First-principles electronic structure investigation of HgBa₂Ca_n−1Cu_nO_2n+2+x with the SCAN density functional

The discovery of high temperature superconductivity at around 40 K in La_2−xBa_xCuO₄ copper oxides (cuprates) in 1986 has opened a new research frontier in condensed matter physics.¹ New superconductor operating above liquid nitrogen temperature (77 K) was soon found in YBa₂Cu₃O₇.² Multilayered compounds, such as Bi₂Sr₂CaCu₂O_8+x^3,4 and HgBa₂Ca_n−1Cu_nO_2n+2+x,^5,6 then earned significant interest from the scientific community^7,8 as adding CuO₂ planes could increase the transition temperature T_c beyond 100 K. The trilayer compound HgBa₂Ca₂Cu₃O_8+x (T_c = 135 K at ambient pressure⁶ and T_c = 164 K at 31 GPa^9,10) is still the highest T_c superconductor outside the electron–phonon mechanism of Bardeen–Cooper–Schrieffer (BCS) theory.¹¹ 

Besides its T_c value, the Hg-based family [Fig. 1(a)] is preferred for its structural simplicity.^12,13 Its tetragonal structure is relatively free from structural phase transitions unlike, for example, La₂CuO₄. Its CuO₂ planes have minimal buckling, and there are no Cu–O chains in its cell structure unlike in YBa₂Cu₃O₇.^17,18 When doped, the dopant O atom is found to reside in the center of the Hg layer, i.e., at the $12,12,0$ position.^19,20 From a theoretical perspective, Ref. 21 has also affirmed, with atomistic simulations, that this dopant location is energetically the most favorable. The disorder effects on the CuO₂ plane from the dopant are minimized with the thick Ba layer between the Hg and CuO₂ planes (see Fig. 1).^12,22 Despite the difficulty in obtaining large single crystals with well-defined composition,²³ these materials serve as ideal benchmarks for assessing theoretical models.

Unlike other prototypical cuprates such as La₂CuO₄, there is no experimental data for the electronic bandgap and other electronic structure features in the parent compound HgBa₂Ca_n−1Cu_nO_2n+2 due to the difficulty to synthesize high-purity samples. Thus, its description is both a challenge and an opportunity for first-principles calculation techniques. Early studies based on local-density or generalized gradient approximations (LDA/GGA) to density functional theory (DFT) were insufficient because an incorrect nonmagnetic metallic ground state was predicted instead of the antiferromagnetic Mott insulating ground state.^24–27 The more sophisticated hybrid exchange-correlation functionals that include a fraction of the nonlocal Fock exchange could provide a reasonable description of the ground state; for example, the HSE06 functional^27,28 opens a finite insulating gap and predicts a magnetic exchange coupling (J) value that is comparable to experiment in the antiferromagnetic ground state of La₂CuO₄.²⁹ However, the hybrid methodology fails to describe the insulator-to-metal transition upon doping for lanthanum cuprates,^30,31 as the energy gap persists even in the doped state. A suitable exchange-correlation functional that can properly describe both the undoped and doped phases of cuprates remains an important research focus until this day.

The multilayered parent compounds (HgBa₂Ca_n−1Cu_nO_2n+2) have been studied using hybrid functionals for n = 1, 2, 3 [we refer to these as Hg-12(n − 1)n, e.g., Hg-1201, Hg-1212, and Hg-1223 for n = 1, 2, and 3, respectively].²⁷ These density of states (DOS) calculations concluded that, unlike the monolayer Hg-1201, which is an insulator with a bandgap slightly above 1 eV, the multilayered Hg-1212 and Hg-1223 compounds are metallic as the Hg–O conduction states are lower in the energy and the Cu–O conduction states remain fixed, thereby closing the bandgap. As this metallicity is not due to Cu states, this would explain the persisting antiferromagnetism in metallic Hg-1212/Hg-1223.

While the presence of low-lying Hg–O states in the conduction band has been suggested since the early calculations of HgBa₂Ca_n−1Cu_nO_2n+2,^32,33 the extent of mercury roles in the multilayered compounds remains unclear. There has neither been experimental evidence of the metallicity due to low-lying Hg–O states nor further studies with hybrid functionals reported on these materials. This stagnancy may be attributed to the difficulties in synthesizing pure samples and the prohibitive cost of hybrid methodology (approx. hundreds of times more expensive^34,35 than using semilocal functionals). Thanks to the wider availability of high-performance computing resources, more extensive calculations have become possible in the past decade. Hence, we would like to investigate this material more comprehensively with the computational resources available at our disposal (see Acknowledgments).

Over the years, there were questions on whether band gaps predicted by density-functional calculations should be compared with experimentally derived energy gaps. These doubts are expected to fade since modern computational packages, such as the Vienna ab initio simulation package (VASP),^36,37 have implemented their calculations in the generalized Kohn–Sham (gKS) scheme. Reference 38 has shown that the gKS bandgap is equal to the fundamental bandgap in the solid, which is defined as the ground-state energy difference between systems with different numbers of electrons. This provides a solid basis for comparing band gaps of gKS formalism with the experimentally observed band gaps, and improvement in the prediction of energies and structures brought by a functional would also indicate improved gKS band gaps.^31,39–41

The bandgap prediction of La₂CuO₄ with hybrid methodology²⁹ served as one of the supporting basis for calculating HgBa₂Ca_n−1Cu_nO_2n+2 in Ref. 27. Citing the 2 eV optical absorption peak in Ref. 42, the computed La₂CuO₄ bandgap of 2.5 eV by HSE06 hybrid functional was considered to concur with experiment. However, Ref. 43 also reported a smaller bandgap of 0.89 eV obtained via Hall transport measurements. It has since been argued^31,40,43,44 that one should estimate the bandgap not from the lowest energy absorption peak but from the leading-edge gap in the optical spectra. This implies that the computed band gaps by density functional theory should be compared to an experimental value around 1 eV from Ref. 42. From a comparative study,³¹ it is apparent that the hybrid functionals overestimate the bandgap of La₂CuO₄, and another functional of the meta-GGA class is better suited for this purpose, which we shall explain in the following paragraph.

The recently devised strongly constrained and appropriately normed (SCAN) meta-GGA exchange-correlation functional³⁵ satisfies all 17 known constraints applicable to meta-GGA. It has successfully described the structural, electronic, and magnetic properties of pristine and doped La₂CuO₄,^40,41 YBa₂Cu₃O₆,⁴⁴ and Bi₂Sr₂CaCu₂O_8+x.⁴⁵ Reference 31 compared SCAN with 12 other functionals spanning across the levels of Perdew–Schmidt hierarchy⁴⁶ in lanthanum cuprates and demonstrated SCAN’s superiority in matching experimental results even when compared to the hybrid functionals. These results indicate that SCAN captures key features of the cuprates and provides a new prospect in describing their correlated electron properties. The computational cost of meta-GGA functionals that is only a few times larger than its LDA/GGA predecessors adds to the viability of more effective studies of cuprates against the cost-prohibitive hybrid methodology or beyond-DFT techniques.

In this work, we present a SCAN density-functional description for the electronic structure of HgBa₂Ca_n−1Cu_nO_2n+2+x series. Using relaxed structures that are in good agreement with the experiment, we show that the undoped (x = 0) compounds remain insulating up to n = 6 with a finite but decreasing, indirect bandgap. The low-lying Hg–O conduction states are noted, but their dominant proportions against the Cu–O states are not apparent in Hg-1212/Hg-1223, and these Hg–O states are only clearly lower in energy at n = 6. In addition, we also investigate the doped phase (x > 0) via supercell construction of Hg-1201, Hg-1212, and Hg-1223 for several representative excess oxygen levels x. We report that SCAN improves an earlier description of the doped phase with semilocal functionals⁴⁷ in capturing lattice contraction and the magnetic moment as a function of x. We confirm the narrow DOS peak due to contributions of the dopant O atom at low doping levels and extract an optimum doping x_o where this feature is located at the Fermi level E_F. Finally, we compute the normal-state, zero-temperature Sommerfeld parameter of electronic specific heat γ from the DOS at E_F and observe a peak across doping levels to support an experimental prediction of it being a universal property among cuprates, which could be a signature of quantum criticality.^48–50 

In general, we followed the computational parameters of the preceding SCAN studies of other cuprates.^40,41,44,45 Ab initio calculations were carried out by using the pseudopotential projector augmented-wave (PAW) method⁵¹ implemented in the Vienna ab initio simulation package (VASP^36,37) with an energy cutoff of 550 eV for the plane-wave basis set. Exchange–correlation effects were treated using the SCAN meta-GGA scheme.³⁵ The crystal structures were relaxed using a quasi-Newton algorithm to minimize energy with an atomic force tolerance of 0.008 eV/̊ and a total energy tolerance of 10⁻⁵ eV. The costlier conjugate gradient algorithm was also utilized in a few cases when the forementioned algorithm encountered convergence problems. The relaxation procedure utilized an 8 × 8 × 4 gamma-centered k-point mesh to sample the Brillouin zone. Denser meshes (20 × 20 × 4 or more) were used to calculate the DOS with the tetrahedron method with Bl̈chl corrections. The band structures are drawn along the path Γ − X − M − Γ − Z − R − A − Z in the first Brillouin zone of the antiferromagnetic $2×2×1$ cell [see Fig. 1(b)]. We utilize PyProCar⁵² and Sumo⁵³ packages to help generate the DOS and band structure plots in this manuscript and its supplementary material.

For investigating the doping-dependent electronic structure, we used a series of supercells containing excess oxygen atoms. We used the same calculation parameters for computing both undoped and doped compounds, except for the additional dopant atoms. Oxygen concentrations of 0.125, 0.25, and 0.5 with cell sizes up to an eightfold single cell are considered for compounds Hg-1201, Hg-1212, and Hg-1223. The corresponding cells used for all compounds are illustrated in the top row of Fig. 2. The bottom row of Fig. 2 includes supercells that are used to compute the x = 0.375 case for Hg-1212 and Hg-1223. An alternative dopant placement is also tested for the x = 0.5 case in Hg-1201 (the bottom right figure in Fig. 2), for which we note only a minor difference in results that do not change the conclusion of this study. For the larger supercells of the multilayered compounds containing 8 formula units and more, the DOS plots used a smaller 10 × 10 × 4 k-point mesh due to the large memory requirements. This change does not affect the resulting DOS plots given the smaller size of the first Brillouin zone for larger supercells. The doping effect on the lattice parameters, atomic positions, and magnetic order was investigated by total-energy and atomic-force calculations. Initial antiferromagnetic order was assumed in our structure, which allowed us to study the interplay between doping levels and the strength of magnetic moments. In this regard, our calculation is an extension of Refs. 47 and 54 in which their calculations were performed on non-magnetic, non-relaxed supercells of Hg-1201 with local-density approximation.

From the DOS calculations, we compute a thermodynamic quantity that can be compared with experiments. The Sommerfeld parameter γ of electronic specific heat is defined as

where the DOS at Fermi energy N(E_F) is defined per atom and per one spin direction.¹⁷ This quantity in the normal state extrapolated to zero temperature can be extracted from N(E_F) of a computational cell containing X formula units,¹⁷ 

which allows us to estimate γ directly from the DOS.

Our SCAN calculations yield antiferromagnetic ground states for the Hg-based compounds with Cu magnetic moments about 0.47–0.49μ_B, comparable with other cuprates.^40,41,45 The single-layered Hg-1201 compound’s magnetic moment agrees with the 0.4 μ_B value predicted from the variational Monte Carlo.⁵⁵ The magnetic moments of the inner plane (IP) are 0.005μ_B smaller than the outer planes (OP) from n ≥ 3 onwards, which concur with the results of the hybrid functional picture.²⁷ 

We plot the in-plane lattice parameter a of HgBa₂Ca_n−1Cu_nO_2n+2 structures relaxed with SCAN in Fig. 3(a). This parameter decreases with the number of copper-oxide planes n in agreement with LDA results⁵⁶ and the trend observed in experiments for doped structures. SCAN predicts values that are closer to experiments than LDA. Moreover, SCAN provides slightly larger parameters for the parent compounds compared to experimental values for doped samples. The LDA results, however, are on the smaller side, reflecting its well-known overbinding issue. Our SCAN results are hence an improvement as it agrees with doping-induced lattice contraction observed in experiments. A similar observation can be made for the out-of-plane lattice parameter c (see Fig. S1 and Table S1 of the supplementary material for the full set of a and c values).

Moving to the electronic structure results, the band gaps are shown in Fig. 3(b). In contrast to the metallic multilayered structure predicted by hybrid functionals in Ref. 27, SCAN predicts a decreasing but finite bandgap from E_g ≈ 0.6 eV at n = 1 to E_g ≈ 0.2 eV at n = 6. Our prediction for Hg-1201 is comparable to the results of variational Monte Carlo $EgVMC≈0.7$ eV⁵⁵ while the HSE06 and B3LYP hybrid functionals in Ref. 27 yielded much larger band gaps of 1.1 and 1.5 eV, respectively.

The projected band structure and density of states are shown in Figs. 4 and 5. These suggest an insulating ground state with indirect band gaps between the X and M points. The top of the valence band is made up of contributions from Cu and O, as commonly expected for cuprates.¹⁷ Starting from n = 3, the top of the valence band splits due to magnetic inequivalence between the inner and outer Cu–O planes [Figs. 4(b) and 5(a)], with the former being higher in energy. The Hg conduction bands move slightly lower in energy, but its energy difference relative to the Cu band remains less than 0.1 eV in Hg-1223. They are not apparently dominant until n = 6 [Figs. 4(c) and 5(b) inset] where the low-lying Hg conduction bands are located 0.2 eV below the Cu bands. In Fig. 5(b), we also show that unlike in La-cuprates, the Cu $dz2$ states in Hg-cuprates are far lower in energy than the Cu $dx2−y2$ and O p_x + p_y valence states, in agreement with Refs. 63 and 64. This is due to the larger separation between Cu ions and the apical O atoms: 2.8–2.9 Å in Hg-cuprates compared to 2.4 Å in La-cuprates. A similar observation of low Cu $dz2$ states was also reported in Bi-cuprates.⁴⁵ 

We conclude this section by stating that in the SCAN picture, the diminishing bandgap is a gradual process, where Cu, Hg, and O conduction states collectively get closer to the valence states. This is different from the hybrid picture (Ref. 27) where the insulator-to-metal transition is immediate and is solely facilitated by the Hg states. The full set of DOS plots for the parent compounds are provided in Fig. S2 of the supplementary material.

In our doped structures, we observe a magnetic moment reduction that trends with x and n as presented in Table I. Our results for Hg-1223 indicate that this reduction is more prominent for Cu atoms in the OPs, which are closer to the dopant situated at the Hg plane. This inhomogeneity agrees qualitatively with the experimental findings of doped Hg-1245,⁶⁵ where the IPs retain their antiferromagnetic order more strongly (μ ≈ 0.6 μ_B) in contrast to the much weaker moments in the OPs (μ ≈ 0.1 μ_B). This suggests that the reduced magnetic order is related to the hole doping concentration p in each copper-oxide plane and demonstrates the capability of SCAN in capturing charge-spin interplay, as also seen in other cuprates.^44,45

We show our relaxed lattice parameters of Hg-1201, Hg-1212, and Hg-1223 alongside their experimental values in Fig. 6. Due to the limited number of experiments, we collect values over a short range of excess oxygen levels x for each compound from multiple sources. The LDA results⁴⁷ for Hg-1201 are included for comparison. Doping-induced lattice contraction is observed in all cases. In the low doping levels (x ≤ 0.125), there is a good agreement between SCAN (closed circles) and experiments (open circles) for Hg-1201. In comparison to LDA,⁴⁷ the discrepancy between theory and experiments is improved with SCAN. On the other hand, the experimental lattice contraction is smaller than our results for the multilayered compounds. Reference 67 noted that their synthesized lattice parameters are very close to the intrinsic size of the CuO₂ plane of 3.855 ̊ in the infinite-layered compound CaCuO₂⁶⁸ such that it is difficult to further reduce the lattice size during their synthesis, which may explain the small contraction. Our supercell calculations are not subject to these experimental complexities, and thus, by varying solely the oxygen concentration, we find that this translates to a bigger lattice contraction. We also note that these quantitative discrepancies in the doped lattice parameters are not unexpected because we perform zero-temperature, normal state density-functional calculations while Ref. 67 measured their samples at finite temperatures in the superconducting phase. Nevertheless, we believe that capturing qualitatively the doping-induced lattice contraction observed in experiments is still good progress for theoretical simulations.

We analyze the electronic structure of the doped structures starting with Fig. 7, where we illustrate the total DOS evolution of Hg-1201 against x. A localized peak that moves toward E_F with increasing x can be identified. This feature is precisely located at E_F for Hg-1201 at x = 0.25. Unfortunately, we could not observe this alignment for Hg-1212/Hg-1223 at the dopant concentration levels investigated in this work. This doping-induced peak is mainly contributed by the additional states provided by the dopant O atom, as we shall see in the following discussion and Figs. S3–S6 of the supplementary material. The peak appears to decrease in height with x, suggesting a delocalization process of the dopant states that eventually leads to its disappearance in the highly doped case at x = 0.5.

We provide the band structure plots of these doped structures in Fig. 8 alongside their undoped counterparts that are computed in the same supercell size. This enables us to compare the doping effects on the same set of folded bands as well as to anticipate possible artifacts arising from comparing results obtained with different supercell sizes. We note the presence of flat bands precisely located at E_F in the x = 0.25 case [Fig. 8(b)], while they are slightly away from E_F in other concentrations [Figs. 8(a) and 8(c)]. These features do not have a direct parallel in the undoped valence bands, which implies that these levels belong mainly to the dopant O atom. Indeed, these flat bands are predominantly of O p orbital character with a minority contribution from Hg d orbitals (see Fig. S7 of the supplementary material), which concurs with the studies in Ref. 47. The presence of flat bands at E_F is, therefore, consistent with our observation of the sharp DOS peak at x = 0.25 in Fig. 7.

Next, we discuss the doping effect to the projected density of states for some orbitals of interest, i.e., the O p_x + p_y and Cu $dx2−y2$ orbitals. Following the band structure study, we compare them with the undoped traces from equal-sized supercells in Figs. 9(a) and 9(b) for the x = 0.25 case. From the O p_x + p_y portion [Fig. 9(a)], we identify a dip in the doped trace at E ≈ 0.75 eV akin to the shape of the top of the valence state from the undoped compound. Hence, the latter is translated by this amount in Fig. 9(a) for clarity (in orange). The doped trace can be divided into portions from the localized dopant state (in red) and the remaining 16 O atoms (in blue). The former led to the enhancement at E_F, in support of the band structure study. Meanwhile, despite some effects induced by doping to the latter contributions, the rigid-band picture seems to remain applicable at low energies (−0.4 eV < E < 1 eV) as reflected by the similarities between the blue and orange-colored traces.

In comparing the $dx2−y2$ orbitals in Fig. 9(b), we keep the undoped traces unshifted in energy to also make connections with Fig. 8. While the rigid band picture may explain the low-energy features to some extent, here doping does not only result in lowering the Fermi level but also in closing the Cu electronic gap. This is prominently reflected in the band structure. The bump in the density of states between 0.5 and 0.75 eV for the doped compound can be attributed to the overlapping bands in this range. We remark that a similar observation of $dx2−y2$ electronic gap closing is also reported in doping studies of other cuprates.⁴⁵ 

In Figs. S5 and S6 of the supplementary material, the density of states is further segregated to contributions from different atomic types, such as apical and planar oxygen atoms. We note the similarities of the dopant O states with atoms closest to it (Hg and apical O atoms) at x = 0.125 and x = 0.25. Together with Fig. 9, these findings thus suggest that the oxygen dopant introduction enhances the density of states over a narrow energy range in a collective interaction with other atoms. Possible interactions between the dopant state with other atomic states have also been discussed in several other works.^47,69–71 Furthermore, if we take the slightly increased DOS contributions at E_F from Cu and planar O in Fig. S6, together with the shape of the dopant state as well as the absence of high peaks in the DOS at x = 0.5 into consideration, we would like to suggest a delocalizing picture of the dopant state that interacts with more atoms and spreads over a wider energy range with increasing doping. This hypothesis would ideally be tested with more supercell calculations to finely sample the doping region 0.25 < x < 0.5 not yet covered in this work.

Our observation in the previous paragraphs encourages us to define an optimum excess oxygen concentration x_o with regard to the total DOS at E_F. For Hg-1201, we can specify x_o ≈ 0.25 based on the peak observed in Fig. 7. However, our few supercell calculations are insufficient to pinpoint the corresponding value for multilayered compounds. Nevertheless, we can estimate based on the sampled concentrations that 0.25 < x_o < 0.375 for Hg-1212 and 0.375 < x_o < 0.5 for Hg-1223, forming an increasing trend with n. This suggests that x_o could be directly related to the amount of hole doping p effectively introduced to each layer. Indeed, this was also the conclusion arrived in Ref. 47 where their optimum value x^LDA = 0.22 is attributed to the point where the number of induced holes on the CuO₂ plane saturates.

The subsequent question is pertaining to how we should interpret the physical role of x_o. For example, it is interesting to relate these values with the optimum concentrations that yield the highest superconducting transition temperature, recorded in Refs. 72–74 for these three compounds (see Table II). The results in this work and Ref. 47 agree with these records within the experimental uncertainties. Meanwhile, we note that the estimation of optimum doping for T_c in cuprates is not unequivocal among experiments. In contrast to Refs. 72–74, Refs. 66 and 67 predicted a different set of values of x that yield maximum T_c for these three compounds. This discrepancy may be caused by different experimental techniques used which we as theorists claim no expertise in, yet we may consider that these conflicting reports exemplify the long-standing question^33,69 on whether the induced hole concentration in the copper-oxide plane deviates from a simple ionic picture whereby two holes are donated for every oxygen dopant x. The results from Refs. 66 and 67 support the simple ionic picture that gives hole concentration p ≈ 2x, while Refs. 72–74 suggest a smaller dependence p ≈ 0.72x. Although there is no experimental consensus yet,⁴⁷ earlier density-functional calculations in Refs. 47 and 69 concur with Refs. 72–74. This is also what we infer from our SCAN calculations. In any case, both ionic and non-ionic pictures suggest that the optimum excess oxygen concentrations for the whole compound follow the ascending order of Hg-1201, Hg-1212, and Hg-1223, which agrees with our results.

Finally, we compute the normal-state, zero-temperature Sommerfeld parameter of the electronic specific heat γ from the DOS at the Fermi level N(E_F) for the doped compounds in Fig. 10 following Eq. (3). Our values for the small and large doping levels agree well with the experimental results from other cuprates,^17,48,49 which lie around 2–7 mJ/K² mol. For Hg-1201, we observe a peak across the doping concentration at x_p = 0.25 in accordance with the DOS peak in Fig. 7 and the subsequent discussions of its properties therein. The peak in γ is of recent interest⁴⁸ as it may be a thermodynamic signature of a quantum critical point, indicated by a logarithmic divergence in $C/T∝log1/x−x*$ where x is some tuning parameter such as the doping concentration. Similar peaks have not been confirmed in our calculations for Hg-1212 and Hg-1223 in Fig. 10 at the concentrations computed in our supercells. If this feature extends to the multilayered compounds, we expect them to materialize at concentrations of 0.25 < x_p < 0.375 for the bilayer Hg-1212 and 0.375 < x_p < 0.5 for the trilayer Hg-1223 compounds based on the dopant state energies computed in our supercells. The peaks in γ have only been experimentally confirmed by direct measurements on lanthanum-cuprate families,^48,75 but recent observations on the bismuth- and mercury-cuprate samples suggest that this is a universal property for all cuprates.^49,50 However, the exact nature of this divergence is still under debate. Besides the quantum criticality argument, there is an alternative explanation without invoking broken symmetries from the two-dimensional Hubbard model,⁷⁶ which associates the feature to arise from the finite-temperature critical end point of the first-order transition between a pseudogap phase with dominant singlet correlations and a metal. Our result for Hg-1201 is, therefore, a positive indicator for density-functional methods to contribute, in the future, a deeper theoretical study to understand the nature of this phenomenon.

We have studied both the parent and doped compounds of HgBa₂Ca_n−1Cu_nO_2n+2+x with first-principles calculations, utilizing the recently devised SCAN meta-GGA density functional. Our results suggest an improvement in the structural characterization of these compounds exemplified by the success in describing doping-induced lattice parameter contraction. SCAN’s description of the electronic structure of the parent compound is distinct from its preceding density-functional studies as SCAN predicts antiferromagnetic insulating ground state even in the multilayered compounds. The diminishing bandgap is described by SCAN as a gradual and collective process contributed by Cu, Hg, and O components as opposed to the immediate process dominantly acted by the Hg states prescribed in previous studies.²⁷ We find this new physical description refreshing and hope it will encourage further advancement in the experimental techniques to synthesize the elusive parent compounds. Meanwhile, we note that doping these compounds with oxygen results in weaker magnetic moments, signaling an interplay between hole carrier concentrations in the CuO₂ planes with antiferromagnetic order. SCAN also correctly captures the magnetic inequivalence between copper planes for n ≥ 3 observed in experiments. The DOS of doped compounds at E_F can be significantly enhanced with some optimum excess oxygen concentration x_o, which manifests in the case of Hg-1201 as a peak in the normal-state, zero-temperature Sommerfeld parameter of the electronic specific heat. As the nature of this feature is currently of active interest, it is likely that modern first-principles density functional calculations can play an important role in unraveling the mysteries of quantum criticality.

See the supplementary material for the complete set of relaxed lattice parameters as well as additional density of states and band structure plots.

The calculations were performed with the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo, and the computational resource of Fujitsu PRIMERGY CX400M1/CX2550M5 (Oakbridge-CX) at the Information Technology Center, The University of Tokyo.

The authors have no conflicts to disclose.

Alpin N. Tatan: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review and editing (equal). Jun Haruyama: Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Osamu Sugino: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Resources (lead); Supervision (lead); Writing – original draft (equal); Writing – review and editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.