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The role of equilibrium volume and magnetism on the stability of iron phases at high pressures
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 046001 (8pp)
doi:10.1088/0953-8984/26/4/046001
The role of equilibrium volume and magnetism on the stability of iron phases at high pressures S Alnemrat1,2 , J P Hooper2 , I Vasiliev1 and B Kiefer1 1 2
Department of Physics, New Mexico State University, Las Cruces, NM 88003, USA Physics Department, Naval Postgraduate School, Monterey, CA 93943, USA
E-mail: [email protected] Received 16 August 2013, revised 26 November 2013 Accepted for publication 26 November 2013 Published 20 December 2013 Abstract
The present study provides new insights into the pressure dependence of magnetism by tracking the hybridization between crystal orbitals for pressures up to 600 GPa in the known hcp, bcc and fcc iron. The Birch–Murnaghan equation of state parameters are; bcc: V0 = 11.759 A3 /atom, K0 = 177.72 GPa; hcp: V0 = 10.525 A3 /atom, K0 = 295.16 GPa; and fcc: V0 = 10.682 A3 /atom, K0 = 274.57 GPa. These parameters compare favorably with previous studies. Consistent with previous studies we find that the close-packed hcp and fcc phases are non-magnetic at pressures above 50 GPa and 60 GPa, respectively. The principal features of magnetism in iron are predicted to be invariant, at least up to ∼6% overextension of the equilibrium volume. Our results predict that magnetism for overextended fcc iron disappears via an intermediate spin state. This feature suggests that overextended lattices can be used to stabilize particular magnetic states. The analysis of the orbital hybridization shows that the magnetic bcc structure at high pressures is stabilized by splitting the majority and minority spin bands. The bcc phase is found to be magnetic at least up to 600 GPa; however, magnetism is insufficient to stabilize the bcc phase itself, at least at low temperatures. Finally, the analysis of the orbital contributions to the total energy provides evidence that non-magnetic hcp and fcc phases are likely more stable than bcc at core earth pressures. Keywords: high pressure, magnetism, iron phases, stability (Some figures may appear in colour only in the online journal)
1. Introduction
the magnetic field that is generated by the convective motion in the outer core, and its implications for the stabilization of the magnetic field against reversal, it is important to understand the magnetic state of iron and its behavior at high pressures [25–28]. Magnetic fluctuations and the reorientation of magnetic moments as well as a changing electronic structure at or near the Fermi energy can also significantly affect the phase stability of iron and its alloys [11, 19, 29–32]. Theoretical and experimental studies show that bcc iron is an equilibrium structure [5, 6, 11, 33–35]. In contrast to the hcp phase, which is non-magnetic (NM) and the ground state at high pressures, the bcc phase carries a finite magnetic moment even at high pressures. Previous theoretical
Several lines of evidence suggest that the composition of the earth’s core is dominated by an iron-rich Fe1−x , Nix alloy with x ∼ 0.1. The crystal structure of iron in the solid inner core has long been assumed to be hcp [1–7]. However, over the past decade, both experiment and theory have shown evidence for the fcc and bcc phases as well [8–17]. Among the proposed phases, the bcc phase stands out in that it retains a finite magnetic moment even at inner core pressures [18]. In contrast, the fcc and hcp phases are paramagnetic at high pressures [18–24]. Due to the different modes of interactions between magnetic and non-magnetic inner core phases with 0953-8984/14/046001+08$33.00
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parameters have been carefully chosen to obtain converged solutions to the Kohn–Sham equations. All degrees of freedom were allowed to relax simultaneously at constant volume during the simulations; the c/a-ratio in the hcp lattice is allowed to change in order to find the lattice of the ground state. The pressure was obtained from the trace of the stress tensor. Equation of state (EOS) parameters were obtained by fitting a third-order Birch–Murnaghan equation of state to our E(V) curves [53, 54]. The COOP, P or orbital-population weighted DOS, can be written as .ij COOP(ε) = DOS(ε), where ε is the energy of interest and the summation must be taken over all possible overlaps between state i and j. Thus, the COOP allows one to distinguish between bonding and antibonding orbital interactions, depending on the sign of the overlap matrix element [55]. The COOP for an atomic centered basis-set, as used in our computations, allows a convenient decomposition of the DOS into distinct overlap contributions between nearest neighbor atoms in the simulation cell. This is particularly useful for the investigation of magnetic properties: for non-magnetic solutions the COOP of the majority and minority spin channels are mirror images of each other. In the case of magnetic solutions the COOP shows a relative shift between majority and minority spins channels, as does the DOS. The information content of the COOP mimics the hybridization between orbitals: bonding orbitals appear with positive amplitude, while antibonding orbitals appear with negative amplitude. A zero overlapping integral results in a non-bonding state between atomic orbitals [56–58]. Therefore the COOP analysis provides a direct method to deconvolve complex interactions in terms of overlap contributions and to follow the changes of the electronic structure under changing conditions. (Anti) bonding states (de)stabilize a given electronic structure and COOP provides a detailed picture, for example, of the evolution of magnetism with pressure. Coincidentally, the same analysis also allows one to decompose the internal energy into orbital contributions through crystal orbital Hamilton population (COHP) analysis, hence providing insights into stabilizing and destabilizing energy contributions not only of the magnetic structures but also of the associated crystal phases. A negative sign of the Hamiltonian matrix of the COHP in the case of positive or bonding COOP means that the total energy of a crystal structure is lowered and hence stabilized.
studies show that the ferromagnetism (FM) of bcc iron is responsible for its stability at ambient conditions [21, 36–39]. The electronic density of states (DOS) of elemental iron shows a significant broadening with pressure due to the direct overlap of the 3d derived electronic bands between neighboring iron atoms. The explanation of the formation of FM structures often relies on the Stoner instability. This condition is given by IN(εf , V) > 1, where N is the density of states at the Fermi level at magnetic moment (µ = 0) and I (µ = 0, V) is the Stoner parameter as a function of magnetic moment and volume (V) [40–42]. On the other hand, the Fermi surfaces and the relative bandwidth of majority and minority spin bands successfully explain FM in bcc iron [43]. The stabilization of the FM state can be attributed to a narrow bandwidth of the majority spin band. Previous ab initio calculations commonly underestimate the equilibrium volume of elemental iron as compared to experiment [3, 22, 44, 45]. However, magnetism depends on orbital overlap and hence on the nearest neighbor distances. Thus, it is not clear if the magnetism in previous computations could be biased toward low values by systematically underestimated volumes. Here we explore the effect of increased volumes relative to experiment on the magnetism in elemental iron. The evolution of magnetism in the bcc, hcp, and fcc phases of elemental iron is explored by tracking the crystal orbital overlapping population (COOP) analysis with increasing compression. The same analysis allows one to deconvolve the total energy in terms of orbital contributions and to gain new insights into the destabilization of the bcc phase itself relative to the close-packed phases—fcc and hcp—at pressure. 2. Computational details
The DFT calculations have been performed using the Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA) [46, 47]. Studies using SIESTA that address the behavior of bulk transition metal systems have been limited to equilibrium structures [19, 48–50]. Electronic exchange and correlation effects are treated within the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) parameterization [51]. Interactions between the nuclei and electrons are described with norm-conserving Troullier–Martins pseudopotentials [52]. A potential with the 4s2 3d6 valence electronic configuration (Rc = 2.0 au) for iron was generated from atomic all-electron computations. A cut-off energy of Ecut = 200 Ryd for the expansion of the band-states and standard Monkhorst–Pack 16 ×16 ×16 and 16 ×16 ×12 k-point grids were used for the cubic phases (fcc and bcc) and the hexagonal (hcp) phase, respectively. A double-zeta plus polarization (DZP) basis-set is used for the expansion of the electronic wavefunctions, with an energy shift of 50 meV. Forces and total energy per atom were converged to better than 0.04 eV A−1 and 1 meV, respectively, and an energy smearing of 25 meV was applied. All computations were conducted in spin-polarized mode. No thermal corrections were considered in this study; all calculations were carried out at 0 K. The computational
3. Results and discussion 3.1. Equation of state
Our results are in good agreement with available experimental values of hcp iron, as shown in figure 1 and table 1. Both, fcc and hcp structures show a magnetic transition from ferromagnetic to non-magnetic, as described previously [22, 45, 59, 60]. The volume reduces by 6% and 3% across this magnetic to non-magnetic transition at pressures of 54 GPa and 50 GPa, for the fcc and hcp phases, respectively. The computed EOS 2
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Table 1. Third-order Birch–Murnaghan equation of state parameters of ferromagnetic bcc, fcc and hcp iron. Values in parentheses refer to the uncertainty of the last digit.
Phase
Magnetic state
V0 (A3 /atom)
K0 (GPa)
K 0 (—)
bcc/this study bcc/this study bcc/expa hcp/this study hcp/this study hcp/this study hcp/expb fcc/this study fcc/this study fcc/this study fcc/expc (model)
FM NM FM NM FM1 FM2 FM NM FM1 FM2 FM
11.759 10.718 11.77(3) 10.505 10.525 12.169 11.16(2) 10.684 10.682 12.465 11.830
177.72 276.52 173.0 (9) 297.13 295.16 157.52 165 (4) 269.05 274.57 192.83 —
4.564 4.153 — 3.934 4.268 4.113 — 4.120 4.2000 4.0000 —
a b c
E0 (eV/atom) −781.890 −781.370 — −781.750 −781.50 −781.710 — −781.630 −781.610 −781.790 —
Reference [15, 16]. Reference [7, 37]. Reference [17].
modulus, due to the correlations between the EOS parameters, V0 , K0 , and K00 . However, the very good agreement between experiment and theory (table 1) may be somewhat fortuitous: the bcc → hcp transition, which is known to be in the range of 13–20 GPa [62], occurs in our calculations at a pressure of 24 GPa, higher than the observed experimental values. The EOS parameters show a large discrepancy between computations and experiment. This discrepancy refers to the underestimation of the lattice parameters even in the GGA approximation, which generally tends to overestimate lattice parameters and volumes. For some properties this may not be of significance, but for properties that rely strongly on the amount of overlap of electronic wavefunction, such as magnetism and compression, this can have a large effect. This distortion can be caused by external pressure, high temperature, and the existence of light element impurities in the crystal structure. Using the GGA approximation in conjunction with an atomic centered basis-set we validate our results by comparison with previous results; this is presented in table 1. This knowledge is difficult to obtain by other methods (for example, from the more widely used plane wave approaches) and gives new insights into the causes for the disappearance of magnetism in hcp and fcc iron and the residual magnetism in the competing bcc phase at high pressure. After this validation we precede to the novel parts of the presented work, namely the evolution of the magnetism in the known phases of elemental iron. The COOP analysis allows us to track the contributions from 4s and 3d–4s hybridizations that give detailed insights into the pressure dependence of magnetism in these phases. Thus, the orbital resolved analysis of magnetism for elemental iron structures at both ambient and high pressures will be discussed in the following sections. The prediction of an intermediate spin state in fcc iron suggests a volume sensitivity of magnetism in the sense that increased volume at constant pressure can be used to stabilize non-trivial spin-states in 3d compounds, or at least in elemental iron. Therefore these analyses extend previous work and advance the knowledge of elemental iron magnetic structures at high pressures.
Figure 1. Comparison of the calculated EOS with available
experimental data. Open squares, experimental data from [7, 30]. Solid line represents ferromagnetic bcc; dashed line ferromagnetic fcc; dotted line ferromagnetic hcp. For fcc and hcp, the lines show the fits of the low-pressure region with finite magnetic moment and the non-magnetic high-pressure region for the EOS parameters given in table 1.
parameters of bcc, hcp, and fcc structures are found to be in very good agreement with experiment (table 1). Previous computations proposed several antiferromagnetic structures—AFM, AFM I, AFM II—and non-collinear spin ordering in order to reach better agreement between the predicted and observed compressional behavior of elemental iron [18, 22, 59, 61]. In general these structures lead to larger equilibrium volumes, reduce the bulk modulus, and achieve a better agreement between theory and experiment. For example, the bulk modulus of the AFM II structure is 209 GPa, which is significantly ∼30% lower than that for NM-hcp iron (292 GPa) [21]. Test calculations show that the AFM II structure is more stable than FM-hcp iron at low pressures; hence, the discrepancies between experiment and theory cannot be attributed to the underestimated volumes in previous computations (relative to experiment). As expected, the increased V0 in our computations decreases the bulk 3
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The high contribution is due to 3d–3d interactions and 4s–3d hybridization, and minor contributions from the s and p orbitals [65]. The shift in majority and minority spin states and the strength of the exchange parameter are the primary reasons for strengthening of the ferromagnetic bcc structure in the Stoner model [41, 60]. The stability of a particular magnetic structure requires a high cumulative density of states (integration of DOS up to Fermi level) and a large amount of splitting between the majority and the minority bands. This is achieved in iron by lowering the energy of bonding states. The corresponding depopulation of antibonding states at the Fermi energy (EF ) increases the exchange energy and decreases the kinetic energy. The COOP curves in figure 4(b) give insights into the contributions of orbital overlap to the DOS. In these curves we notice a large splitting between majority and minority spin bands, the t2g band (lower in energy) contributes strongly to the bonding states while eg band electrons contribute to the antibonding states, the 4s–3d hybridization interaction shows a strong contribution to the bonding states at the Fermi level, while 4s–4s has a small contribution to the bonding states and a somewhat larger contribution to the antibonding states. Overall, the bonding states dominate and a magnetic FM structure is stabilized. At a pressure of 600 GPa, the shifts between spin-up and spin-down components are significantly smaller than at equilibrium, and the overall bandwidth increases, as shown in figures 4(c) and (d). As the bands broaden, the amplitude decreases due to the finite number of available states. According to the Stoner model, this decrease in DOS at the Fermi level and the decrease of the exchange-splitting between majority and minority spins (which is proportional to the energy difference between bonding and antibonding states of the majority and minority bands) decreases the stability of the magnetic structure, and the magnetic moment decreases gradually with increasing compression, as predicted (figure 3(a)). This is an important property of bcc iron, since both fcc and hcp structures show a rapid decrease in magnetic moments at much lower compression, as discussed above. Both majority and minority spin bands (eg as well as t2g bands) contribute to the DOS as the pressure increases, the remaining offset of majority and minority spin bands is the main reason for the magnetic nature of the bcc structure. Deconvolving the orbital contributions of this DOS in figure 4(d) shows that 4s–4s interactions are completely overlapped as bonding states which are approximately twice as high as compared to the values at equilibrium. This suggests that the 4s electrons have no contribution to the magnetic nature of the bcc structure. The splitting of the sd-bands decreases significantly from 1.47 eV at equilibrium (figure 4(b)) to 0.6 eV at 600 GPa (figure 4(d)). Furthermore the bands broaden with decreasing volume (increasing pressure) as expected. In contrast, the dd-bands are weakly affected by pressure and the main cause of magnetism in bcc iron, while the s–d overlap does not increase significantly. In summary, a lower density of states, more antibonding states at the Fermi level, and broadening of majority and minority bands are insufficient to stabilize a non-magnetic structure while a magnetic moment remains.
Figure 2. Energy–volume relationship for iron. Solid line for
ferromagnetic bcc; dashed line for ferromagnetic fcc; dotted line for ferromagnetic hcp iron. The decreasing energy range between V = 9.4 and V = 9.9 A3 /atom corresponds to the compression range of rapidly changing magnetic moment.
3.2. Magnetic moment
In agreement with previous computations and experiment we find that the FM bcc phase is the most stable phase at low pressures, as shown in figure 2. At equilibrium the energy difference between FM and NM bcc iron is about 0.74 eV/atom in favor of the FM phase. This enthalpy difference decreases monotonously as the pressures increases, and the bcc phase is predicted to be NM at pressures in excess of 600 GPa. In contrast to the paramagnetic nature of both fcc and hcp phases, FM is stabilized in the bcc phase to much higher compression, as shown in figure 3. Furthermore, both fcc and hcp structures possess a magnetic transition in the range 50–60 GPa. The present study also predicts that magnetism in the fcc phase disappears via an intermediate spin state in figure 3(b). This feature, which is unique for the fcc phase, is most likely associated with the overestimation of V0 as compared to previous computations and experiment. This finding suggests that it may be possible to stabilize non-trivial intermediate magnetic structures by overextending the crystal structure, a finding that is similar to that observed in compounds such as Fe and FeNi, where an increasing Fe concentration is used to extend the volume of fcc FeNi and to stabilize particular magnetic structures [63]. 3.3. DOS and COOP analysis
The evolution of the total energy of all iron structures with compression can be attributed to shifts between majority and minority spin bands relative to the Fermi energy. The strength of the exchange-splitting parameter between atoms is dominated by the nearest neighbor distance between atoms, the amount of orbital overlap, and the hybridization of orbitals [64]. The density of states in bcc iron is mainly contributed by the triply degenerate t2g and doubly degenerate eg bands, as shown in figure 4(a). 4
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Figure 3. Evolution of magnetic moments with compression in (a) ferromagnetic bcc Fe, (b) ferromagnetic fcc Fe and (c) ferromagnetic
hcp Fe.
Figure 4. (a) Contributions to the DOS of bcc iron at the equilibrium volume. Dash–dot line: d–d contribution, filled area: s–s contribution,
dashed line: s–d hybridization contribution. (b) COOP analysis of majority and minority spin subbands of bcc iron at equilibrium volume. Dash–dot line: d–d electrons, dashed area: s–s electrons, dashed line: s–d electrons. (c) Contributions to DOS of bcc iron at 600 GPa. Dash–dot line: d–d contribution, filled area: s–s contribution, dashed line: s–d hybridization contribution. (d) COOP analysis of majority and minority spin subbands of bcc iron at 600 GPa. Dash–dot line: d–d electrons, dashed area: s–s electrons, dashed line: s–d electrons.
The 4s–4s, 3d–3d, and 4s–3d contributions to the DOS of fcc iron at ambient conditions are shown in figure 5(a). At equilibrium the magnetism in fcc iron is stabilized by exchange-splitting, as in bcc iron. The COOP analysis of fcc iron at ambient conditions in figure 5(b) shows that the
dominant contributions are again due to antibonding ss and dd interactions. Only sd provides bonding contributions to the DOS. As pressure increases, the low density of states of fcc iron at ambient conditions and the highly antibonding nature of ss 5
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Figure 5. (a) Contributions to the DOS of fcc iron at the equilibrium volume. Dash–dot line: d–d contribution, filled area: s–s contribution,
dashed line: s–d hybridization contribution. (b) COOP analysis of majority and minority spin subbands of fcc iron at equilibrium volume. Dash–dot line: d–d electrons, dashed area: s–s electrons, dash line: s–d electrons. (c) Contributions to DOS of fcc iron at 600 GPa. Dash–dot line: d–d contribution, filled area: s–s contribution, dashed line: s–d hybridization contribution. (d) COOP analysis of majority and minority spin subbands of fcc iron at 600 GPa. Solid line: d–d electrons, dashed area: s–s electrons, dashed line: s–d electrons.
and dd interactions result in broadening of the 3d bands, as shown in figure 5(c), and an overall antibonding fcc structure, which leads to a much more rapid destabilization of the magnetic structure as compared to the bcc phase (figure 3). As a result the FM ⇒ NM transition occurs at ∼60 GPa (figures 2 and 3(b)). With increasing pressure antibonding states move above the Fermi energy and the bonding states dominate, which stabilizes the fcc phase relative to the bcc phase. The COOP curves of the fcc structure at high pressure are shown in figure 5(d). Larger s–s interactions are positively overlapped as bonding states, a smaller fraction negatively bonding, and zero states at the Fermi level. s–d interactions are only positively overlapped. Finally, d–d interactions show a larger fraction of negative overlapping at the Fermi level, these antibonding states at the Fermi level destabilize the fcc structure and a larger pressure is needed to remove these antibonding states at the Fermi level. Overall, the COOP analysis results in an overall positively overlapping interaction, which means fcc iron is a strong candidate as a stable structure at high pressures. The density of state of FM-hcp iron at ambient conditions is significantly higher than those of the fcc phase and the exchange-splitting is smaller, as shown in figure 6(a).
In contrast to the fcc structure, the COOP curves for FM-hcp iron in figure 6(b) show a larger antibonding d–d interaction at the Fermi level and a smaller fraction positively bonded. The s–s interactions have positive and negative contributions, which compensate their contribution to the structural stability of hcp iron at equilibrium (figure 6(b)). At 600 GPa, all bands are completely overlapped in bonding and antibonding states, which results in a NM structure, as shown in figure 6(c). The COOP curves of hcp iron at 600 GPa are shown in figure 6(d). As the pressure increases, some of the antibonding states due to d–d interactions at the Fermi level have been removed as compared to the fcc case. Also, here, s–s and s–d are completely positively overlapped, favoring a NM magnetic structure. The striking similarities between fcc and hcp structures are highlighted by the very similar changes of magnetic moments with compression (figure 3(b)), which reflects the fact that the fcc and the nearly ideal hcp structure share the same high coordination number (CN = 12) and a weakly pressure dependent c/a ratio. Comparing the COOP provides an additional tool to distinguish the magnetism and structural stability of the fcc and hcp phases. The comparison of the DOS and COOP analysis of both the fcc and hcp structures show a lower density of antibonding 6
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Figure 6. (a) Contributions to the DOS of hcp iron at the equilibrium volume. Dash–dot line: d–d contribution, filled area: s–s contribution,
dashed line: s–d hybridization contribution. (b) COOP analysis of majority and minority spin subbands of hcp iron at equilibrium volume. Dash–dot line: d–d electrons, dashed area: s–s electrons, dashed line: s–d electrons. (c) Contributions to DOS of hcp iron at 600 GPa. Dash–dot line: d–d contribution, filled area: s–s contribution, dashed line: s–d hybridization contribution. (d) COOP analysis of majority and minority spin subbands of hcp iron at 600 GPa. Solid line: d–d electrons, dashed area: s–s electrons, dashed line: s–d electrons.
states at the Fermi level. The comparisons of the magnitudes of the orbital overlap contributions show that the magnetic fcc and hcp phases are stable at high pressures. However, the fcc and hcp structures are NM at high pressures due to zero splitting between the majority and minority spin bands. Both the fcc and hcp phases show an overall positive (bonding) overlap, which leads to the stabilization of the NM structures at high pressures as compared to the competing bcc phase, at least at low temperatures. Among the close-packed structures, hcp is more favorable than fcc in terms of the DOS and overall positive overlapping, consistent with the enthalpy curves and the known low-temperature phase diagram of iron.
transition from high spin to non-magnetic via an intermediate spin state for the overextended lattice considered here. In agreement with the known phase diagram of iron, the COOP (COHP) analysis shows that the bcc phase is the stable structure at ambient conditions. Furthermore, the COOP decomposition of the DOS shows that magnetism in bcc iron is present to much higher pressures, due to a reduced s–s and s–d antibonding contribution. The results are consistent with the splitting between the majority and minority subbands and with the expansion of bandwidths according to the Stoner model. The COHP analysis of the orbital overlap contributions to the total energy shows that both fcc and hcp are unfavorable at ambient conditions, relative to bcc. In contrast, both NM-fcc and NM-hcp structures are found to be strong candidates for the stable phase at high pressure, due to the higher contribution of bonding states at the Fermi energy. The bonding contributions of the hcp and fcc phase at the Fermi level, showing that the close-packed phases are more favorable at high pressures than the bcc phase. The detailed analysis of the changing magnitude of the orbital overlap shows that the hcp phase is expected to be the more stable iron structure at high pressures, at least up to a pressure of 600 GPa.
4. Conclusion
We performed electronic structure calculations for iron using the atomic centered basis-sets implemented in SIESTA. The results show that the magnetic state in the bcc, fcc, and hcp phases is independent of an under- or overestimated equilibrium volume, at least up to ∼(3–6)% relative to experiment. Therefore, differences between experimental and theoretical EOS at low pressures are likely intrinsic to iron and not an artifact of the underestimated equilibrium volume in previous computations. The fcc phase shows a magnetic 7
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Acknowledgment
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The authors gratefully acknowledge funding from the National Science Foundation under Grant No. EAR-0636075. References [1] Tateno S, Hirose K, Ohishi Y and Tatsumi Y 2010 Science 330 359–61 [2] Sha X W and Cohen R E 2010 Phys. Rev. B 81 094105 [3] Vocadlo L, Dobson D P and Wood I G 2009 Earth Planet. Sci. Lett. 288 534–8 [4] Mao H K, Shu J F, Shen G Y, Hemley R J, Li B S and Singh A K 1999 Nature 399 741–3 [5] Mao H K, Bassett W A and Takahash T 1967 J. Appl. Phys. 38 272–6 [6] Mao H K, Shu J F, Shen G Y, Hemley R J, Li B S and Singh A K 1998 Nature 396 741–3 [7] Mao H K, Wu Y, Chen L C, Shu J F and Jephcoat A P 1990 J. Geophys. Res.-Solid Earth Planets 95 21737–42 [8] Luo W, Johansson B, Eriksson O, Arapan S, Souvatzis P, Katsnelson M I and Ahuja R 2010 Proc. Natl Acad. Sci. USA 107 9962–4 [9] Vocadlo L, Wood I G, Gillan M J, Brodholt J, Dobson D P, Price G D and Alfe D 2008 Phys. Earth Planet. Inter. 170 52–9 [10] Vocadlo L 2007 Earth Planet. Sci. Lett. 254 227–32 [11] Alfe D, Gillan M J, Vocadlo L, Brodholt J and Price G D 2002 Phil. Trans. R. Soc. A 360 1227–44 [12] Morard G et al 2010 Phys. Rev. B 82 174102 [13] Belonoshko A B, Ahuja R and Johansson B 2003 Nature 424 1032–4 [14] Vocadlo L, Alfe D, Gillan M J, Wood I G, Brodholt J P and Price G D 2003 Nature 424 536–9 [15] Rayne J A and Chandrasekhar B S 1961 Phys. Rev. 122 1714–8 [16] Adams J J, Agosta D S, Leisure R G and Ledbetter H 2006 J. Appl. Phys. 100 113530 [17] Krasko G L 1987 Phys. Rev. B 36 8565–9 [18] Cohen R E and Mukherjee S 2004 Phys. Earth Planet. Inter. 143/144 445–53 [19] Garcia-Suarez V, Newman C, Lambert C, Pruneda J and Ferrer J 2004 Eur. Phys. J. B 40 371–7 [20] Sha X W and Cohen R E 2010 J. Phys.: Condens. Matter 22 372201 [21] Steinle-Neumann G, Cohen R E and Stixrude L 2004 J. Phys.: Condens. Matter 16 S1109–19 [22] Steinle-Neumann G, Stixrude L and Cohen R E 2004 Proc. Natl Acad. Sci. USA 101 33–6 [23] Steinle-Neumann G, Stixrude L, Cohen R E and Gulseren O 2001 Nature 413 57–60 [24] Stixrude L and Brown J M 1998 Ultrahigh-Press. Mineral. 37 261–82 [25] Anderson O L 2002 Phys. Earth Planet. Inter. 131 1–17 [26] Anderson O L and Isaak D G 2002 Phys. Earth Planet. Inter. 131 19–27 [27] Buffett B A 2000 Science 288 2007–12 [28] Buffett B A and Glatzmaier G A 2000 Geophys. Res. Lett. 27 3125–8 [29] Lin J F, Fei Y W, Sturhahn W, Zhao J Y, Mao H K and Hemley R J 2004 Earth Planet. Sci. Lett. 226 33–40
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The role of equilibrium volume and magnetism on the stability of iron phases at high pressures
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 046001 (8pp)
doi:10.1088/0953-8984/26/4/046001
The role of equilibrium volume and magnetism on the stability of iron phases at high pressures S Alnemrat1,2 , J P Hooper2 , I Vasiliev1 and B Kiefer1 1 2
Department of Physics, New Mexico State University, Las Cruces, NM 88003, USA Physics Department, Naval Postgraduate School, Monterey, CA 93943, USA
E-mail: [email protected] Received 16 August 2013, revised 26 November 2013 Accepted for publication 26 November 2013 Published 20 December 2013 Abstract
The present study provides new insights into the pressure dependence of magnetism by tracking the hybridization between crystal orbitals for pressures up to 600 GPa in the known hcp, bcc and fcc iron. The Birch–Murnaghan equation of state parameters are; bcc: V0 = 11.759 A3 /atom, K0 = 177.72 GPa; hcp: V0 = 10.525 A3 /atom, K0 = 295.16 GPa; and fcc: V0 = 10.682 A3 /atom, K0 = 274.57 GPa. These parameters compare favorably with previous studies. Consistent with previous studies we find that the close-packed hcp and fcc phases are non-magnetic at pressures above 50 GPa and 60 GPa, respectively. The principal features of magnetism in iron are predicted to be invariant, at least up to ∼6% overextension of the equilibrium volume. Our results predict that magnetism for overextended fcc iron disappears via an intermediate spin state. This feature suggests that overextended lattices can be used to stabilize particular magnetic states. The analysis of the orbital hybridization shows that the magnetic bcc structure at high pressures is stabilized by splitting the majority and minority spin bands. The bcc phase is found to be magnetic at least up to 600 GPa; however, magnetism is insufficient to stabilize the bcc phase itself, at least at low temperatures. Finally, the analysis of the orbital contributions to the total energy provides evidence that non-magnetic hcp and fcc phases are likely more stable than bcc at core earth pressures. Keywords: high pressure, magnetism, iron phases, stability (Some figures may appear in colour only in the online journal)
1. Introduction
the magnetic field that is generated by the convective motion in the outer core, and its implications for the stabilization of the magnetic field against reversal, it is important to understand the magnetic state of iron and its behavior at high pressures [25–28]. Magnetic fluctuations and the reorientation of magnetic moments as well as a changing electronic structure at or near the Fermi energy can also significantly affect the phase stability of iron and its alloys [11, 19, 29–32]. Theoretical and experimental studies show that bcc iron is an equilibrium structure [5, 6, 11, 33–35]. In contrast to the hcp phase, which is non-magnetic (NM) and the ground state at high pressures, the bcc phase carries a finite magnetic moment even at high pressures. Previous theoretical
Several lines of evidence suggest that the composition of the earth’s core is dominated by an iron-rich Fe1−x , Nix alloy with x ∼ 0.1. The crystal structure of iron in the solid inner core has long been assumed to be hcp [1–7]. However, over the past decade, both experiment and theory have shown evidence for the fcc and bcc phases as well [8–17]. Among the proposed phases, the bcc phase stands out in that it retains a finite magnetic moment even at inner core pressures [18]. In contrast, the fcc and hcp phases are paramagnetic at high pressures [18–24]. Due to the different modes of interactions between magnetic and non-magnetic inner core phases with 0953-8984/14/046001+08$33.00
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c 2014 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 046001
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parameters have been carefully chosen to obtain converged solutions to the Kohn–Sham equations. All degrees of freedom were allowed to relax simultaneously at constant volume during the simulations; the c/a-ratio in the hcp lattice is allowed to change in order to find the lattice of the ground state. The pressure was obtained from the trace of the stress tensor. Equation of state (EOS) parameters were obtained by fitting a third-order Birch–Murnaghan equation of state to our E(V) curves [53, 54]. The COOP, P or orbital-population weighted DOS, can be written as .ij COOP(ε) = DOS(ε), where ε is the energy of interest and the summation must be taken over all possible overlaps between state i and j. Thus, the COOP allows one to distinguish between bonding and antibonding orbital interactions, depending on the sign of the overlap matrix element [55]. The COOP for an atomic centered basis-set, as used in our computations, allows a convenient decomposition of the DOS into distinct overlap contributions between nearest neighbor atoms in the simulation cell. This is particularly useful for the investigation of magnetic properties: for non-magnetic solutions the COOP of the majority and minority spin channels are mirror images of each other. In the case of magnetic solutions the COOP shows a relative shift between majority and minority spins channels, as does the DOS. The information content of the COOP mimics the hybridization between orbitals: bonding orbitals appear with positive amplitude, while antibonding orbitals appear with negative amplitude. A zero overlapping integral results in a non-bonding state between atomic orbitals [56–58]. Therefore the COOP analysis provides a direct method to deconvolve complex interactions in terms of overlap contributions and to follow the changes of the electronic structure under changing conditions. (Anti) bonding states (de)stabilize a given electronic structure and COOP provides a detailed picture, for example, of the evolution of magnetism with pressure. Coincidentally, the same analysis also allows one to decompose the internal energy into orbital contributions through crystal orbital Hamilton population (COHP) analysis, hence providing insights into stabilizing and destabilizing energy contributions not only of the magnetic structures but also of the associated crystal phases. A negative sign of the Hamiltonian matrix of the COHP in the case of positive or bonding COOP means that the total energy of a crystal structure is lowered and hence stabilized.
studies show that the ferromagnetism (FM) of bcc iron is responsible for its stability at ambient conditions [21, 36–39]. The electronic density of states (DOS) of elemental iron shows a significant broadening with pressure due to the direct overlap of the 3d derived electronic bands between neighboring iron atoms. The explanation of the formation of FM structures often relies on the Stoner instability. This condition is given by IN(εf , V) > 1, where N is the density of states at the Fermi level at magnetic moment (µ = 0) and I (µ = 0, V) is the Stoner parameter as a function of magnetic moment and volume (V) [40–42]. On the other hand, the Fermi surfaces and the relative bandwidth of majority and minority spin bands successfully explain FM in bcc iron [43]. The stabilization of the FM state can be attributed to a narrow bandwidth of the majority spin band. Previous ab initio calculations commonly underestimate the equilibrium volume of elemental iron as compared to experiment [3, 22, 44, 45]. However, magnetism depends on orbital overlap and hence on the nearest neighbor distances. Thus, it is not clear if the magnetism in previous computations could be biased toward low values by systematically underestimated volumes. Here we explore the effect of increased volumes relative to experiment on the magnetism in elemental iron. The evolution of magnetism in the bcc, hcp, and fcc phases of elemental iron is explored by tracking the crystal orbital overlapping population (COOP) analysis with increasing compression. The same analysis allows one to deconvolve the total energy in terms of orbital contributions and to gain new insights into the destabilization of the bcc phase itself relative to the close-packed phases—fcc and hcp—at pressure. 2. Computational details
The DFT calculations have been performed using the Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA) [46, 47]. Studies using SIESTA that address the behavior of bulk transition metal systems have been limited to equilibrium structures [19, 48–50]. Electronic exchange and correlation effects are treated within the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) parameterization [51]. Interactions between the nuclei and electrons are described with norm-conserving Troullier–Martins pseudopotentials [52]. A potential with the 4s2 3d6 valence electronic configuration (Rc = 2.0 au) for iron was generated from atomic all-electron computations. A cut-off energy of Ecut = 200 Ryd for the expansion of the band-states and standard Monkhorst–Pack 16 ×16 ×16 and 16 ×16 ×12 k-point grids were used for the cubic phases (fcc and bcc) and the hexagonal (hcp) phase, respectively. A double-zeta plus polarization (DZP) basis-set is used for the expansion of the electronic wavefunctions, with an energy shift of 50 meV. Forces and total energy per atom were converged to better than 0.04 eV A−1 and 1 meV, respectively, and an energy smearing of 25 meV was applied. All computations were conducted in spin-polarized mode. No thermal corrections were considered in this study; all calculations were carried out at 0 K. The computational
3. Results and discussion 3.1. Equation of state
Our results are in good agreement with available experimental values of hcp iron, as shown in figure 1 and table 1. Both, fcc and hcp structures show a magnetic transition from ferromagnetic to non-magnetic, as described previously [22, 45, 59, 60]. The volume reduces by 6% and 3% across this magnetic to non-magnetic transition at pressures of 54 GPa and 50 GPa, for the fcc and hcp phases, respectively. The computed EOS 2
J. Phys.: Condens. Matter 26 (2014) 046001
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Table 1. Third-order Birch–Murnaghan equation of state parameters of ferromagnetic bcc, fcc and hcp iron. Values in parentheses refer to the uncertainty of the last digit.
Phase
Magnetic state
V0 (A3 /atom)
K0 (GPa)
K 0 (—)
bcc/this study bcc/this study bcc/expa hcp/this study hcp/this study hcp/this study hcp/expb fcc/this study fcc/this study fcc/this study fcc/expc (model)
FM NM FM NM FM1 FM2 FM NM FM1 FM2 FM
11.759 10.718 11.77(3) 10.505 10.525 12.169 11.16(2) 10.684 10.682 12.465 11.830
177.72 276.52 173.0 (9) 297.13 295.16 157.52 165 (4) 269.05 274.57 192.83 —
4.564 4.153 — 3.934 4.268 4.113 — 4.120 4.2000 4.0000 —
a b c
E0 (eV/atom) −781.890 −781.370 — −781.750 −781.50 −781.710 — −781.630 −781.610 −781.790 —
Reference [15, 16]. Reference [7, 37]. Reference [17].
modulus, due to the correlations between the EOS parameters, V0 , K0 , and K00 . However, the very good agreement between experiment and theory (table 1) may be somewhat fortuitous: the bcc → hcp transition, which is known to be in the range of 13–20 GPa [62], occurs in our calculations at a pressure of 24 GPa, higher than the observed experimental values. The EOS parameters show a large discrepancy between computations and experiment. This discrepancy refers to the underestimation of the lattice parameters even in the GGA approximation, which generally tends to overestimate lattice parameters and volumes. For some properties this may not be of significance, but for properties that rely strongly on the amount of overlap of electronic wavefunction, such as magnetism and compression, this can have a large effect. This distortion can be caused by external pressure, high temperature, and the existence of light element impurities in the crystal structure. Using the GGA approximation in conjunction with an atomic centered basis-set we validate our results by comparison with previous results; this is presented in table 1. This knowledge is difficult to obtain by other methods (for example, from the more widely used plane wave approaches) and gives new insights into the causes for the disappearance of magnetism in hcp and fcc iron and the residual magnetism in the competing bcc phase at high pressure. After this validation we precede to the novel parts of the presented work, namely the evolution of the magnetism in the known phases of elemental iron. The COOP analysis allows us to track the contributions from 4s and 3d–4s hybridizations that give detailed insights into the pressure dependence of magnetism in these phases. Thus, the orbital resolved analysis of magnetism for elemental iron structures at both ambient and high pressures will be discussed in the following sections. The prediction of an intermediate spin state in fcc iron suggests a volume sensitivity of magnetism in the sense that increased volume at constant pressure can be used to stabilize non-trivial spin-states in 3d compounds, or at least in elemental iron. Therefore these analyses extend previous work and advance the knowledge of elemental iron magnetic structures at high pressures.
Figure 1. Comparison of the calculated EOS with available
experimental data. Open squares, experimental data from [7, 30]. Solid line represents ferromagnetic bcc; dashed line ferromagnetic fcc; dotted line ferromagnetic hcp. For fcc and hcp, the lines show the fits of the low-pressure region with finite magnetic moment and the non-magnetic high-pressure region for the EOS parameters given in table 1.
parameters of bcc, hcp, and fcc structures are found to be in very good agreement with experiment (table 1). Previous computations proposed several antiferromagnetic structures—AFM, AFM I, AFM II—and non-collinear spin ordering in order to reach better agreement between the predicted and observed compressional behavior of elemental iron [18, 22, 59, 61]. In general these structures lead to larger equilibrium volumes, reduce the bulk modulus, and achieve a better agreement between theory and experiment. For example, the bulk modulus of the AFM II structure is 209 GPa, which is significantly ∼30% lower than that for NM-hcp iron (292 GPa) [21]. Test calculations show that the AFM II structure is more stable than FM-hcp iron at low pressures; hence, the discrepancies between experiment and theory cannot be attributed to the underestimated volumes in previous computations (relative to experiment). As expected, the increased V0 in our computations decreases the bulk 3
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The high contribution is due to 3d–3d interactions and 4s–3d hybridization, and minor contributions from the s and p orbitals [65]. The shift in majority and minority spin states and the strength of the exchange parameter are the primary reasons for strengthening of the ferromagnetic bcc structure in the Stoner model [41, 60]. The stability of a particular magnetic structure requires a high cumulative density of states (integration of DOS up to Fermi level) and a large amount of splitting between the majority and the minority bands. This is achieved in iron by lowering the energy of bonding states. The corresponding depopulation of antibonding states at the Fermi energy (EF ) increases the exchange energy and decreases the kinetic energy. The COOP curves in figure 4(b) give insights into the contributions of orbital overlap to the DOS. In these curves we notice a large splitting between majority and minority spin bands, the t2g band (lower in energy) contributes strongly to the bonding states while eg band electrons contribute to the antibonding states, the 4s–3d hybridization interaction shows a strong contribution to the bonding states at the Fermi level, while 4s–4s has a small contribution to the bonding states and a somewhat larger contribution to the antibonding states. Overall, the bonding states dominate and a magnetic FM structure is stabilized. At a pressure of 600 GPa, the shifts between spin-up and spin-down components are significantly smaller than at equilibrium, and the overall bandwidth increases, as shown in figures 4(c) and (d). As the bands broaden, the amplitude decreases due to the finite number of available states. According to the Stoner model, this decrease in DOS at the Fermi level and the decrease of the exchange-splitting between majority and minority spins (which is proportional to the energy difference between bonding and antibonding states of the majority and minority bands) decreases the stability of the magnetic structure, and the magnetic moment decreases gradually with increasing compression, as predicted (figure 3(a)). This is an important property of bcc iron, since both fcc and hcp structures show a rapid decrease in magnetic moments at much lower compression, as discussed above. Both majority and minority spin bands (eg as well as t2g bands) contribute to the DOS as the pressure increases, the remaining offset of majority and minority spin bands is the main reason for the magnetic nature of the bcc structure. Deconvolving the orbital contributions of this DOS in figure 4(d) shows that 4s–4s interactions are completely overlapped as bonding states which are approximately twice as high as compared to the values at equilibrium. This suggests that the 4s electrons have no contribution to the magnetic nature of the bcc structure. The splitting of the sd-bands decreases significantly from 1.47 eV at equilibrium (figure 4(b)) to 0.6 eV at 600 GPa (figure 4(d)). Furthermore the bands broaden with decreasing volume (increasing pressure) as expected. In contrast, the dd-bands are weakly affected by pressure and the main cause of magnetism in bcc iron, while the s–d overlap does not increase significantly. In summary, a lower density of states, more antibonding states at the Fermi level, and broadening of majority and minority bands are insufficient to stabilize a non-magnetic structure while a magnetic moment remains.
Figure 2. Energy–volume relationship for iron. Solid line for
ferromagnetic bcc; dashed line for ferromagnetic fcc; dotted line for ferromagnetic hcp iron. The decreasing energy range between V = 9.4 and V = 9.9 A3 /atom corresponds to the compression range of rapidly changing magnetic moment.
3.2. Magnetic moment
In agreement with previous computations and experiment we find that the FM bcc phase is the most stable phase at low pressures, as shown in figure 2. At equilibrium the energy difference between FM and NM bcc iron is about 0.74 eV/atom in favor of the FM phase. This enthalpy difference decreases monotonously as the pressures increases, and the bcc phase is predicted to be NM at pressures in excess of 600 GPa. In contrast to the paramagnetic nature of both fcc and hcp phases, FM is stabilized in the bcc phase to much higher compression, as shown in figure 3. Furthermore, both fcc and hcp structures possess a magnetic transition in the range 50–60 GPa. The present study also predicts that magnetism in the fcc phase disappears via an intermediate spin state in figure 3(b). This feature, which is unique for the fcc phase, is most likely associated with the overestimation of V0 as compared to previous computations and experiment. This finding suggests that it may be possible to stabilize non-trivial intermediate magnetic structures by overextending the crystal structure, a finding that is similar to that observed in compounds such as Fe and FeNi, where an increasing Fe concentration is used to extend the volume of fcc FeNi and to stabilize particular magnetic structures [63]. 3.3. DOS and COOP analysis
The evolution of the total energy of all iron structures with compression can be attributed to shifts between majority and minority spin bands relative to the Fermi energy. The strength of the exchange-splitting parameter between atoms is dominated by the nearest neighbor distance between atoms, the amount of orbital overlap, and the hybridization of orbitals [64]. The density of states in bcc iron is mainly contributed by the triply degenerate t2g and doubly degenerate eg bands, as shown in figure 4(a). 4
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Figure 3. Evolution of magnetic moments with compression in (a) ferromagnetic bcc Fe, (b) ferromagnetic fcc Fe and (c) ferromagnetic
hcp Fe.
Figure 4. (a) Contributions to the DOS of bcc iron at the equilibrium volume. Dash–dot line: d–d contribution, filled area: s–s contribution,
dashed line: s–d hybridization contribution. (b) COOP analysis of majority and minority spin subbands of bcc iron at equilibrium volume. Dash–dot line: d–d electrons, dashed area: s–s electrons, dashed line: s–d electrons. (c) Contributions to DOS of bcc iron at 600 GPa. Dash–dot line: d–d contribution, filled area: s–s contribution, dashed line: s–d hybridization contribution. (d) COOP analysis of majority and minority spin subbands of bcc iron at 600 GPa. Dash–dot line: d–d electrons, dashed area: s–s electrons, dashed line: s–d electrons.
The 4s–4s, 3d–3d, and 4s–3d contributions to the DOS of fcc iron at ambient conditions are shown in figure 5(a). At equilibrium the magnetism in fcc iron is stabilized by exchange-splitting, as in bcc iron. The COOP analysis of fcc iron at ambient conditions in figure 5(b) shows that the
dominant contributions are again due to antibonding ss and dd interactions. Only sd provides bonding contributions to the DOS. As pressure increases, the low density of states of fcc iron at ambient conditions and the highly antibonding nature of ss 5
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Figure 5. (a) Contributions to the DOS of fcc iron at the equilibrium volume. Dash–dot line: d–d contribution, filled area: s–s contribution,
dashed line: s–d hybridization contribution. (b) COOP analysis of majority and minority spin subbands of fcc iron at equilibrium volume. Dash–dot line: d–d electrons, dashed area: s–s electrons, dash line: s–d electrons. (c) Contributions to DOS of fcc iron at 600 GPa. Dash–dot line: d–d contribution, filled area: s–s contribution, dashed line: s–d hybridization contribution. (d) COOP analysis of majority and minority spin subbands of fcc iron at 600 GPa. Solid line: d–d electrons, dashed area: s–s electrons, dashed line: s–d electrons.
and dd interactions result in broadening of the 3d bands, as shown in figure 5(c), and an overall antibonding fcc structure, which leads to a much more rapid destabilization of the magnetic structure as compared to the bcc phase (figure 3). As a result the FM ⇒ NM transition occurs at ∼60 GPa (figures 2 and 3(b)). With increasing pressure antibonding states move above the Fermi energy and the bonding states dominate, which stabilizes the fcc phase relative to the bcc phase. The COOP curves of the fcc structure at high pressure are shown in figure 5(d). Larger s–s interactions are positively overlapped as bonding states, a smaller fraction negatively bonding, and zero states at the Fermi level. s–d interactions are only positively overlapped. Finally, d–d interactions show a larger fraction of negative overlapping at the Fermi level, these antibonding states at the Fermi level destabilize the fcc structure and a larger pressure is needed to remove these antibonding states at the Fermi level. Overall, the COOP analysis results in an overall positively overlapping interaction, which means fcc iron is a strong candidate as a stable structure at high pressures. The density of state of FM-hcp iron at ambient conditions is significantly higher than those of the fcc phase and the exchange-splitting is smaller, as shown in figure 6(a).
In contrast to the fcc structure, the COOP curves for FM-hcp iron in figure 6(b) show a larger antibonding d–d interaction at the Fermi level and a smaller fraction positively bonded. The s–s interactions have positive and negative contributions, which compensate their contribution to the structural stability of hcp iron at equilibrium (figure 6(b)). At 600 GPa, all bands are completely overlapped in bonding and antibonding states, which results in a NM structure, as shown in figure 6(c). The COOP curves of hcp iron at 600 GPa are shown in figure 6(d). As the pressure increases, some of the antibonding states due to d–d interactions at the Fermi level have been removed as compared to the fcc case. Also, here, s–s and s–d are completely positively overlapped, favoring a NM magnetic structure. The striking similarities between fcc and hcp structures are highlighted by the very similar changes of magnetic moments with compression (figure 3(b)), which reflects the fact that the fcc and the nearly ideal hcp structure share the same high coordination number (CN = 12) and a weakly pressure dependent c/a ratio. Comparing the COOP provides an additional tool to distinguish the magnetism and structural stability of the fcc and hcp phases. The comparison of the DOS and COOP analysis of both the fcc and hcp structures show a lower density of antibonding 6
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Figure 6. (a) Contributions to the DOS of hcp iron at the equilibrium volume. Dash–dot line: d–d contribution, filled area: s–s contribution,
dashed line: s–d hybridization contribution. (b) COOP analysis of majority and minority spin subbands of hcp iron at equilibrium volume. Dash–dot line: d–d electrons, dashed area: s–s electrons, dashed line: s–d electrons. (c) Contributions to DOS of hcp iron at 600 GPa. Dash–dot line: d–d contribution, filled area: s–s contribution, dashed line: s–d hybridization contribution. (d) COOP analysis of majority and minority spin subbands of hcp iron at 600 GPa. Solid line: d–d electrons, dashed area: s–s electrons, dashed line: s–d electrons.
states at the Fermi level. The comparisons of the magnitudes of the orbital overlap contributions show that the magnetic fcc and hcp phases are stable at high pressures. However, the fcc and hcp structures are NM at high pressures due to zero splitting between the majority and minority spin bands. Both the fcc and hcp phases show an overall positive (bonding) overlap, which leads to the stabilization of the NM structures at high pressures as compared to the competing bcc phase, at least at low temperatures. Among the close-packed structures, hcp is more favorable than fcc in terms of the DOS and overall positive overlapping, consistent with the enthalpy curves and the known low-temperature phase diagram of iron.
transition from high spin to non-magnetic via an intermediate spin state for the overextended lattice considered here. In agreement with the known phase diagram of iron, the COOP (COHP) analysis shows that the bcc phase is the stable structure at ambient conditions. Furthermore, the COOP decomposition of the DOS shows that magnetism in bcc iron is present to much higher pressures, due to a reduced s–s and s–d antibonding contribution. The results are consistent with the splitting between the majority and minority subbands and with the expansion of bandwidths according to the Stoner model. The COHP analysis of the orbital overlap contributions to the total energy shows that both fcc and hcp are unfavorable at ambient conditions, relative to bcc. In contrast, both NM-fcc and NM-hcp structures are found to be strong candidates for the stable phase at high pressure, due to the higher contribution of bonding states at the Fermi energy. The bonding contributions of the hcp and fcc phase at the Fermi level, showing that the close-packed phases are more favorable at high pressures than the bcc phase. The detailed analysis of the changing magnitude of the orbital overlap shows that the hcp phase is expected to be the more stable iron structure at high pressures, at least up to a pressure of 600 GPa.
4. Conclusion
We performed electronic structure calculations for iron using the atomic centered basis-sets implemented in SIESTA. The results show that the magnetic state in the bcc, fcc, and hcp phases is independent of an under- or overestimated equilibrium volume, at least up to ∼(3–6)% relative to experiment. Therefore, differences between experimental and theoretical EOS at low pressures are likely intrinsic to iron and not an artifact of the underestimated equilibrium volume in previous computations. The fcc phase shows a magnetic 7
J. Phys.: Condens. Matter 26 (2014) 046001
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Acknowledgment
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