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Intermetallics 18 (2010) 1143e1147

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Intermetallics

journal homepage: www.elsevier .com/locate/ intermet

First-principles based investigation on effects of magnetism on lattice dynamicsin Fe72Pd28 alloy

Biswanath Dutta, Subhradip Ghosh*

Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, India

a r t i c l e i n f o

Article history:Received 24 June 2009Received in revised form16 November 2009Accepted 24 February 2010Available online 2 April 2010

Keywords:B. Electronic structure of metals and alloysB. Magnetic propertiesE. Ab-initio calculations

* Corresponding author. Fax: þ91 361 2690762.E-mail address: subhra@iitg.ernet.in (S. Ghosh).

0966-9795/$ e see front matter � 2010 Elsevier Ltd.doi:10.1016/j.intermet.2010.02.052

a b s t r a c t

A first-principles based investigation of influence of magnetism on lattice dynamics of Fe72Pd28 systemnear the Invar composition has been carried by computing the pressure dependence of the phononfrequencies, the Gru€neisen parameters, the disorder-induced widths and the elastic shear constantusing a combination of transferable force constant model, density functional perturbation theory andthe itinerant coherent potential approximation, an analytic tool for performing configuration averagingin disordered alloys. We find that with increasing pressure and collapse of magnetic moment, theTA1[110] phonon frequencies harden along with the elastic shear constant. We do not observe anysignificant variation of the mode Gru€neisen parameter with change in magnetic moment. These resultsindicate that there is no magneto-volume effect on the lattice dynamics and the experimentallyobserved phonon softening with increasing magnetization has to be associated with the martensiticinstability.

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1. Introduction

Fe based alloys such as FePt, FeNi and FePd, in both ordered anddisordered phases have been the subject of extensive theoreticaland experimental investigations for over three decades [1e8]. Theinterest in these systems stem out of the anomalously low thermalexpansion coefficient, the “Invar” property, of these materials forFe-rich compositions. The focus of the research on these systems,therefore, has been the pursuit for understanding the mechanismbehind the Invar behavior. The Invar characteristics are observedfor alloys near Fe75X25 (X ¼ Pt, Ni, Pd) compositions and thus thesesystems have been subjects of extensive investigations. The Invarbehavior in these systems is understood in terms of the instabilityof the magnetic moment i.e. the magneto-volume effect [9e11].During the last two decades, the lattice dynamical behavior of thesesystems was also studied in detail [12e18]. Neutron-scatteringmeasurements in these systems reveal that there is an anomaloussoftening of the TA1[110] phonon branch upon cooling below themagnetic transition temperature, along with a softening of theelastic shear modulus. This anomaly in lattice dynamics wasinterpreted in terms of enhanced electronephonon interactions[19]. However, since these systems also undergo a martensitictransformation around the temperatures where measurements

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were done, attempts were also made to understand whether theanomalous softening in lattice dynamical properties is related tothe instability of the system prior to martensitic transformation.That this could be the case and no coupling between latticedynamics and magnetism is responsible for the anomaly in case ofFe3Pt was established by subjecting the system under pressure sothat it undergoes a transition from a high-volume high-spin state toa low-volume low-spin state [20]. Similarly, for Fe3Ni, band struc-ture calculations attributed the phonon softening upon cooling thesystem, to pre-martensitic transition only [21].

In comparison to FePt and FeNi systems, results on the FeePdsystem, particularly for the disordered phase are much less innumber. There haven’t been any significant investigation of thesystem near the Invar composition exploring the inter-relationsbetween magnetism and lattice dynamics. Fe3Pd, like Fe3Pt andFe3Ni, also undergoes a spin moment collapse upon volumecontraction, as has been observed in first-principles electronicstructure calculations [22,23]. The only extensive study exploringthe effect of magnetism on lattice dynamics in this system wasperformed on Fe72Pd28 and Fe63Pd37 alloys over awide temperaturerange by neutron-scattering experiments [24]. The measurementsshowed that the TA1[110] branch undergoes an anomalous soft-ening near the zone center upon cooling below the magnetictransition temperature. A consequent softening of the elastic shearmodulus is also observed. The investigation was however, incon-clusive about the origin of this anomaly, partly because themeasurements couldn’t be done near the martensitic fccefct

B. Dutta, S. Ghosh / Intermetallics 18 (2010) 1143e11471144

transformation temperature which the system undergoes. Theunderstanding of the phonon anomaly in view of change inmagnetic ordering upon cooling was, thus, incomplete.

In this work, we investigate the inter-relations betweenmagnetism and lattice dynamics in Fe72Pd28. The motivationwas toexplore the possibility of anomalous behavior in lattice dynamicsupon collapse of magnetic moment so that a possible connectionbetween the anomaly in lattice dynamics and magneto-volumeeffect can be established. To our knowledge, a theoretical investi-gation of this aspect has never been carried out for this system. Thereasons could be two fold: first, the computations of latticedynamics at finite temperatures based upon accurate first-princi-ples electronic structure methods are expensive and second, thelack of a suitable analytic, self-consistent theory for addressing theforce constant and environmental disorders in the context of latticedynamics in disordered compositions. In this work, the seconddifficulty is appropriately taken care of by a combination of first-principles based transferable force constant (TFC) approach [25]and the Itinerant Coherent Potential Approximation (ICPA) [26]forconfiguration averaging. The understanding of the anomaly inphonon spectra with change in magnetic moment has beenaddressed by computing the lattice dynamics at various pressuresso that the collapse of the magnetic moment to a low-spin low-volume state from a high-spin high-volume state can be incorpo-rated. Such an approach, as was done in case of neutron-scatteringmeasurements on Fe3Pt, is useful because it helps in gaining insightinto the impact of spin collapse on the lattice dynamics which inturn can help in understanding the experimental observations.

In what follows, we have calculated the variations of phononspectra, the disorder-induced widths, the Gru€neisen parametersand the elastic shear modulus as a function of volume (spin state)for Fe72Pd28. In the following section we briefly discuss the meth-odologies adopted followed by the results. The concluding remarksare presented at the end.

2. Methodology and computational details

Incorporating the randomness of the inter-atomic forceconstants in a disordered alloy environment is themost challengingtask for computation of phonon spectra and related properties. TheICPA method, although, addresses this off-diagonal disorder inrandom alloys, in an analytic and self-consistent way, it requiresaccurate information of inter-atomic force constants betweenvarious pair of chemical species for being applicable to realisticalloys. The most reliable sources of such information are the first-principles electronic structure methods. However, modeling therandom alloy environment in first-principles techniques requiresconstruction of large supercells, thus, increasing the computationtime manifold. This problem is recently alleviated by the TFCmodel. This model is constructed upon the observation that therelation between the stiffness of a given bond connecting a givenspecies pair and the bond distance itself, is transferable acrosscompositions [25]. The advantage of this observation is that itprovides a very simplistic and computationally feasible way todetermine the force constants as the force constants versus bonddistance relationships can be determined from a relatively smallnumber of first-principles calculations on select configurations andthen can be transferred to determine the force constants for otheratomic configurations, once the relevant bond lengths are known.The TFC has found to be extremely successful in computing thevibrational entropies for a number of alloys [25,27,28]. Recently, wehave devised a first-principles based approach for computation ofcomplete phonon spectra of random binary alloys by combining theTFCmodel with the ICPA. The computed phonon spectra and elasticconstants for various concentrations of PdxFe1�x alloys had

excellent agreement with the experimental results [29], thus vali-dating the accuracy of thismethod. In this work, we have, therefore,used this approach to extract the inter-atomic force constants ofvarious pairs of species at various volumes for disordered Fe72Pd28and used as inputs to the ICPA calculations for computation ofphonon spectra and related quantities.

For construction of the transferability relation between thebond stiffness and bond distances, we have used the first-principlesQuantum-Espresso code [30], based upon a Plane wave-Pseudo-potential implementation of the density functional perturbationtheory [31]. Force constants for L12 Fe3Pd and FePd3, L10 FePd andfcc Fe and fcc Pd structures at their respective equilibrium andexperimental lattice parameters have been used for construction ofthe transferability relation. Ultrasoft pseudopotentials [32] withnonlinear core corrections [33] were used. PerdeweZungerparametrization of the local density approximation [34] was usedfor the exchange-correlation part of the potential. Planewaves withenergies up to 55 Ry are used in order to describe electron wavefunctions and Fourier components of the augmented chargedensity with cutoff energy up to 650 Ry are taken into account. TheBrillouin-zone integrations are carried out withMethfesselePaxtonsmearing [35] using a 12 � 12 � 12 k-point mesh. The value of thesmearing parameter is 0.02 Ry. These parameters are found to yieldphonon frequencies converged to within 5%.

After achieving the desired level of convergence for the elec-tronic structure, the force constants are conveniently computed inreciprocal space on a finite q-point grid and Fourier transformationis employed to obtain the real space force constants [36]. Thenumber of unique real-spaced force constants and their accuracydepend upon the density of the q-point grids: the closer theq-points are spaced, the more accurate the force constants are. Inthis work, we have used a 4� 4 � 4 q-point mesh for all structures.

The required configuration averaging is performed by employ-ing the ICPA method. The disorder in the force constants wasconsidered for nearest neighboring shell only and the calculationswere done on 400 energy points. A small imaginary frequency partof �0.05 was used in the Green’s functions. The Brillouin-zoneintegration was done over 356 q-points in the irreducible Brillouinzone. The simplest linear-mixing schemewas used to accelerate theconvergence. The number of iterations ranged from 5 to 15 for allthe calculations. The phonon frequencies are obtained from thepeaks of the coherent scattering structure factors and the disorder-induced widths are obtained from the full-width at half-maxima(FWHM) of the structure factors.

3. Results and discussions

3.1. Magnetic properties

We first look at the magnetic properties of the ordered Fe3Pdalloy. Fig. 1 presents the variation of the total magnetic moment peratom with reduced volume V/V0, where V0 is the volume of theferromagnetic ground state calculated using the Plane wave-Pseudopotential (PW-PP) technique as implemented in theQuantum-Espresso code. The equilibrium lattice constant (volume)and the total magnetic moment are in good agreement with theexperiment and the results of the LMTO calculations [22]. Thequalitative behavior of magnetic moment with contraction involume is also in good agreement with the LMTO results. Theresults show that with increasing pressure (decreasing volume)there is a sudden collapse of the total magnetic moment forV/V0 w 0.83. To explore this behavior in more detail, we calculatedthe total energy variations with volume and spin states using thefixed-spin moment method [37]. Fig. 2 presents the results for thevariation of total energy with volume for the two spin branches.

V/V0

μ B

M

agne

tic m

omen

t (

)/

atom

0

0.5

1

1.5

2

2.5

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

Fig. 1. Magnetic moment (in units of Bohr-magneton (mB) per atom) versus volumeresults for Fe3Pd obtained with Plane wave-Pseudopotential (PW-PP) calculationsusing Quantum-Espresso package. The volume is scaled by the equilibrium volume ofthe ferromagnetic (high-spin) ground state.

0.8 1

V/V0

0

0.5

1

1.5

2

2.5

3

Mag

netic

mom

ent/a

tom

fe3pdfe72pd28

Fig. 3. Variation of magnetic moment (in units of mB) per atomwith volume (scaled bythe volume of the Fe3Pd equilibrium volume) for ordered Fe3Pd and disorderedFe72Pd28 obtained with the EMTO and EMTO-CPA calculations respectively.

B. Dutta, S. Ghosh / Intermetallics 18 (2010) 1143e1147 1145

The results show that with contraction of volume, the systemundergoes a transition from a high-spin to a low-spin state showinga strong magneto-volume coupling. The calculated transitionpressure for the spin-state transition in our case is 34 GPa whichcompares fairly with 24 GPa as obtained in Ref. [23].

Since we are interested in the disordered Fe72Pd28 system, it isimportant to first check whether the magnetization behaves simi-larly in case of the disordered system upon reduction of volume.Since, the modeling of the disordered state within Quantum-Espresso requires large supercell, and thusmaking the computationexpensive, we calculate the magnetization as a function of volumeof Fe72Pd28 using the Exact Muffin-Tin Orbital (EMTO) [38e40] inconjunction with the Coherent Potential Approximation (CPA)which performs the configuration averaging in a fast and efficientway and the accuracy obtained is of the same order as that of thefull-potential based calculations. Fig. 3 presents the results of theEMTO and the EMTO-CPA spin-polarized calculations of variation oftotal magnetic moment per atom for ordered Fe3Pd and disorderedFe72Pd28 respectively. The equilibrium lattice constants for bothsystems are listed in Table 1. The features in the EMTO and theEMTO-CPA results which are of interest to us are that the latticeconstants of ordered Fe3Pd and of disordered Fe72Pd28 are almost

−242.6

−242.55

−242.5

−242.45

−242.4

−242.35

−242.3

−242.25

−242.2

220 240 260 280 300 320 340 360 380

Ene

rgy

(in

Ry)

3V (in a.u. )

High spinLow spin

Fig. 2. Variation of total energies with volume for Fe3Pd obtained with PW-PP andfixed-spin moment calculations using Quantum-Espresso package.

same and the magnetization drastically collapses for V/V0 w 0.85(V0 is the equilibrium volume of the disordered state) for bothsystems. Thus, the qualitative behavior of magnetization withvolume reduction is identical for ordered and disordered phases.This particular result enables us to consider the volume at whichthe magnetic moment of Fe3Pd collapses as the approximatevolume of high-spinelow-spin transition for disordered Fe72Pd28.

3.2. Phonon spectra

In Ref. [24], the neutron-scattering data showed that uponlowering the temperature from above the Curie temperature toroom temperature the [zz0] (z ¼ j q!j=qmax, q! being the wavevector) TA1 branch shows an anomalous softening near the zonecenter. The corresponding elastic constant, the shear modulusC0 ¼ (C11 � C12)/2 also undergoes an anomalous softening; itdecreases by about 80% as one lowers the temperature from 700 Kto the room temperature. Their magnon dispersion results didn’tfind any correlation between the change in magnetic ordering andthe phonon softening. The phonon line-widths obtained by fittingthe phonon line shapes to an oscillator model showed an increaseupon lowering the temperature below the Curie temperature.However, the authors were unable to explain whether this increasewas due to the change in the magnetic ordering.

To gain insight into the relation between magnetism and latticedynamics and thus understand the observations of Ref. [24], wecomputed the phonon spectra, the elastic constants, the disorder-induced widths and the Gru€neisen parameters for the [zz0] TA1phonons as a function of reduced volume V/V0, where V0 is theequilibrium volume of the high-spin state. We reduced the volumesuch that the magnetic moment collapses from a high-spin to

Table 1Results of structural parameters (in a.u.) and magnetic moments (in mB per atom) forordered Fe3Pd and disordered Fe72Pd28.

Fe3Pd Fe72Pd28

PW-PP EMTO Expta PW-PP EMTO-CPA Exptb

Lattice constant 6.85 6.97 7.21 e 6.99 7.10Magnetic moment 2.05 2.07 e e 2.0 e

a Ref. [41].b Ref. [24].

0 0.2 0.4 0.6 0.8 1

ζ

0

2

4

6

8

10

ν (T

Hz)

V/V0

0.7415

0.8156

0.8945

0.9783

[ζζ0]ΤΑ1

Fig. 4. [zz0] TA1 dispersion curves for Fe72Pd28 for various values of reduced volume(V/V0; V0 is the equilibrium volume of the high-spin state) obtained by ICPAcalculations. 0.7 0.8 0.9 1

V(p)/V(0)

0

1

2

3

4

5

ν (Τ

ΗΖ)

ζ0.05

0.1

0.2

0.3

0.4

0.5

γ (ζ)0.95

0.88

0.91

0.91

0.97

1.02

Pc

Fig. 5. Volume dependent change of frequencies of the [zz0] TA1 branch in Fe72Pd28 atdifferent z. pc denotes the transition pressure for high-spinelow-spin transition.Gru€neisen parameters g(z) are also displayed.

Γ(T

Hz)

0.30

0.12

0.14

0.16

0.18

B. Dutta, S. Ghosh / Intermetallics 18 (2010) 1143e11471146

a low-spin state. This, in a way, simulates the transition betweendifferent spin states upon change in volume. In Fig. 4, the dispersioncurves for various volumes are presented. It is observed that uponlowering the volume, the phonon frequencies harden about 30% asone goes from the high-spin ground state to the low-spin one. InRef. [24], a similar softening was observed upon lowering thetemperature from above the Curie temperature down to the roomtemperature. In Table 2, we present the variation of the corre-sponding elastic constant with volume. The elastic shear constanthardens by about 75% on reducing the volume from that of thehigh-spin equilibrium one to the low-spin one. This anomaloushardening upon increasing pressure is qualitatively similar to theanomalous softening upon decreasing temperature as observed inRef. [24].

To understand whether the magneto-elastic effects areresponsible for such behavior upon application of pressure, we nextcompute the Gru€neisen parameter. In Ref. [20], a similar approachwas adopted in the context of ordered Fe3Pt. However, unlike theresults of Ref. [20], we do not see any anomalously large value of theGru€neisen parameter. Fig. 5 presents the variation of the frequen-cies with volume for phonons with various wave vectors. TheGru€neisen parameters are also listed in the figure. We find that theGru€neisen parameters stay close to 1, as is expected for typicalmetals, for up to z ¼ 0.5zmax and for that no significant decrease inthe parameter is observed for the pressure exceeding the criticalpressure pc of the high-spinelow-spin transition. This result speaksagainst the idea that any magneto-elastic effect is responsible forthe strong hardening of the phonon frequencies upon increase inpressure resulting in the collapse of the magnetic moment.

The disorder-induced widths are more sensitive parametersthan the phonon frequencies themselves and thus an inspection ofthe variation of the widths could indicate whether there is anysignificant coupling between lattice dynamics and magnetism. InFig. 6, we show the variation of these widths for the [zz0] TA1branches with volume for various z. We observe that though the

Table 2Variation of elastic shear modulus with reduced volume V/V0; V0 is the equilibriumvolume for the high-spin moment state; for Fe72Pd28 in Mbar units.

V/V0 (C11 � C12)/2

0.74 0.860.82 0.480.89 0.460.98 0.20

widths increase with increase in z, the variation with respect tovolume is not significant. Thus, the absence of any anomalousbehavior of the widths rule out the scope of any strong magneto-volume effect as far as lattice dynamics of the system is concerned.

The results on the variation of frequencies, variations ofGru€neisen parameter and the disorder-induced widths clearlyshow that magnetism, or the variation of it, is not playing any rolein the phonon hardening upon increasing pressure. The only otherreason in such anomalous hardening of the frequencies can,therefore, be the tendency of the system to stabilize itself againsta martensitic fccefct transformation. The pressure, in this case, ismaking the system more stable against such transformation. Basedupon this explanation, we can also explain the reason behind thephonon softening upon reduction of temperature as is observed inRef. [24]. The authors of Ref. [24] could not explore this possibilitybecause they could not perform the measurements near themartensitic transition temperature. Since our calculations showthat the lattice dynamical properties upon reducing pressure(increasing magnetization) are qualitatively similar to thoseobserved in the temperature dependent neutron-scattering results,the mechanism must be the same.

V/V0

0.20

0.25

0.15

ζ 0.04

0.06

0.08

0.1

0.7 0.75 0.8 0.85 0.9 0.95 1

Fig. 6. Volume dependent changes of disorder-induced widths of the [zz0] TA1 branchin Fe72Pd28 for various z values.

B. Dutta, S. Ghosh / Intermetallics 18 (2010) 1143e1147 1147

4. Conclusions

The effect of magnetism on lattice dynamics in disorderedFe72Pd28 alloys has been investigated by a combination of first-prin-ciples density functional perturbation theory, the TFC model foraccurate computation of force constants and the ICPA method forconfiguration averaging in disordered environment. Differentmagnetic states have been simulated by letting the volume of thesystem contract and the relation between magnetism and latticedynamics has been studied by computing the phonon spectra, thedisorder-induced widths, the Gru€neisen parameters and the elasticshear constants. It is observed that the phonon frequencies of [zz0]TA1 branch and the corresponding elastic constants harden (soften)considerably upon decreasing (increasing) volume beyond the tran-sition volume of high-spinelow-spin transition. However, noanomaly in theGru€neisenparametersanddisorder-inducedwidths isobserved, thus ruling out the possibility of anymagneto-elastic effectbeing responsible for such features in the phonon spectra. Theseresults, can, therefore, be explained in terms of the tendency towardsstability of the system under pressure against martensitic transitionfrom fcc to fct phase. The phonon softening in the same branch upondecreasing the temperature from above Curie temperature down tothe room temperature as is observed in neutron-scattering experi-ments can, thus, be understood as an instability due to system’stendency towards martensitic transformation. To understand thisbetter and resolve the issue, carefulmeasurements are required up totemperatures near martensitic transition point.

Acknowledgements

One of the authors (B.D.) would like to acknowledge CSIR, Indiafor financial support under the grant e F.No:09/731(0049)/2007-EMR-I.

References

[1] Hausch G. Phys Status Solidi A 1973;18:735.[2] Hausch G. Phys Status Solidi A 1973;15:501.

[3] Hausch G. Phys Status Solidi A 1974;26:K131.[4] Hausch G. J Phys Soc Jpn 1974;37:819.[5] Hausch G, Warlimont H. Acta Metall 1973;21:402.[6] WassermannEF. Ferromagneticmaterials, vol.V;1990.North-Holland,p.237e322.[7] Podgorny M. Phys Rev B 1991;43:11300;

Phys Rev B 46 (1992) 6293.[8] Khmelevskyi S, Ruban AV, Kakehashi Y, Mohn P, Johansson B. Phys Rev B

2005;72:064510.[9] Entel P, Hoffman E, Mohn P, Schwarz K, Moruzzi VL. Phys Rev B 1993;47:8706.

[10] Hoffmann E, Herper H, Entel P, Mishra SG, Mohn P, Schwarz K. Phys Rev B1993;47:5589.

[11] Wassermann EF, Acet M, Entel P, Pepperhoff J. J Magn Soc Jap 1999;23:385.[12] Tajima K, Endoh Y, Ishikawa Y, Stirling WG. Phys Rev Lett 1976;37:519.[13] Endoh Y, Noda Y, Ishikawa Y. Solid State Comm 1977;23:951.[14] Endoh Y, Noda Y. J Phys Soc Jpn 1979;46:806.[15] Noda Y, Endoh Y. J Phys Soc Jpn 1988;57:4225.[16] Abd-Elmeguid MM, Micklitz H. Physica B 1990;163:412.[17] Kastner J, Petry W, Shapiro SM, Zheludev A, Neuhaus J, Roessel Th, et al. Eur

Phys J B 1999;10:641.[18] Kastner J, Neuhaus J, Wassermann EF, Petry W, Hennion B, Bach H. Eur Phys J B

1999;11:75.[19] Endoh Y. J Magn Magn Mater 1979;10:177.[20] Schwoerer-Bohning M, Klotz S, Besson JM, Burkel E, Braden M, Pintschovius L.

Europhys Lett. 1996;33:679.[21] Herper HC, Hoffmann E, Entel P, Weber W. J De Phys IV 1995;5:C8e293.[22] Kuhnen CA, Da Silva EZ. Phys Rev B 1992;46:8915.[23] Mohn P, Supanetz E, Schwarz K. Aust J Phys 1993;46:651.[24] Sato M, Grier BH, Shapiro SM, Miyajima H. J Phys F 1982;12:2117.[25] Liu JZ, Ghosh G, Van De Walle A, Asta M. Phys Rev B 2007;75:104117;

A. Van De Walle and G. Ceder, Rev Mod Phys 61 (2000) 5972.[26] Ghosh S, Leath PL, Cohen MH. Phys Rev B 2002;66:214206.[27] Van De Walle A. Ph.d. thesis. MIT, Cambridge; 2002.[28] Wu EJ, Ceder G, Van De Walle A. Phys Rev B 2003;67:134103.[29] Dutta B, Ghosh S. J Phys Condens Matter 2009;21:395401.[30] Quantum-ESPRESSO is a community project for high-quality quantum-

simulation software, based on density-functional theory, and coordinatedby Paolo Giannozzi. See http://www.quantum-espresso.org and http://www.pwscf.org.

[31] Baroni S, De Gironcoli S, Dal Carso A, Gianozzi P. Rev Mod Phys 2001;73:515.[32] Vanderbilt D. Phys Rev B 1990;41:7892.[33] Louie SG, Froyen S, Cohen ML. Phys Rev B 1982;26:1738.[34] Perdew JP, Zunger A. Phys Rev B 1981;23:5048.[35] Methfessel M, Paxton AT. Phys Rev B 1989;40:3616.[36] Giannozzi P, De Gironcoli S, Pavone P, Baroni S. Phys Rev B 1991;43:7231.[37] Williams AR, Moruzzi VL, Kubler J, Schwarz K. Bull Am Phys Soc 1984;29:278.[38] Vitos L. Phys Rev B 2001;64:014107.[39] Vitos L, Abrikosov IA, Johansson B. Phys Rev Lett 2001;87:156401.[40] Vitos L, Skriver HL, Johansson B, Kollar J. Comp Mat Sci 2000;18:24.[41] Duplessis RR, Stern RA, Maclaren JA. J Appl Phys 2004;95:11.