Introduction

Strongly correlated materials are the Achilles heel of density-functional theories, where widely used local-density approximation (LDA) and generalized gradient approximation (GGA) fail to open up a band gap and correctly describe localized electrons. These failures are due to the self-interaction error, which tends to overly delocalize electrons in LDA and GGA. The Strongly Constrained and Appropriately Normed (SCAN) functional1, a meta-GGA functional offers at least a partial solution to this problem by correctly opening up band gaps for Mott and charge transfer insulators2,3,4,5 and cuprate superconductors6,7,8,9. In addition to the SCAN, recently developed regularized derivatives (e.g., r2SCAN) exhibit similar accuracy as the SCAN functional but with improved numerical stability10,11,12,13,14,15.

Hubbard-U corrections offer an alternative, albeit parametrized, means to correct for the shortcomings of LDA or GGA, which also have been widely used for Mott and charge transfer insulators16,17,18,19, as well as for cuprate superconductors20,21,22,23,24. The shortcoming of Hubbard-U corrections is that U is an external parameter, and it is not clear how to determine its value, while SCAN meta-GGA functionals are parameter-free.

A known shortcoming of SCAN is its overcorrection for delocalized systems, that is, gaps are opened where there should be none, and spurious electron localization causes unphysical symmetry breakings and magnetizations. For example, SCAN predicts an antiferromagnetic insulator ground state for graphene instead of the well-known non-magnetic semi-metallic Dirac cone bands, and a symmetry-broken antiferromagnetic ground state for benzene25. Similar overstabilization of magnetic phase are also manifested by exaggerated magnetic momenta in ferromagnetic metals (Fe, Co, and Ni) and some magnetic compounds26,27,28. Also, SCAN predicts incorrectly the ground states of Ce-, Mn-, and Fe-based oxides, which however can be improved using SCAN + U.29 In addition, Long et al. show that oxidation enthalpies for several transition metal oxides can be improved using SCAN + U, with the exception of copper oxides, where Hubbard-U corrections offer no improvement over SCAN23,24,28,30 Similar behaviors were also observed with r2SCAN + U functionals for copper and chromium oxides31. Thus, SCAN functionals would be prone to problems in several delocalized systems and transition metal oxides, and may also have problems in copper compounds i.e., SCAN functionals seems to be able to describe well CuO with valence Cu2+ similar to that of many cuprate superconductors, but it is not clear whether it can describe well the valence of non-magnetic Cu+ such as Cu2O.

Fig. 1
figure 1

Lattice structures of (a) CuO, (b) Cu2O, (c) YBa2Cu3O6 (YBCO6) and (d) YBa2Cu3O7 (YBCO7). The arrows of Cu atoms indicate the local magnetic momenta in antiferromagnetic configurations. Three oxygen sites Oplane, Ochain, Oapcial and two copper sites Cuplane and Cuedge in YBCO7 and YBCO6 are categorized. It is noted that removing the oxygens in the Ochain sites in YBCO7 , gives rise to the oxygen-deficient structure of YBCO6.

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Yttrium-based cuprates, namely, YBa2Cu3O7–δ (YBCO) with δ = 0 ~ 1, are particularly interesting superconductors due to their high transition temperature, and the multivalence of copper. Copper in YBCO has two different lattice positions (sites Cuedge and Cuplane shown in Fig. 1); in YBa2Cu3O7 (YBCO7) copper on either site is in the Cu2+ state, whereas in YBa2Cu3O6 (YBCO6) Cuedge is in the Cu+ state, and Cuplane is in the Cu2+ state. Figure 1 shows the structure of YBCO6 and YBCO7. Oxygen has three sites (Oplane, Ochain, and Oapical shown in Fig. 1). All of the oxygen O in the chain site (Ochain) of YBCO7 is removed to form YBCO6, and the valence of Cu changes accordingly, being Cu+ in YBCO6. It has been observed that oxygen vacancies in YBa2​Cu3​O7−δ​ (YBCO) are predominantly formed at the chain site Ochain32,33,34, but also occur at the apical site Oapical35,36, both of which are considered important for understanding the superconducting properties of YBCO. Considering these facts, there is need for a density functional that can consistently describe different valences and correctly predict the sites of oxygen vacancies in YBCO.

In this work, we test the reliability of PBE, SCAN, and r2SCAN functionals with a Hubbard-U correction (PBE + U, SCAN + U, and r2SCAN + U) in CuO, Cu2O and YBCO. We show that the stability of the two copper oxides can be reproduced in PBE + U, but Cu2O is always unstable in SCAN and SCAN + U as well as in r2SCAN and r2SCAN + U. In addition, we show that SCAN and SCAN + U, unlike PBE + U, prefer excessively apical vacancy formation , which is not consistent with the experiment. This similar behavior was also observed for r2SCAN and r2SCAN + U.

Results and discussion

The cupric oxide (CuO)37 and the nonmagnetic cuprous oxide (Cu2O)38 are illustrated in Fig. 1a, b. The formal oxidation state of Cu atoms in CuO and Cu2O are Cu+ 2 and Cu+ 1, resulting in the antiferromagnetism and nonmagnetism of CuO and Cu2O stemming from spin-polarized and spin-unpolarized d9 and d10 electron configurations. YBCO7 (YBa2Cu3O7) and YBCO6 (YBa2Cu3O6) are shown in Fig. 1c, d, which contain different sites of copper (Cu) and oxygen (O) atoms i.e., Cuplane and Cuedge for Cu atoms and Oplane, Ochain, and Oapical for O atoms. By removing Ochain atoms from YBCO7, the valence of Cuedge changes into Cu+, therefore YBCO6 contains both Cu+ and Cu2+ at Cuedge and Cuplane, respectively. This implies that the phase stability of Cu2O and CuO with Cu+ and Cu2+ should be intimately related to the reduction reaction of YBCO7, involving the transition of oxidation state of Cuedge atoms Cu2+ to Cu+. Tables 1 and 2 compare the optimized lattice parameters and local magnetic moment of CuO, Cu2O, YBCO6, and YBCO7 obtained from PBE + U, SCAN, and r2SCAN functionals to the experimental values31,32,35,36,37,38,39,40,41 and shows generally good agreement between experiment and calculation.

Table 1 Lattice constants, volumes, band gaps, and formation enthalpies of CuO and Cu2O obtained from experiments37,38,39,40,41and first-principles calculations with PBE + U (Ueff = 5 eV), SCAN, and R2SCAN functionals including lattice constants (a, b, and c), local magnetic moment m, band gap Eg, formation enthalpy ΔHf, and the bond length d between Cu and O.
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Table 2 Lattice constants of and bond lengths of YBCO6 (YBa2Cu3O6) and YBCO7 (YBa2Cu3O7) obtained with PBE + U (Ueff = 5 eV), SCAN, and R2SCAN functionals compared to experimental values. 35 39
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The chemical stability of CuO and Cu2O explicitly depends on chemical potentials of copper and oxygen, and the predictability of the relative stability of these competing phases has been a bolometer for benchmarking approximate exchange-correlation functionals of DFT calculations22,23,24,28,29,30,38,42. At zero pressure and zero temperature, the formation enthalpy of CuO and Cu2O are calculated as

$$\Delta {H_{\text{f}}}\left( {{\text{CuO}}} \right)\,=\,E\left( {{\text{CuO}}} \right) - {\mu _{{\text{Cu}}}}^{0}\, - \,{\mu _{\text{O}}}^{0},$$
(1)
$$\Delta {H_{\text{f}}}\left( {{\text{C}}{{\text{u}}_{\text{2}}}{\text{O}}} \right)\,=\,E\left( {{\text{C}}{{\text{u}}_{\text{2}}}{\text{O}}} \right)\, - \,{\text{2}}{\mu _{{\text{Cu}}}}^{0}\, - \,{\mu _{\text{O}}}^{0},$$
(2)

where E(CuO) and E(Cu2O) are the total energies per formula unit of CuO and Cu2O, µCu0 and µO0 are the reference chemical potentials of single Cu and O atoms, which are evaluated from total energies of the Cu metal and the spin-triplet state O2 molecule. At finite temperatures and varying pressures, the chemical potentials of Cu and O change from the reference chemical potentials as µi = µi0 + ∆µi (i = Cu, O), and the changes in ∆µi are bound by the following relationships,

$$\Delta {H_{\text{f}}}\left( {{\text{CuO}}} \right){\text{ }}=\Delta {\mu _{{\text{Cu}}}}+\Delta {\mu _{\text{O}}},$$
(3)
$$\Delta {H_{\text{f}}}\left( {{\text{C}}{{\text{u}}_{\text{2}}}{\text{O}}} \right){\text{ }}=\Delta {\mu _{{\text{Cu}}}}\,+\,{\text{2}}\Delta {\mu _{\text{O}}}.$$
(4)

Figure 2 shows the phase stability of CuO and Cu2O predicted with DFT calculations employing PBE + U, SCAN + U and r2SCAN + U functionals. The gray zone in Fig. 2a–e shows the area with ∆µi > 0 where the formation of the Cu metal and O2 molecule are thermally preferred compared to the formation of CuO and Cu2O, which means that the chemical potentials are only allowed to vary in the negative quadrant ∆µi < 0. In Fig. 2, the filled and open circles represent the experimental and calculated intersection points of Eqs. 3 and 4, which represent the chemical environment in which CuO and Cu2O coexist in equilibrium. To the left from the intersection point Cu2O is stable, and to the right CuO. As shown in Fig. 2a, b, PBE gives the intersection point in the allowed area (∆µi < 0) and PBE + U (Ueff = 8 eV) closely reproduces the experimentally observed relative stabilities of Cu2O and CuO, which indicates that the Hubbard-U correction on Cu-3d states can improve the accuracy of the formation enthalpies and phase stability of CuO and Cu2O. In contrast, an opposite behavior is observed in SCAN functional as shown in Fig. 2c, d; SCAN gives the intersection point in the prohibited region (∆µi > 0), which is seemingly caused by a biased energetic preference of SCAN for partially occupied states compared to fully occupied states as previously reported,19–21,25−27 leading to overstabilization of CuO (Cu2+, d9) compared to Cu2O (Cu+, d10). It is important to remark that the absolute values of the SCAN formation enthalpies are much closer to the experimental values than the PBE ones, but for CuO and Cu2O, the errors have opposite signs: there is an overbinding for CuO and an underbinding for Cu2O. Thus, even though the quantitative errors are small, the opposite signs make the error qualitative, such that per SCAN, Cu2O is predicted unstable. The Hubbard-U correction on SCAN functional does not improve the relative phase stability of CuO and Cu2O but instead, leads to further stabilization of CuO with a partially occupied d9 state. This is because the Hubbard-U correction only improves the underbinding of Cu2O but worsens the overbinding of CuO. The overall behavior of PBE + U and SCAN + U functional for a wide range of Ueff parameter is demonstrated in Fig. 2e, which shows that the correct phase stability of CuO and Cu2O cannot be achieved with SCAN + U functional with a positive Ueff parameter. While r2SCAN functionals also exhibit similar overstabilization of CuO over Cu2O, the error is relatively modest compared to SCAN as the intersection point (i.e., the equilibrium chemical potential) in the phase space is closer to the experimental value. Importantly, r2SCAN brings the intersection point of CuO and Cu2O formation enthalpies into the negative quadrant in the phase space, i.e., r2SCAN recovers the chemical stability of Cu2O. We note that a negative Ueff parameter also might be adopted in SCAN + U and r2SCAN + U functionals, however applying such corrections result in severe band gap underestimation of Cu2O. Previous works28 also reported that SCAN and r2SCAN without Hubbard-U correction (Ueff = 0.0 eV) would be the optimal choice to obtain least errors of the formation enthalpy of CuO and Cu2O28,31. The SCAN functionals with nonlocal van der Waals corrections (SCAN + rVV1043 and r2SCAN + rVV1044) were also tested, showing no significant effect on the results (see Supplementary Information).

Fig. 2
figure 2

The phase stability of CuO and Cu2O as function of chemical potential obtained with experiment and DFT calculations with PBE + U and SCAN + U functionals. The phase stabilities with (a) PBE, (b) PBE + U (Ueff = 8 eV), (c) SCAN, and (d) SCAN + U (Ueff = 5 eV). The circles denote intersection points where the relative stability of CuO and Cu2O changes. (e) The Ueff parameter dependence of the intersection points predicted by PBE + U, SCAN + U and r2SCAN + U functionals.

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By comparing the densities of states and experimental ultraviolet photoemission spectroscopy (UPS) spectra45, we examined the predictive power of each functional for the valence band structures of CuO and Cu2O in Fig. 3. Figure 3a shows the UPS spectra and densities of states (DOS) that were obtained using the PBE, PBE + U (Ueff = 5 eV and 8 eV), SCAN and r2SCAN functionals. The primary UPS peak at around 3 eV is attributed to Cu-3d states through comparison of the UPS spectrum and DOS of CuO. Cu-3d states become deeper as Ueff parameter of PBE + U increases, PBE + U (Ueff = 5 eV) gives the closest Cu-3d position compared with the experimental UPS spectrum, whereas the Ueff = 8 eV giving the optimal thermal stability of CuO and Cu2O shifts the Cu-3d position to too low energies. This suggests that a PBE + U (Ueff = 5 eV) might be a reasonable compromise for both the electronic spectrum (the valence band structure) and formation enthalpies. While SCAN and r2SCAN obtain similar valance band structures compared to the experimental UPS spectra, they predict the erroneous phase stability of CuO and Cu2O (cf. Figure 2). The similar behavior of the position of Cu-3d states is also shown for Cu2O in Fig. 3b; the density of states obtained with PBE + U (Ueff = 5 eV) optimally reproduces the main peak of the experimental UPS spectrum around − 3 eV. It is shown that the band gap of Cu2O is less sensitive to the choice of the functional and U parameter, unlike the band gap of CuO, which is a d–d gap originated from the partially occupied 3d9 states and strongly influenced by the choice of Hubbard-U parameter and the type of functional46.

Fig. 3
figure 3

Densities of states of CuO and Cu2O obtained with PBE, PBE + U (Ueff = 5 eV, and 8 eV), SCAN, and r2SCAN functionals compared to experimental spectra of ultraviolet photoemission spectroscopy (UPS) of Ref.45.

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YBCO7 and YBCO6 have two distinct Cuplane and Cuedge sites (cf. Figure 1c, d), the valence of Cuedge atoms is expected to be directly changed from Cu2+ to Cu+ as YBCO7 becomes YBCO6 by removing oxygen atoms at Ochain. As SCAN functional has been found to give an erroneous phase stability of CuO and Cu2O with Cu+(d9) and Cu2+ (d10) due to a spurious energetical preference for partially filled d9 states, the oxygen vacancy formations of YBCO7 seems to be also prone to such a bias of SCAN and r2SCAN functionals.

Notably, it has been confirmed that oxygen vacancies are predominantly formed at the Ochain site compared to the Oapical​ site32,33,34, with both types of oxygen vacancies being experimentally verified35,36. For obtaining the proper description of oxygen-deficient YBCO, we calculate the oxygen vacancies at Ochain and Oapical in YBCO7 with PBE + U (Ueff = 5 eV) SCAN and r2SCAN functionals to investigate the bias of SCAN meta-GGA functionals in oxygen vacancy formation in YBCO.

Fig. 4
figure 4

Oxygen vacancies in YBCO7. (a) Formation energies of the oxygen vacancy at Oapical and Ochain sites in YBCO7−δ with δ = 0.0625 and 0.125 calculated with PBE, PBE + U (Ueff = 5 eV), and SCAN functionals. (b) Local atomic structures of YBCO6.875 around oxygen vacancies at Oapical and Ochain sites. Bond lengths between Cu and O atoms d(Cu-O) are compared with the bond lengths of CuO (d9) and Cu2O (d10). (c) Calculated Cu-2p core level shifts from d9→d10 transition introduced by the oxygen vacancies compared with the core level shifts from CuO (d9) and Cu2O (d10).

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The formation energy of oxygen vacancy in YBCO7 is calculated as Ef =E(VO)+µO0 − E(YBCO7), where E(VO) is the total energy of YBCO7–δ containing an oxygen vacancy (VO), µO0 is the reference chemical potential of oxygen defined as the half of the total energy of the oxygen molecule in the triplet spin state µO0 = 1/2E(O2), and E(YBCO7) is the total energy of pristine YBCO7 supercell. Figure 4a shows the formation energies of VO at the Oapical and Ochain in YBCO7−δ at two concentrations δ = 0.0625 and 0.125 calculated with PBE, PBE + U (Ueff = 5 eV), and SCAN functionals. Whereas PBE and PBE + U predict a higher formation enthalpy of VO at the Oapical compared to VO at the Ochain, SCAN and r2SCAN predict that VO at the Oapical is more stable in contrast to PBE and PBE + U. This contrasting tendency of SCAN functional is originated from the spuriously biased total energy of Cu+(d9) and Cu+(d10) configurations, leading to an inconsistent description compared to experimental observations32,33,34.

We next look at the bond lengths given for the YBCO6.875 with a single oxygen vacancy (VO) at the Oapical and Ochain sites in Fig. 4b. Oxygen vacancies cause electron doping of YBCO, which should provide electrons to neighboring Cu atoms. Indeed, neighboring Cu atoms at Cuedge sites with a Cu2+(d9) configuration are electron-doped into a Cu+(d10)-like configuration, as shown by local distortion of bond lengths surrounding VO, manifested by strongly shrinking Cu-O bonds. As illustrated in Fig. 4b, the bond length shrinking between Cu-O atoms can be attributed to a transition from Cu2+(d9) to Cu+(d10), based on comparison wirh Cu-O bond lengths in CuO (d9) and Cu2O (d10) of 1.95 Å and 1.86 Å, respectively. While the VO at Oapcial site displaces one neighbor Cu atom by providing electrons, the VO at Ochain site displaces two surrounding Cu atoms, which means the VO at Ochain site more strongly drives the reduction from Cu2+(d9) to Cu+(d10) compared to the VO at Oapical site. This is in line with the observed phase stability of CuO and Cu2O, where SCAN and r2SCAN overly stabilize CuO (d9) against Cu2O (d10).

Figure 4c shows the calculated core level shift of Cu-2p compared between Cu-d9 Cu-d10 configurations calculated by PBE + U (Ueff= 5 eV). As demonstrated in the core level shift from CuO (d9) to Cu2O (d10), providing electrons into d9 states results in a positive core level shift (i.e., weakening the binding energy of core electrons), which is due to an enhanced electron screening effect with more valence electrons in 3d states. Oxygen vacancies at Oapical and Ochain sites also induce positive core level shifts in the Cu-2p state, which indicates an effective the d9 to d10 transition of neighboring (displaced) Cu atoms around oxygen vacancies (VO). Since the VO at Ochain site induces larger core level shifts compared to the VO at Oapical site, this further clarifies the increased d10 character of the VO at Ochain which is evaluated as more unstable than the VO at Oapical by SCAN functionals.

In summary, we clarify a systematic failure of SCAN meta-GGA density-functionals (SCAN and r2SCAN), which spuriously stabilizes the Cu2+(d9) configurations rather than the Cu+(d10) configurations. This bias leads to the failure to stabilize Cu2O at any allowed chemical potential for the SCAN functionals, while PBE-GGA functional with a Hubbard-U correction reasonably reproduces the experimental relative phase stability of CuO and Cu2O. Meanwhile, both PBE + U and SCAN functionals predict the valence band structures of CuO and Cu2O well in agreement with the ultraviolet photoemission spectroscopy (UPS) spectra. The energetic bias of SCAN functionals on d9 and d10 configurations results in different relative stability of oxygen vacancies in YBCO7 at Oapical and Ochain sites: SCAN functionals prefers Ochain vacancies as opposed to PBE and PBE + U functionals, which contradicts a series of experimental observations32,33,34. This work confirms a systematic bias in the SCAN meta-GGA functional regarding the energetics of partially and fully occupied 3d states, as observed by the overstabilization of spin-polarized states in various compounds26,27,28. Addressing this bias is essential for providing accurate predictions for copper-based compounds, particularly those containing Cu+ and Cu2+ states.

Methods

The first principles calculations were performed by using density-functional theory (DFT) with the projected augmented-wave (PAW) method as implemented in the VASP package47,48,49. The kinetic cutoff energy was set to 500 eV. The calculations were carried out with PBE-GGA50, SCAN11, and r2SCAN10 exchange correlation functionals and SCAN functionals with the revised VV10 kernel51 (SCAN + rVV1043 and r2SCAN + rVV1044) with applying the Hubbard-U correction on the Cu-3d states based on Dudarev’s method52. The antiferromagnetic cupric oxide CuO (tenorite)37 and the nonmagnetic cuprous oxide Cu2O (cuprite)38 were calculated by using DFT with 7 × 10 × 7 and 8 × 8 × 8 k-point grids generated with the Monkhorst-Pack meshes53. The YBa2Cu3O7 (YBCO7) and YBa2Cu3O6 (YBCO6) were calculated within antiferromagnetic configurations sampled with the 4 × 4 × 3 k-point grids. Oxygen vacancies at chain and apical sites (Ochain and Oapcial, see Fig. 1) in YBCO7 were simulated using the supercell approach adopting 4 × 4 × 1, and 2 × 2 × 1 supercells, which correspond to YBCO6.9375, and YBCO6.875, respectively. Cu-2p core levels are evaluated from a postprocess of DFT results with PAW method with an initial state approximation, as implemented in the VASP package54.