PHYSICAL REVIEW B 66, 205109 ~2002!

Density-functional investigation of magnetism ind-Pu

Per So¨derlind, Alex Landa, and Babak SadighLawrence Livermore National Laboratory, University of California, P.O. Box 808, Livermore, California 94550

~Received 26 July 2002; published 27 November 2002!

We present density-functional results ofd-Pu obtained from three electronic-structure methods. These meth-ods have their individual strengths and are used in combination to investigate the magnetic and crystal stabilityof d-Pu. An all-electron, full potential linear muffin-tin orbitals~FPLMTO! method, that includes correctionsfor spin-orbit coupling and orbital-polarization effects, predictsd-Pu to be an antiferromagnet at zero tempera-ture with a volume and a bulk modulus in very good agreement with experiment. The site-projected magneticmoment is smaller than expected (;1.5 mB) due to large cancellation of spin and orbital moments. Thesecalculations also predict a mechanical instability of antiferromagnetic~AF! d-Pu. In addition, techniques basedon the Korringa-Kohn-Rostoker~KKR! method within a Green’s-function formalism and a projector aug-mented wave~PAW! method predict the same behavior ofd-Pu. In order to study disordered magnetism ind-Pu, the KKR Green’s-function technique was used in conjunction with the disordered local-moment model,whereas for the FPLMTO and PAW methods this was accomplished within the special quasirandom structuremodel. While AFd-Pu remains mechanically unstable at lower temperatures, paramagneticd-Pu is stabilizedat higher temperatures where disordered magnetic moments are present and responsible for the crystal struc-ture, the low density, and the low bulk modulus of this phase.

DOI: 10.1103/PhysRevB.66.205109 PACS number~s!: 71.15.Mb, 64.10.1h, 71.27.1a, 75.10.Lp

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I. INTRODUCTION

Plutonium metal is located in the series of actinides inperiodic table of elements, does not occur naturally, buman-made for nuclear power purposes. With the practimportance of this material it is essential to understandfundamental properties. Pu, however, is an extremely cplicated metal with a most anomalous phase diagram1 show-ing as many as six distinct phases, see Fig. 1. The groustatea-Pu is relatively well understood with cohesive astructural properties governed by narrow 5f bands contain-ing about five electrons. Density-functional electronstructure calculations predict the density, bulk modulus, crtal structure, thermal expansion, and magnetic propertiegood agreement2 with experiment. This gives us strong cofidence in density-functional theory~DFT! because many othe above properties are in fact quite anomalous and upected for a metallic element. Also, the fact that the cloneighbor metal, uranium, is very accurately describedDFT including delicate details of the crystal structure as was elastic constants3 supports this notion. This ground-stateaphase is, however, very brittle and not suitable for engineing applications whereas the face-centered-cubic~fcc! d-Puis more ductile and useful in practice. The pure Pud phase isstable at about 600 K but can be alloyed with a small amoof, for example, Ga, Al, Sc, Ce, or Am which will stabilizd-Pu at lower temperatures.

Although technologically more important,d-Pu and itsalloys are less understood thana-Pu. For instance, DFTwhich relatively accurately describesa-Pu, underestimatethe volume ofd-Pu by about 20%, overestimates the bumodulus by 300%, and predicts negative elastic consta4

To remedy this, severalad hoctechniques5–7 have been ap-plied tod-Pu attempting to reproduce experimental data. Tmost recent publications are based on the dynamical mfield picture of Savrasovet al.8,9 and conventional DFT with

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spin and orbital corrections.4 Both these approaches reprduce several properties ofd-Pu but the former theory is noaccurate enough to deal with thea phase of Pu. The lattecalculations4 accurately describea-Pu but predict orderedmagnetism ind-Pu which has not been confirmed by expement.

In the present paper we follow the ideas put forwardSoderlind4 but expand the investigation to study sevemagnetic configurations ofd-Pu including models for disor-dered magnetic structure. Also, the structural stability offcc d phase is investigated. To this end, full potential linemuffin-tin orbitals ~FPLMTO!, Korringa-Kohn-Rostoker

FIG. 1. The experimental~Ref. 1! phase diagram of plutonium

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SODERLIND, LANDA, AND SADIGH PHYSICAL REVIEW B 66, 205109 ~2002!

~KKR!, and projector augmented wave~PAW! methods havebeen employed and these are described in Sec. II. Themainder of the paper is organized as follows. We presentresults from studying stability of the magnetic structuresSec. III and structural stabilities in Sec. IV. Also, in an atempt to compare our electronic structure with experimand other models, we present the calculated electronicsity of states in Sec. V. Lastly, in Sec. VI, we offer oconclusions and ideas for future work.

II. COMPUTATIONAL DETAILS

In this paper we calculate the electronic structure ofd-Puusing a combination of three methods which each havetinct advantages. The most accurate method, which isthe most computer intensive, is an all-electron FPLMTmethod that will be described first. Second, we give thetails of a KKR method which is computationally efficienwith the added advantage of being able to treat ordereddisordered alloys within the coherent-potential approximtion ~CPA!. This latter feature enables the disordered locmoment ~DLM ! method to be used to describe disordermagnetism without the use of a super cell. Lastly, wescribe a PAW method with a plane-wave basis set, whenables efficient calculations of forces and relaxations. Ththree methods all rely upon the density-functional thewith recent approximations to the electron exchancorrelation functional. In addition, the FPLMTO method iclude corrections due to spin-orbit coupling and orbipolarization.

A. FPLMTO

The parameters of the calculations are the same as tgiven earlier,4 but the more important details are repeathere. This electronic-structure method is an implementaof density-functional theory as applied for a bulk materia10

It is a first-principles method; no experimental numbersused in the calculations except for the nuclear charge whis 94 for Pu. The approximations in this approach are limito the approximation of the exchange/correlation enefunctional, cutoffs in the expansion of basis functionk-point sampling in integrations over the Brillouin zon~BZ!, and the Born-Oppenheimer approximation. Forexchange/correlation approximation we used the generalgradient approximation~GGA!,11 which has proven to bebetter for f-electron metals than the more commonly uslocal-density approximation~LDA !. Spin-orbit coupling andspin/orbital polarization were allowed for, in the same was has been described earlier.4

The use of full nonsphericity of the charge-density aone-electron potential is essential for accurate total energThis is accomplished in our method by expanding chadensity and potential in cubic harmonics inside nonoverlping muffin-tin spheres and in a Fourier series in the intstitial region. In all calculations we used two energy taassociated with each basis orbital and for 6s, 6p, and thevalence states (7s, 7p, 6d, and 5f ) these pairs were different. With this ‘‘double basis’’ approach we used a total of senergy tail parameters and a total of 12 basis functions

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atom. Spherical harmonic expansions were carriedthroughl max56 for the bases, potential, and charge densTwo types of crystal structures were considered here. Fthe antiferromagnetic, ferromagnetic, and nonmagnetic sttures were accounted for in a two atom/cell simple tetragostructure. Within this structure the axialc/a ratio distin-guishes between fcc (c/a5A2) and body-centered cubi~bcc! (c/a51). Otherc/a values were also considered whestudying mechanical instabilities. A special quasirandstructure model with an eight atom super cell was usedapproximate a disordered magnetic structure with zero tspin moment described below~Sec. III!. The sampling of theirreducible BZ was done using the specialk-point method12

and the number ofk points used were up to 800 in the twatom/cell calculation and about 10 for the eight atom/ccalculation. To each energy eigenvalue a Gaussian was aciated with 20-mRy width to speed up convergency.

B. KKR

The calculations we have referred to as KKR are pformed using the scalar relativistic~no spin-orbit coupling!Green’s-function technique based on the KKR method witthe atomic sphere approximation.13,14 Here the ASA is im-proved by addition of higher multipoles14 of the charge den-sity, and the so-called muffin-tin correction15 to the electro-static energy is also included. The calculations weperformed for a basis set including valencespd f orbitals andthe semicore 6p state whereas the core states were recalated at each iteration~soft-core approximation!. For theelectron exchange/correlation energy functional the mrecent16 local Airy gas approximation has been used. Integtion over the irreducible wedge of the BZ was performusing the specialk-point method12 with 916 k points. Themoments of the density of states needed for the kineticergy and the valence charge density were calculated by igrating the Green’s function on a complex energy cont~with a 2.5-Ry diameter! using a Gaussian integration tecnique with 30 points on a semicircle enclosing the occupstates. The equilibrium density of Pu was obtained fromMurnaghan17 fit to about ten total energies calculated asfunction of lattice constant.

The calculations were performed for nonmagnetic~NM!~spin degeneracy!, ferromagnetic~FM!, and antiferromag-netic ~AF! magnetic configurations. Also, magnetic disordwas modeled within the CPA.18 This was accomplished byusing the so-called disordered local-moment model. In orto calculate the DLM state, one uses a random mixturetwo distinct magnetic states, namely, the spin-up and sdown configurations of the same magnetic species insystem.19 Thus, the total energy for a system with completerandomly oriented local moments can be calculated. Hererepresent a paramagnetic state ofd-Pu as a Pu50

u Pu50d alloy.

C. PAW

The norm-conserving pseudopotential scheme20 hasshown to be a powerful technique for performing large-scstatic as well as dynamic DFT calculations using a pla

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DENSITY-FUNCTIONAL INVESTIGATION OF . . . PHYSICAL REVIEW B 66, 205109 ~2002!

wave basis set. Transferable pseudopotentials of this khowever, can become computationally very expensive wapplied to transition orf-electron systems, because of trequired small core radius. A remedy to this problem wproposed by Vanderbilt in the ultrasoft pseudopotenscheme,21 where the norm-conserving condition is relaxand the core radius can be moved out to approximatelyof the nearest-neighbor distance. In this approach, localatom-centered augmentation charges need to be introduBlochl22 developed a generalization of the Vanderbilt ultrsoft pseudopotential and the linear augmented plane-wa23

methods, i.e., the projector augmented wave techniqWithin the PAW method, the all-electron wave functions arelated to the pseudowave functions via a linear transfortion.

We have performed scalar relativistic spin-polarized PAcalculations for Pu using theVASP code.24 The electronicexchange/correlation energy functional was representeding the GGA approximation.11 The calculations include 16valence electrons, including the semicore 6s and 6p stateswith a plane-wave cutoff of 23.4 Ry. The Brillouin zone wsampled with the same grid ofk points for all the spin con-figurations, equivalent to an 83838 fcc Monkhorst-Packgrid.25

The PAW method was checked and compared toFPLMTO method with the same exchange/correlation futional and no spin-orbit coupling. The two methods ganearly the same equilibrium lattice constants~within about1%! for both the spin-polarized and spin-degenerate trements of fcc Pu, suggesting that the PAW method is perfoing as well as FPLMTO for this property of Pu.

III. MAGNETIC STRUCTURE

As has been mentioned earlier,4 DFT predicts a spontaneous formation of an ordered spin moment ind-Pu, whereasin the ground-statea-Pu a nonmagnetic state is favored. Tquestion of magnetism in Pu is somewhat controversial,the most recent experiments known to these authors, adding magnetism in Pu, are magnetic-susceptibility measuments which show a Curie-Weiss-like behavior26,27 for d-Pu.The susceptibility ofa-Pu, however, is significantly differenand essentially temperature independent.26,27 In the latterwork27 attempts were made to fit the susceptibility of allostabilized d-Pu to a modified Curie-Weiss law. For P6-at. % Ga the best fit gave an effective moment of 1.2mB ,whereas the renormalized value for Pu 6-at. % Ce w1.7 mB .

From a DFT standpoint, the exchange splitting that givrise to the formation of a large spin moment is very fundmental and cannot be ignored because if the theory is cstrained to be spin degenerate~nonmagnetic!, it fails to re-produce any known properties ofd-Pu. In the following weaddress the question of magnetism ind-Pu and study severamagnetic configurations and also try to represent the pmagnetic state via disordered local moments. This lamagnetic state, if stabilized, would fulfill the requirementlarge exchange splitting produced by the DFT and also

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consistent with the experimentally suggested Curie-Wemagnetic susceptibility ofd-Pu.

In Fig. 2 we show FPLMTO@Fig. 2~a!#, KKR @Fig. 2~b!#, and PAW@Fig. 2~c!# total energies as a function of volumfor four magnetic configurations, namely, NM, FM, AF, andisordered~D!. the AF configuration is type I here, i.e., spmoments are aligned in100& ferromagnetic sheets with adjacent sheets having antiparallel spins and the spin quantion along the@001# direction. This type of AF ordering waspredicted to be the most favorable in a previous study4 andthis fact is also confirmed here. For the FPLMTO and PAcalculations the disordered moment model consists of acial quasirandom structure with eight atoms/cell. This strture is designed28 to mimic a perfectly random structure breproducing its behavior for the closest neighbors aroueach site, deferring periodicity errors to more distant nei

FIG. 2. FPLMTO ~a!, KKR ~b!, and PAW ~c! total energies~mRy/atom! for fcc Pu in nonmagnetic~NM!, ferromagnetic~FM!,antiferromagnetic~AF! type-I, and disordered~D! configurations asa function of atomic volume. Dashed vertical line represents expmental equilibrium volume ford-Pu. ~Ref. 33!

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TABLE I. Theoretical equilibrium volume (V, in Å3), bulk modulus (B, in GPa!, and site-projected spinmoment (ms , in mB). For the FPLMTO calculations the orbital moment is given in parentheses. Experital ~Ref. 33! atomic volume and bulk modulus are 25.0 Å3 and 30–35 GPa, respectively.

FPLMTO KKR PAWMagnetic order V B ms V B ms V B ms

FM 25.6 26 4.4 (23.0) 29.0 30 5.7 27.1 24 5.2D 25.0 42 4.3 (23.0) 25.0 42 5.0 23.9 34 4.7AF 25.6 35 4.5 (23.0) 24.6 51 4.8 23.3 41 4.8

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Reviewing Fig. 2, we realize that despite the great diffences in numerical implementations, our three DFT methare in good quantitative agreement and that AF is the egetically most stable configuration closely followed by a dordered state about 2–5-mRy (;320–800 K) higher in en-ergy. Considering the fact that the calculations assumetemperature andd-Pu is stable at about 600 K, this smaenergy difference is not sufficient to rule out any magneconfiguration. In fact, spin entropy (Sspin) strongly favorsthe disordered moment state at higher temperatures becof the large spin moment,ms , involved (ms;5mB). As arough estimate of this contribution we could useSspin5kBln(ms11) for a complete disorder of spins, and thamounts to about 7 mRy atT5600 K.

The effect of spin-orbit coupling in fcc Pu can be appciated by comparing the FPLMTO results with that of KKand PAW. The NM equilibrium volumes, shown in Fig.obtained from KKR and PAW calculations are significansmaller than that of the FPLMTO calculation, i.e., the sporbit interaction expands that lattice in the NM treatmeThis effect was first discovered and explained by Brook29

and was later confirmed by So¨derlind et al.30 Some morerecent studies have come to the same conclusion.31,32 Theeffect of spin-orbit coupling in the FM case is the opposihere it leads to a smaller lattice constant as is also clear fFig. 2. Although important for the finer details of the resulspin-orbit coupling is not essential for the quantitative bhavior of d-Pu.

In Table I we summarize the equilibrium volumes, bumoduli, and magnetic moments obtained from our calcutions. For the FPLMTO treatment the equilibrium volumbulk modulus, and site-projected magnetic momentsabout the same for the FM, AF, and D configurations andgood agreement with experiment33 for the atomic volumeand the bulk modulus. The total magnetic moment is zeroAF and D, but also the site-projected moment is rather smdue to cancellation of spin and orbital contributions. TKKR calculations are more sensitive to the actual spin cfiguration, but for the two lowest-energy states, AF andthe results are very similar. FPLMTO and KKR agree wwith each other for the AF and D configurations althoughspin moment is about 10% larger for the KKR treatmeAlso the PAW equilibrium volumes, bulk moduli, and spmoments are close to both the FPLMTO and the KKRsults. It is important to note that although the numerical teniques are quite different, all three methods are able to r

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tively accurately reproduce the anomalousd-Pu volume andbulk modulus, especially for the AF and D configurations

Supercell PAW calculations for several other spin cofigurations were performed in addition to the above invegations. Some of these included spin-restricted~nonmag-netic! sites and others a mixture of local ferromagnetic aantiferromagnetic order. These studies can be summarizefollows. ~i! Configurations containing nonmagnetic sitesways increase the DFT total energies significantly and pdict too small volumes ford-Pu. ~ii ! Local ferromagneticalignment is always less favorable than an antiferromagnalignment. ~iii ! The AF type-I configuration, describeabove, is predicted to be the lowest-energy configuration

IV. CRYSTAL STRUCTURE

In this section we study the structural stability of the twlowest-energy configurations, AF and D. Experimentally34

d-Pu is known to have a very small elastic constantC8which could be interpreted as a tendency towards a tetranal distortion of the fcc structure. Also, some fcc actinicompounds are known to show tetragonal lattice distortinduced by antiferromagnetism.35 To investigate this ford-Pu we display FPLMTO total energies, in Fig. 3, asfunction of thec/a axial ratio at the theoretical equilibriumvolume (25.6 Å3) for AF Pu. Note that AF Pu is unstablwith respect to a tetragonal distortion and that it is predicto be body-centered tetragonal~bct! with a c/a axial ratio ofabout 1.5. Similar FPLMTO calculations were done~notshown! at other volumes with essentially the same result, ivolume relaxation effects are small and do not change

FIG. 3. FPLMTO total energy~mRy/atom! as a function ofc/aaxial ratio for AFd-Pu at 25.6 Å3.

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DENSITY-FUNCTIONAL INVESTIGATION OF . . . PHYSICAL REVIEW B 66, 205109 ~2002!

main result. This tetragonal instability of AF Pu was cofirmed by both KKR and PAW calculations~not shown! per-formed at their respective equilibrium volumes. Next, wanalyze the disordered configuration, D. Here we choseshow in Fig. 4 the results from the KKR calculations withthe DLM model, of total energy as a function of thec/a axialratio. In this figure, the volume is kept fixed to its fcc equlibrium value (25.0 Å3), but KKR calculations at other volumes display a similar behavior. In contrast to AF Pu,d-Puis mechanically stable assuming disordered magnetism.result was also confirmed by FPLMTO and PAW calcutions, where the disorder was modeled by the special qurandom structure described above.

In Fig. 4 an approximate tetragonal shear constant (C8)can be calculated from the curvature of the energies closc/a5A2. By extracting the coefficient to the harmonic terof a least-squares fit to these energies, an approximateC855.0 GPa is obtained. Although thisC8 only serves as anestimate, it is in good agreement with the measured34 andvery smallC8 ~4.78 GPa!. Another interesting insight fromFigs. 3 and 4 is that thed8 phase~see Fig. 1!, which is a bctstructure withc/a51.33, has a total energy less than 1-m~160 K! greater than thed energy. This is consistent with thexperimental phase diagram which separates these phasamount less than 200 K. Figure 3 suggests, however, thad8 phase is unstable with respect to a tetragonal distortbecause there is no local minimum in the energy atc/a51.33.

At even higher temperature, Pu melts from the bccephase!. Our calculations suggest the energy differencetween d-Pu ande-Pu to be about 6–8 mRy which is idisagreement with estimates based on the experimental pdiagram. Clearly entropy or other effects, not accountedin the present theory, are important in stabilizing thee phasein Pu.

V. ELECTRONIC STRUCTURE

There are ongoing efforts to study the electronic structof Pu experimentally by means of photoelectrspectroscopy.36,37Although not straightforward, these can bcompared to calculated electronic density of states~DOS!

FIG. 4. KKR total energy~mRy/atom! as a function ofc/a axialratio for disordered~D! d-Pu at 25.0 Å3.

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close to and below the Fermi level (EF). If the DOS andphotoemission~PE! are incompatible, one could question thaccuracy and validity of the theory. This is the case forLDA1U method, in which there is a pseudogap7 close toand below theEF , where in fact the PE shows a peak in thintensity. Also a nonmagnetic LDA calculation producesDOS in poor agreement with PE.36 The ‘‘constrained LDA’’technique,6,36 as well as the DMFT,9 however, are more consistent with PE, although neither compares perfectly wexperiment.

For comparison, we show in Fig. 5 our calculated DOobtained from FPLMTO for AF Pu at 25.0 Å3. Here we usedlifetime broadening parameters identical to those chosenArko et al.36 who also measured the PE shown. Notice ththe broadened FPLMTO DOS compares quantitatively wwith photoemisson with a peak just below theEF and aminimum at 20.2 eV (EF is at zero!. In the PE,36 thisminima is located at about20.25 eV whereas in the PE bHavelaet al.37 it is located at about20.3 eV. This feature isimportant because Havelaet al. have shown that the weasatellite centered at about20.8 eV is absent in the spectrfor a-Pu and therefore indicative of thed phase. At lowerenergies, below21 eV, the convoluted FPLMTO DOScompares rather well with PE and that of the ‘‘constrainLDA’’ model,36 whereas here the DMFT model9 is in severedisagreement with the measurement.

We note that the DOS for AF, D, and FM are all vesimilar and that the comparison with photoelectron spectrcopy is not dependent upon the actual spin configurationis clear, however, that neglecting magnetic interactions girise to a DOS that is not at all compatible with the PE,also pointed out by Arkoet al.

VI. CONCLUSIONS

In conclusion, DFT is able to reproduce many of tanomalous properties ofd-Pu, when magnetic exchange interaction is taken into account. This interaction is essentiathe theory, and if ignored severely erroneous behavior of

FIG. 5. Comparison of spectra~Ref. 36! ~dashed line! toFPLMTO DOS ~solid curve!. Calculated curve is total electronidensity of states that has been convoluted with lifetime broaden~Ref. 36!.

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SODERLIND, LANDA, AND SADIGH PHYSICAL REVIEW B 66, 205109 ~2002!

metal is predicted. The lowest zero-temperature DFT eneis obtained for antiferromagnetic type-I ordering of spin mments with a magnitude close to 5mB . Energetically verycompetitive is a disordered magnetic structure with a sowhat smaller spin moment. FPLMTO also predicts a laantiparallel orbital moment that nearly cancels the spin mment. The resulting total moment is about 1.3mB–1.5mB ,which is close to an experimentally estimated effectmoment,27 but it is unclear if these can be directly compare

We suggest that the AF ordering does not occur at teperatures whered-Pu is stable because we have shown tthis ordering destabilizesd-Pu with respect to a tetragonadistortion. Instead, we propose thatd-Pu is a disorderedmagnet. At zero temperature the DFT energies for AF arandom ordering are close but at higher temperaturesgreat magnitude of the spin moments implies that spintropy would strongly disfavor ordering of the spins. In addtion, and in contrast to the AF ordering, random ordering

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the spins supports mechanical stability ofd-Pu.Experimentally, d-Pu is unstable below about 600 K

where fcc transforms to a lower-symmetry structure (a-Pu!.This behavior is here explained by the inherent tendetoward magnetic ordering ind-Pu at lower temperatures followed by a mechanical instability of the fcc phase. It is wknown, however, that a small amount of Al or Ga~for ex-ample! can stabilized-Pu to lower temperatures. In the nefuture, we plan to study the effects of alloying on the manetic properties ofd-Pu and the stability of the fcc phase.

ACKNOWLEDGMENTS

We would like to thank A.V. Ruban, P.A. Korzhavyi, anO. Eriksson for helpful discussions. This work was peformed under the auspices of the U.S. Department of Eneby the University of California Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.

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