Earth and Planetary Science Letters 276 (2008) 198–206
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Earth and Planetary Science Letters
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l
Spin transition in (Mg,Fe)SiO3 perovskite under pressure
Koichiro Umemoto a,⁎, Renata M. Wentzcovitch a, Yonggang G. Yu a, Ryan Requist b,1
a
b
Minnesota Supercomputing Institute and Department of Chemical Engineering Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA
Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA
a r t i c l e
i n f o
Article history:
Received 28 February 2008
Received in revised form 18 September 2008
Accepted 22 September 2008
Editor: R.D. van der Hilst
Keywords:
Spin transition
Ferrous iron
Atomic and magnetic configurations
Perovskite
Lower mantle
First principles
a b s t r a c t
We present a density functional study of the pressure-induced spin transition in ferrous iron (Fe2+) at the A
site in MgSiO3 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.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
The lower mantle of the Earth is believed to consist mainly of ironbearing magnesium silicate perovskite, (Mg,Fe)SiO3, with a little
Al2O3, and smaller amounts of (Mg,Fe)O and CaSiO3. The presence of
iron in perovskite affects several lower mantle properties: elastic and
seismic properties (Kiefer et al., 2002; Li et al., 2005; Tsuchiya and
Tsuchiya, 2006; Stackhouse et al., 2007), the post-perovskite transition pressure (Caracas and Cohen, 2005; Mao et al., 2005; Ono and
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 (Fe2+) and/or ferric
(Fe3+), 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
⁎ Corresponding author. Fax: +1 612 626 7246.
E-mail addresses: [email protected] (K. Umemoto),
[email protected] (R.M. Wentzcovitch), [email protected] (Y.G. Yu),
[email protected] (R. Requist).
1
Present address: Theoretische Festkörperphysik, Universität Erlangen-Nürnberg,
Staudtstrasse 7, 91058 Erlangen, Germany.
0012-821X/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2008.09.025
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 spectroscopy
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 and 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 and 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 and Oganov, 2006;
K. Umemoto et al. / Earth and Planetary Science Letters 276 (2008) 198–206
199
Fig. 1. Atomic arrangements of irons and magnesiums in configuration 0 for various iron concentrations x. Red and green spheres
pffiffiffi pffiffiffidenote iron and magnesium. Blue polyhedra
represent SiO6 octahedra. The unit cells denoted by black lines contain 80 atoms (2 × 2 × 1 supercell) for x = 0.0625, 40 atoms ( 2 2 1 supercell) for x = 0.125 and 20 atoms for
other x. The k point grids used for these supercells are 2 × 2 × 4 for x = 0.0625 and x = 0.125 and 4 × 4 × 4 for others. (For interpretation of the references to color in this figure legend, the
reader is referred to the web version of this article.)
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 Mg1 − xFexSiO3 perovskite. For
ferrous iron, HS, LS, and IS states have magnetic moments per iron of
4 µB (5 majority-spin and 1 minority-spin electrons, S = 2), 0 µB (3 and 3,
S = 0), and 2 µB (4 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.
captions of figures showing atomic configurations. We used variablecell-shape molecular dynamics (Wentzcovitch, 1991; Wentzcovitch
et al., 1993) for structural optimization under arbitrary pressures.
3. Results and discussion
3.1. Effect of iron concentration
First we investigate the effect of iron concentration on the spin
transition. For each iron concentration from 6.25% (x = 0.0625) to 100%
2. Computational method
Calculations were performed using LDA and GGA (Perdew and
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 3s23p63d6.54s14p0, 3s23p1,
and 2s22p4 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, 3s23p0,
3s13p1, 3s13p0.53d0.5, 3s13p0.5, and 3s13d1 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 1 mRy/iron in the total energy. They are described in the
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. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
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K. Umemoto et al. / Earth and Planetary Science Letters 276 (2008) 198–206
Fig. 3. Local atomic structure around iron in configuration 0 at 120 GPa optimized by
LDA. Numbers denote Fe–O bond lengths in Å.
(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
FeSiO3 (calculated to be 284 GPa) was higher than in Mg0.875Fe0.125SiO3.
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 destabilizes iron in the high symmetry
site (with zFe = ±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 2.186 Å to
1.830 Å at 120 GPa and x = 0.125 (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 3.3.
Fig. 4. Five atomic configurations of iron and magneiums for x = 0.5. Red and green spheres denote iron and magnesium respectively.
pffiffiffi For
pffiffiffi 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 ( 2 2 1 supercell), respectively. The k point grids
used for these supercells are 4 × 4 × 4 for the 20-atoms unit cells and 2 × 2 × 4 for the 40-atoms unit cells. (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
K. Umemoto et al. / Earth and Planetary Science Letters 276 (2008) 198–206
3.2. 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 p
cells
ffiffiffi of
p20
ffiffiffi atoms, while
configurations 4 and 5 are generated in the 2 2 1 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 19 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 2 × 2 × 1 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
0.5.
10
extensions of configurations 3, 4, and 5 for x =p
ffiffiffi In configurations
pffiffiffi
(1 × 1 × 4 supercell with 80 atoms) and 11 ( 2 4 2 1 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,
201
the dependence becoming more accentuated with increasing pressure
(Fig. 7 (a) and (b)). Configuration 11 has the lowest enthalpy for AFMHS 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 ~10–15 GPa at most.
Variation in the transition pressures for 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.
3.3. 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 = 0.125 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
Fig. 5. Calculated (a) enthalpies at 0 GPa and (b) at 150 GPa and (c) spin transition pressures for each atomic configuration with x = 0.5.
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K. Umemoto et al. / Earth and Planetary Science Letters 276 (2008) 198–206
Fig. 6. Fourteen atomic configurations with x = 0.125. 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 2 × 2 × 4 for configurations 1 through 10, 4 × 4 × 1 for configuration
10, 4 × 1 × 4 for configurations 11 and 12, 1 × 4 × 4 for configuration 13, and 4 × 1 × 4 for configuration 14. (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
shown in Fig. 8, we see that iron d states do not simply split into lower
eg and higher t2g states as usually assumed (Li et al., 2004; Hofmeister,
2006). This is because the A site 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 t2g-derived and the higher empty eg-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 2500 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 (Mg0.9,Fe0.1)SiO3 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.
3.4. 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: MgSiO3 and
FeSiO3. In fact, static enthalpy calculation for x = 0.125 indicates that
the dissociated phases have lower enthalpy than any configuration
generated so far (Fig. 9). But at high temperature the configurational
entropy stabilizes Mg0.875Fe0.125SiO3 against the dissociation products.
For reference we may assume Mg0.875Fe0.125SiO3 is an ideal solid
solution of MgSiO3 and FeSiO3, the configurational entropy Sconf per
Mg0.875Fe0.125SiO3 is given by Sconf = −kB(x log x + (1 − x) log (1 − x))
where kB is Boltzmann constant and x = 0.125. The configurational
entropy of the dissociated products is 0 (x = 0 and 1). The difference in
Gibbs free energy between Mg0.875Fe0.125SiO3 in configuration 0 and
the dissociated products, ΔG,
ΔG ¼ ΔH−TΔSconf
¼ HMg0:875 Fe0:125 SiO3 − 0:875HMgSiO3 þ 0:125HFeSiO3 −TSconf :
ð1Þ
When ΔG b 0, Mg0.875Fe0.125SiO3 is energetically stable with respect to
the dissociation. Fig. 9 shows that ΔG is negative at low pressure and
high temperature; at 26 GPa, configuration 0 should be stable with
respect to the dissociation over ~ 1300 K by LDA (~ 600 K by GGA). This
is consistent with the synthesis of (Mg,Fe)SiO3 perovskite with ~ 10%
iron concentration at ~ 26 GPa and ~2000 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
K. Umemoto et al. / Earth and Planetary Science Letters 276 (2008) 198–206
203
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 FeSiO3 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
FeSiO3 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 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. 7. Calculated (a) enthalpies at GPa and (b) 150 GPa and (c) spin transition pressures
for each atomic configuration with x = 0.125.
to Mg0.875Fe0.125SiO3 (ΔG N 0), 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
Mg0.875Fe0.125SiO3 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 FeSiO3: 56 or
16 GPa (116 or 76 GPa) by LDA (GGA), respectively. However, these
configurations assume (110) plane segments and FeSiO3 islands of
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 = 0.125. States filled
with electrons are hatched. Energy is measured from the top of the MgSiO3 valence
band. (x, y, and z) axes used for the assignment of orbitals are taken locally at the iron
site. (a, b, and c) axes are those of the orthorhombic unit cell with 20 atoms.
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K. Umemoto et al. / Earth and Planetary Science Letters 276 (2008) 198–206
Fig. 9. LDA and GGA relative enthalpies of configuration 0 with respect to the aggregation of MgSiO3 and FeSiO3: ΔH = H(Mg0.875Fe0.125SiO3) − (0.875H(MgSiO3) + 0.125H(FeSiO3)).
Cusps in the lines are caused by spin transitions in Mg0.875Fe0.125SiO3 or in FeSiO3. Relative enthalpies of other configurations are intermediate values between blue and red lines.
Horizontal dotted lines denote TSconf values in Rydberg at several temperatures, where Sconf = −kB(x log x + (1 − x) log(1 − x)),x = 0.125 (see text). The mantle geotherm was derived by
Brown and Shankland (1981). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9 indicates that ΔG by LDA becomes positive beyond ~40 GPa
in the lower mantle, i.e., that Mg0.875Fe0.125SiO3 with completely
random iron distribution is no longer stable beyond 40 GPa. On the
other hand, GGA indicates that Mg0.875Fe0.125SiO3 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)SiO3 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
FeSiO3 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 facilitated 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 and
Constable, 2004; Kuvshinov and Olsen, 2006). Clustering of irons
implies separation of (Mg,Fe)SiO3 into Fe-rich and Fe-poor segments
with lower spin transition pressures. Iron-rich segments can exist
under pressure because the maximum solubility of FeSiO3 into MgSiO3
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).
3.5. 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
nðP; T Þ ¼
1
;
ðP Þ
1 þ mð2S þ 1Þexp ΔHkLS−HS
B xT
ð2Þ
where kB is the Boltzmann constant, x is the iron concentration
(x = 0.125), ΔHLS–HS(P) is the relative enthalpy of the LS state per Mg1 − x
FexSiO3 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
Fig. 10. Spin transition through mixed-spin state from the HS to the LS states for configurations 0 and 11 for x = 0.125. n is the LS iron fraction. Figures for GGA configuration 0 are
omitted because n is ~ 0 in this pressure and temperature range. Dashed lines denote a mantle geotherm derived by Brown and Shankland (1981).
K. Umemoto et al. / Earth and Planetary Science Letters 276 (2008) 198–206
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 between the FM and the AFM states (ΔHAFM − FM = HAFM −
HFM) in each configuration increases with increasing pressure (Fig. 7).
In configuration 11, whose ΔHAFM − FM is largest among the 14
configurations, the LDA values of ΔHAFM − 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)SiO3 exists in a paramagnetic state in the lower mantle, i.e., above
~ 2000 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). Fig. 10
shows n(P,T) for x = 0.125, where ΔHLS − HS is HLS − HFM for configuration
0 and HLS − (HAFM + HFM)/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 = 0.125. 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 ΔHAFM − 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.
4. Conclusions
The spin transition from the HS to the LS state in ferrous irons at
the A site in (Mg,Fe)SiO3 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 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 postperovskite 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 calculations.
205
Note added to proof
Very recently, ferrous iron was reported to exist predominantly in the
IS state at the lower mantle condition by Mössbauer/nuclear forward
scattering and XES/SMS experiments (McCammon et al., 2008; Lin et al.,
2008).
Lin, J.F., Watson, H., Vankó, G., Alp, E.E., Prakapenka, V.B., Dera, P.
Struzhkin, V.V., Kubo, A., Zhao, J., McCammon, C., Evans, W.J., 2008.
Intermediate–spin ferrous iron in lowermost mantle post-perovskite
and perovskite. Nature Geosci. 1, 688–691.
McCammon C., Kantor, I., Narygina, O., Rouquette, J., Ponkratz, U.,
Sergueev, I., Mezouar, M., Prakapenka, V., Dubrovinsky, L., 2008. Stable
intermediate-spin ferrous iron in lower-mantle perovskite Nature
Geosci. 1, 684–687.
Acknowledgments
We would like to thank Professor D. Yuen for helpful comments.
Calculations in this work were performed using the QuantumESPRESSO package (Baroni et al.) at the Minnesota Supercomputing
Institute and at Indiana University's BigRed system. The spatial
distribution of wave functions in Fig. 8 was visualized by XCrySDen
(Kokalj, 2003). This research was supported by NSF/EAR-0230319, ITR0426757 (VLab), NSF/DMR-0325218 (ITAMIT), the UMN-MRSEC, and
the Minnesota Supercomputing Institute.
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