The spectral densities for 3d and 4d transition metals are calculated using the simplified version of the self-consistent GW method employing the local (one-site) approximation and the self-consistent quasiparticle basis set. The results are compared with those given by the traditional local density approximation (LDA) and also with experimental x-ray photoemission and inverse photoemission spectra. While no systematic improvements over LDA are observed, this fully self-consistent many-body technique generates quite reasonable results and can serve as a practical prototype for further development of the many-body electronic structure theory

Antropov, Vladimir P.

Belashchenko, Kirill D.

Zein, N.E.

English

DigitalCommons@University of Nebraska

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications from Nebraska Center for Materials and Nanoscience Materials and Nanoscience, Nebraska Center for (NCMN) 2-7-2006 Self-consistent local GW method: Application to 3d and 4d metals Kirill D. Belashchenko University of Nebraska-Lincoln, belashchenko@unl.edu Vladimir P. Antropov Ames Laboratory, US Department fo Energy, antropov@ameslab.gov N.E. Zein Russian Research Center “Kurchatov Institute,” Moscow 123182, Russia Follow this and additional works at: https://digitalcommons.unl.edu/cmrafacpub  Part of the Nanoscience and Nanotechnology Commons Belashchenko, Kirill D.; Antropov, Vladimir P.; and Zein, N.E., "Self-consistent local GW method: Application to 3d and 4d metals" (2006). Faculty Publications from Nebraska Center for Materials and Nanoscience. 8. https://digitalcommons.unl.edu/cmrafacpub/8 This Article is brought to you for free and open access by the Materials and Nanoscience, Nebraska Center for (NCMN) at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications from Nebraska Center for Materials and Nanoscience by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Self-consistent local GW method: Application to 3d and 4d metalsK. D. BelashchenkoDepartment of Physics and Astronomy and Center for Materials Research and Analysis, University of Nebraska-Lincoln,Lincoln, Nebraska 68588, USAV. P. AntropovCondensed Matter Physics, Ames Laboratory, Ames, Iowa 50011, USAN. E. ZeinRussian Research Center “Kurchatov Institute,” Moscow 123182, RussiaReceived 1 September 2005; revised manuscript received 14 November 2005; published 7 February 2006The spectral densities for 3d and 4d transition metals are calculated using the simplified version of theself-consistent GW method employing the local one-site approximation and the self-consistent quasiparticlebasis set. The results are compared with those given by the traditional local density approximation LDA andalso with experimental x-ray photoemission and inverse photoemission spectra. While no systematic improve-ments over LDA are observed, this fully self-consistent many-body technique generates quite reasonableresults and can serve as a practical prototype for further development of the many-body electronic structuretheory.DOI: 10.1103/PhysRevB.73.073105 PACS numbers: 71.10.w, 71.20.Be, 71.15.QeI. INTRODUCTIONThe electronic properties of many solids are describedreasonably well by the local density approximation LDA ofthe density functional theory DFT,1 which is essentially asingle-particle theory based on a drastic simplification of thefull many-body problem. However, the LDA fails in materi-als exhibiting strong spatial electronic correlations, temporalcorrelations retardation associated with Coulomb screen-ing, etc. These fundamental difficulties are extremely hard toovercome within the DFT without sacrificing the first-principles approach.One of the methods used to correct the deficiencies ofLDA in the description of excited states in metals and semi-conductors is the GW approximation GWA.2,3 The GWAusually improves the band gaps in semiconductors andinsulators;3 in metals it may provide information on the qua-siparticle lifetimes and renormalization which is absent inthe DFT.3,4 However, in most treatments used until recently,the GWA was employed in a non-self-consistent fashion, byusing unrenormalized Green’s functions constructed from theKohn-Sham eigenstates obtained in LDA. This approach iseasier to implement, but it is internally inconsistent: since itsself-energy cannot be obtained by variation of any Luttinger-Ward functional,5 it violates basic conservation laws,6 andthe results depend on the approximation used to solve theKohn-Sham equations. On the other hand, studies of the ho-mogeneous electron gas have shown that self-consistencyworsens the agreement of the GWA results with experimentat typical metallic densities,7 highlighting the limitations ofGWA which is formally accurate only in the high-densitylimit.Recently, several self-consistent realizations of the GWAwere demonstrated. One of them8 was tailored for transitionmetals and employed the one-site approximation OSA forthe self-energy which is justified by the localized characterof d-electron wave functions and by the fact that the staticCoulomb interaction is efficiently screened. Some spin-selective diagrams beyond GW were also included. The re-sults for Fe and Ni were quite reasonable, although no im-provement was obtained compared to LDA. Later, theaccuracy of the OSA for transition metals was demonstratedexplicitly by cluster GW calculations.9 Although the choiceof GW diagrams is unjustified for the homogeneous electrongas at typical metallic densities and likewise for semiconduc-tors the diagrams that are left out do not contain any smallparameter, it was noted that the situation may be better intransition metals, because high orbital degeneracy providesan additional enhancement to the diagrams with the largestnumber of closed electron loops, thereby favoring the GWset.8 On the other hand, OSA makes self-consistent calcula-tions much easier.Another realization10 using GW set with the full k depen-dence of the self-energy was applied to elemental semicon-ductors. It was found that self-consistency and accurate treat-ment of core electrons improve the agreement withexperiment for the band gaps in Si and Ge.A simplification of GWA neglecting the renormalizationfactor Z was also suggested.11 For many insulators and semi-conductors this method predicts band gaps in very goodagreement with experiment.12 Self-consistency was found tobe essential for this agreement.While the adequacy of GWA for the studies of metals andsemiconductors has not been firmly established from thepoint of view of the many-body theory, this method may beregarded as a practical step toward a consistent Green’sfunction-based scheme. Therefore, it is important to ascertainthe degree of accuracy of this approximation for differentmaterials. This is especially desirable for transition metalswhere, as noted above, there are reasons why GWA maywork better than in the homogeneous electron gas. In thispaper we calculate the conduction-band spectral densities forPHYSICAL REVIEW B 73, 073105 20061098-0121/2006/737/0731054/$23.00 ©2006 The American Physical Society073105-1all elemental 3d and 4d metals using the self-consistent one-site GW approach and compare them both with experimentalspectroscopic data and with standard LDA results.II. SELF-CONSISTENT GW METHOD IN THE ONE-SITEAPPROXIMATIONThe technique used in this paper was introduced in Ref. 8and further tested in Ref. 9. Here we describe some points inmore detail.The self-energy = /G is obtained from theLuttinger-Ward generating functional  defined by the set ofskeleton graphs.5 The Hartree diagram gives the local contri-bution VHr, and the exchange diagram contributes x=Vr−r Im Gr ,r ,d /, where V is the Coulomb po-tential. The total contribution of the remaining GW sequencethe “correlation term” iscr,r, = − Gr,r, − Vr − r1	r1,r2,Wr2,r,dr1dr2d2i. 1Here W is the effective screened Coulomb interaction de-fined by the Dyson equationW = V − Vr − r1	r1,r2,Wr2,r,dr1dr2, 2and 	 is the polarization operator	r,r, = − Gr,r,Gr,r, +  d2i . 3The integration contour in Eqs. 1 and 3 is directed alongthe imaginary axis and embraces the cut on the real axis fromthe Fermi energy EF to the external energy. The GW approxi-mation may be modified by the inclusion of vertex correc-tions in the polarization operator 3 which may sometimesimprove the results.The calculations are drastically simplified by the use ofthe one-site approximation8 OSA, in which the self-energyis calculated only on one lattice site neglecting all matrixelements connecting different sites. This approximation isconceptually similar to the highly successful single-site dy-namical mean-field theory DMFT.13 It is particularly rea-sonable for transition metals for two reasons: first, becausethe d-electron wave functions are fairly localized, whichmakes the intersite exchange integrals small already at theHartree-Fock level and, second, because static Coulomb in-teraction in metals is effectively screened at the interatomicdistance. The validity of OSA was checked for different ma-terials, and it was found to be very accurate for transitionmetals.9In OSA the self-energy depends on energy and on thecoordinates r, r belonging to the same unit cell. In order toimplement OSA we need to choose an appropriate on-sitebasis. Considering the successful description of closelypacked solids by the atomic sphere approximation within thelinear muffin-tin orbital method LMTO-ASA, we use theminimal set with one basis function for each angular momen-tum l and its projection m, along with its energy derivativebelow we denote L lm. The radial basis functions lrsatisfy the equationl + l2 − VH − ˆ lllr = 0, 4where l is the radial part of the Laplasian, and ˆ l is anintegral radial operator whose kernel is obtained from Reˆby projecting onto the l subspacelr,r, =12l + 1m  YLrˆRe xcr,r,YLrˆdodo,5where we denoted xcx+c, YL are the spherical har-monics, and integration is over the directions of r , r. Theoperator ˆ l may be represented as the sum of its local partllr similar to an external potential, and a nonlocal partˆ lnwhose operation on l gives a linear combination of lwith l l.Similar to the LMTO-ASA method, the solutions of Eq.4 for each l are only found at one energy l chosen at thecenter of gravity of the occupied part of the given band. Wemay safely discard the imaginary part of the self-energy op-erator because it is small in the vicinity of EF where l isusually chosen. Thus, our radial basis functions are real. Thenonlocal equation 4 is solved by iterations. The n+1-thiteration ln+1 for the solution is obtained using auxiliaryfunctions f l and gl defined asl + l2 − VH − llr f lr = ˆ lnlln, 6l + l2 − VHglr = 0 7according to ln+1r= f lr+Aglr, where A is found by nor-malizing ln+1. Just as in the LMTO method, we also com-pute the energy derivative ˙ ll /:l + l2 − VH − ˆ ll˙ l = − l + ˆ lll l. 8To stabilize the solution of the Schrödinger equation 4,we added the exchange with the nearest-neighbor cells toxc. The corresponding matrix element was subtracted fromGreen’s function 9 below.With the on-site self-energy xcr ,r , defined for r , rwithin the same unit cell, we can calculate its matrix ele-ments between the muffin-tin “eigenfunctions” k that arelinear combinations of l and ˙ l. Due to the k dependence ofk, the self-energy also acquires k dependence, just as thematrix elements of the local potential in LDA. Finally, on-site Green’s function is found by integration using the tetra-hedron methodBRIEF REPORTS PHYSICAL REVIEW B 73, 073105 2006073105-2Gr,r, = kkR rkL r − k, 9where kR and kL are right and left eigenvectors, and k theeigenvalues of the non-Hermitian operator H0+xc, whereH0 is the Hartree Hamiltonian. Equation 9 imposes the lo-cality condition and is equivalent to the self-consistency re-lation of the DMFT.13III. CONDUCTION BAND SPECTRAL DENSITIES OFTRANSITION METALSUsing the GW technique described above, we calculatedthe spectral densities N=−Tr Im G / for elemental 3dand 4d metals, where LDA has long been the only appropri-ate approximation. We used the fcc structure for hcp ele-ments Sc, Ti, Y, Zr, Tc, Ru, Co and the bcc structure forMn. All atomic volumes were taken from experiment. Cr wastreated within the single bcc cell, and hence was nonmag-netic. The Brillouin zone integrations were performed by tet-rahedron method on a grid with 16 points along each recip-rocal lattice vector.The first iteration was started from the LDA potential, andat each iteration the nonlocal self-energy was mixed with thispotential with a gradually increasing weight, leaving onlyxc in the end. In the final state the magnitude of xc differsby about 40% from its initial LDA value. The iterationalprocedure is rather stable in all metals except Ni where themagnetic moment is very sensitive to the details of the cal-culation.The results are shown in Figs. 1–3 along with the LDAdensities of states DOS and the experimental XPS and BISspectra taken from Refs. 14 and 15. Strictly speaking, com-parison with experiment requires the calculation of the cor-responding matrix elements, but we believe that in thepresent context some qualitative conclusions can be drawnbased on the spectral densities alone.The difference between the LDA DOS and GW spectraldensity may be generally summarized as follows. The con-duction band widens as all DOS features are pushed awayfrom EF; this outward shift is roughly proportional to thedistance from EF. Moreover, all DOS features are increas-ingly smeared out due to the decreasing quasiparticle life-time as the distance from EF increases. Substantial spectralweight is transferred from the quasiparticle states to the in-coherent “tail” extending far below EF.The LDA DOS for d metals is typically too small to ac-count for the measured electronic specific heat C=T. Asseen from Figs. 1–3 the GW spectral density at EF is gener-ally smaller compared to LDA. Although to obtain  we haveto remove the renormalization factor Z from Im G, the GWmethod does not improve the overall agreement with specificheat measurements.FIG. 1. Color online Spectral densities obtained in GW-OSAsolid red lines, LDA densities of states dashed black lines, andexperimental XPS and BIS spectra dotted blue lines shifted up fornonmagnetic 3d metals. Elements for which the crystal structurewas not experimental are marked by a star in the upper left corner.In all graphs the x axis shows energy referenced from EF in eV, andthe y axis denotes the spectral density in eV−1 for the calculatedcurves.FIG. 2. Color online Same as in Fig. 1 but for magnetic 3delements. The data for majority-spin and minority-spin electrons areplotted as positive and negative values, respectively.FIG. 3. Color online Same as in Fig. 1 but for 4d metals.BRIEF REPORTS PHYSICAL REVIEW B 73, 073105 2006073105-3The difference between the GW and LDA spectra is mostnotable for early transition metals treated here in the fccstructure, where the strongest unoccupied peaks given byLDA are almost completely smeared out in GW. Notably, theworst agreement with experiment for the LDA spectra is ob-served for the same elements. In particular, the position ofthe Fermi level obtained in LDA for hcp Sc and Y appears tobe off by nearly 1 eV.14The agreement between LDA and GW improves as wemove to later transition metals and as the fcc structure isreplaced by the bcc one. In V and Nb the GW and LDAcurves are already quite similar except for the shift of theunoccupied d peak by 1.5–2 eV to higher energies.As in LDA, the 3d metals from Mn to Ni were magneticin our calculation. For simplicity, we used the bcc structurefor Mn, and also the fcc structure for Co this structure isstable in thin films. As it is seen in Fig. 1, the general fea-tures of the GW spectral density described above are ob-served in these metals as well. In general, the GW descrip-tion also gives a reasonable exchange splitting and magneticmoment. We obtained the moments of 0.9B for Mn com-pared to 1.03B in LDA, 2.3B for Fe compared to 2.25Bin LDA, and 1.85B for fcc Co compared to 1.62B in LDA.From these results it is clear that there is no systematic trendfor GW to give larger or smaller magnetic moments com-pared to LDA. The magnetic moment in Ni is rather sensitiveto various details of the calculation, and proper convergenceturned out to be problematic. We believe that the approachbased on Matsubara Green’s functions is necessary to avoidthis problem. Apart from the exchange splitting, the shape ofthe spectrum is quite stable. We also note that the magneticmoment is expected to be sensitive to the choice of the skel-eton graph set.As an example of the general trend of band dilatation offthe Fermi level, the distance from EF to the upper edge of thefully occupied d-band in Cu and Ag is notably larger in GWcompared to LDA, which results in a better agreement withexperimental XPS spectra. On the other hand, we also ob-serve a rather strong upward shift of the unoccupied peak inV, Nb, Fe, and Mo in obvious disagreement with the BISspectra. This shift is more pronounced compared to thedownward shift of the occupied states at a similar distancefrom EF.The results presented above demonstrate that the self-consistent GW approach with one-site approximation pro-vides a reasonable description of transition metals. For 3dand 4d systems the GW spectral density is generally similarto the LDA density of states, while the GW approach in-cludes typical Fermi-liquid effects such as finite quasiparticlelifetime and self-consistent renormalization. The preliminarycomparison of the GW-OSA results with experiment is sat-isfactory and clearly indicates its problems that need im-provement, namely, the exchange splitting in ferromagnets,value of NEF, and the unoccupied peak position which istoo high for some metals. The first two deficiencies are com-mon with LDA. In general, the presented technique based onthe Luttinger-Ward functional is a practical alternative toDFT and can serve as a reasonable starting point for moresophisticated methods.ACKNOWLEDGMENTSThis work was completed utilizing the Research Comput-ing Facility of the University of Nebraska-Lincoln. Thiswork was partially carried out at Ames Laboratory, which isoperated for the U.S. Department of Energy by Iowa StateUniversity under Contract No. W-7405-82. This work wassupported by the Director for Energy Research, Office ofBasic Energy Sciences of the U.S. Department of Energy.N.E.Z. acknowledges support from the Council for the Sup-port of Leading Scientific Schools of Russia under Grant No.NS-1572.2003.2.1 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 1964; W.Kohn and L. J. Sham, Phys. Rev. 140, A1133 1965.2 L. Hedin, Phys. Rev. 139, A796 1965.3 F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 2371998.4 V. M. Silkin, E. V. Chulkov, and P. M. Echenique, Phys. Rev. B68, 205106 2003, and references therein.5 J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417 1960.6 G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 1961; G.Baym, ibid. 127, 1391 1962.7 B. Holm and U. von Barth, Phys. Rev. B 57, 2108 1998.8 N. E. Zein and V. P. Antropov, Phys. Rev. Lett. 89, 1264022002.9 N. E. Zein, S. Y. Savrasov, and G. Kotliar, cond-mat/0511064unpublished.10 W. Ku and A. G. Eguiluz, Phys. Rev. Lett. 89, 126401 2002.11 S. V. Faleev, M. van Schilfgaarde, and T. Kotani, Phys. Rev. Lett.93, 126406 2004.12 M. van Schilfgaarde, T. Kotani, and S. V. Faleev, cond-mat/0510408 unpublished.13 A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod.Phys. 68, 13 1996.14 A. G. Narmonev and A. I. Zakharov, Phys. Met. Metallogr. 65,315 1988.15 W. Speier, J. C. Fuggle, R. Zeller, B. Ackermann, K. Szot, F. U.Hillebrecht, and M. Campagna, Phys. Rev. B 30, 6921 1984.BRIEF REPORTS PHYSICAL REVIEW B 73, 073105 2006073105-4

Self-consistent local GW method: Application to 3\u3ci\u3ed\u3c/i\u3e and 4\u3ci\u3ed\u3c/i\u3e metals

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Self-consistent local GW method: Application to 3\u3ci\u3ed\u3c/i\u3e and 4\u3ci\u3ed\u3c/i\u3e metals

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