The Electronic CorrelationStrength ofPu A.Svane,1,∗ R.C.Albers,2 N.E.Christensen,1 M.vanSchilfgaarde,3A.N.Chantis,4 andJian-XinZhu2 1Department of Physics and Astronomy, Aarhus University, DK 8000 Aarhus C,Denmark 2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 USA 3Department of Physics, King’s College London, The Strand, London WC2R 2LS, UK 4American Physical Society, 1 Research Road, Ridge, New York 11961, USA (Dated:June21,2012) Anewelectronicquantity,thecorrelationstrength,isdefinedasanecessarystepforunderstandingtheprop- ertiesandtrendsinstronglycorrelatedelectronicmaterials.Asatestcase,thisisappliedtothedifferentphases 2 ofelementalPu. WithintheGWapproximationwehavesurprisinglyfounda“universal”scalingrelationship, 1 where the f-electron bandwidth reduction due to correlation effects is shown to depend only upon the local 0 densityapproximation(LDA)bandwidthandisotherwiseindependentofcrystalstructureandlatticeconstant. 2 n PACSnumbers:71.10.-w,71.27.+a,71.30.Mb u J Many technologically important materials have strong this theory, which is a Green’s function G times a screened 0 2 electron-electron correlation effects. They exhibit large Coulomb interaction W. We also demonstratea scaling rela- anomalies in their physical properties when compared with tionship that is universal in that it is independent of crystal ] materials that do nothave these effects, and have significant structure and atomic volume. The ideas in this paper could l e deviationsintheirelectronic-structurefromthatpredictedby certainlybemodifiedandgeneralizedtobeabletotreatother - r conventionalband-structuretheorybasedonthelocal-density types of correlated materials (e.g., spin-fluctuation or high- st approximation (LDA). Because these anomalies and devia- temperature superconductingmaterials) by using other elec- . tionsarecausedbyelectroniccorrelationeffects,whichoften tronic properties to determine a correlation strength and by t a dominatethephysicsofthesematerials,inthispaperwede- usingmoresophisticatedtheoreticaltechniquesthanarecon- m fine a new quantity, which we call “correlation strength,” or sideredhere. d- C, as a necessary step in order to be able to describe trends Our meaning of correlation makes it necessary to use a n andbringorderintoourunderstandingofthesematerials. We theory that includes correlation effects that go beyond those o emphasize the word “quantity” since a quantitative measure included by the LDA approximationin order to determine a c is needed to answer the question: “how strong are the elec- theoreticalcorrelationstrength. Thisischallenging,sincethe [ troniccorrelations?”Withoutsomeunderstandingofhowbig mostsophisticatedtreatmentsofcorrelationeffectsaremainly 2 this is, it is not possible to make sense of the properties of confined to abstract theoretical models. These parameterize v these materials. In this context, “correlation”is defined in a theelectronicstructureinsuchanoversimplifiedmannerthat 9 waysomewhatdifferentfromhowissometimesused(e.g.,in theconnectionwithactualmaterialsexaminedexperimentally 3 1 the term “exchange-correlationpotential”). By “correlation” can be somewhat vague. Even recent methods, such as dy- 2 wespecificallymean“correlationbeyondLDAtheory”. This namicalmean-fieldtheory(DMFT)[5–8],stillrequirefitting . usagereflectsthewaythetermisoftenlooselyusedincom- Hubbardparameters[9],andthemodelpart,whichisadhoc 1 0 monterminologyintheareaofstronglycorrelatedelectronic and cannot be precisely defined or derived, rather than the 2 systems. first-principles part of the Hamiltonian often dominates the 1 Tocreateanewquantityrequiresdetermininga“scale”by physics of the material. Attempts to calculate Hubbard U’s : v which to measure its size. In principle, any experimentalor orotherparametersofthe modelsare basedonintuitionand Xi theoretical property (e.g., specific heat) that monotonically do not provide any solid foundation for these models since increases or decreases over the full range of correlation ef- any connection between the calculations and the models are r a fects, where we define correlation strength to lie between 0 tenuousatbest. As far as we are aware, the onlyreasonable fornoneand1 forfullcorrelation,canbeused asa measure first-principles method for calculating electronic correlation of this quantity. Hence correlation strength is an indetermi- effectsinmetalsbeyondLDAistheGWapproximation. Al- nant quantity and depends on the property used to define it. thoughthis is a low-orderapproximationthat definitely fails However, this doesnotmatter since onlyrelative ratherthan forverystrongcorrelationeffects,itissufficientforourpur- anyabsolutestrengthisimportantforcharacterizingthesema- poses as a way to estimate correlation deviationsfrom LDA terialsandforpredictingtrendsintheirproperties. Anymea- band-structure theory. The main purpose of our work is to sure based on one property can easily be converted to that show that it is possible and useful to define a new quantitiy, based on another property. In this paper we develop a the- whichwecallcorrelationstrength,forbothexperimentaland oreticalcorrelationstrengthbasedontheGWapproximation theoreticalworkonnewmaterialsinordertoplacethesesys- [1,2]toelectronic-structuretheoryandapplyittoplutonium temsintheirproperphysicscontext. [3,4], whichisknowntohavesignificantcorrelationeffects. As a theoretical method for estimating correlation effects The GW approximation is named for the correction term in we have used the quasi-particle self-consistent GW approx- 2 imation (QSGW) [10–12]. The GW approximation can Tosetanappropriatecorrelationscale,wedefineourtheo- be viewed as the first term in the expansion of the non- reticalC by: localenergy-dependentself-energyΣ(r,r′,ω)inthescreened CoulombinteractionW. Fromamorephysicalpointofview C =1−w , (2) rel itcan alsobe interpretedasa dynamicallyscreenedHartree- FockapproximationplusaCoulombholecontribution[1,2]. which ranges from C = 0 (no bandwidth reduction) in the Therefore, GW is a well defined perturbation theory. In LDAlimittoC =1inthefullylocalizedoratomiclimit(the its usualimplemention, sometimes called the “one-shot”ap- bandwidthbecomeszero). proximation,itdependsontheone-electronGreen’sfunctions As mentioned above, our test case for correlation is ele- which use LDA eigenvalues and eigenfunctions, and hence mental Pu, an actinide metal, which exhibits large volume the resultscan dependonthis choice. Unfortunately,ascor- changes compared to predictions from band structure the- relationsbecome strongerserious practicaland formalprob- ory that are clearly due to correlation effects [18–22]. The lemscanariseinthisapproximation[11]. However,Kotaniet largevariationinvolumesiscontrolledbytheamountofvery al.[12]haveprovidedawaytosurmountthisdifficulty,byus- strong f-bonding, which is due to direct f-f wave-function ingaself-consistentone-electronGreen’sfunctionthatisde- overlap. The f-bonding for many of the different phases is rivedfromtheself-energy(thequasi-particleeigenvaluesand greatlyreducedleadingtoanomalousvolumeexpansionsdue eigenfunctions)insteadofLDAasthestartingpoint.Inthelit- to the narrowingof the f-bandsthatresults fromcorrelation erature,ithasbeendemonstratedthattheQSGWformofGW effects[22].Ifnocorrelationwerepresent,thef-bondswould theoryreliablydescribesawiderangeofsemiconductors[13], havetheirfullstrengthandarelativelysmallvolumeperatom spd [10, 14, 15] and rare-earth systems [16]. It should be for all phases would be accurately predicted by LDA band- noted that the energy eigenvalues of the QSGW method are structuremethods.Inthelimitofextremelystrongcorrelation thesameasthequasiparticlespectraoftheGWmethod.This thebandswouldhavenarrowedsomuchthatthef electrons captures the many-body shifts in the quasiparticle energies. wouldbefullylocalized,andtheywouldnotcontributetothe However,whenpresentingthequasiparticleDOS,thisignores bonding. The volume per atom would then be much larger the smearing by the imaginary part of the self-energyof the andclosetothatofAm,whichhasfullylocalizedf electrons spectraduetoquasiparticlelifetimeeffects,whichshouldin- thatdonotextendoutsidetheatomiccore. creaseasquasiparticleenergiesbecomefartherawayfromthe Using the QSGW approximation we have calculated [23] Fermienergy. thequasi-particlebandstructuresofthefcc,bcc,simplecubic Todefineatheoreticalcorrelationstrengthsomeelectronic- (sc),γ,andthepseudo-αphasesofPuasafunctionofvolume. structurequantitythatscaleswithanintuitivenotionofcorre- Thepseudo-αisatwo-atomperunitcellapproximation[24] lationstrengthisneeded.InourapplicationtoPu,wepropose to the true α structure of Pu that preserves the approximate to considerthe f-bandwidth,W , and use the relativeband- nearest-neighbordistancesandotheressentialfeaturesneeded f widthreductioninQSGWcomparedtoLDA, fortheelectronic-structure. Inthiswayweavoidperforming an extremely large and expensive 16 atom-per-unit-cell cal- w =W (GW)/W (LDA), (1) culation for the α-structure. We are unfortunatelyunable to rel f f present GW results for the β-structure, which is even more asthekeyquantity,whereW (GW)andW (LDA)arethef- complex than the α structure, since no pseudo-structure for f f bandwidths as obtained from QSGW and LDA calculations, this crystal structure is available and a QSGW calculation is respectively. This is consistent with the correlation-induced presentlynotfeasibleforsomanyatomsperunitcell. QSGW f-bandwidth reduction in Pu that was demonstrated Tocalculatethef-electronbandwidthsfromthef-electron inRef.3. projected density of states (DOS), D (E), an algorithm is f Usingaquasiparticlecalculationisimportantsincelifetime neededtodeterminethewidthofthemainpeakinthisDOS. effects, which are absentin the LDA calculationswould ob- A simple first guess is to choose a rectangular DOS and to scure the band narrowing in GW relative to LDA. We also use a least-squares fit to the GW or LDA f-DOS to deter- needa measurethatisrobustatthe hightemperaturesof the minethebestheightandwidthoftherectangle. Adrawback strongly correlated phases of Pu, where any low energyfea- ofthismethodisthatanartificialbroadeningoftheeffective turesintheelectronicstructurearelikelytobethermallyav- f-bandwidth appears, which is due to a significant d-f hy- eragedaway.[17]. Inthisregard,itshouldbenotedthat, al- bridization at the bottom of the f-DOS that creates an extra though temperature certainly plays an important role in pre- peakatlowenergies.Thismasksthecorrelationinducedband dicting the correct equilibrium crystal structure, we believe narrowing.Sincethispeakhasrelativelylowerheightthanthe that it is the resulting volumeper atom of anyPu phase that mainf peak,wemayavoidthiscomplicationbygeneratingan determines the amount of correlation, since this is an elec- algorithmthatemphasizesthe“high-peak”partofthef-DOS. tronicproperty. In particular,we don’texpectthatthe band- Thealgorithmwe haveusedis thereforethe secondmoment widthpredictedbyourzero-temperatureGWcalculationswill ofthef-DOS besensitivetoanytemperatureintherangesetbythePusolid phases. W =2(hE2i−hEi2)1/2. (3) 3 α β γ ε δ δ εγ β α 1.0 1.0 0.9 Pu δ Pu ε 0.8 0.8 sc ps−α el0.7 γ el 0.6 γ wr0.6 wr δ 0.4 0.5 ε 0.2 sc 0.4 ps−α 0.3 0 14 16 18 20 22 24 26 28 .5 1.0 1.5 2.0 2.5 3 V (Å ) W(LDA) (eV) f FIG. 1. (Color online) Plot of w = W (GW)/W (LDA) versus rel f f FIG. 2. (Color online) Plot of w = W (GW)/W (LDA) versus volume, V, per atom, for the γ, fcc, bcc, sc, and ps-α (pseudo-α, rel f f W (LDA) for the γ, fcc, bcc, sc, and ps-α. The dashed red line anapproximateα-phase[24])crystalphasesofPu. Notethatthesc f representsthefitofEq.(5)Thesmall,verticalbarsatthetopofthe (simplecubic)isahypotheticalstructureforPu. Thesmall,vertical figuremarkthevalues of W (LDA)calculated attheexperimental barsatthetopofthefiguremarktheexperimentallyobservedatomic f volumesofthefivePuphases[26]. volumes[26]. i.e., The factor of two is needed because the bandwidth extends aboveandbelowthemeanenergyandisnotjusttheaverage w (x)=0.15+0.43x−0.07x2, (5) rel deviationfromthemeanenergy. Toemphasizethemainpart ofthef-DOSpeak,thesquareofthef-DOSisusedasweight where x = W (LDA) in eV. From Eq. (2) we can use these f function[25]: resultstodetermineacorrelationstrengthC. Itisremarkable thatthemany-bodypropertiesofastronglycorrelatedsystem 2 2 can be tuned with what is normally considered to be a one- hf(E)i≡ dEf(E)D (E)/ dED (E). (4) Z f Z f electronproperty. In Fig. 3 we show [27] that our definition of theoretical InFig.1weillustratehoww varieswithvolumeforthe rel correlation strength does indeed fulfill our expectations and fivedifferentphasesconsideredhere[26]. Largevolumevari- can be used to bring order into the trends for various ex- ationsrangingbetweenabout14–28A˚3 peratomareconsid- perimentalproperties,includingvolume,soundvelocity,and ered,withbandwidthsthatspanalmostanorderofmagnitude, resistivity. These properties exhibit an approximately 25%, from≈0.5eVto≈2.5eV.AlthoughtheLDAbandwidthde- 50%,and35%changeoverthecorrelationrange(about0.2to creases with increased volumedue to reductionin f-f over- 0.6)betweentheαandδ phasesofPuand,withsomescatter lap of the wavefunctions, the QSGW bandwidth decreases thatmightpartiallydependonsamplequality,fallonsmooth even faster illustrating increased correlation effects with lat- curveswhen plotted as a function of our theoretical correla- tice expansion. Thebandwidthata specific volumedepends tionstrength. Itisremarkablethatallofthisdatashouldcol- on crystal structure (due to differences in coordination and lapse to a single curve for each property that is independent bondlengths),asdoesalsothecorrelationstrength. ofanyexplicitconsiderationoftemperature,crystalstructure, Although we expect electronic-structure calculations to or other variable. However, more generally, we would only strongly dependon the crystal structure and lattice constant, expect this to be true for a property that was predominantly we surprisingly found that correlation effects were approxi- affectedbycorrelationeffects. mately independent of these. Indeed, Fig. 2 shows that all In terms of theoretical trends, various theories have often of our different calculations for our measure of correlation attempted to estimate the amount of correlation in terms of strength, the reduced bandwidth, collapse to a single “uni- theZ-factor, versal” curve when plotted as a function of the LDA band- width. In making this plot, it is likely that the effective −1 screenedCoulombinteractionbetweenthe5f electronsisap- Znk = 1−hΨnk|∂Σ(ǫnk)|Ψnki , (6) (cid:18) ∂ω (cid:19) proximatelyconstantandthatthecorrelationeffectsarebeing tuned by the effective average kinetic energy of these elec- whereΨnk arethe(LDA)electroniceigenfunctionswithen- tronsasreflectedintheirLDAbandwidth.IntherangeofWf ergies ǫnk, and Σ denotes the self-energy. We have found valuesconsideredhere the curveis approximatelyquadratic, thatthevolumedependenceoftheZ-factorsfollowsthetrend 4 α β γ ε δ 0.6 γ 0.4 δ C ε Co 0.2 sc ps−α Rh Ir d 0 0.0 0.5 1.0 1.5 −1 −1 W(LDA) (eV ) f FIG.4.(Coloronline)CfromGWtheoryversus1/W (LDA).The f data for Co, Rh and Ir are for the 3d, 4d, and 5d bandwidths, re- spectively. Thesmall,verticalbarsatthetopofthefiguremarkthe valuesofW (LDA)−1calculatedattheexperimentalvolumesofthe f fivePuphases[26]. withtwoparameters:theHubbardparameterU whichinduces correlation,andaneffectivet,whichcanberelatedtotheun- correlated bandwidth W. When W dominates, the system is in a weakly correlated limit and, when U dominates, the system is in a strongly correlated regime. Hence, one can study the solutions as a function of U/W to go from one limit to another. In more realistic electronic-structurecalcu- lations,thesamephysicsisintuitivelyexpectedtocarryover. The Hubbard U can then be thoughtof as a screened onsite Coulombinteractionandthebandwidthasduetothenormal band-structurehybridization.Inourcontext,thissuggeststhat thecorrelationstrengthC shouldalsobeafunctionofU/W. To test this, in Fig. 4 we plot C versus 1/W (LDA). If the f effectiveU wereapproximatelyconstant,wehadhopedtoob- servesomeapproximatelinearbehavioratweakcorrelations, butanysuchbehaviorisunclearinFig.4. Toshowwhatmight happenatweakercorrelationstrengthswehavealsoincluded FIG.3. (Coloronline)TrendsinPupropertiesasafunctionofcor- inFig.4alsotheequilibrium-volumeresultsforCo,Rh,andIr relationstrength C, including (a) volumeper atom[26], (b) sound velocity[28],and(c)resistivity[28]. forthed-electronprojectedDOS.Interestinglyenough,thed- electronresultsseemtofollowthesameoveralltrendtolarge bandwidths(smallcorrelation). Amongthe transitionmetals of the f-bandwidth reduction in Fig. 1, i.e., our measure includedintheplot,Co(3d)hasthemostnarrowd-band,and of correlation strength, albeit with variations due to k- and thecorrelationvalueisclosetothelowestvaluesforPuinthe hybridization-dependence. However, it should be noted that figure. the relation between Z and bandwidth reduction is not the In summary, we have introduced the idea of a “correla- sameinallmaterials,especiallyforweaklycorrelatedbroad- tion strength” quantityC, which must be taken into account bandsystems,whichseemverydifferentfromstronglycorre- in orderto explainthe propertiesofstronglycorrelatedelec- latedmaterialslikePu. tronicmaterials. As an example,we have shownhowto use The simplest Hubbard-like Hamiltonian to describe the GW method to define a theoretical C for metallic Pu, stronglycorrelatedelectronsystemshasaform and that various experimental physical properties, including anomalous volume expansion, sound velocity, and resistiv- H = t c† c +U n n . (7) ity, for the differentphases of Pu follow well defined trends ij iσ jσ i↑ i↓ Xij,σ Xi when plotted versusour theoretical correlationstrength. We 5 havealsodemonstratedauniversalscalingrelationshipforthe as2000Kintemperatureinmanyoftheseproperties,wellinto correlation-reducedbandwidthasafunctionoftheLDAband- thehigh-temperatureliquidphaseofPu.Also,seeRef.[29]. width. [18] O.J.Wick,PlutoniumHandbook: AGuidetotheTechnology (GordonandBreach,NewYork,1967). This work was carried out under the auspices of the Na- [19] S.S.Hecker,MRSBull.26,672(2001). tional Nuclear Security Administration of the U.S. Depart- [20] S.S.Hecker,Met.Mat.Trans.A35A,2207(2004). ment of Energy at Los Alamos National Laboratory under [21] S.S.Hecker, D.R.Harbur, andT.G.Zocco,Prog.Mat.Sci. ContractNo. DE-AC52-06NA25396,theLosAlamosLDRD 49,429(2004). Program,andtheResearchFoundationofAarhusUniversity. [22] R.C.Albers,Nature410,759(2001). The calculations were carried out at the Centre for Scien- [23] We have not included spin-orbit effects, which can be safely tificComputinginAarhus(CSC-AA),financedbytheDanish ignored for the purposes of this paper. The Pu f-DOS splits Centre for Scientific Computing (DCSC) and the Faculty of intoapairofclearlyseparatedj=5/2and7/2peaks.Toinclude spin-orbit, we would need to calculate thebandwidth of each ScienceandTechnology,AarhusUniversity. peakseparatelyandusethatcorrespondingtoj=5/2.Byignor- ingspin-orbitcoupling,wearesavedfromthisadditionaltrou- ble,whichisnotexpectedtochangetheeffectivef-bandwidths. Recentspin-orbitGWcalculationshavebeencalculatedinPu [4].Howeverthesehavebeendoneinthefullyself-consistent ∗ Electronicaddress:[email protected] GW method, which usually isa poor approximation in solids [1] L.Hedin,J.Phys.:Condens.Matter11,R489(1999). duetoanincorrecttreatmentofplasmoneffects.SincetheDOS [2] F.AryasetiawanandO.Gunnarsson,Rep.Prog.Phys.61,247 inthispaperincludesbroadeningeffectsduetotheimaginary (1998). partoftheself-energyinallofthedifferentapproximationthat [3] A.N.Chantis,R.C.Albers,A.Svane, andN.E.Christensen, wereused, itisalsounclear how bandwidth narrowingwould Phil.Mag.89,1801(2009). separatelybeaffectedbyspin-orbiteffects. [4] A. Kutepov, K. Haule, S. Y. Savrasov, and G. Kotliar, Phys. [24] J.Bouchet, R.C.Albers, M.D.Jones, andG.Jomard,Phys. Rev.B85,155129(2012). Rev.Lett.92,095503(2004). [5] A.Georges,G.Kotliar,W.Krauth, andM.J.Rozenberg,Rev. [25] Choosing instead for example D (E) as the weight function f Mod.Phys.68,13(1996). doesnotservethepurposeofemphasizingthecentraloverthe [6] G.Kotliar,S.Y.Savrasov,K.Haule,V.S.Oudovenko,O.Par- “hybridizationwings”inthef-DOS. collet, andC.A.Marianetti,Rev.Mod.Phys.78,865(2006). [26] For theatomicvolumes wehaveignored anythermal volume [7] K. Held, O. K. Andersen, M. Feldbacher, A. Yamasaki, and expansion.Eachphaseisrepresentedbyavolumecorrespond- Y.-F.Yang,J.Phys.:Condens.Matter20,064202(2008). ingtoafixedtemperaturewithinthatphase.Wehaveusedthe [8] J.Kunesˇ, I.Leonov, M.Kollar,K.Byczuk, V.Anisimov, and originaldataofZachariasenandEllinger[30–33]correspond- D.Vollhardt,Eur.Phys.J.SpecialTopics180,5(2010). ingtothevolumesatthetemperatures: 21,190,235,320,477, [9] R.C.Albers,N.E.Christensen, andA.Svane,J.Phys.: Con- and490◦C,fortheα,β,γ,δ,δ′,andǫphases,respectively. dens.Matter21,343201(2009). 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