Theory of L-edge spectroscopy of strongly correlated systems
Johann Lüder, Johan Schött, Barbara Brena, Maurits W. Haverkort, Patrik Thunström, Olle Eriksson, Biplab Sanyal, Igor Di Marco, Yaroslav O. Kvashnin
TTheory of L -edge spectroscopy of strongly correlated systems Johann L¨uder,
1, 2
Johan Sch¨ott, ∗ Barbara Brena, Maurits W. Haverkort, PatrikThunstr¨om, Olle Eriksson,
1, 4
Biplab Sanyal, Igor Di Marco, and Yaroslav O. Kvashnin Department of Physics and Astronomy, Uppsala University, Box-516,Uppsala SE-751 20 Sweden Department of Mechanical Engineering, National University of Singapore,21 Lower Kent Ridge Rd, Singapore 119077, Singapore Institute for Theoretical Physics, Heidelberg University,Philosophenweg 16, D-69120 Heidelberg, Germany School of Science and Technology, ¨Orebro University, SE-701 82 ¨Orebro, Sweden (Dated: December 19, 2017)X-ray absorption spectroscopy measured at the L -edge of transition metals (TMs) is a powerfulelement-selective tool providing direct information about the correlation effects in the 3 d states. Thetheoretical modeling of the 2 p → d excitation processes remains to be challenging for contemporary ab initio electronic structure techniques, due to strong core-hole and multiplet effects influencing thespectra. In this work we present a realization of the method combining the density-functional theorywith multiplet ligand field theory, proposed in Haverkort et al. [Phys. Rev. B , 165113 (2012)]. Inthis approach a single-impurity Anderson model (SIAM) is constructed, with almost all parametersobtained from first principles, and then solved to obtain the spectra. In our implementation weadopt the language of the dynamical mean-field theory and utilize the local density of states and thehybridization function, projected onto TM 3 d states, in order to construct the SIAM. The developedcomputational scheme is applied to calculate the L -edge spectra for several TM monoxides. A verygood agreement between the theory and experiment is found for all studied systems. The effect ofcore-hole relaxation, hybridization discretization, possible extensions of the method as well as itslimitations are discussed. I. INTRODUCTION
The interaction of x rays with matter gives insightsinto structural, chemical and electronic properties ofmaterials. For instance, spectroscopy is used nowa-days to find materials for catalysis purposes[1, 2], hy-drogen storage[3, 4], batteries[5] and many more. De-velopment of the synchrotron radiation facilities allowedto make spectroscopic analyses with an unprecedentedspeed and resolution. The recently achieved increase inthe experimental resolution, e.g. with the use of crys-tal analyzers[6], allowed to resolve more detailed featuresin the absorption spectra [7]. This advancement of ex-perimental tools constantly challenges the existing theo-retical methods aimed at calculating X-ray spectroscopy,and forces the researchers to refine the approximationsthey use.The L -edge x-ray absorption spectroscopy (XAS) ofthe transition metals (TMs) is particularly interestingand informative. In this type of experiments one pri-marily probes the possible transitions from the core 2 p to unoccupied 3 d levels. A variety of exotic phenom-ena such as charge, spin and orbital orderings[8], Motttransitions[9, 10], Kondo resonances, multiferroicity[11]and superconductivity[12] originate from the correlationeffects in the 3 d states. Thus, the 3 d states play a deci-sive role in defining many important properties of the TMcompounds and the L -edge XAS directly provides theirfingerprints. Another serious advantage of L -edge XAS is ∗ [email protected] the applicability of the so-called sum-rules [13, 14]. Thismakes it possible to extract the element-resolved spin andorbital moments in ferromagnetic materials.As to the theoretical modeling of XAS, initially therewere two main groups of methods: the ones based ondensity functional theory (DFT) [15] and those utiliz-ing atomic multiplet theory[16]. For a review of bothclasses of methods, see Refs. 17 and 18. The methodsbased on DFT are in principle parameter-free and pro-vide a very detailed description of the chemical struc-ture. On the other hand, they drastically simplify theelectron-electron correlations and therefore work best foritinerant electron systems. The atomic multiplet theoryis completely the opposite, as it is intended to describeexactly all many-body interactions within an isolated ion.The theory best describes very localised electronic states,like 4 f orbitals of rare-earth elements. All effects likecrystal field splitting and hybridisation have to be takeninto account by introducing ad hoc parameters, whichresulted in the development of the multiplet ligand fieldtheory (MLFT)[19]. The main drawback of the latter isan ambiguity in the choice of the parameters, which candrastically affect the final results (see, e.g., Ref. [20]).This work concerns the study of TM monoxides, sincethey exhibit a correlation-driven insulating state andhave been extensively studied both theoretically and ex-perimentally [10, 21–25]. In order to be able to calculatethe L -edges of TMs in these systems, a method whichtakes into account both multiplet and band-structure ef-fects is required. There are three main reasons for that.First of all, the crystal field strength is estimated to beof the order of few eVs[26] and the metal-oxygen hy- a r X i v : . [ c ond - m a t . s t r- e l ] D ec bridization is appreciable, giving rise, for example, tothe super-exchange interactions, that lead to a magneticorder at low temperature [27]. Second, even the de-scription of the ground state of TM oxides (TMO) us-ing conventional first-principles DFT-based methods re-mains a problem, which manifests itself in the wronglypredicted metallic character. The complexity of TMOslays in the fact that the 3 d orbitals are neither com-pletely localised (like the 4 f states in rare-earth met-als), nor itinerant (like sp -states of Al). In order to cap-ture both atomistic and band characters, several meth-ods accounting for strong on-site correlation effects havebeen proposed, such as DFT+ U [28] and DFT plus dy-namical mean field theory (DFT+DMFT)[29]. SinceDFT+DMFT takes into account the multiplet effects, itwas shown to provide a good description of valence bandspectra of TMOs [24, 25]. The final reason why correla-tions are important in the XAS process is that the L -edgeexcitation involves the presence of a 2 p core-hole in the fi-nal state. This creates an additional attractive potentialfor the valence electrons, which tends to further localisethe valence states. In addition, the created core-hole hasa certain symmetry, which applies additional restrictionson the allowed transitions, giving rise to very distinctmultiplet features in the XAS[30].Several attempts to include all above-mentioned effectswithin a single computational framework for calculat-ing L -edges of TM’s have been suggested. The state-of-the-art methods include time-dependent DFT [31, 32],multiple scattering[33], DFT+DMFT with the final-state approximation [34], configuration interaction[35]and Bethe-Salpeter equation-based [36, 37] methods. Inspite of theoretical complexities, that often require heavycomputational efforts, most of the methods do not de-liver a sufficiently good description of XAS of TMOs.An efficient computational scheme combining DFT andMLFT has been proposed a few years ago in Ref. [38].The method is based on construction of a compact tight-binding description of the DFT band structure by using aprojection onto Wannier functions. As a second step, thisinformation is used to parametrize the single-impurityAnderson model (SIAM) [39], which is the core of MLFT.The obtained Wannier orbitals are also explicitly used tocalculate the onsite electron-electron Coulomb interac-tions. In Ref. [38] the suggested approach was appliedto several TMOs and showed systematically good agree-ment between theory and experiment.In this study we adopt an alternative realization of theDFT+MLFT method by using a different set of local-ized orbitals and concepts from DMFT. We present thedetails of the implementation and apply the developedmachinery on the series of TM monoxides: MnO, FeO,CoO and NiO. The theoretical spectra are evaluated overa parameter space defined by the relevant interactions ofthese compounds, and a detailed comparison is made toexperimental results. E d L L j = 3 / j = 1 / p / ⇣ p E b ✏ F FIG. 1: Schematic picture of the L , -edge XA process.An electron is excited from the TM-2p core states tothe TM-3d valence states. Note that the energy scale isschematic. In the presently investigated compounds thebinding energy E b of the 2 p state is of the order of 700eV, the spin-orbit splitting ζ p of the 2 p level is of theorder of 10 eV and the width of the unoccupied part ofthe 3 d valence band is a few eV. II. THEORETICAL ASPECTS
In the following, we describe a combined DFT+MLFTapproach to compute x-ray absorption (XA) spectra.The basic process leading to the XA L , spectra isschematically depicted in Fig. 1, in which an incomingphoton excites an electron from the TM-2 p core states tothe TM-3 d valence states. The aim of this section is totreat this process in a proper theoretical framework, andwill result into the final expression for XA spectrum inEq. (9).In principle, if one had the exact density functional de-scribing the electron-electron interaction, it would not bepossible to calculate XA spectra since Kohn-Sham quasi-particles, obtained in DFT, are only meant to describethe ground state. Furthermore, in reality, an approxi-mate exchange-correlation functional has to be employed.The local density approximation (LDA) is derived for adensity of a homogenous electron gas, giving rise to re-sults, if applied to complex materials, of approximatecharacter. In the approach presented in this work, weinstead treat the Coulomb interaction on a many-bodylevel within MLFT, by solving the SIAM for valence 3 d electrons. All other quantities, such as crystal field andhybridisation are assumed to be well-described by DFT.The solution of the SIAM is used to obtain the excitationspectrum.Our theory is a generalization of the DFT+MLFT pre-sented in Ref. [38] by using the language of DMFT. Thisallows us to go beyond the cluster model, where theTM 3 d orbitals hybridize only with nearest-neighbor lig-ands [40]. The formulated approach is rather general anddoes not depend on the particular implementations of itssteps. Nevertheless, the calculated XA spectra will de-pend on the choice of local orbitals used in the SIAM.This issue is however present for all first-principle meth-ods based on the construction of a set of localized or-bitals, e.g. DFT+U, DFT+DMFT and many other, sincethere is no unique definition for those in a solid [29]. A. One-particle Hamiltonian and the hybridizationfunction
In this section, we explain how the single-particle (i.e.non-interacting) Hamiltonian of the impurity and bathorbitals are constructed from DFT results.The one-particle Green’s function of the lattice encodesthe ligand-field contribution and the crystal-field split-ting of the TM 3 d orbitals, and is defined as the resolventof the lattice-momentum dependent Hamiltonian ˆ h DFT k :ˆ g k, ( ω ) = (( ω + µ )ˆ1 − ˆ h DFT k ) − , (1)where µ is the chemical potential and ˆ1 is the identityoperator. The local Green’s function of the impurityorbitals situated at site R is constructed by projecting g k, ( ω ) on the set of selected impurity orbitals:ˆ G R, ( ω ) = (cid:88) k ˆ P R,k ˆ g k, ( ω ) ˆ P R,k , (2)where ˆ P R,k is the projection operator from lattice mo-mentum k to orthonormal orbitals on the impurity site R . More details about the choice of projection will begiven in the next section. The hybridization functionˆ∆ R ( ω ) = ( ω + µ )ˆ1 − ˆ G R, ( ω ) − − ˆ H R (3)gives information at which energies and how strong theimpurity interacts with its surrounding. In Eq. (3) thelocal Hamiltonian ˆ H R is calculated from ˆ h DFT k by usingthe same projection as in Eq. (2). Formally one can writethe operator in Eq. (3) in matrix form∆ R,dd ( ω ) = (cid:88) b | V bd | ω − (cid:15) b , (4)where V bd are the impurity-bath hopping parameters and (cid:15) b are the positions of the bath orbitals. In a genericsystem the sum over bath states in Eq. (4) is infinite.However, for localized impurity states, the imaginarypart of the hybridization function often consists of sev-eral distinct peaks. In this case one can approximate the sum in Eq. (4) by including only a finite (and usu-ally small) number of bath states. This approximation isroutinely done in the exact diagonalization (ED) solverin DMFT [41]. This name stems from the fact that thefinite-size SIAM can be exactly solved by direct diago-nalization of the full many-body Hamiltonian, or at leastits relevant sectors. In the presented technique, the pro-jection is performed only on the TM 3 d orbitals, whichare relatively well-localised in TMOs. The influence frommore delocalized O-2 p states to the XAS process is in-corporated in the hybridization function for the TM 3 d orbitals. B. Multiplet theory
The single-particle Hamiltonian, obtained from DFT,can be combined with Coulomb interaction terms and theresulting many-body Hamiltonian (including the core 2 p -states) corresponds to a SIAM of the form:ˆ H = (cid:88) ij (cid:15) d i,j ˆ d † i ˆ d j + (cid:88) i (cid:15) b i ˆ b † i ˆ b i + (cid:88) j V i,j ( ˆ d † j ˆ b i + h.c.) + ζ d (cid:88) ij (cid:104) d i | (cid:126) ˆ l · (cid:126) ˆ s | d j (cid:105) ˆ d † i ˆ d j + (cid:88) ijkl U ddijkl ˆ d † i ˆ d † j ˆ d l ˆ d k + (cid:88) i (cid:15) p ˆ p † i ˆ p i + ζ p (cid:88) ij (cid:104) p i | (cid:126) ˆ l · (cid:126) ˆ s | p j (cid:105) ˆ p † i ˆ p j + (cid:88) ijkl U pdijkl ˆ d † i ˆ p † j ˆ p l ˆ d k , (5)where the annihilation operators ˆ d i , ˆ b i and ˆ p i respectivelyremove an electron from a 3 d , a bath and a 2 p -core spin-orbital state. The super-indices i, j, k, l run over for allspin-orbitals within the 3 d -shell, the bath or the 2 p -core.The non-relativistic single-particle energies are (cid:15) d i,j , (cid:15) b i and (cid:15) p . The spin-orbit coupling of the 3 d (2 p ) statesis described in Eq. (5) by the coupling-constant ζ d ( ζ p ),the angular momentum operator (cid:126) ˆ l and the spin operator (cid:126) ˆ s . The strong spin-orbit coupling for the 2 p -states re-sults in the splitting of the L and L -edges in the XAspectrum. The on-site Coulomb repulsion between the3 d -electrons is described by the U dd tensor, which can beexpressed via Slater-Condon integrals F dd , F dd and F dd ,and the Coulomb interaction between the 2 p -core holeand 3 d electrons is described by the interaction tensor U pd , which can be expressed via Slater-Condon integrals F pd , F pd , G pd and G pd . Slater-Condon integrals are spe-cial cases of the more general equation R k ( n l , n l , n l , n l ) = (cid:90) ∞ drr (cid:90) ∞ dr (cid:48) r (cid:48) R n l ( r ) × R n l ( r (cid:48) ) r k< r k +1 > R n l ( r ) R n l ( r (cid:48) ) , (6)where R nl is the radial wave function, n is the principalquantum number and l the angular momentum, namely F k ( nl ; n (cid:48) l (cid:48) ) = R k ( nl, n (cid:48) l (cid:48) , n (cid:48) l (cid:48) , nl ) G k ( nl ; n (cid:48) l (cid:48) ) = R k ( nl, n (cid:48) l (cid:48) , nl, n (cid:48) l (cid:48) ) . (7)In a cubic harmonics basis the non-spin polarized en-ergies (cid:15) d i,j , (cid:15) b i are reduced to (cid:15) d,t = (cid:15) d + α t Dq and (cid:15) b,t = (cid:15) b + α t δ b , with t ∈ { e g , t g } , (cid:15) d ( (cid:15) b ) the e g - t g aver-aged 3 d (bath b ) energy, 10 Dq the crystal-field splittingbetween the e g and t g , δ b the e g - t g splitting of bathstate b , α e g = and α t g = − . In this basis also thehybridization parameter V i,j simplifies to V b,t , where V b,t describes hopping between a 3 d orbital and a bath b or-bital of character t . Appendix A describes how V b,t isobtained via a fitting to the hybridization function.The solution of Eq. (5) results in the set of many-bodyeigenstates | i (cid:105) , each expressed through a sum of Slaterdeterminants, and the corresponding eigenenergies E i :ˆ H | i (cid:105) = E i | i (cid:105) (8)Once the | i (cid:105) corresponding to the few lowest energies arefound, all statistical properties such as occupation num-bers and spin moments can be directly obtained. TheXA intensity is computed from I ( ω ) = 1 Z (cid:88) i − Im (cid:34) (cid:104) i | ˆ D † ω − ( ˆ H − E i ) + iΓ / D | i (cid:105) (cid:35) × exp( − βE i ) , (9)where the dipole operator ˆ D = (cid:15) · ˆ r describes the excita-tion of a 2 p -core electron to the 3d-shell, with (cid:15) being thelight polarization, ˆ r the position operator, Γ the imag-inary offset from the real axis which gives a Lorentzianbroadening of the spectra, Z the partition function, and β the inverse temperature [42, 43]. III. COMPUTATIONAL DETAILS
The isotropic XAS calculations were conducted forNiO, CoO, FeO and MnO. These TM oxides all havethe same rock-salt crystal structure and experimental lat-tice parameters have been used [27]. Self-consistent non-spin polarized DFT calculations were performed with a26 × ×
26 k-point mesh sampling the Brillouin zone. Weused a linear muffin-tin orbital method (LMTO) with afull-potential as well as a warped LDA potential (see be-low) as implemented in the ”RSPt” code[44, 45] to solvethe DFT problem. The set of localized impurity orbitalsis constructed by projecting the total electron density ona set of L¨owdin orthogonalized LMTOs for the TM 3 d orbitals, denoted as ”ORT” in Ref. [25, 46]. The dis-cretization of the hybridization function is described inAppendix A.Most of the Slater-Condon integrals are calculatedusing the projected 3 d wave functions. However, the TABLE I: Summary of the charge-transfer energycorrection, Slater-Condon integrals and spin-orbitcoupling parameters used in the MLFT calculations. δ CT , F pd and F dd are treated as free parameters ( F dd from Ref. [25]) while the other parameters arecalculated within RSPt [44, 45]. Values are in eV. δ CT F pd F dd F pd G pd G pd F dd F dd ζ p ζ d MnO 1.5 7.5 6.0 5.6 4.0 2.3 9.0 6.1 6.936 0.051FeO 1.5 7.5 6.5 6.0 4.3 2.4 9.3 6.2 8.301 0.064CoO 1.5 8.0 7.0 6.4 4.6 2.6 9.6 6.4 9.859 0.079NiO 1.5 8.9 7.5 6.8 5.0 2.8 9.9 6.6 11.629 0.096 screened values of F dd and F pd are difficult to calculateand are treated as tuneable parameters. It is worth men-tioning that methods which allow to estimate their valuesfrom complementary experimental techniques exist [47].From the theory side, several methods for calculating thescreened value of F dd from first-principles have been pro-posed [48–50]. However, one should bear in mind thatthe screened values of both F dd and F pd depend on thechoice of the projected low-energy subspace and of thecorrelated orbitals, and are therefore not directly trans-ferable from one code to another [51]. On the other side,since the XA involves a charge-neutral excitation, thespectrum is not very sensitive to F dd , F pd [38].Another important aspect is the double counting (DC)correction, which has to be subtracted from the DFT-derived Hamiltonian. This is done in order to removethe contribution of the Coulomb repulsion that is alreadytaken into account at the DFT level. The DC correctionis not uniquely defined and its choice is known to influ-ence the DFT+U and DFT+DMFT results [52]. In thiswork, we apply a DC that is normally used in MLFTby considering the relative energy for different configu-rations. The charge-transfer energy is the energy differ-ence between configurations d n d +1 b and d n d and can beexpressed as ∆ CT b = (cid:15) (0) d − (cid:15) b + δ CT [53], where (cid:15) (0) d is theon-site 3 d energy before the double-counting correction.In this work, the charge transfer energy correction δ CT is treated as a parameter. For all four studied systemswe use δ CT = 1 . δ CT . See Appendix Bfor more details.A temperature of 300 K is used in Eq. (9), which isabove the experimental Neel temperature for all studiedsystems, except for NiO [27]. The paramagnetic phase isstudied by having no exchange field present in Eq. (5).The solution of the SIAM is attained using the Quantysoftware [38, 54–57]. The basis vectors for the groundstate (GS) are obtained by using a Lanczos algorithmstarting with a random d n d configuration, where n d isthe (initial) occupation of d -orbitals, and generate theso-called tridiagonal Krylov basis.TABLE II: Summary of the 3d occupation. The firstrow contains occupation used as input for the D.C.calculations presented in Sec. III, and the followingrows contain occupations obtained from solving Eq. (8). MnO FeO CoO NiO n d n calc. d , 0 bath 5 6 7 8 n calc. d , 1 bath 5.116 6.179 7.187 8.194 n calc. d , 2 bath 5.118 6.175 7.176 8.176 n calc. d , 3 bath 5.144 6.209 7.197 8.177 IV. RESULTS AND DISCUSSION
Slater-Condon integrals, charge-transfer energy correc-tion and spin-orbit couplings are summarized in Table I,impurity occupation (both initial guess and calculatedvalue) are summarized in Table II, hybridization param-eters are summarized in Table III and on-site energies inTable IV. The calculated Slater-Condon values of F dd and F dd have been multiplied with screening factors 0.82 and0.88, respectively [58], while the calculated unscreenedvalues of F pd , G pd , G pd are used. In this work, the onlyparameters relevant for the XAS process that are not cal-culated from DFT are F pd , F dd and δ CT . The eigenstatesin Eq. (8) are superpositions of different configurations,e.g. d n d , d n d +1 b and d n d +2 b , where b represents a lig-and hole. On top of this, also the temperature averagewill mix occupation numbers. This results in an effectiveoccupation number n calc. d , which is slightly higher than n d , see Table II.The XA spectra of the TMOs are computed accordingto Eq. 9. The effective broadening parameter Γ, whichgives a uniform Lorentzian broadening of the spectra,was set to 0 . L is broader than L dueto its shorter core-hole lifetime, primarily due to Coster-Kronig decay [59]. However, we use a constant broaden-ing of the theoretical spectra. To facilitate comparisonwith the experimental spectra, all theoretical spectra areshifted in energy.In Fig. 2, the theoretical spectra are compared to XASmeasurements [60–63]. Note that Fig. 2 contains theoret-ical information of various degrees of accuracy, as shownby the curves with different number of bath orbitals, ob-tained from the fitting of the hybridization function ∆( ω )(see Appendix A for more details). The results in Fig. 2clearly show that all the experimental features are basi-cally reproduced if one bath orbital per impurity orbitalis employed. The addition of more bath orbitals does notchange the overall pictures and mostly redistributes theintensities and shifts certain features in the final XASresults. We will discuss the convergence in detail below, but first we make a comparison between the higher levelof theory (three bath states in Fig. 2) to experimentalobservations. NiO − The resemblance between the XA spectrum andthe theoretical data of NiO is very good. The branch-ing ratio and the peak positions are reproduced withhigh accuracy by the calculation. The experimental in-tensity around 867 eV can be ascribed to excitationsof the 2 p / electrons to free-electron-like conductionstates [38], which were ignored in the calculated curve.The two L peaks have similar intensities both in exper-iments and in the theory and it is known including anexchange field in the theory will further improve the rel-ative intensity [63]. FeO − A very good agreement is obtained for FeO. Themain L peak with its spread out shoulder at energieshigher than 710 eV is well reproduced and the three-peakstructure in the L -edge shows good similarities to themeasurement. Regardless of the background contribu-tions to the measured spectrum, the computed branchingratio matches well to the experimental one. FeO is proneto be off-stoichiometric [64, 65], which leads to point andcluster defects. Possible off-stoichiometry present in theactual samples may further contribute to the remainingdifferences between theory and experiment. MnO − The L , XAS line profile for MnO has rich fea-tures. The L edge contains three distinct peaks and awide high-energy shoulder region. The L edge is broadwith no main peak visible. Also for this case, the pre-sented approach resolves most of the details of the ob-served spectral shape. We can see indications of a slightunderestimation of the crystal-field splitting provided byour approach. CoO − A less pronounced agreement is found for the L edge in CoO. The main peak structure is well repro-duced and the agreement with experiment for the pre-peak position improves with increasing number of bathstates used. The shoulder region between 775 and 779eV is underestimated. This underestimation could bea consequence of charging effects in the experimentaldata. See spectra of thin CoO films and CoO mixed withAg in Ref. [43]. The position of the L -edge for CoOdoes not meet the quantitative similarity of the otherTMOs’ L sub-edge. This could be related to the inter-play of spin-orbit coupling and local non-cubic distortionsnot included in the current calculations. These effectsare known to be more important for CoO than for theother studied TMOs [43, 66]. It is also established thatthe XA spectrum for CoO is sensitive to the tempera-ture [42, 43, 66].An advantage with the presented approach is the possi-bility to investigate the impact that separate terms in theHamiltonian [Eq. (5)] have on the XA spectrum. Figure 3illustrates as an example the NiO L , spectra obtainedby neglecting selected Hamiltonian terms, e.g. crystal-field splitting, hybridization and Slater-Condon integrals.The top panel is the spectrum obtained by including allterms in the Hamiltonian and using one bath orbital percorrelated orbital. By removing both the energy splittingof the e g and t g orbitals (10 Dq = 0) and the bath split-ting ( δ b =0), several changes occur. The L edge gets anew peak next to the main peak, the peak around 855 eVshifts up in energy, the experimental shoulder around 856eV is absent, and the relative intensity ratio between thetwo L peaks is in worse agreement with the experimentaldata. The importance of the d-electrons to dynamicallyinteract with its environment is shown in the third panelfrom the top by setting the hopping strengths ( V ) to zero.In the L edge a new shoulder around 854 eV arises andthe peak at 855 eV is shifted up too far in energy. Alsohere, without hopping, the correct relative intensity ofthe two peaks in the L edge is not captured. A more se-vere approximation in shown in the fourth panel, namely,the atomic limit. Here, all effects from the environmentare removed, thus no hopping and no crystal-field split-ting are considered. This corresponds to combining thetwo approximation steps above. Within this approxima-tion, only one peak exists in the L edge. The small peakat 856 eV is absent and a new peak around 852 eV arises.In the fifth panel (DFT limit), all Slater-Condon integralsare zero (no many-body physics) and as expected, onlyone peak per edge is obtained. In the last panel the ap-proximations in the DFT and atomic limit are combined.The spin-orbit coupling generates states with j = 5 / j = 3 / j = 3 / j = 1 / j = 3 / j = 0 , ± j = 1 / L edge is expected withinthis approximation. A final comment in this section isthat the strength of the terms of the Hamiltonian enter-ing Eq. (9) depends to some extent on the presence ornot of a core hole in the 2p shell. We analyze the effectof a core hole in Appendix C, and find that the influenceis only marginal when it comes to the spectral properties. V. CONCLUSIONS
In summary, we present an approach to compute the L , -edge XA spectra and we have applied it to 3 d TMOs. The calculations rely on DFT ground state cal-culations and a projection to localized orbitals to obtainthe projected density of states, the hybridization func-tion, Slater-Condon integral values and spin-orbit cou-pling parameters, using the RSPt software [44–46]. Thesedata are used as input parameters by the Quanty soft-ware [54, 57], which is used to diagonalize the Hamilto-nian and include core-hole interaction with the valenceband states. The DC is formulated using the concept ofcharge-transfer energy.With this approach, the computed XA spectra forMnO, FeO, CoO and NiO agree very well with experi-mental data, with respect to branching ratios and lineshape profiles. We have also investigated the sensitivity of the calculated spectra with respect to the number ofbath states, and find that this sensitivity is small, butrather non-linear.We also address peculiarities related with the non-sphericity of the potential, which appears in full-potentialrealizations of the DFT [38]. We have used a warped po-tential for the present calculations, which seems to im-prove the estimates of the crystal-field splitting, com-pared to using a full-potential. These results are pre-sented in Appendix B. Furthermore, we discuss how thepresence of the core hole on the DFT level influences thevalence band parameters. Our results clearly show thatthe core-hole induced changes in all calculated parame-ters are relatively small and do not dramatically influencethe simulated XA spectra.The approach adopted here relies on the informationextracted from the first principles electronic structure,which can be obtained by different means. Within DFT,the choice of the functional is important. In this workwe employed warped-potential LDA, assuming that itgives an adequate description of the crystal-field split-ting and TM-O hybridization. To get a more accuratedescription of the hybridization one could also use DFTcombined with DMFT including more bath states, whichrecently was used to calculate core-level x-ray photoe-mission spectra [40]. Extension of the present methodfor clusters containing non-local Coulomb interactions isanother promising direction.In order to get a complete first-principles theory ofXA spectra, the current approach has to be augmentedwith an ability to calculate the charge-transfer energyand screened values of F dd and F pd . The choice of charge-transfer correction δ CT is related with the DC problemand also needs to be solved. The F dd can be calculated bymeans of various methods [48–50] and it would be veryuseful to generalize these methods for the evaluation of F pd . Further possible improvement would be to predictthe screening effects on the higher order Slater-Condonintegrals. This paper provides a tool to predict measur-able quantities (i.e., core-level XAS and photoemissionspectroscopy ) that are sensitive to the double-countingscheme used as well as the screened value of F dd . As such,this scheme can be used to test the accuracy of differentimplementations. ACKNOWLEDGMENTS
We acknowledge the Knut and Alice WallenbergFoundation (KAW) (Projects No. 2013.0020 and No.2012.0031) for financial support. B.B. and O.E. thankthe Swedish Research Council (VR) and eSSENCE. Y.Kand J.L. thank the EUSpec COST for the financial sup-port of the research visit to Dresden. We also appreci-ate the Swedish National Infrastructure for Computing(SNIC) which has provided computing time on the clus-ters Abisko at Ume˚a University, Triolith at Link¨opingUniversity, and Beskow at KTH, Stockholm. For finan-
640 645 650 655 I n t e n s i t y ( a r b . un i t s ) MnO exp3 bath2 bath1 bath0 bath
705 710 715 720 725
FeO
775 780 785 790 E (eV) I n t e n s i t y ( a r b . un i t s ) CoO
855 860 865 870 875 E (eV) NiO
FIG. 2: (Color online) Comparison between the computed and the experimental XA spectra (red curves) ofMnO [60], FeO [61], CoO [62] and NiO [63]. A different number of bath states are considered in the theoreticalspectra.cial support we also acknowledge the Carl Trygger Foun-dation, Sweden. We would like to thank L. Nordstr¨omand T. Bj¨orkman for discussions of the muffin-tin poten-tial.J.L and J.S. contributed equally to this work.
Appendix A: Discretization of the hybridizationfunction ∆( ω ) The hybridization functions for the studied systems(MnO, FeO, CoO and NiO) are similar, due to the mean-field like treatment with DFT. The e g orbitals hybridizemore than the t g orbitals with their surrounding of Ostates. The bath energies are picked by inspecting theDFT hybridization function and the hopping parame-ters are obtained by considering the weight of the hy-bridization intensity in the vicinity of each bath energy.Table III contains the discretized hybridization functionparameters. The hybridization weight close to the Fermilevel for the t g orbitals is picked up by the third set ofbath orbitals. Note, in the hybridization function of atruly insulating state, this weight is suppressed [40].To compensate for the discretization approximation of the hybridization function in Eq. (4), we adjust the 3 d on-site energy (cid:15) (0) t such that the discretized Green’s func-tion [67] G t ( ω ) = ( ω − (cid:15) (0) t − (cid:88) b | V b,t | ω − (cid:15) b,t ) − , (A1)with t ∈ { e g , t g } , resemble the local DFT Green’s func-tion in Eq. (2). In practice, we achieved this by demand-ing that the imaginary part of G t ( ω ) to have the samecenter of gravity as the imaginary part of the local DFTGreen’s function in a restricted energy window. For allconsidered TMOs, we selected the energy window to be[ − ,
3] eV around the Fermi level, as the electron bandswith predominantly TM-3 d character exist in this inter-val. The obtained on-site energies are presented in Ta-ble IV. The DFT and the discretized hybridization func-tion using 3 bath states per 3 d orbital is shown in Fig. 4for FeO. For the e g orbitals, the hybridization at around-19 eV is too far away from the Fermi level to enable acharge transfer in the SIAM. Therefore, no e g bath stateis placed at around -19 eV and this hybridization is in-stead (implicitly) compensated by the adjustment of (cid:15) (0) e g .TABLE III: Summary of the bath parameters extracted from the hybridization functions obtained using the RSPtsoftware [44, 45]. Values are in eV. Figure 4 shows both the DFT and the fitted hybridization function of FeO usingthree bath states. (cid:15) b,e g , (cid:15) b,t g , V b,e g , V b,t g (cid:15) b,e g , (cid:15) b,t g , V b,e g , V b,t g (cid:15) b,e g , (cid:15) b,t g , V b,e g , V b,t g (cid:15) b,e g , (cid:15) b,t g , V b,e g , V b,t g
855 860 865 870 875E (eV) I n t e n s i t y ( a r b . un i t s ) NiO exp
FIG. 3: (Color online) Calculated XA spectra of NiO.Various terms in the Hamiltonian in Eq. (5) areremoved. Description of the label names, from top tobottom: 1bath: all terms are present and one bathorbital per correlated orbital is used. 1bath, no CF:degenerate e g and t g (both 10 Dq and δ b is zero).0bath: no hopping to the environment. atomic limit: nohopping to the environment and degenerate e g and t g .DFT limit: All Slater-Condon integrals are set to zero. Appendix B: Double counting
To avoid double counting the monopole part of theCoulomb interaction, we consider three configurations: TABLE IV: On-site energies for three different bathdiscretizations. Values are in eV. (cid:15) (0) e g , (cid:15) (0) t g (cid:15) (0) e g , (cid:15) (0) t g (cid:15) (0) e g , (cid:15) (0) t g (cid:15) (0) e g , (cid:15) (0) t g - I m ( e V ) e g - I m ( e V ) t g
20 180100
FIG. 4: (Color online) Comparison between the DFThybridization function (red curves) and its fitting datausing three bath states (black curves) of FeO. For sakeof compactness an inset is used to show thehybridization peak corresponding to O-2s states. p b d n d , p b d n d +1 and p b d n d +1 and their corre-sponding energies E b, , E b, +∆ CT b and E b, +( (cid:15) (0) d − (cid:15) (0) p ).These energies can be expressed as [22] E b, = 6 (cid:15) p + 10 (cid:15) b + n d (cid:15) d + (cid:18) n d (cid:19) U dd + 6 n d U pd ,E b, + ∆ CT b = 6 (cid:15) p + 9 (cid:15) b + ( n d + 1) (cid:15) d + (cid:18) n d + 12 (cid:19) U dd + 6( n d + 1) U pd ,E b, + ( (cid:15) (0) d − (cid:15) (0) p ) = 5 (cid:15) p + 10 (cid:15) b + ( n d + 1) (cid:15) d + (cid:18) n d + 12 (cid:19) U dd + 5( n d + 1) U pd , (B1)where ∆ CT b is the charge transfer energy and (cid:15) (0) d ( (cid:15) (0) p ) isthe DFT on-site energy for the 3 d (2 p ) states. By solv-ing these equations for the double-counting corrected en-ergies (cid:15) d and (cid:15) p , which enter in the SIAM in Eq. (5),and expressing the charge-transfer energy as ∆ CT b = (cid:15) (0) d − (cid:15) b + δ CT [53], where δ CT is a charge-transfer cor-rection parameter, one obtains (cid:15) d = (cid:15) (0) d + δ CT − n d U dd − U pd (cid:15) p = (cid:15) (0) p + δ CT − (1 + n d ) U pd . (B2)Note that (cid:15) p only will shift the XA spectrum. The av-erage Coulomb repulsion energies used in Eq. (B1) andEq. (B2) are expressed in Slater-Condon integrals by U dd = F dd −
263 ( F dd + F dd ) (B3) U pd = F pd − G pd − G pd . (B4)In order to avoid double counting the multipole partof the Coulomb interaction, we have used a warped LDApotential instead of a full LDA potential, as suggestedin [38]. This means the non-spherical part of the poten-tial inside the muffin-tin is zero. Removing non-sphericalparts of the potential, to improve the crystal-field split-ting, has been discussed in the past for d -electrons [38] interms of double-counting and for f -electrons [68] in termsof self-interaction. The main difference in the band struc-ture between the warped and the full potential is thatthe e g and t g bands are further apart in energy withthe warped potential. All other parameters extractedfrom DFT, such as bath energies, hoppings, and Slater-Condon integrals, are barely affected by this approxima-tion of the potential. For comparison, the XA spectraobtained using the warped and full potential of CoO isshown in Fig. 5. It may be seen that the differences be-tween the two theoretical curves are small, but clearlynoticeable. The calculation based on warped potentialseems to reproduce observations better, especially thefeatures in the low-energy region.
775 780 785 790 E (eV) I n t e n s i t y ( a r b . un i t s ) CoO expfull pot.warped pot.
FIG. 5: (Color online) Comparison between thecomputed XA spectra using warped and full potentialLDA of CoO. Experimental spectrum (red line) isshown as reference [62]. Three bath states are used. - I m ( e V ) e g no core-holecore-hole8 6 4 2 0 2E (eV)02 - I m ( e V ) t g
20 180100
FIG. 6: (Color online) Hybridization function includingthe effect of core-hole relaxation, obtained from DFT,for FeO.
Appendix C: Presence of a core-hole within DFT
In Ref. [38], the parameters of a single-particle Hamil-tonian are extracted from a ground state DFT calculationwith no core-hole. The 2 p core-hole only enters the calcu-lation on the stage of MLFT, participating in a Coulombinteraction with the valence 3 d states. Thus, the influ-ence of a core-hole on the valence band electronic struc-ture is not explicitly considered. In this work we aim toquantify these so far neglected effects.We have constructed a supercell and created a 2 p corehole on one of the TM atoms. The created excess chargewas either added as a uniform background or simplyadded to the valence, to maintain charge neutrality. Both0 P D O S MnOMnO no hole, e g no hole, t g hole, e g hole, t g FeOFeO P D O S CoOCoO
NiONiO
FIG. 7: (Color online) PDOS including the effect ofcore-hole relaxation, obtained from DFT. schemes resulted in identical sets of results, which givescredence to the chosen size of the supercell.The imaginary part of the hybridization function, withand without the core hole, is shown in Fig. 6 for FeO. ForMnO, CoO, and NiO, the changes in the hybridizationfunctions are similar. The hybridization is suppressed,which roughly corresponds to a 7%, 7%, 8% and 9% de-crease in the hopping parameters for MnO, FeO, CoO,and NiO, respectively. The Slater-Condon integrals F pd , G pd , G pd , F dd and F dd are increased a few percentagepoints (see Table V). Both findings are consistent withthe idea that the core hole creates a potential, which fur-ther tends to localize the 3 d orbitals. The crystal-fieldsplitting, 10 Dq , extracted from the discretized hybridiza-tion function and the PDOS in Fig. 7, changes -18%,-18%, -16% and -12% for MnO, FeO, CoO and NiO re-spectively.In Fig. 8, we show the simulated XA spectra takinginto account the renormalization of the parameters, de-scribed above. One can see that the resulted spectra areslightly modified with respect to the theoretical resultsobtained with no explicit core hole considered in DFT.The changes in the spectra are almost entirely due to theincrease of F pd , G pd and G pd . [1] J. Sa, High-Resolution XAS/XES: Analyzing ElectronicStructures of Catalysts (CRC Press, 2014).[2] B. Weckhuysen,
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