Theoretical description of optical and X-ray absorption spectra of MgO including many-body effects
Vijaya Begum, Markus E. Gruner, Christian Vorwerk, Claudia Draxl, Rossitza Pentcheva
TTheoretical description of optical and X-ray absorption spectra of MgO includingmany-body effects
Vijaya Begum, Markus E. Gruner, Christian Vorwerk, Claudia Draxl, and Rossitza Pentcheva Department of Physics and Center for Nanointegration Duisburg-Essen (CENIDE),University of Duisburg-Essen, Lotharstr. 1, 47057 Duisburg, Germany Institute f¨ur Physik and IRIS Adlershof, Humboldt-Universit¨at zu Berlin, Berlin, Germanyand European Theoretical Spectroscopy Facility (Dated: December 17, 2020)Here we report the optical and x-ray absorption (XAS) spectra of the wide-band-gap oxide MgOusing density functional theory (DFT) and many-body perturbation theory (MBPT). Our compre-hensive study of the electronic structure shows that while the band gap is underestimated withthe exchange-correlation functional PBEsol (4.58 eV) and the hybrid functional HSE06 (6.58 eV)compared to the experimental value (7.7 eV), it is significantly improved (7.52 eV) and even over-compensated (8.53 eV) when quasiparticle corrections are considered. Inclusion of excitonic effectsby solving the Bethe-Salpeter equation (BSE) yields the optical spectrum in excellent agreementwith experiment. Excellent agreement is observed also for the O and Mg K-edge absorption spec-tra, demonstrating the importance of the electron-hole interaction within MBPT. Projection of theelectron-hole coupling coefficients from the BSE eigenvectors on the band structure allows us todetermine the origin of prominent peaks and identify the orbital character of the relevant contribu-tions. The real space projection of the lowest energy exciton wavefunction of the optical spectrumindicates a Wannier-Mott type, whereas the first exciton in the O K-edge is more localized.
I. INTRODUCTION
MgO is one of the most extensively studied oxideswhich is used as a substrate material and in vari-ous heterostructures with applications related to tun-neling magnetoresistance [1–3]. Recently, this mate-rial was employed in transient x-ray spectroscopy andtime-dependent density-functional theory (DFT) calcu-lations aiming to unravel the propagation of excitationsacross the interface in metal-insulator heterostructures[4–7]. Understanding spectroscopic features from first-principles requires accurate modeling beyond the groundstate properties including excitations of different originand energy scale.The structural and electronic properties of this wideband gap material with a measured band gap between 7.7[8] and 7.83 eV [9] have been widely studied with first-principles calculations [10–12]. Unsurprisingly, DFT cal-culations with semilocal functionals significantly under-estimate the band gap [10, 12, 13]. Many-body perturba-tion theory (MBPT) calculations employing Hedin’s GW approximation [14] yield an increased band gap [11, 12],which is still lower than the experimental one. The op-tical spectrum, calculated by Wang et al. [13] usingthe local density approximation (LDA) as the exchange-correlation functional for the DFT calculation and sub-sequently including GW and excitonic corrections agreeswith experiment [15] w.r.t. peak positions up to 12 eVwhereas the amplitude of the peaks beyond the first oneis overestimated due to the limited number of unoccu-pied bands employed in the BSE corrections. Schleife etal. [10] studied the frequency-dependent dielectric func-tion for different MgO polymorphs – wurzite, zinc blende,and rocksalt – in the independent particle (IP) approxi-mation using the generalized gradient approximation in the PW91 parametrization [16]. Good agreement withexperiment concerning the peak positions was obtainedby including excitonic corrections with BSE, based on theKohn-Sham (KS) eigenenergies and a scissors operator todescribe the QP eigenenergies [17, 18].While optical spectroscopy probes excitations from va-lence bands, x-ray absorption spectroscopy (XAS) probesthose from the strongly localized core states. A commonapproach to model XAS is the final state rule (FSR) [19]based on Fermi’s Golden rule, where the effects of screen-ing of the core-hole (the so-called final-state effects ) arecalculated in a supercell. Alternatively, XAS can be de-scribed by considering quasiparticle and excitonic effectswithin MBPT by using GW and solving the BSE. Rehr et al. [20] showed that while both approaches led to sim-ilar overall features in the O and Mg K-edge spectra ofMgO, BSE calculations result in better agreement withexperiment at high transition energy due to the non-localtreatment of the exchange interaction. Recent implemen-tations of BSE in all-electron codes [21, 22] with explicittreatment of core states have demonstrated very goodagreement with experiment for the XAS spectra of TiO (rutile and anatase), PbI , and CaO [22]. The latter ap-proach is adopted in this work.Here we describe both the optical and x-ray absorptionspectra of bulk MgO including many-body effects. As afirst step, we perform the G W corrections starting fromKohn-Sham (KS) wavefunctions. We show that carefulconsideration of the electron-hole interaction with BSE isessential to achieve agreement with experiment for bothvalence and core excitation spectra. In particular, the op-tical spectrum calculated with two different DFT func-tionals (PBEsol and HSE06) including the G W andBSE corrections are consistent with experiment [8, 23]and previous theoretical work [17, 18] and yield an im- a r X i v : . [ c ond - m a t . o t h e r] D ec proved agreement regarding the intensity of the peaks athigher energies, highlighting the importance of quasipar-ticle and excitonic effects.Previous studies have shown that a dense k -mesh is re-quired for the sampling of the Brillouin zone to describesufficiently the localization of the excitonic wave functionand the fine structure in the vicinity of the absorptionedge [11]. Here, we use a model for the static screen-ing with parameters fitted to the G W calculation, tosolve the BSE (model BSE [11, 24]) starting directly fromDFT wavefunctions on a denser k -mesh, which improvesin particular the low energy range (7 −
11 eV). Beyondprevious work we provide a thorough analysis of inter-band transitions contributing to the peaks in the opticalspectrum. Further insight into the nature of the firstbound exciton is given by the real-space visualization ofits wavefunction.Employing the exciting code, the O and Mg K-edgeXAS spectra calculated with BSE show very good agree-ment with the experimental spectra [25] and with pre-vious theoretical results using the FSR [20]. Knowledgeof the origin of peaks is essential for the interpretationof x-ray spectra. The main incentive of this study is toidentify the nature of transitions which contribute to thepeaks and analyze the character of the first exciton inthe O K-edge both in real and reciprocal space.The paper is structured as follows: the details of thecalculations are presented in Section II, followed by thediscussion of the results in Section III. We start with theelectronic properties of MgO in III A and then comparethe optical spectra calculated with two different startingexchange-correlation functionals in III B. Subsequently,we analyze the transitions in reciprocal space to derivethe origin of contributions to the peaks in the spectrum.In subsection III C, we present the XAS spectra of the Oand Mg K-edge and identify the underlying transitions inreciprocal space for the prominent peaks. Finally, sub-section III D is dedicated to the real-space visualizationof the first exciton of the optical and the O K-edge x-ray absorption spectrum. The results are summarized inSection IV, followed by two appendices showing a com-parison of the optical spectra obtained with VASP and exciting and the optical spectrum with the model BSE.
II. COMPUTATIONAL DETAILS
The DFT calculations are performed with the VASPcode (version 5.4.4) [26, 27], using pseudopotentials incombination with the projector augmented wave (PAW)method [28], and the exciting code [29] (version Nitro-gen) employing the all-electron full-potential (linearized)augmented planewave + local orbital [(L)APW+lo]method. For the exchange-correlation functional wechose the generalized gradient approximation (GGA) inthe implementation of Perdew, Burke, and Ernzerhof(PBE96) [30], PBEsol [31, 32], and the hybrid functional,HSE06 [33, 34]. The equilibrium lattice constant deter- mined with the different functionals amounts to 4.24 ˚A(PBE96), 4.21 ˚A (PBEsol), and 4.20 ˚A (HSE06), the ex-perimental one being 4.212 ˚A [35].For the calculation of the optical spectrum with VASP,we have performed single-shot G W on top of theKS wavefunctions obtained with two DFT functionals,PBEsol and HSE06 and subsequently included excitoniccorrections by solving the BSE. For all the BSE calcu-lations the Tamm-Dancoff-approximation (TDA) [36] isadopted. The calculations are performed for a two-atomunit cell with a Γ-centered 15 × × k -mesh (unless oth-erwise specified) with a plane-wave cut-off energy of 650eV. GW PAW pseudopotentials for excited propertieswere employed in all the calculations with two valenceelectrons for Mg: 3 s and six for O: 2 s , 2 p . 192 unoc-cupied bands are used for both the DFT and single-shot G W calculations with 100 frequency-grid points. Forthe optical spectrum a Lorentzian broadening of 0.3 eVis used.Single-shot G W calculations are also performed withthe exciting code [29] together with BSE [37] for theoptical and x-ray absorption spectra [22]. A Γ-centered11 × ×
11 mesh shifted by (0.09, 0.02,0.04) is employedfor the calculations. Muffin-tin radii of 1.058 and 0.767 ˚Afor Mg and O, respectively, are used with a basis set cut-off R MT | G + k | max = 7, and the lattice constant is setto the PBEsol value of 4.21 ˚A. The energy threshold toinclude the local field effects in the excited properties, | G + q | max , is set to 4.5 a.u. − for the optical and O K-edge and 1.5 a.u. − for the Mg K-edge absorption spec-tra. The exchange-correlation functional PBEsol [31, 32]is employed for the Kohn-Sham (KS) states and a totalof 192 unoccupied bands are considered in the groundstate and G W calculation for the optical and O andMg K-edge x-ray absorption spectra. In the BSE cal-culation, for the optical spectrum four occupied and fiveunoccupied bands are considered, while eight unoccupiedbands were taken into account for the XAS spectra. ALorentzian broadening with a width of 0.55 eV is ap-plied to the spectra to mimic the excitation lifetime. Theatomic structures and isosurfaces are visualized with theVESTA software [38] and the band structure is calculatedwith the Wannier90 [39] package in VASP. III. RESULTSA. Electronic properties
We start our analysis by comparing the electronicproperties obtained from DFT calculations with threedifferent functionals, namely PBE96, PBEsol andHSE06. Table I presents the band gap calculated withVASP. With PBE96 (4.49 eV) and PBEsol (4.58 eV), theband gaps are considerably underestimated, consistentwith previous calculations [10–12]. On the other hand,HSE06 renders a band gap of 6.58 eV closest but still be-low the experimental value of 7.7 and 7.83 eV [8, 9]. The
FIG. 1. (a) Kohn-Sham and G W band structure and (b-d) total and projected density of states (PDOS) of MgO calculatedwith PBEsol within VASP. G W band gap obtained with PBEsol (7.52 eV) is clos-est to experiment, whereas a somewhat lower value (7.26eV) is obtained with PBE96 which is in agreement withRef. [12]. In contrast, with HSE06 (8.53 eV) the G W band gap is overcorrected. The trend highlights the start-ing point dependence of the G W band gap[40, 41].Since PBEsol and HSE06 provide better electronicproperties as compared to PBE96 we continue the analy-sis with those. In Fig. 1a the Kohn-Sham and G W band structure with the PBEsol functional is plottedalong high-symmetry points, showing a direct (Γ − Γ)band gap. The inclusion of quasiparticle effects in the G W calculation leads to a nearly rigid shift of the un-occupied Kohn-Sham bands to higher energies. The topof the valence band (VB) consists mainly of O 2 p states(cf. the projected density of states in Figs. 1b-d) withlow dispersion along the L − Γ − K direction, whereas thelower bands are more dispersive. Further insight into theorbital-resolved contributions of O and Mg on the bandstructure is provided in Fig. 2. The bottom of the con-duction band (CB) comprises hybridized O 3 s , 3 p , andMg 3 s states that are highly dispersive along the L − Γ − X and K − Γ directions (cf. Figs. 1c, d and Figs. 2a, b,and d). In the range of 4.5 - 11 eV beyond the CB mini-mum, 3 s , 3 p , and Mg 3 s states prevail, whereas above 11eV O 3 p states become predominant, followed by Mg 3 p and 3 d states above 15 eV (cf. Fig. 1d and Figs. 2e, f).We will further analyze the ion- and orbital projectionsin the band structure in Section III B 3 and Section III Cto correlate the contributions with the optical and XASspectra. B. Optical properties
Starting from the electronic structure presented in thelast section, we determine the optical spectrum includingalso many-body effects. We discuss the effect of approx-
TABLE I. Comparison of the band gap from the DFT andthe G W calculation with different starting functionals.E xc IP G W Experiment E g (eV)(Γ − Γ) PBE96 4.49 7.26 7.7 a ,7.83 b PBEsol 4.58 7.52HSE06 6.58 8.53 a Reference 8 b Reference 9 imations to the exchange-correlation functional, namelyPBEsol and HSE06, on the spectra and the role of inclu-sion of G W and excitonic corrections with BSE. In ad-dition, the interband transitions responsible for the spec-tral features are analyzed in reciprocal space.
1. Optical spectrum within IP approximation and inclusionof G W corrections The calculated optical spectra are plotted in Fig. 3together with the experimental ones [8, 23]. The imag-inary part of the experimental dielectric function showsfour prominent peaks (marked in Fig. 3b): the first twoat ∼ ∼ FIG. 2. Oxygen (a-c) and Mg (d-f) orbital-resolved contributions projected on the ground state band structure within VASP. band transitions that coincide with points of inflection inthe real part of the spectrum.Inclusion of many-body effects within the G W ap-proximation results in a blue shift by ∼ G W . This strong effectis attributed to the weak dielectric screening in MgO [13].In Figs. 3b, d sharper features emerge in (cid:15) with peaksat ∼ ∼ (cid:15) ∞ =Re (cid:15) ( ω = 0) obtained with PBEsoland HSE06 is presented in Table II. Within the IP ap-proximation, (cid:15) ∞ is overestimated for PBEsol (3.29) com-pared to the experimental value 2.94 [8], similar to pre-vious results with GGA-PW91 (3.16) [17]. On the otherhand, with the hybrid functional, (cid:15) ∞ is underestimated(2.76). Upon including quasiparticle effects ( G W ), thevalues are substantially reduced to 2.78 (PBEsol) and2.53 (HSE06).
2. Optical spectrum with excitonic corrections
Additional to the quasiparticle corrections, we considerthe effects arising from electron-hole interaction by solv-
TABLE II. Comparison of the macroscopic static electronicdielectric constant (cid:15) ∞ in the IP approximation and after G W and BSE with different DFT functionals.E xc IP G W BSE ExperimentPBEsol 3.29 2.78 3.08 2.94[8]HSE06 2.76 2.53 2.81 ing the Bethe-Salpeter equation. The calculations areperformed with four occupied and five unoccupied bandswhich are sufficient to evaluate the optical spectrum upto 30 eV.The inclusion of excitonic effects leads to a redistribu-tion of the spectral weight to lower energies w.r.t. the G W spectrum and the emergence of a sharp peak atthe absorption onset. With PBEsol as the starting pointin Fig. 3b, the agreement with experiment w.r.t. spectralshape is improved, but the onset of the imaginary part ofthe dielectric function is ∼ ∼ FIG. 3. Optical spectrum of bulk MgO obtained with VASP: (a), (c) real part and (b), (d) imaginary part of the dielectricfunction for PBEsol and HSE06 as the starting functional, respectively. A Lorentzian broadening of 0.3 eV is employed for allthe calculated spectra. The IP, IP+ G W , and G W +BSE results are shown by brown solid, green dash-dotted, and red solidlines, respectively. Additionally, the experimental data from Exp. 1 [8] (black solid line), and Exp. 2 [23] (black dashed line)are displayed. [23]. The four peaks of (cid:15) at ∼
8, 10.5, 13, and 17 eVare largely aligned with experiment, as shown in Figs. 3cand d. Further analysis of the origin of the peaks in re-ciprocal space and the real-space projection of the firstexciton are provided in Section III B 3 and Section III D,respectively. The improved description w.r.t. the ener-getic positions and, to a lesser extent, intensity of thepeaks can be attributed to the description of the groundstate with a hybrid functional HSE06, the larger numberof unoccupied bands considered in the BSE calculationand to performing BSE on top of G W . Furthermore,the agreement to experiment concerning the macroscopicstatic electronic dielectric constant (cid:15) ∞ is improved afterBSE to 3.08 (PBEsol) and 2.81 (HSE06), respectively(cf. Table II), also consistent with a previous value of3.12 [17], where excitonic corrections were included us-ing the KS eigenenergies and a scissor shift approach. Wenote that increasing the number of unoccupied bands to9, leads to slightly higher values for (cid:15) ∞ et al. [11] employing the KS eigenenergies (GGA)with a scissor shift of 2.98 eV and subsequently includingexcitonic corrections. The overestimation of the bind-ing energy w.r.t experiment (80 meV [8]) may be at-tributed to the fact that the ionic contributions to thestatic screening is not considered [42, 43].
3. Analysis of spectral features in reciprocal space
In order to identify the origin of the most promi-nent peaks, we have performed calculations with the all-electron exciting code. The real and imaginary part ofthe dielectric function for the G W +BSE with PBEsolas the DFT functional and similar parameters (four oc-cupied and five unoccupied bands) are plotted in Figs.4a, b, and show good agreement with experiment as wellas the VASP result w.r.t. the energetic positions of thepeaks (a comparison of the spectra obtained with thetwo codes is provided in Appendix A and Fig. 8). Themost prominent peaks are marked in Fig. 4b and thecorresponding e-h contributions are studied in Figs. 4c-f.We recall that the Bethe-Salpeter equation represents aneigenvalue problem for an effective two-particle Hamilto-nian [37, 44]: (cid:88) v (cid:48) c (cid:48) k (cid:48) H vc k ,v (cid:48) c (cid:48) k (cid:48) A λv (cid:48) c (cid:48) k (cid:48) = E λ A λvc k . (1) FIG. 4. Optical spectrum with PBEsol including many-body corrections calculated with the exciting code: (a) real and(b) imaginary part of the dielectric function. A Lorentzian broadening 0.3 eV is employed for the calculated spectrum for the G W +BSE corrections (red line). The direct band gap at 7.26 eV is marked by a vertical green line. Experimental spectrafrom Roelssler et al. [8] (black solid line) and Bortz et al. [23] (black dashed line) are shown for comparison. (c − f) electron-holecoupling coefficients represented as circles in reciprocal space for the peaks at different energies marked in (b), where the sizeof the circle is proportional to the magnitude of the e-h contribution. where E λ are the transition energies and A λvc k are thecorresponding states in terms of v k → c k transitions.The e-h coupling coefficients for a particular transitiondisplayed as circles in Figs. 4c-f are calculated from theBSE eigenvector A λ as: w λc k = (cid:88) c | A λvc k | , w λv k = (cid:88) v | A λvc k | . (2)The first exciton at 6.82 eV has a binding energy of 435meV, close to the value obtained with VASP, as discussedin the previous section. This bound exciton contributesto the shoulder at the onset of the spectrum. The in-terband transitions responsible for the exciton and itsreal-space distribution are discussed in detail in SectionIII D.The first peak at 7.3 eV (cf. Fig. 4c) arises due totransitions from the top of the valence band (VB) tothe bottom of the conduction band (CB) around the Γ-point in reciprocal space. A comparison with the site andorbital- projected DOS (Fig. 1) and band structure (Fig.2) reveals a mixed O 3 s , 3 p , and Mg 3 s character. Thesecond peak at 9.4 eV involves interband transitions fromthe topmost VB to the lowest CB along L − Γ − X andΓ − K . The CB is more dispersive along L − Γ and hasmixed O 3 s and 3 p character with Mg 3 p contributions near L . The next peak at 10.4 eV stems from transitionsto the CB from deeper-lying valence bands along L − Γ − X and Γ − K . The final peak at 12.2 eV, plotted inFig. 4f, results from transitions from the top of the VBto the higher-lying CB around X as well as along K − Γ.In this energy range, the CB consists of O 3 p and Mg 3 s and 3 d xy , d xz states along Γ − X and K − Γ. C. X-ray absorption spectra
We now turn to the x-ray absorption spectra of theO and Mg K-edge of bulk MgO calculated with the exciting code.
1. O K-edge
The theoretical XAS spectrum of the O K-edge is plot-ted in Fig. 5d together with the experimental spectrumfrom Luches et al. [25], who performed x-ray absorp-tion measurements on MgO films of varying thicknessgrown epitaxially on Ag(001) as well as on polycrystallinebulk samples. The G W +BSE spectrum is character-ized by six prominent peaks with high oscillator strength FIG. 5. XAS spectrum of the O K-edge using G W +BSE calculated with the exciting code: (a) calculated absorption spectrawith G W +BSE (red line) and IP (brown shaded area) are compared with experimental spectra from Luches et al. [25] (blackline). A shift of 34.4 eV was applied to the calculated spectra to align to the first peak of the experiment and a Lorentzianbroadening of 0.55 eV is adopted to mimic the excitation lifetime. The direct band gap at 535.2 eV is marked by a verticalgreen line. (b-g) excitonic contributions to the final states in the CB of the peaks marked in (a). (cf. Fig. 5a). Their origin in terms of transitions to theconduction bands is visualized in Fig. 5b-g.While the IP spectrum captures the overall shape withfour peaks, a very good agreement to experiment con-cerning the relative positions of the three prominentpeaks at ∼ G W +BSE corrections. The spectrum is also con-sistent with earlier work of Rehr et al. [20] using FSRand BSE. The reduced intensity of the third peak in the G W +BSE spectrum can be attributed to the limitednumber of unoccupied bands considered in the calcula-tion.The first bound exciton is found at 534.5 eV with abinding energy of 691 meV, its real-space distribution isdiscussed in Section III D. Analysis of the transitions atthe onset of the spectrum at 535.3 eV shows that theseare localized around Γ at the bottom of the CB (Fig. 5b)and comprise predominantly O 3 p character hybridizedwith O 3 s (cf. Figs. 2a-c), and Mg 3 s character (cf.Fig. 2d). The second peak at 536.9 eV also arises fromtransitions to the lowest CB, but is more dispersive along L − Γ − X and K − Γ. The subsequent peak at 537.7 eVstems from transitions to the lowest CB, but is localizedmidway along the L − Γ with some contribution along K − Γ and has hybridized O 3 s , 3 p , and Mg 3 s character. Furthermore, the peak at 539.5 eV results from transi-tions to the second lowest unoccupied band localized at X and dispersive along K − Γ with Mg 3 d xz (cf. Fig.2f) as well as O 3 p character. Transitions to higher un-occupied bands around W and Γ with mixed O 3 p and3 d and Mg 3 p , t g character result in a peak at 546.5eV. The final peak at 557.2 eV arises from transitions toCB at energies above 25 eV with O 3 p and Mg 3 p , e g contributions along X − W − K − Γ.
2. Mg K-edge
Fig. 6a displays the Mg K-edge from the G W +BSEcalculation and the experimental spectrum from Luches et al. [25]. The experimental spectrum has four promi-nent peaks at 1308.3, 1314.4, and 1316.2 eV, followed bya broader peak at 1326.6 eV, with a noticeable differ-ence in peak intensities for normal and grazing incidenceof the MgO film on Ag(001) and for the polycrystallinesample. While the IP spectrum for the O K-edge showsoverall agreement with the G W +BSE result, for theMg K-edge the IP spectrum fails to describe the gen-eral features of the experimental spectrum. On the otherhand, including the core-hole - electron interaction leads FIG. 6. XAS spectrum of Mg K-edge including G W +BSE corrections calculated with the exciting code: (a) calculatedabsorption spectra with G W +BS (red line) ) and IP (brown shaded area) are compared with experimental spectra fromLuches et al. [25] on a MgO film on Ag(001), grazing/normal incidence of photon beam (black dashed-dotted/solid line);polycrystalline MgO (black dashed line). A shift of 58.8 eV was applied to the calculated spectra to achieve coincidence withthe first peak of the experiment and a Lorentzian broadening of 0.55 eV is adopted for the theoretical curve to mimic theexcitation lifetime. The direct band gap at 1306.6 eV is marked by a vertical green line. (b-k) excitonic contributions to thefinal states in the CB of the peaks marked in (a). to a large redistribution of the spectral weight, accom-panied by the emergence of a high intensity peak at theonset of the BSE spectrum. Overall, the G W +BSEMg K-edge is in very good agreement with the experi-mental spectrum of [25] and with previous BSE and FSRcalculations [20] concerning peak positions and relativeintensity.Ten prominent peaks with high oscillator strength aremarked which are analyzed further in Figs. 6b-k. Thefirst peak at 1307.4 eV arises from transitions to the bot-tom of CB with Mg 3 p character, hybridized with Mg 3 s states (cf. Figs. 2a,b). The second peak at 1308.2 eVcomprises transitions along the L − Γ − X with Mg 3 p character and along K − Γ with mixed Mg 3 s and 3 p character. The third peak at 1310.1 eV includes tran-sitions to the lowest CB concentrated halfway betweenΓ − X with hybridized Mg 3 s and O 3 s and 3 p character(cf. Figs. 2a,b). Moreover, the peaks at 1313.8, 1315.6,and 1316.7 eV arise from transitions to higher energy CB( >
10 eV) and are dispersive along the whole k-path withhybridized O 3 p and Mg 3 s and 3 p as well as Mg 3 d xz character (cf. Figs. 2d-f). The peaks at 1323.6, 1325.7, FIG. 7. Analysis of the first exciton in the optical spectrum (a) and O K-edge XAS (b) in reciprocal space. The lower panelsshow the density associated with the electronic part of the excitonic wavefunctions for a selected cross section in real space:In (c) along (56¯1) for the optical excitation shown in (a) and (d) along (¯51¯6) for the O K-edge XAS in (b). The color codevisualizes the spacial extension of the wave functions: Blue colors refer to vanishing or low densities, orange to red colors toelevated densities. The hole is fixed near the oxygen (fractional coordinate: 0.52, 0.52, 0.52) and is marked by a white cross.
E >
20 eV predominantly along X − W − K with prevailing hybridized Mg 3 p and e g character. D. Real-space projection of the first exciton
The real-space wavefunction of the excited electron fora given exciton can be obtained from the BSE eigenvec-tors A λ as:Ψ λ ( r h , r e ) = (cid:88) vc k A λvc k ψ ∗ v k ( r h ) ψ c k ( r e ) . (3)For more details see Ref. [37, 45] and references therein.For the analysis we fixed the hole slightly off the oxygenposition (0.52,0.52,0.52) and plotted the electronic partof the wavefunction in real-space for the first exciton ofthe optical and the O K-edge XAS spectrum in Fig. 7.The first bound exciton of the optical spectrum (Fig. 7a) consists of transition from the valence band maxi-mum (VBM) to the conduction band minimum (CBM)that are strongly localized around Γ. Since the excitedelectron is distributed solely over the lowest, highly dis-persive conduction band, the bound exciton was previ-ously described in the Wannier-Mott two-band modelby Fuchs et al. [11]. In Fig. 7c we display a cut alongthe (56¯1) plane through the center of the spread of thewave function near the fixed position of the hole, thatshows that the exciton is delocalized over several unitcells which supports the Wannier-Mott character. More-over the intensity of the spread has a maximum at theoxygen sites and is weaker at the Mg sites. The recip-rocal space projection in conjunction with the orbitallyprojected band structure (cf. Fig. 2) shows a main con-tribution of hybridized O 3 s and Mg 3 s states at theCBM.For comparison, we have also analyzed the real spaceprojection of the first exciton in the O K-edge XAS spec-trum. As shown in Fig. 7b this exciton involves tran-0sitions to the CBM, but is more dispersive in reciprocalspace along L − Γ − X and K − Γ. This goes hand inhand with a stronger localization in real space, visiblefrom the real space projection in Fig. 7d along the (¯51¯6)plane that exhibits a significant decrease in the spread ofthe wavefunction. Compared to the exciton of the opti-cal spectrum, here the spread is confined to two to threeunit cells only. The 2D cut through the center of thewavefunction spread also illustrates the orbital contribu-tions with s and p character near the O sites, whereas thecontributions around the Mg sites have s - like character.This can be attributed to the strong hybridization of theO 3 s , 3 p , and Mg 3 s states around the CBM, discussedabove. IV. SUMMARY
We have provided a comprehensive study of the opti-cal and x-ray absorption spectra of bulk MgO with theVASP and exciting codes. The results indicate thatthe quasiparticle, and in particular excitonic effects arecrucial to describe the spectra, concerning peak positionsand to a lesser extent intensity.For the optical spectrum, the effect of two differentfunctionals (GGA-PBEsol and the hybrid HSE06) arestudied: an excellent agreement with the experiment isobtained with HSE06 w.r.t. the energetic positions of thepeaks. Analysis of the electron-hole coupling coefficientsin reciprocal space allows us to identify the valence toconduction band transitions contributing to the peaks inthe spectrum. In particular, the peak at 7.3 eV arises dueto transitions localized around Γ from the top of the VBto the bottom of the CB with mixed O 3 s , 3 p , and Mg 3 s character, followed by a peak at 9.4 eV stemming fromsimilar interband transitions but along L − Γ − X andΓ − K with mixed O 3 s , 3 p , and Mg 3 p character near L . The third peak at 10.4 eV is from transitions to thebottom of CB from deeper lying valence bands and thefinal peak at 12.2 eV results from a transition to higherlying conduction bands with hybridized O 3 p and Mg 3 s and 3 d xy , d xz character.The inclusion of core-hole electron interaction by solv-ing the BSE is found to be essential also for the XASMg and O K-edge. By visualizing the transitions to theunoccupied bands in reciprocal space, we determine theorigin of the relevant peaks in the spectra. In the O K-edge spectrum, the peak at ∼
537 eV originates fromthe transitions to unoccupied bands with hybridized O3 s , 3 p , and Mg 3 s character, the peak at 546 eV stemsfrom O 3 p , 3 d hybridized with Mg 3 p and t g states,and the peak at 557 eV emerges from transitions to theCB with hybridized O 3 p and Mg 3 p and 3d character.The real space projection of the electronic part of thewavefunction of the first exciton in the optical spectrumshows it has a delocalized Wannier-Mott character, con-sistent with previous studies in reciprocal space [11]. Onthe other hand, the wavefunction of the first exciton in the O K-edge spectrum is stronger localized and spreadsup to only three unit cells. We believe that our de-tailed analysis of the optical and x-ray excitations in thisparadigmatic oxide material regarding their orbital char-acter and extension in real and reciprocal space basedon state-of-the-art many-body approaches serves as animportant benchmark and provides useful backgroundinformation for the interpretation of experimental databoth from static but also time-dependent investigations. ACKNOWLEDGMENTS
We thank Caterina Cocchi, Andr´e Schleife, HeikoWende, Andrea Eschenlohr, Katharina Ollefs, NicoRothenbach, and Okan K¨oksal for fruitful discussions.We wish to acknowledge funding by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) within collaborative research center CRC1242(project number 278162697, subproject C02) and com-putational time at the Center for Computational Sci-ences and Simulation of the University of Duisburg-Essen on the supercomputer magnitUDE (DFG grantsINST 20876/209-1 FUGG, INST 20876/243-1 FUGG).C. V. and C. D. appreciate funding from Leibniz-ScienceCampus GraFOx.
FIG. 8. Comparison of the G W +BSE spectrum calculatedwith PBEsol as the starting functional with exciting (redsolid line) and VASP (blue solid line). The spectra are com-pared with the experiments of Roelssler et al. [8] (black solidline) and Bortz et al. [23] (black dashed line). A Lorentzianbroadening of 0.3 eV is employed for the theory spectra. Appendix A: Comparison of the optical spectrumcalculated with exciting and VASP
The optical spectra obtained with exciting and VASPwith the same exchange-correlation functional (PBEsol)in Fig. 8 show very good agreement in the overall shapeand peak positions with some smaller differences in peakheights.
Appendix B: DFT+model-BSE (mBSE)
FIG. 9. Comparison of the spectrum obtained with mBSE ontop of DFT ( k -mesh of 21 × ×
21) and with G W +BSE cor-rections ( k -mesh of 15 × × et al. [8] (black solid line) and Bortz et al. [23] (black dashed line). A Lorentzian broadening of0.3 eV is employed for the theory spectra. Fuchs et al. [11] have pointed out the importance ofusing a dense k -mesh for BSE in order to obtain conver-gence of the optical spectrum in the lower energy range.However, this goes hand in hand with a high computa-tion cost in the G W calculation. One way to circum-vent this is to perform a model BSE calculation directlyusing the DFT wave functions and considering only therequired number of bands which cover the energy rangeof interest [11]. The omission of the G W step allowsus to use a higher k -mesh and thus improve the conver-gence. Here, we discuss the result obtained by using amodel for the static screening with parameters fitted tothe screened Coulomb kernel diagonal values obtainedfrom G W calculation, a detailed description can befound in Ref. [11, 24, 46, 47] and was previously used inRef. [24, 46]. Good agreement between mBSE and the full BSE ( G W +BSE) was recently obtained for bulkSrTiO [41].In our study, mBSE is performed on a 21 × × k -mesh with a range separation parameter λ = 1 .
44 ˚A − ,ion-clamped static dielectric function (cid:15) − ∞ = 0 . k -mesh of 15 × ×
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