Phase stability of Fe and Mn within density-functional theory plus on-site Coulomb interaction approaches

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Journal of Magnetism and Magnetic Materials 320 (2008) 100–106 www.elsevier.com/locate/jmmm

Phase stability of Fe and Mn within density-functional theory plus on-site Coulomb interaction approaches N.L Stojic´a,b,,1, N.L Binggelia,b a

Abdus Salam International Centre for Theoretical Physics, Trieste 34014, Italy b Democritos National Simulation Center, INFM-CNR, Trieste I-34014, Italy Received 24 April 2007; received in revised form 11 May 2007 Available online 24 May 2007

Abstract Approaches based on the density-functional theory and including the on-site Coulomb interaction U have been extensively used to describe strongly correlated systems. Moreover, it has been shown that even in the case of moderate correlations, present for example in some of the 3d transition metals, this and similar methods can improve upon the local-density and generalized-gradient approximation (LDA and GGA) results. We investigate, by means of the LDA þ U and GGA þ U approaches, the phase stability of Fe and Mn, for which it is known that the LDA predicts a wrong ground state. In particular, we compare two different double-counting corrections, the so called ‘‘fully-localized limit’’ (FLL) and ‘‘around mean-field’’ (AMF). We find that the LDA and the LDA þ U AMF do not yield the correct ground state, while the LDA þ U FLL , GGA þ U AMF and GGA þ U FLL for specific ranges of effective U values, typically around 1 eV, give the correct phase stability, and in general, an improved description of the equilibrium volume and magnetic moment, compared to the GGA values. r 2007 Elsevier B.V. All rights reserved. Keywords: Fe; Mn; Phase stability; Density-functional theory; LDA+U

1. Introduction A generalization of the density-functional methods based on the local density approximation (LDA) or generalized gradient approximation (GGA) to improve the description of the strongly correlated systems by adding an on-site Coulomb interaction, U, has been widely used.2 This so-called LDA þ U approach [1] has been applied with great success in the late-transition-metal oxides and rare-earth metal compounds, where the LDA could not describe well the effects of the high degree of localization of the valence d (or f) electrons. This method includes an orbital dependence of the Coulomb and exchange interacCorresponding author. Tel.: +39 040 3787 317; fax: +39 040 3787 528.

E-mail address: [email protected] (N.L. Stojic´). Present address: SISSA- Scuola Internazionale Superiore di Studi Avanzati, 34014 Trieste, Italy. 2 For simplicity of the notation, in this paper we use ‘‘LDA’’ and ‘‘GGA’’ also for their generalization to the spin-polarized case: the localspin density approximation (LSDA) and the spin-polarized GGA (sGGA), respectively. 1

0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.05.011

tions which is not present in the regular density-functional potential. It gives a better description of the band-gap, magnetic and spectroscopic properties than the LDA and GGA alone. Recently, it has been applied to some less correlated systems [2–7], also often yielding an improved description of the magnetic and spectroscopic properties of those systems. In such cases, the LDA þ U ðGGA þ UÞ may also be viewed as an approach to correct for the selfinteraction within the LDA (GGA). In particular, Yang et al. [2] found that the addition of a small on-site parameter within the LDA þ U corrects the previous density-functional results on the magnitude of magnetic anisotropy energy and direction of magnetization in the cases of Fe and Ni, bringing them in better agreement with experiments. Also for Fe and Ni, Chioncel et al. [8] went beyond the LDA þ U investigating the spectroscopic properties of Fe and Ni within the LDA plus dynamical mean field theory ðLDA þ DMFT Þ, which takes into account dynamical fluctuations and of which the LDA þ U method is the static limit. For Ni, in particular, they explained a satellite observed at energies well below

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the Fermi level in the photoemission spectrum, which was not accounted for by the LDA or GGA. Cococcioni and de Gironcoli [6] found that the Fe bandstructure, obtained by the LDAðGGAÞ þ U method, agrees better with the experimental data, compared to the LDA and GGA alone. Finally, we note that Mohn et al. [3] also reported that for values of U larger than 3.7 eV, they were able to obtain a nonmagnetic ground state for FeAl, as found experimentally, whereas the LDA and GGA yield only a ferromagnetic ground state. This explanation, however, is controversial, and the FeAl paramagnetism has been interpreted more recently in terms of critical spin fluctuations within the LDA+DMFT [9]. Surprisingly, despite a relatively large number of studies showing improvement of various properties of 3d-transition metals using the LDA þ U (or LDA þ DMFTÞ method, the influence of the on-site parameter on the stability of the different structural and magnetic phases of 3d-transition metals has not been investigated so far, to the best of our knowledge. This is, however, an important issue to achieve a consistent description of these systems. The issue of phase stability is especially relevant in the cases of Fe and Mn, where the ground state is known to be sensitive to the exchange-correlation functional used. For example, the LDA incorrectly predicts a nonmagnetic ground state for both Fe and Mn [10,11], while the GGA correctly yields the magnetic ground state of these systems [12–15]. In this paper, we investigate the influence of the on-site Coulomb correction on the phase stability, magnetic, and structural properties of Fe and Mn within the LDAðGGAÞ þ U, using two different flavors of the on-site correction, originally designed for moderately and strongly correlated systems. 2. Method The LDA þ U method goes beyond the LDA (or GGA) by treating exchange and correlation differently for a chosen set of states, in the present case, the 3d orbitals. These orbitals are treated with an orbital-dependent potential, derived from the Hubbard model, and defined in terms of the on-site Coulomb and exchange interaction parameters U and J, respectively. To avoid double counting of the electron–electron interactions, which are already taken into account in an average way within the LDA (GGA), one needs to apply a double counting correction, so that the energy functional is of the following form: E LDA þ U ¼ E LDA þ E ee  E dc ,

(1)

where E ee is the intra-atomic electron–electron interaction among the 3d electrons and E dc is the double-counting correction. Most commonly, two types of double-counting correction are used: one focuses on the limit of uniform occupancy, the so-called ‘‘around mean field’’ correction (AMF) [16], and the other one, for which E dc term satisfies the atomic-like limit of the LDA, is also called ‘‘fully-

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localized limit’’ (FLL) [17]. For the E ee as given by Dudarev et al. [18] in the spherically averaged form of the rotationally invariant LDA þ U, the on-site correction DE LDA þ U to E LDA reads: U JX Trðdrs  drs Þ, (2) DE AMF LDAþU ¼  2 s DE FLL LDAþU ¼ 

U JX ½Trðrs  rs Þ  ð2l þ 1Þns , 2 s

(3)

where drsm;m0 ¼ rsm;m0  ns dm;m0 , with rsm;m0 being the orbital occupation matrix and ns the average occupation per orbital of spin s given by N s =ð2l þ 1Þ, with N s ¼ Trðrs Þ and l the orbital momentum. In this paper, we will use U eff ¼ U  J [19,18]. Notwithstanding the great success of the LDA þ U method, its correct parametrization remains problematic, as the determination of U and J is in large portion empirical, method-dependent and with large error bars (around 1 eV). In the case of Fe, an analysis of the experimental Auger spectra [20,21] provides a value for U around 1 eV. Theoretically determined values of U range from rather unrealistic values of 5–6 eV [22] to more moderate values of U eff ¼ 2:1 eV [6] from first-principles calculations and values of U ¼ 1:2 eV, J ¼ 0:7 eV [23] and U ¼ 1:2 eV, J ¼ 0:8 eV [2] obtained empirically from comparison of calculated and experimental photoemission spectra, in the former case, and magnetic anisotropy, in the latter case. In the present work, we will consider values for U eff of 1 and 2 eV, both for Fe and Mn. Our LDA þ U calculations, using the AMF and FLL corrections described above, have been performed with the WIEN2k implementation [24] of a full-potential linearized augmented plane-wave (FLAPW) method. We used the tetrahedron method for Brillouin zone integrations. For the exchange and correlation energies, we used the GGA as given by the Perdew–Burke–Ernzerhof parametrization [25] and for the LDA the interpolation formula, given by Perdew and Wang [26] (in their spin-polarized version). The muffin-tin radius was set to 2.20 a.u. for Fe and to 2.17 a.u. for Mn, and the product of the muffin-tin radius and the maximum reciprocal space vector K max , RMT K max was equal to 9.5. For Fe, we used 406 k-points in the irreducible Brillouin zone for the BCC and FCC lattice, and 413 for the HCP lattice. For Mn, we used up to 1200 for the FCC and up to 1240 k-points for the HCP lattice. The total energy was always converged to better than 6 meV/atom and the moment was converged to better than 0:03 mB. We included 3s and 3p states in the semicore. In calculations to determine the equilibrium volume, we also used an additional local orbital for the high-lying 4d-states. 3. Results 3.1. Iron Experimentally, iron has a body-centered-cubic (BCC) ground state, which is ferromagnetic at normal pressure

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Table 1 DE in meV/atom from the lowest energy phase for Fe using LDAðGGAÞ þ U, U 1 ¼ 1 eV, U 2 ¼ 2 eV.

BCC FM BCC NM FCC FM FCC NM HCP FM HCP NM

LDA

LDA þU FLL 1

LDA þU FLL 2

LDA þU AMF 1

LDA þU AMF 2

GGA

GGA þU FLL 1

GGA þU FLL 2

GGA þU AMF 1

GGA þU AMF 2

144 400 – 74 – 0

0 540 155 194 214 118

0 870 106 526 172 451

115 433 77 76 – 0

73 413 54 75 – 0

0 488 143 164 – 96

0 840 92 495 160 427

0 1200 60 905 123 838

0 509 122 197 – 129

0 538 55 244 – 178

‘‘–’’ denotes that the magnetic state is unstable and relaxes towards the NM solution.

Table 2 Lattice constants and magnetic moments for FM BCC Fe in LDAðGGAÞ þ U, U 1 ¼ 1 eV, U 2 ¼ 2 eV.

a(bohr) mðmB Þ a

LDA

LDA þU FLL 1

LDA þU FLL 2

LDA þU AMF 1

LDA þU AMF 2

GGA

GGA þU FLL 1

GGA þU FLL 2

GGA þU AMF 1

GGA þU AMF 2

exp

5.209 2.02

5.238 2.23

5.317 2.58

5.215 1.98

5.212 1.89

5.356 2.18

5.452 2.57

5.504 2.73

5.355 2.12

5.350 2.02

5.405a 2.22b

Experimental lattice constant is extrapolated to T ¼ 0 K [28]. Ref. [29].

b

below 1183 K (a phase) and nonmagnetic above 1663 K, up to the melting point of 1807 K (d phase) [27]. In Table 1 we compare our results for the energies of different phases of Fe, calculated using the LDA, GGA, LDA+U and GGA+U, for both double-counting corrections. The energies are compared at the equilibrium volume, calculated for each phase. For this procedure, we applied the same U eff for each volume. The phases considered were ferromagnetic (FM) and nonmagnetic (NM) BCC, FCC and ideal HCP structure. In the LDA, the lowest energy states are nonmagnetic HCP and FCC, while with an addition of U in FLL scheme, the ground state becomes the experimentally found FM BCC structure. Even an U eff of 1 eV suffices to change the ground state into a stable FM BCC, ð120 meV below NM HCP) and induces the existence of metastable FM FCC and HCP phases. With a larger U eff (2 eV), all three FM structures (BCC, FCC, HCP) become more stable relative to their NM counterparts. In the case of the AMF scheme, for U eff up to 2 eV, the ground state remains the NM HCP.3 We note that increasing U eff tends to decrease the energy of the FM BCC phase relative to the other phases. In contrast, GGA finds the FM BCC phase to be the lowest in energy, followed by the NM HCP. Adding a small U eff in the FLL scheme, the ordering of the phases changes in the same 3 We also checked for the antiferromagnetic HCP phase (spins in each (0 0 1) plane parallel and opposite between the planes), since it was found to be the most stable HCP phase [30], and we found that the NM and antiferromagnetic phase were virtually degenerate, i.e. their energy difference was inside the numerical uncertainty of the calculation. However, since HCP is not the experimental ground state, we did not compare its antiferromagnetic phase to other structural phases with antiferromagnetic order.

sense as in the LDA þ U FLL ðU eff ¼ 2 eVÞ, stabilizing even more the FM phases for all the structures. For GGA þ U AMF , the ground state remains the FM BCC, which is, for both considered U eff in this scheme, followed by the FM FCC. Table 2 gives the equilibrium lattice constants and magnetic moments (sum of the moments inside the muffintin radius and the interstitial contribution) for the FM BCC phase, calculated within the different schemes. The LDA lattice constant and magnetic moment are too small compared to the experimental value. The LDA þ U FLL for U eff ¼ 1 eV increases the lattice constant, and gives the correct value for the moment. However, further increase of U eff results in too large a moment and a lattice constant still too small, although improved with respect to a smaller U eff . The LDA þ U AMF has different trends with increase of U eff . The lattice constant does not change significantly with respect to the LDA value (3:5% too small), while the moment decreases with increasing of U eff in this scheme. The GGA’s lattice constant is 1% too small and the moment is in close agreement with the experimental value. With the inclusion of U FLL the lattice constant and the magnetic moment increase. For U eff ¼ 2 eV, the lattice constant is 2% and the moment 23% too large. GGA þ U AMF yields a similar lattice constant and a slightly smaller magnetic moment, compared to the GGA. Further increase of U eff does not alter significantly the lattice constant and slightly reduces the magnetic moment. Comparing our results with the previously published calculations in the LDA [10,13] and GGA, [12,31,13,32,33], we find no significant difference. A large number of previous calculations have been performed using a different GGA parametrization, the one by Perdew and

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Table 3 DE in meV/atom from the lowest energy phase for Mn using LDAðGGAÞ þ U, U 1 ¼ 1 eV, U 2 ¼ 2 eV

FCC AFM FCC NM FCC FM HCP AFM HCP NM

LDA

LDA þU FLL 1

LDA þU FLL 2

LDA þU AMF 1

LDA þU AMF 2

GGA

GGA þU FLL 1

GGA þU FLL 2

GGA þU AMF 1

GGA þU AMF 2

– 36 – – 0

0 29 – 29 21

0 350 352 179 308

22 21 – – 0

9 7 – – 0

0 43 – 12 14

0 334 331 228 273

0 948 203 137 860

0 55 49 29 39

0 73 71 48 75

‘‘–’’ denotes that the magnetic state is unstable and relaxes towards the NM solution.

Wang [34]. Using this GGA for the FM BCC phase, we find almost identical lattice constant and 1% smaller magnetic moment, compared to the values presented in Table 2. Concerning the effect of the U within the GGA þ U, we find similar trends for the change in the lattice constant and magnetic moment as in the pseudopotential study by Cococcioni and de Gironcoli [6].4 Overall, for Fe, we can say that the LDA and LDA þ U AMF are unsatisfactory, as they give a wrong ground state and too small lattice constants and magnetic moments. The LDA þ U FLL , instead, corrects the LDA, predicting the right ground state and increasing the values of the lattice constant and magnetic moment. However, based on the comparison with experiment, U eff should be around 1 eV, because for a larger value, the magnetic moment is significantly greater than the experimental one. For the GGA, the addition of a U eff within the FLL scheme is improving the lattice constant, but causing too large magnetic moment. For this reason, in this case, U eff should be rather small, less than 1 eV (for U eff ¼ 0:5 eV, a ¼ 5:380 bohr and m ¼ 2:36 mB ). Finally, for the GGA þ U AMF , with U eff 1 eV, the agreement with experiment of the lattice constant and magnetic moment is comparable to that of the GGA. We note, however, that improvement in the description of other properties, such as the bandstructure, has been found using this method [6]. 3.2. Manganese Experimentally, the low-temperature, low-pressure phase of manganese is the a phase, which exists up to 1073 K and has a complex structure with 58 atoms per unit cell [36,27]. The high-temperature g phase (1373– 1407 K) is face-centered-cubic (FCC) and can be quenched to room temperature. Below the Nee´l temperature of about 540 K, it is a tetragonal antiferromagnet [face-centered tetragonal (fct) structure] [35]. Upon addition of a small amount of impurities, it can be quenched also in the FCC structure [35]. In Table 3, we compare the calculated energies of different Mn phases with respect to the lowest-state energy, for the LDA þ U and GGA þ U methods with the two 4 In Table I of Ref. [6], the label LDA þ U refers to GGA þ U calculations (M. Cococcioni, private communications).

double-counting corrections. For manganese, we considered antiferromagnetic (AFM) and NM HCP, FCC, FM FCC and AFM fct phases. The AFM phase assumes the same direction of spins inside a (0 0 1) plane and opposite in the next plane along the [0 0 1]-direction. This AFM arrangement corresponds to the a-Mn phase and quenched g-Mn. Here, we restrict ourselves to the g-Mn phase. The LDA gives a wrong ground state, NM HCP structure, and for FCC predicts only NM phase, while the AFM ordering exists for larger volumes. The LDA þ U FLL , in contrast, predicts the AFM FCC structure as the ground state, followed by the NM HCP (for U eff ¼ 1 eV). Increase of U eff lowers the energy of the AFM HCP phase relative to the NM HCP phase. The LDA þ U AMF , like the LDA, finds NM HCP to be the lowest in energy. We notice that the AFM and NM FCC phases are very close in energy and that FM FCC phase can be stabilized only in LDA þ U FLL with U eff ¼ 2 eV (although with a small magnetic moment m ¼ 0:9 mB ). The GGA yields the AFM FCC structure as the ground state, and the second lowest is the AFM HCP. In GGA þ U FLL the energy differences increase, and the FM FCC gets stabilized. For U eff ¼ 2 eV in this scheme, the energy differences between magnetic and NM phases increase further. GGA þ U AMF , with U eff ¼ 1 eV, also gives the same ordering of phases, as the GGA, but for U eff ¼ 2 eV the two highest lying phases change order. We have also calculated the AFM fct phase of Mn, finding that for the LDA, LDA þ U AMF and LDA þ U AMF , this phase 1 2 is the highest-lying among the considered phases. For the other methods, it was either the lowest or the second lowest phase with negligible energy differences (a few meV), relative to the AFM FCC. In Table 4 we present the lattice constant and magnetic moment for the AFM FCC phase, obtained using the same schemes as in the previous table. As mentioned before, the LDA yields only the NM FCC phase, while the LDA þ U FLL method for U eff ¼ 1 eV gives lattice constant 5% and magnetic moment 14% too small. Applying U eff ¼ 2 eV results in lattice constant 2% too small and magnetic moment 25% too large. The LDA þ U AMF method gives lattice constant significantly smaller and the FCC phase is barely magnetic, with moments 0.02 and 0:04 mB for U eff ¼ 1 and 2 eV, respectively. The GGA underestimates both the lattice constant (4%) and the moment (20%), while applying U FLL increases both values. For U eff ¼ 1 eV the

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Table 4 Lattice constant and magnetic moment for AFM FCC Mn in LDAðGGAÞ þ U, U 1 ¼ 1 eV, U 2 ¼ 2 eV.

a(bohr) mðmB Þ a

LDA

LDA þU FLL 1

LDA þU FLL 2

LDA þU AMF 1

LDA þU AMF 2

GGA

GGA þU FLL 1

GGA þU FLL 2

GGA þU AMF 1

GGA þU AMF 2

expb

–a –

6.639 1.97

6.880 2.88

6.491 0.02

6.490 0.04

6.768 1.85

7.045 2.91

7.561 3.9

6.784 1.90

6.790 1.87

7.042 2.1

In the LDA, at zero pressure, we converged only the NM FCC phase. For that phase, lattice constant was 6.490 bohr. Experimental data are from Ref. [35]

b

lattice constant is virtually identical to the experimental one, while the moment is considerably larger than the experimental value. Increasing U eff to 2 eV causes transition to a high-spin phase ð3:9 mB Þ and the volume is further increasing. GGA þ U AMF improves both the lattice constant and the moment for U eff ¼ 1 eV, with respect to the GGA results. With the increase of U eff the volume does not change significantly, while the value of the moment slightly decreases. We also note that the magnetic moments for the AFM fct phase are almost identical to the AFM FCC phase, except in the cases of the LDA, LDA þ U AMF and 1 LDA þ U AMF , for which it is larger ð0:06 m for the LDA, B 2 0:27 mB for the LDA þ U AMF and 0:44 mB for the a LDA þ U AMF ). 2 Previous calculations of FCC and HCP Mn [11,37,14] obtained very similar results in the LDA, but in the GGA, they have different ordering of the Mn phases and values of the magnetic moments. A linear-muffin-tin orbital (LMTO) calculation [11] in the GGA finds, with lowest total energy, an AFM HCP with m ¼ 0:02mB , while a pseudopotential study [14] finds AFM FCC to be the lowest in energy with a moment of 2:4 mB . AFM FCC is also the ground state in a study using projector-augmentedwaves (PAW) method [38], with a smaller moment of 1:6 mB . Whereas we also find in the GGA that the AFM FCC is the ground state, the energy difference between this and the next lowest state (AFM HCP) in our case is smaller than in the pseudopotential study [14] and the moment at the equilibrium lattice spacing is smaller than the experimental extrapolation to 0 K. The differences possibly could be attributed to approximating the HCP phase by the ideal HCP structure (c=a ¼ 1:633) and not including the semicore states5 in the pseudopotential in Ref. [14]. In addition, in different DFT implementations (pseudopotentials, FLAPW and LMTO) magnetic moments are calculated in different ways.6For example, in our case, an increase of the muffin-tin radius from 1.95 to 2.39 bohr is causing a change of 8% in the antiferromagnetic moment, i.e. from a value of 2.05 to 2:21 mB . 5 In Ref. [32] the authors show that one of the effects of not including the 3p6 semicore in Fe, is larger magnetic moment. 6 For example, the moments of Fe, at the experimental volume, as calculated in a pseudopotential code [39] significantly vary depending on the way the moment is calculated: from the total magnetic moment m ¼ 2:39mB , from the absolute magnetization m ¼ 2:54 mB and from the projection on the atomic d-state basis, the moment is 2:22 mB .

For Mn it is more difficult than for Fe to assess the most successful approaches on the basis of the structural and magnetic properties, as its ground state is complicated and we concentrated on the quenched g-Mn. However, with this limitation, we conclude that, similarly to the case of Fe, the LDA and LDA þ U AMF are not adequate for the description of the lowest-energy states. The LDA þ U FLL for U eff about 1.5 eV would describe instead the lattice constant and magnetic moment of the g phase rather well compared to experiment. Compared to the Fe case, the GGA describes the Mn ground state a bit poorer, significantly underestimating the magnetic moment. The GGA þ U FLL helps, but only for values of U eff somewhat smaller than 1 eV. The GGA þ U AMF , for U eff  1 eV is slightly improving the lattice constant and magnetic moment. We can say that both schemes improve the GGA results, although the optimal U eff is different, it is definitely smaller for FLL, than AMF.

4. Discussion We find thus that, both for Fe and Mn, the schemes which yield a correct lowest-energy state are the LDA þ U FLL , GGA, GGA þ U FLL , and GGA þ U AMF . Furthermore, based on the comparison of the calculated volume and magnetic moment with the experimental values, we infer that, both for Fe and Mn, the optimal U eff should be approximatively: U eff  121:5 eV within the LDA þ U FLL , U eff  0:5 eV within the GGA þ U FLL , and U eff  1 eV within the GGA þ U AMF . Clearly, in the case of 3d bulk metals, such as Fe and Mn, one expects to be closer to the AMF than to the FLL limit (see appendix) and U eff  1 eV agrees well with the experimental estimate of U  J for Fe, around 1 eV [13,14]. In the light of the above results, the value of 4 eV for U, with J ¼ 0:95 eV, used by Mohn et al. [3] for Fe in FeAl appears very unrealistic. It should be noted, however, that for isolated transition-metal impurities the situation is very different, and the calculated values of U eff can be as large as 5 eV for Fe and Mn impurities in Rb [40]. Similarly, for lowdimensional systems, one expects U to increase with reduced dimensionality, and hence increased values of U eff at surfaces [7] and nanowires [5]. From our results on the effects of the LDA þ U and GGA þ U in Fe and Mn, it can be observed that the FLL scheme produces larger effects (and, in particular, a larger

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magnetic moment) than the AMF scheme, and hence that the AMF scheme allows somewhat larger values of U eff than the FLL, for a similar level of correction. This can be understood by considering the contributions to the Stoner parameter, I, due to the addition of U and J: DI ¼ 2q2 DE LDAþU =qM 2 , where M ¼ m=mB is the local atomic magnetic moment per Bohr magneton [9]. The Stoner parameter is invoked in the criterion for magnetic stability: I  DF 41, where DF is the density of states (DOS) at the Fermi energy. In the limit of uniform occupancy and using the force theorem [9], in the FLL case, DI FLL ¼ ðU  JÞ=ð2l þ 1Þ and in the AMF case, DI AMF ¼ 0. To get a nonzero DI for the AMF scheme, one needs to consider the case of nonuniform occupancy. In particular, when the d electrons are in a cubic crystal field, one obtains DI AMF ¼ 2 2 6 1 5ðU  JÞðDD=DF Þ and DI FLL ¼ 5ðU  JÞð6ðDD=DF Þ þ 1Þ, where DD is the difference between the DOS per orbital of the eg and t2g electrons at the Fermi energy. Obviously, DI in the AMF scheme is always smaller than in the FLL scheme, and from there it can be concluded that the AMF scheme can allow larger U  J in order to result in the same DI. Our results also indicate that the U eff in the LDA þ U is larger than in the GGA þ U. This can be understood from the fact that the GGA, in general, yields larger equilibrium lattice constants, which tend to favor magnetism, compared to the LDA. As mentioned earlier, all results presented so far have been obtained by assuming the same value of U eff for all volumes. This is, however, an approximation. Cococcioni and de Gironcoli calculated U eff for Fe at different volumes using the ab initio pseudopotential method [6]. Their calculations indicated a significant increase in U eff under pressure. For comparison, we used their values for Fe FM BCC structure and found out that regarding the lattice constant and magnetic moment, there are no significant differences between the calculations with constant U eff and the volume dependent U eff .7 LDA+U FLL yields for lattice constant (magnetic moment) the value of 5.457 bohr ð2:78 mB Þ, LDA þ U AMF 5.178 bohr ð1:86 mB Þ, GGA þ U FLL 5.570 bohr ð2:87 mB Þ and GGA þ U AMF 5.326 bohr ð2:01 mB Þ. Actually, from comparison with Table 2, for U eff ¼ 2 eV, it can be seen that lattice constants and magnetic moments, calculated by volume-dependent U eff give slightly worse agreement with experiment than the values obtained with a constant U eff . A part of a reason for that lies in the fact that the lowest value of U eff from Ref.[6] is 2.1 eV, which is larger than our U eff of 2 eV, which we have already found to be too large for a good agreement with experiment. 7 The influence of U eff is much larger on the bulk modulus. The precise knowledge of U eff as a function of lattice constant should be available in order to calculate it. So far, this behavior has been studied only with pseudopotentials and for a single phase of Fe. It is known that the values of U eff have large uncertainties which strongly depend on the method of calculations. These are highly extensive calculations, from which it is hard to obtain accurate information, and those are beyond the scope of this paper.

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Finally, we note that the general formulation of the LDA þ U method is based on an expansion of the matrix elements of the Coulomb potential in E ee (the Hartree– Fock terms) into Slater integrals F k [41]. This formulation involves two independent parameters: U ¼ F 0 and J ¼ ðF 2 þ F 4 Þ=14, and reduces to Eqs. (2) and (3) when the m angular-momentum dependences of the Coulomb and exchange matrix elements in E ee [41] are neglected [18,42]. To test the effect of this approximation, we have performed calculations for Fe using the general formulation of the LDA þ U [41]. First, we chose J ¼ 0:8 eV and U ¼ 0:8 eV in the GGA þ U FLL formalism, to compare with the GGA results. We find a ¼ 5:347 bohr and m ¼ 2:09 mB , which is indeed very close to the GGA values in Table 2. Similarly, for J ¼ 0:8 eV and U ¼ 2:8 eV in GGA þ U FLL , a ¼ 5:498 bohr and m ¼ 2:71 mB , which is again corresponding well to the values for U eff ¼ 2 eV in Table 2. For the GGA þ U AMF , the effect is slightly larger, for J ¼ 0:8 eV and U ¼ 2:8 eV, a ¼ 5:341 bohr and m ¼ 1:80 mB , i.e., for lattice constant there is a negligible difference, whereas for the moment, the difference is 11%. Similar trends are found for the LDA þ U. 5. Summary We investigated the influence of the LDA þ U and GGA þ U on the phase stability and on the structural and magnetic properties of a-Fe and g-Mn in two doublecounting schemes. We found that, both for Fe and Mn, the LDA þ U FLL , GGA, GGA þ U FLL , and GGA þ U AMF yield a correct magnetic ground state, unlike the LDA and LDA þ U AMF . Based on the comparison of the calculated volume and magnetic moment with the experimental values, we estimated the optimal U eff , both for Fe and Mn: U eff  121:5 eV within the LDA þ U FLL , U eff  0:5 eV within the GGA þ U FLL , and U eff  1 eV within the GGA þ U AMF . The trends for the stability of the magnetic phase and U eff values in the various schemes were understood in terms of the Stoner criterion and the influence of U eff on the effective Stoner parameter. Acknowledgments We are grateful to M. Altarelli for helpful discussions and the critical reading of the manuscript. We acknowledge support for this work by the INFM within the framework ‘‘Iniziativa Trasversale Calcolo Parallelo’’. Appendix A. AMF versus FLL schemes Petukhov et al. [9] have proposed a correction for moderately correlated systems, which is an interpolation between the two schemes: E LDAþU ¼ aE FLL LDAþU þ ð1  aÞE AMF , where a is determined by imposing that LDAþU the total energy does not change upon the addition of the U eff (E LDAþU ¼ E LDA ) [9]. Applying their procedure in the WIEN2k code, we find that a  0:05 for U eff ¼ 2 eV for

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