Interplay of charge-transfer and Mott-Hubbard physics approached by an efficient combination of self-interaction correction and dynamical mean-field theory

TL;DR

This paper enhances the DFT+DMFT method by incorporating self-interaction correction for oxygen, improving the description of charge-transfer and Mott-Hubbard physics in transition-metal oxides like NiO.

Contribution

It introduces an efficient SIC implementation into DFT+DMFT, explicitly including oxygen correlations and improving intersite $p-d$ correlation treatment.

Findings

Spectral features of NiO are well described with local DMFT self-energies.

In-gap states in Li$_x$Ni$_{1-x}$O are confirmed theoretically.

The method effectively captures charge-transfer and Mott-Hubbard physics interplay.

Abstract

Late transition-metal oxides with small charge-transfer energy $\Delta$ raise issues for state-of-the-art correlated electronic structure schemes such as the combination of density functional theory (DFT) with dynamical mean-field theory (DMFT). The accentuated role of the oxygen valence orbitals in these compounds asks for an enhanced description of ligand-based correlations. Utilizing the rocksalt-like NiO as an example, we present an advancement of charge self-consistent DFT+DMFT by including self-interaction correction (SIC) applied to oxygen. This introduces explicit onsite O correlations as well as an improved treatment of intersite $p-d$ correlations. Due to the efficient SIC incorporation in a pseudopotential form, the DFT+sicDMFT framework is an advanced but still versatile method to address the interplay of charge-transfer and Mott-Hubbard physics. We revisit the spectral features of stoichiometric NiO and reveal the qualitative sufficiency of local DMFT self-energies in describing spectral peak structures usually associated with explicit nonlocal processes. For Li$_x$Ni$_{1-x}$O, prominent in-gap states are verified by the present theoretical study.

Equations

$$\mathcal{H}=\frac{1}{2}\sum _{\begin{array}{c}{m}_1{m}_2\\ m_3{m}_4\end{array}}\sum _{\sigma {\sigma }^{\prime }}{U}_{m_1{m}_2{m}_3{m}_4}{c}_{m_1\sigma }^{†}{c}_{m_2\sigma {}^{\prime }}^{†}{c}_{m_4\sigma {}^{\prime }}^{\text{hfill}}{c}_{m_3\sigma }^{\text{hfill}},$$

$$U_{m_1{m}_2{m}_3{m}_4}=\sum _{k=0}^{2l}{a}_k(m_1,m_2,m_3,m_4)F^k,$$

$$a_k(m_1,m_2,m_3,m_4)=$$

$$\times {\left(\begin{array}{ccc}l& k& l\\ 0& 0& 0\end{array}\right)}^2\left(\begin{array}{ccc}l& k& l\\ -m_1& q& m_3\end{array}\right)\left(\begin{array}{ccc}l& k& l\\ -m_2& -q& m_4\end{array}\right),$$

$$F^0=U,F^2=\frac{14}{1+r}{J}_{\mathrm{H}},F^4=r{F}^2.$$

Full text

I Introduction

An understanding of the electronic structure and phase diagrams of transition-metal (TM) oxides still poses a major challenge. The realistic modeling of strong electronic correlations in this family of compounds has been tremendously improved since establishing the combination of density functional theory (DFT) with dynamical mean-field theory (DMFT). Especially early-TM oxides with a partially-filled(empty) $t_{2g}$($e_{g}$) sub-manifold of the TM-$d$ shell, showing dominant Mott-Hubbard physics, often may readily be addressed by such an approach. Late-TM oxides display a more intriguing interplay of hybridization and correlation effects and remain demanding. Reason is that in the latter systems with usually completely-filled $t_{2g}$ subshell and partially-filled $e_{g}$ subshell, the ligand oxygen $2p$ states are much closer in energy to the TM $d$ states. The charge-transfer energy $\Delta$, required to transfer a ligand electron to a TM $d$ orbital, eventually becomes smaller than the energy cost for a doubly-occupied $d$ orbital, i.e. the Hubbard $U$. In late TMs, the $d$ states are lower in energy, hence $\Delta$ shrinks, but $U$ on the other hand increases due the reduced orbital extension with larger nuclear charge Imada et al. (1998). According to the famous Zaanen-Sawatzky-Allen (ZSA) scheme Zaanen et al. (1985), correlation-induced insulators with $\Delta<U$ are of charge-transfer kind.

While e.g. Cu oxides are typical charge-transfer compounds, the rocksalt-like NiO, a material of paramount importance for the development of quantum solid-state theory de Boer and Verwey (1937); mot37, is located in the intermediate regime of the ZSA diagram. This renders NiO particularly interesting, and more generally, marks nickelate compounds as exceedingly affected by various competing instabilities (see e.g. Ref. Imada et al., 1998 for a review). Stoichiometric nickel oxide is insulating with a sizeable charge gap of $\sim$4.3 eV Newman and Chrenko (1959); Powell and Spicer (1970); Sawatzky and Allen (1984); Hüfner et al. (1984) and becomes antiferromagnetic below $T_{\rm N}=523$ K Roth (1958). Substitutional doping of Li+ for Ni2+* is effective in providing holes to cause conductivity Heikes and Johnston (1957); Goodenough et al. (1958); Kuiper et al. (1989); van Elp et al. (1992); Reinert et al. (1995); Lany et al. (2007); Zhang et al. (2018).

Numerous theoretical studies examined the interacting electronic structure of NiO. Advanced descriptions need to rely on a sophisticated treatment of $\Delta$, $U$ and further key quantum characteristics of the compound. In addition to cluster calculations Fujimori and Minami (1984); van Elp et al. (1992); van Veenendaal and Sawatzky (1993); Taguchi et al. (2008); Kuo et al. (2017), DFT+DMFT Ren et al. (2006); Kuneš et al. (2007); Yin et al. (2008); Miura and Fujiwara (2008); Nekrasov et al. (2012); Thunström et al. (2012); Leonov et al. (2016); Panda et al. (2016) and variational-cluster-approximation Eder et al. (2005) studies provided already a good account of the electronic spectrum, but challenges remain Kuo et al. (2017). For instance, to a certain degree, the correct determination and understanding of the specific nature/positioning of peaks within the valence-band spectrum is still a matter of debate. Furthermore, the increased importance of the role of ligand orbitals and their hybridization with TM orbitals in materials which lack a pure Mott-Hubbard character, renders an investigation of defect properties very demanding.

Therefore, the intention of this work is twofold. Using NiO as a test case, we first show that an improved description of the intriguing interplay between Mott-Hubbard and charge-transfer physics at stoichiometry is achieved by treating electronic correlations on the TM as well as on the ligand sites. This is realized by an efficient combination of the self-interaction correction (SIC) with the charge self-consistent DFT+DMFT framework. Thereby, SIC is applied on O and Ni marks the DMFT impurity problem. This theory advancement enables us to shed novel light on the intriguing features of the paramagnetic NiO spectrum. Second, the usefulness of this DFT+sicDMFT scheme for advanced correlated materials science is demonstrated by the application to the even more challenging case of Li-doped NiO. We straightforwardly reproduce the long-standing experimental finding of in-gap states at $\sim 1.2$ eV above the valence-band maximum.

II Theoretical Approach

II.1 Problem and general framework

In simplest approximation Imada et al. (1998), the charge-transfer energy is given by the difference between the TM$(d)$ and O$(2p)$ single-particle levels, i.e. $\Delta=\varepsilon_{d}-\varepsilon_{p}$. In early TM oxides, such as e.g. certain titanates or vanadates, the respective level separation is rather large with TM$(d)$ well above O$(2p)$. Then usually, electronic correlations matter most within the partially filled $t_{2g}$ subshell of TM$(d)$, since O$(2p)$ states are much deeper in energy than the scale of the lower-Hubbard-band formation. The local Hubbard $U_{dd}:=U$ within TM$(d)$ is the smaller energy and thus governs the correlation effects. On the contrary for late TM oxides, $\Delta$ becomes the smaller energy and O$(2p)$ are often located between the lower- and upper-Hubbard-band formation scales.

In the latter case, the correlation physics is more subtle. Of course, $U$ remains a vital player, since it triggers strong correlation. However, the charge-transfer energy and more generally also the Coulomb interactions of local O$(2p)$ kind and of intersite TM($d$)-O$(2p)$ kind gain significant impact. Extended model Hamiltonians, besides $U_{dd}$ terms furthermore including additional onsite $U_{pp}$ terms on oxygen and intersite $U_{pd}$ terms, are believed to become relevant for a generic description of the correlated electronic structure Han et al. (2011); Hansmann et al. (2014); Panda et al. (2017). But especially in a realistic context, as e.g. DFT+DMFT, such extensions will raise issues: at least two new Coulomb parameters have to be quantified and questions concerning the quality of the many-body treatment adequate for the new terms arise. Moreover, the technical/numerical effort increases significantly, particularly for low-symmetry structures, heterostructure problems, defect properties, etc..

Therefore, we here introduce an efficient extension to the state-of-the-art charge self-consistent DFT+DMFT framework, geared to treat charge-transfer physics with a minimum of additional Coulomb parametrization and essentially without any further increase in technical/numerical effort. First, our DFT and DMFT parts remain structurally unmodified, i.e. a mixed-basis pseudopotential framework Elsässer et al. (1990); Lechermann et al. (2002); Meyer et al. (1998) is utilized for the former and a continuous-time quantum Monte Carlo technique Rubtsov et al. (2005); Werner et al. (2006) for the latter (see Ref. Grieger et al., 2012 for further details). Besides these established building blocks, a third one is intergrated. A self-interaction-correction (SIC) formalism Perdew and Zunger (1981); Filipetti and Spaldin (2003) is applied to cope with the correlations explicitly originating from the ligand sites. Importantly, the SIC procedure is performed on the pseudopotential level Vogel et al. (1996); Körner and Elsässer (2010) and enters the calculational scheme simply as a modified ligand pseudopotential. Hence during the self-consistency cycle, the SIC effects on the crystal lattice adapt to the system-dependent characteristics. Those effects, without the coupling to DMFT here described on the effective single-particle level, capture both, the impact of the ligand-onsite $U_{pp}$ and of the ligand-TM-intersite $U_{pd}$. Yet effectively, only a single additional parameter, namely the degree of SIC renormalization for O$(2p)$, may be sufficient. Last but not least, DFT+sicDMFT allows to use the identical double-counting approach as for standard DFT+DMFT: the SIC formalism is double-counting free by definition, and the remaining TM-onsite double-counting representation remains unaffected. This is favorable as the issue of double counting for charge-transfer systems has been a matter of debate Karolak et al. (2010); Sakuma and Aryasetiawan (2012).

II.2 Calculational details

For the DFT part, the local density approximation (LDA) is utilized. Norm-conserving pseudopotentials according to Vanderbilt Vanderbilt (1985) are constructed for Ni, O and Li. The mixed basis consists of localized orbitals for Ni($3d$), O($2s$) and O($2p$), as well as plane waves with an energy cutoff $E_{\rm cut}$. The self-interaction correction for the O pseudopotential employs orbital weight factors $w({\rm O})=(w_{s},w_{p},w_{d})$ and a global SIC scaling parameter $\alpha$ (see Refs. Vogel et al., 1996; Körner and Elsässer, 2010 for further details). While by default the O($2s$) potential is 100% corrected and the O($3d$) potential is 0% corrected, key freedom is provided by the correction $w_{p}$ of the crucial O($2p$) potential, i.e. $w({\rm O})=(1.0,w_{p},0.0)$. In the case of an isolated atom $\alpha=1$ (atomic SIC) holds. A detailed study of the variation of $\alpha$ in different kind of solids is given by Pemmaraju et al. Pemmaraju et al. (2007). Throughout the work we choose $\alpha=0.8$, shown to be reliable in previous calculations for TM oxides Körner and Elsässer (2010, 2014); Urban et al. (2016). If not otherwise stated, we also pick $w_{p}=\alpha$ and thus effectively, a single additional parameter setting enters here the advanced formalism. Note that no self-interaction correction is applied to Ni, since the whole localization/correlation effects originating from the nickel site are here described within DMFT.

Projected local orbitals Amadon et al. (2008); Anisimov et al. (2005); Haule et al. (2010) are employed to define the correlated subspace for the DMFT part. The five Ni($3d$) atomic-like orbitals are projected onto eight Kohn-Sham valence states of NiO, namely onto the dominant Ni$(3d)$ and O$(2p)$ dispersions. Remaining states are of course included in the complete charge self-consistent framework, but do not enter explicitly the correlated subspace. The general Slater-Condon Hamiltonian

[TABLE]

with $m_{i}=1\ldots 5$ and $\sigma,\sigma^{\prime}$=$\uparrow$,$\downarrow$, is used for the electron-electron interaction in the correlated subspace. Coulomb matrix elements for $l=2$ are expressed in spherical symmetry via standard Slater integrals $F^{k}$ through

[TABLE]

with expansion coefficients $a_{k}$ given by

[TABLE]

and parametrized using the Hubbard $U$ and Hund’s exchange $J_{\rm H}$ via

[TABLE]

The $F^{4}/F^{2}$ Slater-integral ratio is chosen as $r=0.625$, which is adequate for transition-metal atoms. For the present NiO study, $U=10$ eV and $J_{\rm H}=1.0$ eV are employed. While the latter value is standard for this compound, the Hubbard interaction is somewhat larger than the usual value of $\sim$8 eV Ren et al. (2006); Kuneš et al. (2007). But charge self-consistent DFT+DMFT often enforces an enhanced local Coulomb interaction compared to one-shot calculations Lechermann et al. (2018) because of the increased number of screening channels. A recent computation Panda et al. (2017) of NiO Coulomb parameters for a $dp$ Hamiltonian within the constrained random-phase approximation yields also a Hubbard $U\sim 10$ eV. The DMFT problem is solved by hybridization-expansion continuous-time quantum Monte Carlo Werner et al. (2006), as implemented in the TRIQS package Parcollet et al. (2015); Seth et al. (2016). A double-counting correction of fully-localized-limit (FLL) type Anisimov et al. (1993) is applied. All calculations are performed in the paramagnetic regime at a system temperature of $T=580$ K. In order to obtain the spectral function $A(\omega)=-1/\pi\,{\rm Im}\,G(\omega)$, analytical continuation of the Green’s function $G$ from the Matsubara axis to the real-frequency axis is performed by the maximum-entropy method. To reveal k-resolved spectra, we use the Padé method applied to the Ni self-energy for the analytical continuation. If not otherwise stated, the shown spectra is based on the maximum-entropy method.

The NiO lattice constant is set to the experimental value Brentano (1925) $a=4.17$ Å. Doping with lithium is realized by means of a supercell approach. Each symmetry-inequivalent Ni site poses a different single-site DMFT problem; all are coupled within the general multi-site many-body scheme Potthoff and Nolting (1999). As DFT convergence parameters we used $E_{\rm cut}=16(13)$ Ryd and a k-point mesh of $13(5)\times 13(5)\times 13(5)$ for the pristine(Li-doped) case.

III Results

III.1 Self-interaction corrected oxygen in NiO

Before dvelving into the results of the new DFT+sicDMFT scheme, it is illustrative to inspect the impact of the self-interaction correction to oxygen in nonmagnetic NiO from the effective single-particle LDA+SIC viewpoint.

Figure 1 displays the site- and orbital-projected density of states (DOS) within LDA and for two different SIC parameters $w_{p}=0.7,0.8$. The Ni-$t_{2g}$ states are completely filled and the strongly hybridized $\{$Ni-$e_{g}$,O($2p$)$\}$ states are partially filled. Hence holes are located on the TM site as well as on the ligand sites. Not surprisingly, each scheme renders the system metallic. It is well known that for gap opening on the static DFT(+U) level, symmetry breaking in the form of magnetic ordering is indispensable and the same applies here. Note that we could apply SIC also on the TM site Körner and Elsässer (2010); Urban et al. (2016) and additionally allow for antiferromagnetic order to investigate the insulating state in that effective-single-particle approximation. The result would be qualitatively similar to the one obtained from a DFT+U treatment Anisimov et al. (1991), but we do not want to follow this route in the present work.

Instead, let us concentrate on the principal effects of SIC applied on the oxygen site. There are in essence two main effects. First, the $pd$ splitting and hence the charge-transfer energy is increased by SIC. Using $\Delta=\varepsilon_{d}-\varepsilon_{p}$, the value reads, respectively, $\Delta_{\rm LDA}=3.13$ eV, $\Delta_{w_{p}=0.7}=3.98$ eV and $\Delta_{w_{p}=0.8}=4.50$ eV. Fits to experimental data yield $\Delta$ values in the range $\sim[4,5]$ eV Fujimori and Minami (1984); van der Laan et al. (1986); Kuo et al. (2017). Thus LDA severely underestimates $\Delta$ and the effect of SIC on oxygen brings the charge-transfer energy in line with experimental estimates. Second, band narrowing takes place with SIC, roughly on the order of $Z\sim 0.8$ for $w_{p}=0.8$, as also visible from Fig. 1. Importantly, the band-narrowing effects are not only encountered for the O($2p$) contribution, but also for the dominant Ni($3d$) bands. Hence a nonlocal ligand-TM correlation effect occurs as a result of self-interaction correction applied to the oxygen pseudopotential. Those two key effects originate from the effective inclusion of $U_{pp}$ and $U_{pd}$ terms within the present LDA+SIC treatment.

Let us remark again that the parameter setting $\alpha=0.8$ is not specifically adjusted to NiO, but this $\alpha$ value turns out to be a proper choice for various TM oxides Körner and Elsässer (2010, 2014); Urban et al. (2016). In other words, a present SIC parametrization with $w_{p}=\alpha$ is much less case sensitive as the usual choice/calculation of the Hubbard interaction(s).

III.2 NiO many-body spectrum

III.2.1 DFT+sicDMFT examination

We now discuss the spectral results from the complete DFT+sicDMFT approach, employing $w_{p}=0.8$. Figure 2 shows the main outcome for stoichiometric NiO together with the combined experimental data Sawatzky and Allen (1984) from photoemission and inverse-photoemission.

Let us first briefly recall the state-of-the-art interpretation Fujimori and Minami (1984); Sawatzky and Allen (1984); van Veenendaal and Sawatzky (1993) of the experimental spectrum. NiO is a correlation-induced insulator subject to the interplay of Mott-Hubbard and charge-transfer physics. The Ni ground state in the $3d$-shell amounts to $d^{8}$, and $\underline{L}$ describes a hole in the ligand O($2p$) states. The crucial charge-transfer process, associated with the energy $\Delta$, is described by the transition $d^{8}\,\rightarrow\,d^{9}\underline{L}$, i.e. electron transfer from O($2p$) to Ni($3d$). This energy scale sets the NiO charge gap of $\sim 4.3$ eV. An added electron conclusively enters the $d^{9}$ state associated with peak E. On the other hand, adding a hole to the system either results in the high-energy $d^{7}$ state (i.e. lower Hubbard band) associated with peak C, or gives rise to $d^{8}\underline{L}$ at a much lower energy of peak A. The shoulder D is usuallySawatzky and Allen (1984) interpreted as originating from the nonbonding part of O($2p$). The most-controversially discussed peak B is build from a substructure of the $d^{8}\underline{L}$ state and often associated with nonlocal excitations van Veenendaal and Sawatzky (1993) (see Ref. Kuo et al., 2017 for a detailed discussion).

Let us note that Taguchi et al. Taguchi et al. (2008) suggested an alternative scenario for the lower-energy peaks A and B. Namely, peak A should originate from a $d^{8}\underline{Z}$ state and only peak B builds up from $d^{8}\underline{L}$, without the need of invoking explicit nonlocal processes. Here, $d^{8}\underline{Z}$ refers to a Zhang-Rice (ZR) bound state, a strongly correlated TM-O-hybridized low-energy entity, presumably most relevant for low-energy cuprate physics Zhang and Rice (1988). In NiO, the ZR (doublet) bound state is based on the interaction of the O($2p$) hole with both Ni($3d$) holes. Loosely speaking, $d^{8}\underline{Z}$ is the tightly-bound ’collective’ counterpart of the weakly-bound $d^{8}\underline{L}$ excitation.

The upper part of Fig. 2 displays the total k-integrated spectral function $A_{\rm tot}(\omega)=\sum_{\nu}A_{\nu}(\omega)$, with $\nu$ as the Bloch (band) index, from DFT+sicDMFT compared to the experimental data. Additionally, the projection of $A_{\rm tot}(\omega)$ onto Ni($3d$) and O($2p$) is depicted. Note that this site- and orbital-resolved spectrum is strictly not identical to the true local spectral function in a many-body sense. Since the projection is performed from the Bloch-resolved $A_{\nu}(\omega)$ it carries the complete hybridization on the lattice and moreover results from an analytical continuation of the Bloch Green’s function $G_{\nu}$. Overall, the agreement with experiment is quite remarkable: the theoretical charge gap of $\sim 4$ eV matches perfectly, and also the further features A$-$D are well reproduced. The principal charge-transfer character is obvious from the fact that the dominant part of O($2p$) is located between the lower Hubbard band at $\sim-9$ eV and the upper Hubbard band at $\sim 4$ eV. As already expected from the previous discussion, the lower-energy region $\Lambda=[-3.5,0]$ eV indeed asks for a deeper analysis.

Therefore, the lower part of Fig. 2 shows the Ni($3d$) local spectral function $A_{\rm loc}(\omega)$ as obtained from analytical continuation of the local Green’s function $G_{m}$. Locally, the $t_{2g}$ manifold is completely filled and the upper Hubbard band is exclusively of $e_{g}$ character. Within $\Lambda$ the $e_{g}$ part displays a three-peak structure, whereas the $t_{2g}$ part a two-peak structure. The first sharp resonance closest to the valence-band maximum (VBM) at $\sim-0.1$ eV is of exclusive $e_{g}$ kind, in line with previous studies van Elp et al. (1992); Kuneš et al. (2007); Yin et al. (2008); Kuo et al. (2017). A second sharp peak resonates in both cubic $3d$ sectors roughly at the same energy $\sim-0.85$ eV, while a slightly broader peak occurs at $\sim-1.8(-2.5)$ eV for $t_{2g}$($e_{g}$). Albeit a bit speculatively, we interpret this intriguing structure as follows: the first sharp substructure at $\geq-1.2$ eV belongs to $d^{8}\underline{Z}$, whereas the higher-energy substructure in $\Lambda$ belongs to $d^{8}\underline{L}$. Then the experimental peak A is a $\{d^{8}\underline{Z},d^{8}\underline{L}\}$ superposition (with larger $d^{8}\underline{Z}$ content) and peak B results from a superposition of $d^{8}\underline{L}$ with part of the nonbonding O($2p$) spectrum. Explicit nonlocal correlations, e.g. through an intersite Ni-Ni self-energy, appear not necessary to qualitatively account for a sizeable Ni spectral weight within peak B. As our many-body method is (charge) self-consistent on the lattice, implicit features of nonlocality are included. It still may be that explicit nonlocal self-energies beyond DMFT enhance the Ni weight in peak B. Thus in essence, the present study highlights the intricate entanglement between the more basic aspects of charge-transfer physics and its highly-correlated ZR ramifications.

Finally, we display in Fig. 3a the k-resolved spectral function $A({\bf k},\omega)$ obtained by analytical continuation of the Matsubara self-energy via the Padé method. It can be seen that the upper Hubbard band still displays some dispersion. Note that we did non include higher-energy states above the Ni-$e_{g}$-based bands in our construction of the spectral function. Additionally, Fig. 3 provides a comparison between spectral data obtained via the Padé and the maximum-entropy method. Naturally, the former method produces sharper details. Concerning the total spectrum, all peaks A-E are also visible with Padé, albeit the peaks A and B are slightly shifted to higher energies. Very close to the VBM, both analytical-continuation methods display spectral features in the k-integrated spectrum. Those features are supported by the still visible dispersion, e.g. around the $L$ point, up to the VBM in Fig. 3a. However, this structure is not observed in the experimental spectrum Sawatzky and Allen (1984). This might be due to different reasons, e.g. to the suppression of spectral weight close to the VBM/Fermi level because of Fermi-function effects at the boundary of the occupied part and/or resolution issues in experiment. Novel experimental examinations with focus on the Ni-O hybridization in bulk NiO close to the VBM region would be helpful to clarify this matter. Last but not least, Fig. 3b shows a clearer projected-Ni$(3d)$ contribution to peak B at $\sim$ 3 eV within the Padé-based spectrum compared to the more-smeared maximum-entropy spectrum.

III.2.2 Comparing with standard DFT+DMFT

Let us now comment on the methodological aspect of our approach. In order to compare the present scheme with the traditional DFT+DMFT method for NiO, Fig 4 shows the total spectral function as well as the Ni-$e_{g}$,$t_{2g}$ local spectral function with and without SIC for oxygen. The most obvious and striking difference concerns the gap size: without SIC, the charge gap turns out only of order $\sim 1.5$ eV. A similar qualitative observation has been made by Panda et al. Panda et al. (2016), who noted that they reached only a small NiO charge gap with the FLL double counting in charge self-consistent DFT+DMFT. On the occupied high-energy side, the lower Hubbard band is shifted to more negative energies without SIC. On the other hand, the structure in the $\Lambda$ region is not dramatically altered, albeit the fine structure appears less detailed without SIC. For instance, the local Ni-$t_{2g}$ spectrum exhibits only a single peak in standard DFT+DMFT. Altogether, even if focusing only on the occupied spectrum, the agreement with experiment concerning peak signature and position is less satisfactory than with SIC. We also performed DFT+DMFT calculations for smaller $U=8$ eV, but the gap size did not change.

The very fact that charge self-consistent DFT+DMFT with standard double counting fails in reproducing the correct paramagnetic gap size for a charge-transfer(-like) compound, is not that surprising. As observed, the main Mott-Hubbard physics, i.e. the formation of Hubbard bands and their splitting in energy, is rather similar with and without SIC. However without SIC, the O($2p$) level is too strongly shifted in the direction of the upper Hubbard band, rendering the final charge gap small. Because energy-beneficial charge fluctuations are suppressed on the Ni site, the formalism tries to shift O($2p$) towards the upper Hubbard band to enable as much as possible the (virtual) charge fluctuations between O($2p$) and Ni-$e_{g}$. Since there is no Coulomb penalty from SIC, the charge-transfer energy shrinks. Note that there is a recent charge self-consistent NiO calculation by Leonov et al. Leonov et al. (2016), providing a reasonable spectrum with only minor gap-size reduction. But in that calculation, only density-density interactions in the local interacting Hubbard Hamiltonian have been taken into account.

Generally, one-shot DFT+DMFT calculations Ren et al. (2006); Kuneš et al. (2007) obtain a charge gap somewhat smaller than in experiment, but still of sensible size. This is understandable, as the O($2p$) level remains essentially fixed in those calculations, and the DFT charge-transfer energy of $\sim 3.2$ eV is nearly unaltered. Hence when promoting the method to charge self-consistency, it is essential to include also the ligand-based Coulomb interactions. Of course, it may be that some other double-counting protocol can “fix” the problem. But the present approach is more physical, it is applicaple to general charge-transfer problems and allows one the use of the identical standard double-counting form for Mott-Hubbard and charge-transfer systems without any further adjustments. Last but not least, we are confident that the correct interplay of Mott-Hubbard, charge-transfer and charge self-consistent processes is very well captured by our DFT+sicDMFT framework.

III.3 Li-doped NiO

Lithium doping of NiO has first been studied in detail in the 1950s Heikes and Johnston (1957); Goodenough et al. (1958) and remained of research significance ever since Kuiper et al. (1989); van Elp et al. (1992); Reinert et al. (1995); Kuneš et al. (2007); Lany et al. (2007); Zhang et al. (2018). Recently, it furthermore gained interest in the context of hybrid organic-inorganic perovskite solar cells Jeng et al. (2014); Chen et al. (2015).

The alkali element enters as a substitutional Li*+* defect replacing Ni2+ and thus providing holes to the compound. The LixNi1-xO system is stable for a wide $x$ range up to LiNiO2 Goodenough et al. (1958). Importantly, Li doping does not render NiO metallic, but semiconducting with in-gap states appearing at $\sim 1.2$ eV above the valence-band maximum van Elp et al. (1992); Zhang et al. (2018). In the following, LixNi1-xO is studied for $x=0.125$ by our DFT+sicDMFT method within a supercell approach. We reduced the stoichiometric lattice constant for the 16-atom cell to $a_{\rm dop}=4.15\,$Å  in the doped case Goodenough et al. (1958) and relaxed the atomic positions in antiferromagnetic LDA+U (see Fig. 5). There are three symmetry-inequivalent Ni classes in the supercell. Note that structural relaxation including local Coulomb interactions is important, since a LDA-based relaxation leads to a too strong shift of the O sites next to Li towards Ni. This results in artificial resonances in the low-energy part of the spectral function. Identical local Coulomb parameters as chosen for stoichiometric NiO are used for the defect problem. The number of Kohn-Sham projection states of is properly scaled from 8 at $x=0$ to 59 $(=7\,{\rm Ni}\times 5\,d$-${\rm orbitals}+8\,{\rm O}\times 3\,p$-${\rm orbitals})$ at $x=0.125$.

III.3.1 Many-body spectrum with in-gap states

Figure 6 depicts the collected spectral results together with inverse-photoemission data from van Elp et al. van Elp et al. (1992). The total spectrum in the upper part of Fig. 6a shows minor shifts and changes of intensity for the occupied states compared to the stoichiometric case. A shift of the spectrum to smaller binding energies with Li doping is also observed in experimental data van Elp et al. (1992); Reinert et al. (1995); Zhang et al. (2018). Peak E in the conduction states is shifted to higher energy, which qualitatively coincides with available experimental data from Reinert et al. Reinert et al. (1995), yet the present theoretical shift appears somewhat too strong. Note in addition that the inverse-photoemission data in Ref. van Elp et al., 1992 were aligned at 5 eV. There is some minor occupation just above the VBM, i.e. the spectrum is not of perfectly strict insulating kind. Without disorder mechanisms and/or without a local structural/electronic mechanism Lechermann et al. (2018), avoiding residual metallicity in a doped correlated insulator is very challenging in theoretical electronic structure methods. We still believe that our spectrum provides a reasonable account of experimentally semiconducting Li0.125Ni0.875O.

Concerning the joint appearance of the $d^{8}\underline{Z}$ and $d^{8}\underline{L}$ states, the inspection of the local spectral function again turns out to be useful (see lower part of Fig. 6a). In the energy window $\Lambda=[-3.5,0]$ eV the lowest-energy peak of the $t_{2g}$ spectral weight is increased and the other one of the two-peak substructure is weakened compared to the stoichiometric case. The changes in the $e_{g}$ spectrum within this region have an even larger bearing: the stoichiometric peak at $\sim 2.6$ eV is nearly smeared out for $x=0.125$. This means that the vital Ni($3d$) peak contribution to peak B in the total spectrum is weakened. Thus the subtle balance between $d^{8}\underline{Z}$ and $d^{8}\underline{L}$ as observed in the stoichiometric case becomes qualitatively disturbed with Li doping. The spectral weight of $d^{8}\underline{Z}$ states close to the VBM is increased.

Finally, let us focus on the in-gap states (IGS) visible in the range $[0.7,1.7]$ eV of the total spectrum. The energy location of the IGS is in excellent agreement with inverse-photoemission data van Elp et al. (1992); Reinert et al. (1995); Zhang et al. (2018). In the local Ni spectrum, as already seen for the upper Hubbard band, there is no $t_{2g}$ contribution to the IGS. The question arises at the expense of which spectral weight do the IGS appear? Inspection of the weight transfers among the Bloch-resolved spectral functions $A_{\nu}(\omega)$ may help to answer this question. Figure 6b displays all the 59 $A_{\nu}$ functions and highlights the four ones with a major contribution to the IGS. From these, one observes that the ’spectral growth’ of the IGS corresponds with spectral weight at the VBM and a ’spectral shrinkage’ of weight that is just somewhat deeper than peak A in the occupied spectrum. It is therefore tempting to directly associate the IGS with the weakening of the original $d^{8}\underline{L}$ subpeak in the local Ni-$e_{g}$ spectrum as noted above.

III.3.2 Where are the holes?

The rigorous real-space location of the holes introduced by Li doping of NiO is a tenacious matter of debate Heikes and Johnston (1957); Kuiper et al. (1989); van Elp et al. (1992); Reinert et al. (1995); Zhang et al. (2018). Originally, data have been interpreted via the formation of Ni3+, yet more recent studies favor the picture of the holes being located in the O($2p$) shell. From a theoretical point of view it is indeed notoriously difficult to uniquely associate valence charges in a condensed matter system with a specific lattice site. Depending on the choice of local orbitals and the kind of charge analysis, often rather different results are obtained.

From a general point of view, underlined by the previous discussions of spectra, the relevant charge-transfer physics in NiO renders the tight entanglement of Ni($3d$) and O($2p$) in the immediate neighborhood of the VBM obvious. The Zhang-Rice doublet state is a direct consequence thereof. Quantitatively, doping of $x$ Li ions introduces a hole doping of $\delta=x/(1-x)$ on the remaining Ni sites. Thus in the present case of $x=0.125$, this amounts to $\delta=0.143$. The Ni($3d$) occupation from the local Green’s function reads $n_{d}=8.175$ at stoichiometry and $n_{d}=8.064$ with Li doping. Hence numbers $\delta_{d}=0.111$ of effective nickel holes and $\delta_{p}=\delta-\delta_{d}=0.032$ of effective oxygen holes results here. The mixed character of hole formation is in line with doping into the entangled Ni($3d$)-O($2p$) VBM states. Note that the real-space hole distribution is of course also subject to the actual choice of the supercell.

IV Summary

A methodological advancement of the combination of density functional theory and dynamical mean-field theory, geared to especially address materials problems with substantial charge-transfer character, has been presented. The combination of self-interaction correction on the ligand sites within the state-of-the-art charge self-consistent DFT+DMFT framework proves to be a powerful tool. Not only to approach long-standing ’basic’ problems of late TM oxides, but due to its efficient and easily scalable structure also for application to novel problems e.g. occuring from nanoscale structuring. There are also other theoretical approaches that may deal with electronic correlations originating directly from ligand states. Most notably, the GW+DMFT scheme Biermann et al. (2003); Choi et al. (2016); Panda et al. (2016) has the potential to tackle such physical issues. But that framework is computationally much heavier than the DFT+sicDMFT scheme and has not yet been applied to materials with defects or dopants.

We exemplified our method for the case of stoichiometric and Li-doped NiO, two persistently problematic correlated materials. A faithful description of the NiO spectrum at stoichiometry with good accordance to experimental findings was given. The interplay of different forms of Ni($3d$) and O($2p$)-hole states, in the form of $d^{8}\underline{Z}$ and $d^{8}\underline{L}$, (re)emerged Taguchi et al. (2008) from this analysis. Importantly, the present scheme based on still local DMFT self-energies is sufficient to account for the experimentally-derived hypotheses of Ni($3d$) contributions to the double-peak structure (i.e. peak A and B) below the valence-band maximum. The introduction of explicit nonlocal Ni-Ni self-energies (e.g. via cluster-DMFT) is not needed for a qualitative appearance of such contributions. In the case of LixNi1-xO, we theoretically verified the long-standing experimental observation of in-gap states at $\sim 1.2$ eV above the VBM. Our examination suggests that spectral-weight transfer from $d^{8}\underline{Z}$,$d^{8}\underline{L}$ into the gap region plays a relevant role in the formation of the IGS.

Our extended DFT+DMFT treatment of NiO does however not provide an end to the enduring theoretical investigation of this challenging material. For instance, in order to improve on the classification of the competing states, interacting many-body and multiplet resolution on Ni, on O and inbetween, within a translational-invariant DMFT construction is needed. Still, the successful state-of-the-art DFT+DMFT approach for early/middle-row transition-metal oxides is thus complemented with the DFT+sicDMFT framework for late-row TM oxides. Various problems of materials with interacting electrons, such as e.g. rare-earth nickelates or high-$T_{\rm c}$ cuprates, await (renewed) investigation.

Acknowledgements.

We gratefully acknowledge financial support from the German Science Foundation (DFG) via the project LE-2446/4-1 and thank I. Leonov, L. F. J. Piper and G. A. Sawatzky for helpful discussions. Computations were performed at the University of Hamburg and the JUWELS Cluster of the Jülich Supercomputing Centre (JSC) under project number hhh08.