap and magnetic satellite observations that the electric conductivity ceases to increase in the mantle at depths greater than 900 km ( 35 GPa) (Constable & Constable, 2004; Kuvshinov & Olsen, 13

14 ). Clustering of irons implies separation of (Mg,Fe)SiO 3 into Fe-rich and Fe-poor segments with lower spin transition pressures. Iron-rich segments can exist under pressure because the maximum solubility of FeSiO 3 into MgSiO 3 increases with increasing pressure (Mao et al., 1997; Tateno et al., 2007) while it is up to 12% only at low pressure, i.e., 26 GPa as measured by Fei et al. (1996) Effect of magnetic entropy In addition to configurational entropy, magnetic entropy is another important factor at high temperatures. It is known that in magnesiowüstite magnetic entropy leads to a gradual spin transition through a MS state (Sturhahn et al., 2005; Tsuchiya et al., 2006; Lin et al., 2007) The LS iron fraction, n, in paramagnetic MS state above the Curie or Néel temperatures is estimated by 305 n(p, T ) = 1 + m(2s + 1) exp ( H LS HS (P ) k B xt 1 ), (2) where k B is the Boltzmann constant, x is the iron concentration (x = 0.125), H LS HS (P ) is the relative enthalpy of the LS state per Mg 1 x Fe x SiO 3 with respect to the HS state, S is the iron spin quantum number (S = 2 for HS and S = 0 for LS), and m is the electronic configuration degeneracy (Tsuchiya et al., 2006). In the present case, m = 1 for both HS and LS states, since the degeneracy is lifted even in the HS state due to the low symmetry of the iron site (see Fig. 8 for x = 0.125) We do not know the Curie or Néel temperatures of this system, but we can discuss the validity of eq. 2 in the lower mantle. The enthalpy difference be- 315 tween the FM and the AFM states ( H AFM FM = H AFM H FM ) in each 14

15 configuration increases with increasing pressure (Fig. 7). In configuration 11, whose H AFM FM is largest among the 14 configurations, the LDA values of H AFM FM at 26 and 125 GPa are 0.27 and 8.9 mry per iron. These values correspond to 45 K and 1350 K, respectively. Since these values are smaller than lower mantle temperatures at these pressures, it is reasonable to assume that (Mg,Fe)SiO 3 exists in a paramagnetic state in the lower mantle, i.e., above 2,000 K, even when the irons are close to each other. Therefore we can estimate the LS iron fraction n in the mantle using eq. 2. Figure 10 shows n(p, T ) for x = 0.125, where H LS HS is H LS H FM for configuration 0 and H LS (H AFM + H FM )/2 for configuration 11. Like in magnesiowüstite, the MS state region between HS and LS states is found to be wider at higher temperatures. In the lower mantle, even in iron-ordered configurations, the spin transition should not be complete at the CMB for x = At room temperature, the pressure range of the MS state estimated by eq. 2 is less than 10 GPa. If eq. 2 is not adequate at room temperature and high pressure (in fact H AFM FM in configuration 11 at the static transition pressure is over 300 K), this pressure range should be narrower. Therefore, at room temperature the occurrence of the MS state should be mainly due to atomic disorder, not due to the magnetic entropy effects Conclusions The spin transition from the HS to the LS state in ferrous irons at the A site in (Mg,Fe)SiO 3 perovskite has been investigated by first principles. We found that a displacement of irons from the preferred magnesium site leading to a change in iron coordination number is vital for the occurrence of the spin 15

16 transition. We also found a strong dependence of the transition pressure on types of exchange-correlation functionals, iron concentration, and atomic and magnetic orderings. Our calculations suggest a gradual spin transition with MS state between the HS and the LS state in the pressure range of the lower mantle of the Earth as observed in previous experiments. We also showed there is a significant blueshift in the electronic gap across the spin transition. This property is crucial for understanding electrical and radiative conductivities There are several important factors we have not considered in the present study: vibrational entropies, the effect of ferric iron, impurities and defects, such as aluminum and oxygen vacancy, the strongly correlated nature of irons (the Hubbard U), and the post-perovskite transition. The vibrational entropy is necessary for calculating phase boundaries at finite temperatures. Phonon frequencies involving iron displacements should change significantly across the HS-LS transition, because the atomic environment around iron changes drastically. Inclusion of the Hubbard U should be important. In the case of magnesiowüstite, the Hubbard U is essential to predict its electronic properties; without U, the HS state was calculated to be metallic (Tsuchiya et al., 2006). In the present case, however, our calculations without the Hubbard U already showed both HS and LS states have nonmetallic band structure at lower mantle pressure and hence the charge density calculated from the fully occupied states should be reliable. Still the Hubbard U might change our results. The smaller band gap of the HS state than the LS state is expected to give a higher U value. Then, the inclusion of the Hubband U could lower the transition pressure further. The band gap of the HS state might be increased with the Hubbard U. These important factors still remain to be investigated. Nevertheless, the effects uncovered here should still be important in all future 16

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22 Fig. 1. Atomic arrangements of irons and magnesiums in configuration 0 for various iron concentrations x. Red and green spheres denote iron and magnesium. Blue polyhedra represent SiO 6 octahedra. The unit cells denoted by black lines contain 80 atoms (2 2 1 supercell) for x = , 40 atoms ( supercell) for x = and 20 atoms for other x. The k point grids used for these supercells are for x = and x = and for others. 22

23 Fig. 2. Calculated spin transition pressures. Blue and red points denote calculated values with configuration 0 and the iron-(110) plane configuration. Dashed lines and color bands are guides to the eye. 23

24 Fig. 3. Local atomic structure around iron in configuration 0 at 120 GPa optimized by LDA. Numbers denote Fe-O bond lengths in Å. 24

25 Fig. 4. Five atomic configurations of iron and magneiums for x = 0.5. Red and green spheres denote iron and magnesium respectively. For the AFM state, red and white spheres denote irons with up and down spins. The smaller and larger black rectangles represent the unit cell with 20 atoms and with 40 atoms ( supercell), respectively. The k point grids used for these supercells are for the 20-atoms unit cells and for the 40-atoms unit cells. 25

26 Fig. 5. Calculated (a) enthalpies at 0 GPa and (b) at 150 GPa and (c) spin transition pressures for each atomic configuration with x =

27 Fig. 6. Fourteen atomic configurations with x = Red and green spheres denote iron and magnesium. The unit cells denoted by the black lines contain 80 atoms in all configurations except for configuration 11, which contains 160 atoms. The k point grids used for these supercells are for configurations 1 through 10, for configuration 10, for configurations 11 and 12, for configuration 13, and for configuration

28 Fig. 7. Calculated (a) enthalpies at GPa and (b) 150 GPa and (c) spin transition pressures for each atomic configuration with x =

29 Fig. 8. Electronic density of states (DOS) and probability densities corresponding to groups of wave-functions at each peak of the DOS at 120 GPa for x = States filled with electrons are hatched. Energy is measured from the top of the MgSiO 3 valence band. (x, y, z) axes used for the assignment of orbitals are taken locally at the iron site. (a, b, c) axes are those of the orthorhombic unit cell with 20 atoms. 29

30 Fig. 9. LDA and GGA relative enthalpies of configuration 0 with respect to the aggregation of MgSiO 3 and FeSiO 3 : H = H(Mg Fe SiO 3 ) (0.875H(MgSiO 3 )+0.125H(FeSiO 3 )). Cusps in the lines are caused by spin transitions in Mg Fe SiO 3 or in FeSiO 3. Relative enthalpies of other configurations are intermediate values between blue and red lines. Horizontal dotted lines denote T S conf values in Rydberg at several temperatures, where S conf = k B (x log x + (1 x) log(1 x)), x = (see text). The mantle geotherm was derived by Brown and Shankland [1981]. 30

31 Fig. 10. Spin transition through mixed spin state from the HS to the LS states for configurations 0 and 11 for x = n is the LS iron fraction. Figure for GGA configuration 0 is omitted because n is 0 in this pressure and temperature range. Dashed lines denote a mantle geotherm derived by Brown and Shankland [1981]. 31