GW lifetimes of quasiparticle excitations in paramagnetic transition ...

PHYSICAL REVIEW B 76, 245125 共2007兲

GW lifetimes of quasiparticle excitations in paramagnetic transition metals I. A. Nechaev,1,* E. V. Chulkov,1,2 and P. M. Echenique1,2 1

Donostia International Physics Center (DIPC), P. de Manuel Lardizabal, 4, 20018 San Sebastián, Basque Country, Spain 2 Departamento de Física de Materiales, Facultad de Ciencias Químicas and Centro Mixto CSIC, UPV/EHU, Apdo. 1072, 20080 San Sebastián, Basque Country, Spain 共Received 14 September 2007; published 28 December 2007兲 We present ab initio calculations of the quasiparticle lifetime ␶ in cubic transition metals 共V, Nb, Ta, Mo, W, Rh, and Ir兲 performed within the GW approximation. By analyzing the behavior of the lifetime as a function of the exciting energy, we demonstrate that upon moving along the d periods and within the d groups of the Periodic table, the changes in d-band width, filling, and shape have a strong effect on quasiparticle lifetimes and result in trends fundamentally distinct from those predicted within the homogeneous electron gas model. We show that ␶共E − EF兲 is mainly determined by the density of d states localized in the vicinity of the Fermi level EF and by the Coulomb interaction screened by d electrons. By making use of this partially screened interaction instead of the fully screened one, we estimate quasiparticle lifetimes in Ir3Nb with L12 structure and Rh3Ir with D022 structure. DOI: 10.1103/PhysRevB.76.245125

PACS number共s兲: 71.10.⫺w, 71.15.Ap, 71.20.Be

I. INTRODUCTION

During the past decades, ultrafast dynamics of quasiparticles in metals has received great attention both from experimental and theoretical sides.1 It is reasoned by the fact that quasiparticle dynamics plays an important role in a rich variety of physical and chemical phenomena.2 Among experimental methods developed for such kind of investigations, one of the most suitable is the technique of time-resolved two-photon photoemission 共TR-2PPE兲 spectroscopy, which enables one to measure quasiparticle excitation lifetimes directly in time domain at a temporal resolution of a few femtoseconds.3–9 Theoretically, a study of quasiparticle properties, including the lifetime ␶ determined by the imaginary part of the quasiparticle self-energy ⌺, is based on the well-known closed set of Hedin’s coupled integral equations.10 The first cycle of the iterative solution of Hedin’s equations ends up by modeling the quasiparticle self-energy as the product of the zeroth order 共free-particle兲 Green function and the dynamic screened Coulomb interaction obtained within the random phase approximation 共RPA兲. Such a product constitutes the so-called GW approximation 共GWA兲 to ⌺ that makes ab initio calculations of the quasiparticle self-energy feasible.11 In the past, the GW-lifetime calculations have been performed for simple,12–15 noble,12,14–17 3d-ferromagnetic,18 some 4d-transition,9,13,19,20 and 5d-transition metals, such as tantalum8 and platinum.21 Based on different band theory methods, these calculations have obviously shown that, in the case of simple metals, taking the real band structure into account leads to the lifetime that demonstrates practically the same behavior as it is predicted for a homogeneous electron gas 共HEG兲, i.e., ␶ ⬀ ␻−2 with the exciting energy ␻ = E − EF, where EF is the Fermi energy. Somewhat different situation is observed in the case of noble metals due to the presence of filled d band under EF 共⬃2 eV兲. However, owing to the fact that simple and noble metals possess qualitatively similar22 band structures and densities of states 共DOSs兲 in the vicinity of EF, here again, the lifetime exhibits the expected energy 1098-0121/2007/76共24兲/245125共5兲

dependence ␶ ⬀ ␻−2. Fundamentally distinct case is transition metals, where EF is positioned within localized states of d band, which in its turn varies strongly upon moving along the d periods of the Periodic Table.22 As was shown within GWA evaluations,19,20 the lifetimes also strongly vary along the d periods following trends in band structure. Nevertheless, a systematic study of transition metals in order to reveal regularities in changes of quasiparticle lifetimes upon moving both along the periods and within the groups has not been undertaken. In this work, to find out the factors that mainly influence ␶共␻兲 in transition metals, on the basis of GWA calculations, we give a comparative analysis of the ␶共␻兲 behavior in cubic paramagnetic transition metals: V, Nb, Mo, Rh, Ta, W, and Ir. The choice of these metals is accounted for by avoiding additional complications related to spin 共ferromagnets兲 and lattice 共hexagonal兲 effects. Already, this choice allows us to highlight the remarkable differences occurring when moving from metal to metal both within the group and period. In particular, we find how filling, width, and shape of d band govern the quasiparticle lifetime. Compared to Refs. 8, 9, and 20 in the present work, we perform the GWA calculations with higher accuracy 共752 k points in the irreducible 1 / 48th part of the Brillouin zone instead of 256 k points used in the cited works兲 that allows us to more precisely average lifetimes over momentum k. One-electron wave functions ␺kn and energies ⑀kn are calculated by using the linear muffin-tin orbital method23 within the local density approximation. Instead of plane waves, in expanding quantities invariant with respect to lattice translations, a set of product functions24 兵Bki共r兲其 satisfying Bloch’s theorem is used. For details, we refer the reader to Refs. 16 and 25. Unless stated otherwise, atomic units, e2 = ប = m = 1, are used throughout this work. II. RESULTS AND DISCUSSION

In Fig. 1, we show the calculated DOS in the considered transition metals combined into groups of the Periodic table.

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©2007 The American Physical Society


NECHAEV, CHULKOV, AND ECHENIQUE

FIG. 1. 共Color online兲 DOS in the considered transition metals.

As is evident from the figure, when one moves along the period, e.g., from Nb to Mo, the DOS undergoes only slight modifications due to the fact that the crystal structure 共bcc in the case兲 remains unchanged. The same situation is observed in going down within any group, e.g., from Mo to W. Cardinal changes in the DOS are found out when one compares metals of the center and the end of d periods. Actually, on the one hand, a profound minimum halves the bcc d band, and, on the other hand, fcc metals show the continuous d band. A distinguishing feature of all the metals is the DOS shape in the vicinity of the Fermi level. In this sense, a fundamental difference between, e.g., Nb and Mo 共metals with similar DOS and nearly half-filled d bands兲 is that the former has EF in immediate proximity to a DOS peak, whereas the latter is characterized by EF in the minimum of the DOS due to greater number of valence electrons. For metals of the end of 4d and 5d periods with almost-filled d band 共Rh and Ir, respectively兲, again one can see relatively high DOS and peak structure near EF. At that, the Fermi level is situated quite close to the edge of d band. Since the DOS directly depends on dispersion of ⑀kn that, along with ␺kn, determine quasiparticle lifetimes, the mentioned changes in filling, width, and shape of d band should affect the lifetime as a function of energy. Such an influence is confirmed by the results presented in Fig. 2, where the calculated lifetime ␶kn is averaged over momentum for a given exciting energy. As follows from Fig. 2, metals of the V-Nb-Ta group demonstrate close values of the averaged lifetime. In that sense, as well as in Fig. 1, niobium and tantalum are most similar to each other. Possessing notably more narrow d band and higher density of states, vanadium shows shorter quasiparticle lifetimes. On moving from tantalum to tungsten, where due to changes in band filling the Fermi level is shifted to the DOS minimum, on the whole, the lifetime increases drastically. This becomes particularly evident in the region of hole excitations. Analyzing the dependencies shown in Fig. 2, one can note that, first, lifetimes of quasiparticle excitations in metals of the same group are of the same order of magnitude. Second, when moving from the

FIG. 2. 共Color online兲 Momentum-averaged lifetimes ␶ of excited 共e兲 electrons and 共h兲 holes in all the considered metals. Symbols are results of ab initio GW calculations. Solid lines are polynomial fit curves 共see the text兲.

bcc metals of the centers of 4d and 5d periods to the fcc metals of the ends of these periods, an effect, which is contrary to that observed in going from Ta to W, occurs. Actually, comparing Mo and Rh, one can easily see that the averaged lifetime becomes notably shorter. Table I contains our results on the polynomial fit for the N calculated scaled lifetime ␶ ⫻ ␻2 = a + 兺n=1 bn␻n, with N = 4 in the case of the V-Nb-Ta group. Values for the polynomial fitting parameters for the case of Mo, W, Rh, and Ir are listed in Table II with the index of polynomial N = 3. Such values for the indices ensure rather well fitting 共see Fig. 2兲 under conditions of a minimum set of parameters and a positive free term a, whose value allows one to estimate an effective electron density parameter rs with the help of the QuinnFerrell formula,26 TABLE I. Polynomial fit parameters a and bn for V, Nb, and Ta with the index of polynomial N = 4. Here, e and h indicate the E ⬎ EF and E ⬍ EF cases, respectively. The effective electron density parameter rs is obtained from a within the Fermi-liquid model by Quinn and Ferrell 共Ref. 26兲. Metal

a

b1

b2

b3

b4

rs

V

e h

1.387 5.025

1.435 4.589

18.181 9.277

−8.207 3.262

0.941 0.331

8.15 4.87

Nb

e h

7.673 6.047

−9.676 2.487

25.481 7.469

−7.912 2.358

0.677 0.202

4.11 4.52

Ta

e h

9.187 6.673

−3.950 1.416

10.512 4.712

−1.482 1.313

−0.044 0.096

3.83 4.35

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GW LIFETIMES OF QUASIPARTICLE EXCITATIONS… TABLE II. Same as in Table I but for Mo, W, Rh, and Ir with the index of polynomial N = 3. Metal

a

b1

b2

b3

rs

Mo

e h

33.957 33.805

7.497 −36.578

−4.036 −14.645

0.447 −1.562

2.27 2.27

W

e h

38.303 34.510

11.846 −51.397

0.416 −18.290

−0.626 −1.717

2.16 2.25

Rh

e h

7.166 5.669

−4.077 −3.208

4.005 −0.402

−0.444 −0.046

4.23 4.64

Ir

e h

29.319 27.096

−23.211 15.573

8.636 7.578

−0.771 0.910

2.41 2.48

␶QF共E − EF兲2 = 36共3␲2/2兲−1/3rs−5/2 .

共1兲

At that, we keep in mind that this formula is exact at 共E − EF兲 EF and rs 1. However, in actual practice,3,4,6,7,12 Eq. 共1兲 is often used to estimate fast the quasiparticle lifetime even in the case of rs that corresponds to typical metallic densities 共from ⬃2 to ⬃6兲. In the present work, we use Eq. 共1兲 to evaluate the effective rs that agrees with the parameter a of the fit to ␶ ⫻ ␻2 dominating at small exciting energy. The energy interval for the polynomial fit is defined by 兩␻兩 ranging from 0.2 to 5.0 eV. Values listed in Table I reflect the aforementioned situation: Nb and Ta are close to each other, and V stays slightly apart in the 共E − EF兲 ⬎ 0 range due to more narrow d band. Note that these d metals have quite large estimated rs ⲏ 4.0 that according to the HEG model ensures the low DOS at EF, which disagrees with Fig. 1. Moreover, as is seen from Table II, in the case of Mo and W, where at EF, the DOS is notably lower than in V, Nb, and Ta, the parameter rs is substantially smaller. When moving further along 4d period to Rh, at EF, the DOS runs up close to its value in, e.g., Nb. As a result, we have practically the same rs’s. As compared with rhodium, iridium has wider d band and lower density of states. This entails rs values similar to those estimated for Mo and W. Thus, regarding the free term a, which dominates at small exciting energies, as a function of the DOS at EF, one can infer that the transition metals exhibit a tendency opposite to that peculiar to the HEG. As is known,2,8,9 theoretical GW lifetimes are longer than experimental TR-2PPE relaxation times. Nevertheless, the GWA allows one to describe qualitatively 共at some energies, quantitatively兲 the changes in dependencies ␶共␻兲 upon moving from one metal to another. Figure 3 shows the energy dependencies of the ratio between the calculated averaged lifetimes in different metals. As follows from the figure, the ratio ␶Mo / ␶Rh is consistent with experimental data taken from Ref. 9. This means that the revealed tendencies with respect to changes in lifetime values upon moving from metal to metal bear a fundamental character. To all appearances, possible modifications of the lifetimes when considering beyond the GWA9,27 will not lead to an essential transformation of the ratios.

FIG. 3. 共Color online兲 Ratios of the averaged lifetimes for indicated metals. Dots depict experimental data taken from Ref. 9.

Regarding the ratios of averaged lifetimes shown in Fig. 3, one can note that, in spite of the common tendency toward changes in the DOS shape, modifications of ␶共␻兲 occurring in going along 4d and 5d periods are quite different. The ratios ␶Mo / ␶Rh and ␶W / ␶Ir can be considered as striking examples of the difference. As a result, when moving from Mo to Rh 共bcc→ fcc兲, an effect that is practically the same as in the case of W and Ta 共bcc→ bcc兲 occurs. As compared with the ratios of other metals from different groups, the ratio ␶W / ␶Ir does not demonstrate such considerable electron-hole 共e-h兲 asymmetry.28 Thus, the DOS should be regarded as a quantity that plays an important, but not determinative, role in forming the ␻ behavior of ␶. In other words, a description of quasiparticle lifetimes cannot be limited merely by analyzing the DOS shape at the vicinity of EF. Actually, within a scattering theory approach,14,20,29 the inverse lifetime is approximated by a product of the convolution C共␻兲 of the density of occupied and unoccupied states and the matrix element 兩M共␻兲兩2 of the screened interaction. This means that a DOS shape governed by features of oneelectron energy spectrum ⑀kn influences quasiparticle lifetimes indirectly 共by means of the convolution兲, and the matrix element is an additional quantity accounting for dielectric screening effects 共see, e.g., Ref. 30兲. Dielectric screening properties of a system are determined by characteristics of initial and final states of possible intra and interband transitions within the energy range 2␻ centered at EF. In the considered metals, these states are predominantly the localized d states. Thus, one can suppose that a matrix element behavior is mainly determined by transitions from d to d states. In order to examine the contribution of such transitions to ␶共␻兲 within the GWA, the RPA irreducible polarizability P0 that defines the screened Coulomb interaction can be rewritten as the sum P0 = P0d + Pr0, where P0d includes only the d-d transitions and Pr0 accounts for all the other possible transitions.31,32 As a result, the screened inter-

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FIG. 4. 共Color online兲共a兲 Quasiparticle decay rate ⌫ calculated with the use of both the fully screened interaction 共all transitions兲 and the interaction screened by d electrons only 共d only兲. 共b兲 Ratios of the lifetimes for several metals.

action can be presented as W = Wd关1 − Pr0Wd兴−1, with Wd = ␷c关1 − P0d␷c兴−1, where ␷c is the bare Coulomb interaction. Regarding the difference W − Wd as a negligible quantity, we can use Wd in place of W in the GW formula modeling the self-energy.33 Some representative results of such calculations are shown in Fig. 4. Inspecting Fig. 4共a兲, one can see that taking into account the Coulomb interaction screened by d electrons only results in inverse lifetime values close to those obtained with the use of the fully screened interaction, except for the case of vanadium 关especially at 共E − EF兲 ⬍ 0兴. In search of obvious correlations, we can note that the observed discrepancy between different calculations is mainly governed by ratios between partial densities of d and sp states. Actually, in vanadium, as E below EF approaches 共quite fast兲 a region with relatively high sp-partial DOS, the discrepancy between W and Wd values becomes greater. Also at E ⬍ EF, iridium with a wide and continuous band 共full of d states兲 does not show any notable deviation induced by substitution of W by Wd. The same situation is observed in the case of tungsten at E ⬎ EF. Regarding the question of what effect the substitution of W by Wd has on the revealed trends, we compare the lifetime ratios obtained from respective calculations. As follows from Fig. 4共b兲, for excited electrons, the “d-oriented” calculations catch rather well the trend in change of ␶ upon moving both along the d periods and down within the d groups. By this, we demonstrate that already at the Wd level, it is possible to semiquantitatively predict quasiparticle lifetimes and their ␻ dependencies in more complex system such as transition alloys and compounds. As an example, Fig. 5 shows the evaluated quasiparticle lifetimes in Ir3Nb with L12 structure and in Rh3Ir with D022 structure. According to Fig. 5共a兲, the intermetallic compound Ir3Nb has the DOS with a deep minimum in the vicinity of EF, what makes it similar to the DOS, e.g.,

FIG. 5. 共Color online兲共a兲 DOS in Ir3Nb and Rh3Ir. 共b兲 The scaled lifetime calculated for these compounds with the use of the Coulomb interaction screened by d electrons only. For comparison, the scaled lifetime calculated with the fully screened interaction for the compound constituents and tungsten is also presented.

in W. As a consequence 关see Fig. 5共b兲兴, the scaled lifetime values in the compound Ir3Nb are closer to those in tungsten than to those in iridium or niobium. In the case of Rh3Ir, only small deviation from the DOS in Rh is observed. This results in the scaled lifetime which is very close to its values in rhodium. On the whole, as before, lower density of states corresponds to longer lifetimes. III. CONCLUSIONS

In conclusion, within the GW approximation, we have analyzed the effect of width, shape, and filling of d band on energy dependence of quasiparticle lifetimes in cubic paramagnetic transition metals. On considering the metals with nearly half-filled 共V, Nb, Ta, Mo, and W兲 and almost-filled 共Rh and Ir兲 d band, we have ascertained that taking the real band structure into account leads to the energy dependence of quasiparticle lifetimes, which differs from that in the HEG model. When moving along the d periods from the V-Nb-Ta group to the Mo-W one, a notable decrease in the DOS in the vicinity of the Fermi level results in a quite sharp increase of lifetime values. Moving further to the Rh-Ir group, the DOS at EF becomes higher and runs up to values close to those in, e.g., Nb, what entails shortening quasiparticle lifetimes. This tendency is also contrary to the well-known dependence of lifetime values on density of states at the Fermi level in the HEG. We have shown that the band broadening that occurs upon moving down within the d groups mainly affects lifetimes of electron excitations in the case of the bcc metals with nearly half-filled d band and lifetimes of hole excitations in the case of the fcc metals with almost-filled d band. We have noted that in a quantitative sense, changes in an energy dependence of ␶ upon moving along the 4d and 5d

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GW LIFETIMES OF QUASIPARTICLE EXCITATIONS…

We believe that an important area for further studies is an extension of the present consideration to ferromagnetic transition metals and their alloys. The influence of the composition and degree of atomic ordering in ferromagnetic alloys on the lifetime is an essential part of such an extension.

periods are different in many respects, in spite of common tendency in DOS shape modifications. It is caused by differences in dielectric screening that in transition metals is predominantly realized by electrons in d states. This means that, in the considered metals, as in the case of crystal phase stability governed by features in the energy spectrum of valence d electrons 共see, e.g., Ref. 34兲, lifetimes of quasiparticle excitations are mainly defined by density of d states localized in the vicinity of the Fermi level and by the Coulomb interaction screened by d electrons. This fact allowed us to evaluate the quasiparticle lifetime in such compounds of transition metals as Ir3Nb with L12 structure and Rh3Ir with D022 structure. The obtained lifetimes have confirmed the aforementioned tendencies.

The work was partially supported by Departamento de Educación del Gobierno Vasco 共Grant No. 9/UPV 00206.215-13639/2001兲 and the Spanish Ministerio de Ciencia y Tecnología 共Grant No. FIS 2004-06490-C03-01兲.

*Also at Theoretical Physics Department, Kostroma State Univer-

17 F.

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ACKNOWLEDGMENTS

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