Ann. Physik 5 (1996) 247-257 Annalen der Physik 0 Johann Ambrosius Barth 1996 Electronic excitations in NiO Manabu Takahashi and Jun-ichi Igarashi Faculty of Engineering, Gunma University, Kiryu, Gunma 376, Japan Received 15 December 1995, accepted 22 February 1996 Dedicated to Peter Fulde on the occasion of his 60th birthday Abstract. A many-body theory for a multi-orbital tight-binding model is developed by taking account of local three-body correlations. The theory is applied to the antiferromagnetic NiO. The multiplet structures of hree-particle states are fully taken into account. It is shown that the self-energy correction leads to the correct value of the band gap, the orbital character of the valence band top which is consistent with the charge-transfer insulator, and the presence of the satellite intensities. The quasiparticle band is obtained in good agreement with the angular-resolved photoemission data of NiO. Keywords: Electron correlation; Transition metal compounds: Angular-resolved photoemission. 1 Introduction The electronic structure of transition-metal compounds has been a subject of active research for a long time [l]. Extensive studies have been done in particular on NiO as a Prototype material. In this paper, we concentrate our attention on NiO. The band calculations based on the local-density approximation (LDA) usually give good results in agreement with photoemission and inverse photoemission experiments. The LDA, however, does not work when it is applied to strongly-correlated electron systems such as high-Tc materials, transition-metal compounds. For transition-metal oxides it underestimates the size of band gaps by an order of magnitudes, and in other cases where large gaps exist predicts metals [2], although it gives good results for the lattice parameter and phonon frequencies. In addition, it does not give the satellite intensity below the valence band which is observed in the photoemission Spectra [3]. Another approach is to use the cluster or impurity models [MI.By fitting the Parameters to experimental data, the model calculations reproduced successfully the Spectra of photoemission [3, 7, 81, inverse photoemission [7, 81, resonant photoemission f9], and core-level photoabsorption [S, 81. These analyses support for NiO the Picture of charge transfer insulator [lo] instead of the conventional Mott-Hubbard insulator [ 111: the 3d bands are split into the upper and lower Hubbard bands separated by the Coulomb energy U,and the oxygen band resides in between the lower and Upper Hubbard band, thereby forming a charge transfer gap with the upper Hubbard 248 Ann. Physik 5 (1996) band. The first ionization state has accordingly a strong 0 2p-character, which is also confirmed by the analysis of 0 K-edge absorption spectra in Li doped NiO [8, 121. This finding contradicts with LDA’s prediction that the oxygen bands are located below the Ni 3d bands. On the other hand, the angular-resolved photoemission spectroscopy (ARPS) has revealed clearly the itinerant character of quasiparticles [13], which the cluster or impurity models are difficult to describe. A puzzling fact is that the energy dispersions for the valence band are fairly fitted by the LDA bands. Therefore we need a manybody theory by which we calculate the energy dispersions of quasiparticles. It is not clear to what extent the LDA is including electron correlation. For making clear the role of electron correlation, it may be better to start from the Hartree-Fock (HF) approximation. The HF calculation has been done on the multi-orbital tight-binding model with parameters determined by the cluster or impurity model analyses of experiments [ 141. The states around the top of the valence band have a strong 0 2p-character in accordance with the charge transfer insulator. The band gap is obtained with a reasonable magnitude. The HF results, however, have a weak point that the quasiparticle dispersions are quite different from the ARPS data. Another weak point is that it does not give rise to satellite intensities, Recent ab initio HF calculations using local Gaussian basis sets [ 151 and the “LDA+U’ method [ 16, 171 give the similar results to the abovementioned HF calculation on the tight-binding model. Note that this fact indicates the appropriateness of using the tight-binding model mentioned above. Since the Coulomb interaction U between 3d electrons is larger than the electron transfer energy, we expect that the self-energy correction due to electron correlation is large and improves the results of the HF calculation. The self-energy correction due to the second-order perturbation with respect to U has recently been calculated [14]. Although it improves to some extent the HF results, the large value of U indicates that higher order corrections to the self-energy is too important to be neglected. In this paper we develop a many-body theory to calculate the self-energy on the tight-binding model with parameters determined by the cluster model analysis of experiments. The theory takes account of the correlation of three particles created in the intermediate state, and use a local approximation to deal with the complexity of the multi-orbital model. Preparing local states in which three particles are excited out of the HF ground state, we treat exactly the correlation within the states [18]. The multiplet structures are fully taken into account. The self-energy is approximated to be local, working only on Ni sites. This approximation is partly justified by the fact that the Coulomb interaction, which is the origin of the self-energy correction, is the strongest on nickel sites. This type of local three-body correlation theory has already been applied to ferromagnetic nickel in a good agreement with the photoemission experiment [ 19, 201. We will show that the self-energy correction improves considerably the HF results. It pushes up the 3d levels in the valence band, and leads to the quasiparticle dispersions in a good agreement with the ARPS data [131. Around the top of the valence band, the weight of 0 2p-character is comparable to that of Ni 3dcharacter, although the former is to a certain degree suppressed from the HF value. This is consistent with the insulator of charge-transfer type. The self-energy correction also gives rise to sufficient satellite intensities in a good agreement with the photoemission experiments [3, 81. In Section 2, the three-body correlation theory is formulated in the multi-orbital tight-binding model of NiO. In Section 3, the numerical calculation is shown in comparison with experiments. Section 4 contains the concluding remarks. 249 M. Takahashi, J. Igarashi, Electronic excitations in NiO 2 FormuIation 2. I Hamiltonian We start with a multi-orbital tight-binding model, H = Ho + H I , where dim, and pjruare the annihilation operators for an electron in 3d-orbital m and spin r~ at Ni-site i and 2p-orbital I and spin r~ at b i t e j, respectively, The part Ho represents the kinetic energy. The transfer integrals, tgJt, are written in @PO), @pn), ( d h ) , terms of the Slater-Koster two-center integrals, @do), (ddx), (ddd). We have included a point charge crystal field splitting (lODq), which splits the 3d orbital energies as Ed(eg)= Ed - 6 0 q , Ed(tzg)= Ed 409. The d-level position relative to p-levels is given by the charge-transfer energy A, which for the d" configuration is defined by A = Ed - Ep nU. Here U is the multiplet-averaged d-d Coulomb interaction defined below. The part HI represents the intraatomic Coulomb interactions, which is neglected in 0 sites. Abbreviation v is used for * (m,0 ) . The coefficient g(v1,v2;v3,v4) is written in terms of the Racah parameters A , B, and C. The quantity U is defined by U = A - 14B/9 7C/9. The Racah parameters B and C are expected to be close to their atomic values little screened by solidstate effects, while A may be much reduced from the atomic value. We use their atomic values for B and C, while we regard A as an adjustable parameter to give a reasonable satellite position; we use A = 5.6 eV (u = 5.9 eV) smaller than the value given by the cluster model analysis. The parameters (dda), (ddn), (ddd) are not determined by the cluster model analysis of photoemission spectra. We use Mattheiss' values estimated by the LDA calculation [211. The other parameter values in the model are taken from the cluster model analysis of photoemission spectra 14-61. The parameter values we use for NiO are listed in Table 1. q.ji, tg,itml, @dh, + + + 2.2 The Hartree-Fock approximation First we divide the Ni sites into two sublattices A and B by assuming the second kind antiferromagnetic structure for NiO. Then we make a decoupling approximation to H I , Corresponding to the HF approximation. Defining the 3d states by orbitals rn = xy, JZ, zx, x2 - ~ 23.9 , - r2 with x, y, z referring to the crystal axes, we write it as 250 Ann. Physik 5 (1996) Table 1 Parameter values for the tight-binding model of NiO in units of eV. Parameter Value 5.6 0.13 0.60 5.0 1.4 -0.63 0.60 0.15 -0.227 0.103 -0.01 0.70 with ZZr@)= Z:!?. The coefficients Z ~ ~ ( A ~ contain E ) ’ s the ground-state averages (dhddim,d),which are self-consistently determined. Using a unit cell containing two Ni atoms and two 0 atoms, we diagonalize the Hamiltonian in the momentum representation. The on-site HF energies of an electron and a hole are defined by Here Eq(k) represents the HF quasiparticle energy for the state of wavefunction[$. For the electron (hole) energy, the sum over ( q ,k) is restricted within the unoccupied (occupied) states, that is, outside (inside) the Fermi volume F . 2.3 Selj-energy correction We introduce the single-particle Green’s functions in the conventional definition. For 3d and 2p electrons, they are given by Here dim,,(t) = ei(H-~Ne)fdirnae-i(H-pN,)r with p and N , denoting the chemical potential and the number operator of electrons, respectively. In order to calculate these functions, we follow the three-particle correlation theory discussed in [191. Taking Ho H y F as the unperturbed Hamiltonian, we consider the self-energy correction. We take account of only the site-diagonal part of the self-energy on Ni sites. This part of the self-energy has the largest contribution, since the Coulomb interaction is assumed to work only on Ni sites. We divide the self-energy into a retarded and an advanced part as shown in Fig. 1: + 25 1 M.Takahashi, J. Igarashi, Electronic excitations in NiO t + t Fig. 1 Diagrams for the self-energy in the local approximation: (a) Z(')(i; t - t')and (b) Z(")(i;t - f"). Upward and downward lines represent the propagation of electrons and holes, respectively. Broken lines represent the Coulomb interaction. In the region with slanting lines, the multiple-scattering processes are taken into account. (b) where (2.10) (2.1 1) we introduce local three-particle states for the intermediate states; for the retarded part, two electrons and one hole are created out of the HF ground-state, while for the advanced part two holes and one electron are generated. For site i they are defined by where (nju)= (d!udiu).We require v1 > v2 to avoid redundancy. The second and third terms on the right hand side of eqs. (2.12) and (2.13) are for eliminating the single-particle states created by d/uldkdiw and diuldiy d!u, respectively. These threeparticle states are orthogonal to each other, but not normalized, that is, ( Ri , v1,y,~ 3 1 i ,~J;, J , I/>= xR (i)&,d,<&,~. ( A ;i, ~ i Y, ,YIA; i, 4 ,J Z I V ; ) 6y 4gu3<, 1 2 with 3 ~ ! : : ( i ) = (1 - (niw\)(l- (niy))(niy),x ~ l y u , ( l=) -xu,yw(i)~,l< - A (niw ) (niy ) ( 1 - ( n i y) 1. Within the space expanded by these states, we express the site-diagonal self-energy as (2.14) 252 Ann. Physik 5 (1996) The second line in Eq. (2.16) represents the on-site HF energy of two electrons and oqe hole. The other lines represent interaction contributions. The Hamiltonian matrix [ H A ] i vv2 v 3 ; i J J J for the advanced part is given by a form similar to Eq. (2.16) but with the re$ldckments ei, t-1 -&, v tt J , ni, tt 1 - njd, and 1 - ni, tt nig. Since the dimensions of the Hamiltonian matrices are less than 225x225, we can numerically evaluate the resolvents in (2.14) and (2.15). For cubic symmetry the self-energy becomes a diagonal matrix, where the superscript A ( B ) designates sublattice A ( B ) of site i . The diagonal elements are triply and doubly degenerate for each spin direction. Due to the antiferromagnetic order, the following relations are satisfied: + Figure 2(a) and (b) show the self-energy Zmg,m,,(i; w ) (= ( w ) Z$r (a)) on the real w axis for m E Eg and for m E T2g.They consist of discrete poles on the real o axis. The retarded part vanish in all the states except the states m E ER with minority spin. The non-vanishing contribution to the retarded part comes from the poles located at the higher energy region o > 5 eV. It is negative for w < 5 eV. By adding it to the advanced part, the self-energy is found to be extremely small values (see the dotted line in Fig. 2(a)). Note that the contribution of the advanced (retarded) part to the quasiparticle energy of electron (hole) is interpreted as the suppression of the 253 M. Takahashi, J. Igarashi, Electronic excitations in NiO 10.0 W -5.0 -1 0.0 -1 5.0 -1 0.0 a -5.0 0.0 5.0 w (eV> Fig. 2 Self-energy Zmn,mo(i; o)on the real w axis for (a) m E Eg and (b) m E Tzg.The origin of the w axis, the Fermi level, is set at the top of the valence band. The solid and broken lines represent the self-energy for the majority spin states and for the minority spin states, respectively. The vertical lines indicate the presence of poles. ground-state fluctuation due to the presence of the electron (hole) [19, 201. Inserting the self-energy given above into the Dyson equation and solving the equation, we obtain the single-pmicle Green’s functions. They consist of poles on the real o axis. 3 Calculated results in comparison with experiments we first discuss the dispersion relation of quasiparticles, in comparison with the ARPS data 1131. Figure 3 shows the calculated results for momentum along r (O,O,0) to x 0,O)direction with a denoting the lattice constant. Within the LDA, several energy bands are found with relatively flat dispersion around the top of the valence band, as shown on the left panel of Fig. 3. They have strong Ni 3d-chsacter. Below these flat bands, strongly-dispersive bands appear, which have dominant 0 2p-character. Their dispersion relations agree rather well with the experiment. This agreement looks surprising, since the LDA gives only a very small band gap and no satellite. Also the strong 3d-character around the top of 254 Ann. Physik 5 (1996) LDA and data of ARPS HF HF+SELF 1 4.0 2.0 [ 1 .:" .................................................. ........... ...... ............ ::::::I.....:........... ::::....... ....................... ........ ......... ' ...............*.'.""'.'.""'. .......... ......................... ............................... '.. ...... ...... ............... -......,., .................- ............. '.. ................. .......... .... ... ..... ....*......_ ........ ...... 1 :::..*I::: b :l:s'z -6.0 xr xr r 8 l,,.*t::::~.ly., -a-- I - -- . - -- ---*w w-a '5.""-L I 0 X Fig. 3 Dispersion relation for quasiparticles for momentum along r (0, 0,O) to X (n/u,0,O) direction. The left panel shows LDA's results. Full dots are the ARPS data [13]. The central panel shows the results by the HF approximation. The right panel shows the results with the self-energy correction. The origin of quasiparticle energies is set at the top of the valence band. the valence band contradicts with a widely-accepted view that NiO is of charge-transfer type. The HF calculation on the tight-binding model described before gives the quasiparticle bands quite different from those of the LDA, as shown on the central panel of Fig. 3. The conduction band is obtained with very flat dispersion. It consists mainly of Ni 3d states with Eg symmetry and minority spin, corresponding to the transition from the 3d8 to the 3d9 configuration. The band gap is as large as -4.5 eV. Around the top of the valence band, the quasiparticle bands have a considerable weight of 0 2p states. This is consistent with the insulator of charge-transfer type. On the other hand, around the middle and the lower part of the valence band, it is rather difficult to identify the quasiparticle bands with the ARPS data. No satellite intensity is obtained. The self-energy correction due to the three-body correlation improves considerably the HF results. The conduction band hardly shifts, since the self-energy for the minority spin states with rn E Eg symmetry is extremely small for o > 0 eV as shown in Fig. 2(a). On the other hand, in the valence band, 3d levels are pushed up, since the self-energy is positive in the region -5 < w < 1 eV. Thereby, the band gap is reduced to be 3.5 eV, mainly due to the upward shift of the top of the valence band. The gap determined experimentally is 4.0 eV [7, 221. Around the middle of the valence band, quasiparticle dispersions are considerably modified; the quasiparticle bands marked a, b, c, c' on the right panel of Fig. 3 are in good agreements with the ARPS data. A deviation from the experiment is that the quasiparticle bands marked c and and c' are not smoothly connected. In addition to these quasiparticle bands, many bands appear with nearly flat dispersion for o < -6 eV. They constitute the satellite intensity observed in the photoemission experiment [3]. - N M.Takahashi, J. 255 Igarashi, Electronic excitations in NiO )r + .cn c a, + -c -15.0 -10.0 -5.0 0 (ev) a 0.0 5.0 -1 b Fig. 4 Density of states on Ni sites and on 0 sites. The ongin of the w axis is set at the top of the valence band. (a) Results by the LDA, the HF approximation, and the self-energy correction. (b) Results by the self-energy correction in comparison with the experimental spectra of XPS [8] and 0 Ko XES [23]. For 0 Ka XES, the origin of the w axis corresponds to 527.5 eV. The density of states (DOS) on Ni sites defined by C m a ( l / n ) s g n o I m G$&,,o(i; o)is observed by the x-ray photoemission spectroscopy (XPS), which uses x-ray of h o > 1000 eV. In such energy region, the cross section of Ni 3d states is an order of magnitude larger than that of 0 2p states. Also the DOS on 0 sites defined by 1 / n ) sgn w Im Gf'Jj; o)may be observed by 0 K a x-ray emission spectroscopy (XES) [23], as discussed by Anisimov, Kuiper, and Nordgren [24]. Figure 4(a) and (b) show the DOS's on Ni and on 0 sites, in comparison with the experimental spectra of XPS and 0 K a XES. The 0 K a XES data are shifted to agree with the calculation. In the LDA calculation, the DOS on Ni sites is concentrated around the upper part of the valence band, while the DOS on 0 sites is concentrated around the lower part of the valence band. In the HF approximation, the situation is quite different; the DOS on Ni sites is distributed widely in the region -6 eV < o < -1 eV. The most prominent peak is at the upper part of the valence band. This is different from the previous model calculation by the HF approximation [14], since the present model is including the d-d transfer energy. The DOS curve on 0 sites has broad peaks around the top and the bottom of the valence band. At the vicinity of the top of the valence band, the weight of DOS on 0 sites is stronger than that on Ni sites. By including the self-energy correction, the weight of DOS on Ni sites is transferred from the lower part to the upper part of the valence band. The DOS on 0 sites increases in the lower part of the valence band. At the vicinity of the top of the valence band, the DOS on 0 sites is a little suppressed, but is still comparable to that on Ni sites. This is consistent with the insulator of charge-transfer type. Below the valence band, the DOS on Ni sites has considerable intensity. This explains the satellite intensity of XPS. Note that the shape of the DOS on Ni sites in the upper part of xla( 256 Ann. Physik 5 (1 996) the valence band is sensitively affected by including the d-d transfer energy. All these changes lead to good agreements with the experiments of XPS [8] and 0 K a XES [23], as shown in Fig. 4(b). 4 Concluding remarks We have developed a three-body correlation theory for a multi-orbital tight-binding model of NiO. We have used a local approximation in order to deal with the complexity of the multi-orbital model. Thereby we have taken full account of the multiplet structures of the three-particle states. We have succeeded in describing the itinerant as well as the local character of quasiparticles; the quasiparticle bands are found in good agreements with the ARPS data. We have also found that the weight of 0 2p-character is comparable to that of Ni 3d-character at the vicinity of the top of the valence band, being consistent with the insulator of charge-transfer type. The satellite spectra are obtained with sufficient intensities on reasonable energy positions. The obtained DOS on Ni sites and on 0 sites are also in good agreements with the data of XPS and 0 K a XES. The present theory can not describe life-time broadening of spectral peaks. This drawback is due to the use of the local approximation. The original three-body correlation theory [IS] can describe such spectral widths. In the local approximation, the three-particle states in the intermediate state are localized in a single Ni site. Since the charge transfer energy is smaller than the Coulomb interaction, the holes created in the intermediate state may have easily transfer to neighboring 0 sites. In order to take account of this effect, we need to treat the self-energy extending over the 0 sites neighboring to the concerned Ni site. Recently Manghi, Calandra, and Ossicini [25] have calculated a single-particle Green’s function for NiO by making a “local approximation” directly on the Faddeev equation for the three particles created in the intermediate state. This theory is similar to the present one, since both consider local three-body correlations. An important difference is that in their calculation the self-energy is added to the quasiparticle energies given by the LDA, not by the HF approximation. The quasiparticle energies given by the LDA are partly including the effect of electron correlation. Therefore, in the situation that LDA’s results are different from the HF calculation, the self-energy correction is better to be added to the HF calculation. In their calculation, the self-energy correction gives rise to a large band gap from a metallic state given by the LDA. The effect of the self-energy correction is opposite in the present paper; it reduces the band gap from the HF value. Finally we mention that there is an alternative approach, the strong coupling theory of a generalized spin-fermion model [26]. Although the theory explains semi-quantitatively the ARPS data, the model is too simple to give quantitative discussions. It is our pleasure to dedicate this paper to Professor Dr. Peter Fulde on the occasion of his sixtieth birthday. Also we would like to thank H. Akai, V.I. Anisirnov, N. Harnada, T. Jo, and K. Terakura for valuable discussions. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture, Japan. M.Takahashi, J. Igarashi, Electronic excitations in NiO 257 References N.F. Mott, Proc. Phys. SOC.London, Sec. A 62 (1949) 416; The metal insulator transition, Taylor& Francis, London 1974 K. Terakura, T. Oguchi, A.R. Williams, J. Kiibler, Phys. Rev. B 30 (1984) 4734 G. K. Wertheim, S. Hiifner, Phys. Rev. Lett. 2 (1972) 1028 A. Fujimori, F. Minami, Phys. Rev. B 30 (1984) 957 G . van der Laan, J. Zaanen, G.A. Sawatzky, R. Karnatak, J.-M. Esteva, Phys. Rev. B 33 (1986) 4253 K. Okada, A. Kotani, J. Phys. SOC.Jpn. 61 (1992) 4619; A. Tanaka, T. Jo, J. Phys. SOC.Jpn. 63 (1994) 2788 G.A. Sawatzky, J.W. Allen, Phys. Rev. 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