Quasiparticle self-consistent GW study of cuprates: electronic structure, model parameters, and the two-band theory for Tc

Despite decades of progress, an understanding of unconventional superconductivity still remains elusive. An important open question is about the material dependence of the superconducting properties. Using the quasiparticle self-consistent GW method, we re-examine the electronic structure of copper oxide high-Tc materials. We show that QSGW captures several important features, distinctive from the conventional LDA results. The energy level splitting between and is significantly enlarged and the van Hove singularity point is lowered. The calculated results compare better than LDA with recent experimental results from resonant inelastic xray scattering and angle resolved photoemission experiments. This agreement with the experiments supports the previously suggested two-band theory for the material dependence of the superconducting transition temperature, Tc.

clearly exhibits a linear dependence on Δ E while the other parameters are shown to be less important. Further, this theory can be extended to the bilayer case 29 , which explains the correlation between the Fermi surface shape and T c 24 .
A possible experimental test to verify this two-band theory is to examine the correlation between T c and Δ E, the Fermi surface shape, or the partial density of states of the − d z r 3 2 2 orbital, which can be measured by recent techniques such as resonant inelastic xray scattering (RIXS), and (angle resolved) photoemission spectroscopy (ARPES). However, while the theoretical Δ E or the Fermi surface shape was obtained from the LDA and used as the "inputs" for the many-body FLEX calculation in , the experimentally determined Δ E and the Fermi surface shape should be regarded as the "outputs" or "results" after the consideration of the many-body correlation effects beyond LDA/GGA. In fact, while a RIXS study reports that T c is higher for larger Δ E 32 , the actual experimental value of Δ E is larger than the theoretical evaluation, presumably due to this "input vs. output" problem. One possible way to resolve this problem, at least partially, is to re-evaluate Δ E as an output of the FLEX calculation. However, this approach would suffer from various ambiguities regarding the Hubbard interaction strength and the definition of the renormalized Δ E. It is problematic since a quantitative comparison is required in between the theory and experiment, while only the qualitative comparison was made regarding T c in  In the present paper, we use a first-principles approach, exploiting the quasiparticle self-consistent GW (QSGW) method. It enables us to take into account the correlation effects beyond LDA/GGA. In this way, we can obtain a well-defined renormalized Δ E without introducing adjustable parameters.
In Ref. 17, a quantum chemical approach was adopted to evaluate the energy level offset between Cu-− d x y 2 2 and − d z r 3 2 2 orbitals, where the correlation effects were taken into account within a cluster-based configuration-interaction-type calculation. A good agreement with the RIXS experiment was found by assuming the energy difference of ferromagnetically and antiferromagnetically ordered states to be 2J, where J is the antiferromagnetic coupling constant. Our approach is fairly different and is along the line of the first-principles band calculation as in  In the sense mentioned above, the calculated Δ E can be compared to the experiments, while it should not be regarded as an input parameter for the many-body calculation, because doing so would result in a partial double counting of the correlation effects. Still, the present approach can also provide a first-step hint toward obtaining a better "non-interacting" Hamiltonian that can be used as an input for the many-body calculation of superconductivity. In fact, it is known that the non-interacting Hamiltonian obtained from LDA has a problem when used as an input for the FLEX calculation, and the LDA/GGA estimation of Δ E for La 2 CuO 4 is too small to account for the maximum T c of 40 K in the La 2 CuO 4 33 . In this context, it is worth pointing out that the GW method has been successfully applied to the many of strongly correlated materials in combination with, for example, dynamical mean field theory (DMFT) [34][35][36][37] .

Results and Discussion
To our knowledge, there is no previous QSGW study for the cuprate band structure although it has been discussed conceptually 38 . Here we first examine the electronic structure and the two-band theory for the material-dependent T c of a single layer cuprate. While the QSGW calculation produces notable differences in the band structure and Fermi surface from LDA, the two-band explanation for T c still remains valid. QSGW results of model parameters are presented and compared to the RIXS data as well as the LDA calculations. It clearly shows that the parameters produced by QSGW are in better agreement with the experiment. Finally, we investigate the epitaxially strained La 2 CuO 4 whose noticeable T c increase has been previously reported. Two-band theory also works well for this situation.
Electronic structure and the T c of single layer compounds. Figure 1 In QSGW, the ΔE e g of both La 2 CuO 4 and HgBa 2 CuO 4 is enhanced compared to LDA/GGA. How does this affect the theoretical estimation of T c ? First, it should be noted that these parameters cannot be directly adopted as the inputs for the FLEX calculation. This is because, in principle, the QSGW self energy should be partially subtracted before we put it into any of many-body calculations. While there is no well-defined prescription yet for this kind of 'double-counting' problem 39 , the "best" ΔE e g that should be adopted in the FLEX evaluation of T c may be lying somewhere in between the QSGW and LDA/GGA values. This can provide better quantitative agreement with the experiment, especially in La 2 CuO 4 , for which the LDA/GGA value of ΔE e g is found to be too small to account for T c = 40 K.
Oxygen states are also affected. Compared to LDA results, the O-2p levels obtained by QSGW are significantly lowered in energy, as indicated in Fig. 2. As summarized in Table 1, the center position of in-plane oxygen PDOS is located at − 4.44 (− 3.55) eV in LDA and at − 5.06 (− 4.05) eV in QSGW for La 2 CuO 4 (HgBa 2 CuO 4 ). The same feature is found for the apical O-p z PDOS. As a result, the energy difference, Δ E p = E apical − E inplane , is changed from 1.68 (1.00) in LDA to 0.55 (− 0.86) in QSGW for the case of La 2 CuO 4 (HgBa 2 CuO 4 ), see Table 1. The correct estimation of Δ E p is also important for understanding T c since it is an underlying quantity to determine ΔE e g (≈ Δ E ≈ Δ E d + Δ E p ) in combination with other parameters. Note that our ΔE e g is different from Δ E d and Δ E in Ref. 26 , and all of the parameters are calculated from the maximally localized Wannier orbital analysis. The Δ E contains the contribution from oxygen hybridization. We note, however, that our ΔE e g becomes effectively quite similar with Δ E in Ref. 26-29 since we set our E min,max to cover only the anti-bonding band complex. Actually one can make it almost equal, i.e., Δ ≈ Δ E E e g , by fine-tuning the E min,max range. Some other changes produced by QSGW are also noted. The − d z r 3 2 2 components in the bands below − 1.5 eV in Fig. 1(a) are reduced in QSGW, and the free-electron-like bands at Γ and Z points above the Fermi energy are shifted upward. As the position of the t 2g complex is lowered (red color), the − d x y 2 2 band has almost no mixture with other bands below the Fermi energy. Higher-lying La-4f bands (not shown) move further upward as has been previously noted in the nickelate systems 40 .
Fermi surface.  between the experimentally observed T c at the optimal doping (T c max ) and the Fermi surface warping has been identified by Pavarini et al. 24 . Here we discuss the Fermi surface calculated by QSGW in comparison to the LDA result and experiment.
The calculated Fermi surfaces are presented in Fig. 3; LDA ((a, c)) and QSGW ((b, d)). The hole doping is simulated by the rigid band shift method so that the electron occupation in e g orbitals is reduced by 0.15e per unit cell. Notable features are found in the QSGW Fermi surface for La 2 CuO 4 . Contrary to the LDA result of Fig. 3(a), Fig. 3(b) has the pocket centered at (π, π) point as in HgBa 2 CuO 4 Fermi surface (see Fig. 3(c,d)). This feature is in good agreement with ARPES data 41 which also reports the pocket centered at (π, π) point. Further, the In the case of HgBa 2 CuO 4 , the difference between LDA and QSGW is less pronounced, see Fig. 3(c,d). While the QSGW Fermi surface is slightly more rounded, the overall shape is not much different. Since the Our results are summarized in Table 1 slightly overestimates, which is related to the tendency that Cu-t 2g bands are pushed down relative to e g , as was also observed in the previous QSGW calculations for other transition-metal oxides 40,42,43 . It is important to note that overall the QSGW result is in better agreement with experiment, as clearly seen in Fig. 4(b). As noted in the above, according to the two-band theory by Sakakibara et al. 26,27 , the important parameter that governs T c is ΔE e g (or 4Ds + 5Dt in Ref. 32) . Figures 4(b) and 5 clearly show that the calculated values of ΔE e g by QSGW are in excellent agreement with those from RIXS spectra; the difference is 2-8%. The LDA values are noticeably smaller than the experiments although the difference gets reduced in the higher T c materials, CaCuO 2 and HgBa 2 CuO 4 (see Fig. 5). This can be taken as a strong support for the two-band theory in the sense that the LDA value of ΔE e g as an input for FLEX provides qualitative information of material dependence, while the ΔE e g by QSGW already contains the correlation effect beyond LDA, being consistent with RIXS.
Another parameter deduced from RIXS in Ref. 32 is 3Ds-5Dt, the energy level difference between d xy and d yz,zx . In this case, the LDA results are not much different from QSGW and experiment (see Fig. 4(b)).
The effect of epitaxial strain. An interesting aspect found in the T c trend of the cuprates is its significant enhancement in the thin film form. Locquet et al. reported 44 that T c can be controlled by epitaxial strain by about factor of two 45 . The underdoped La 2 CuO 4 with its bulk T c of 25 K exhibits a higher and lower T c of ~49 K and 10 K when it is grown on SrLaAlO 4 (SLAO) and SrTiO 3 (STO) substrates,   respectively 44 . It is therefore important to check whether the two-band theory is also consistent with this observation.
In order to simulate the tensile and compressive strain produced by STO and SLAO, we first optimized the c lattice parameter with two different in-plane lattice constants, a STO = 3.905 and a SLAO = 3.755 Å, for La 2 CuO 4 , which originally has a 0 = 3.782 Å and c 0 = 13.25 Å. As expected, the optimized out-of-plane parameters get smaller and larger under the tensile and compressive strain, respectively; = . . As a result, the ratio between the out-of-plane and in-plane Cu-O distance, r = d apical /d inplane , is found to be 1.32, 1.28, and 1.24, for a SLAO , a 0 and a STO , respectively.
The calculated values of ΔE e g are plotted in Fig. 6. Both LDA and QSGW predict that ΔE e g gets enhanced and reduced under compressive and tensile strain, respectively, which is consistent with the experimental observation 44 . The reduction of ΔE e g at a = a STO is about 0.16 eV in both LDA and QSGW, and the enhancement at a = a SLAO is 0.29 (LDA) and 0.47 eV (QSGW).

Summary and Conclusion
Using the QSGW method, we re-examined the electronic structure of copper oxide high temperature superconducting materials. Several important features were found to have been captured by the GW procedure, such as effective mass enhancement. The shape and orbital character of the Fermi surface were also notably changed, especially for the case of La 2 CuO 4 , and they are in good agreement with the ARPES data 41 . Important model parameters including the key quantity for the two-band theory of T c , ΔE e g , were examined, and the QSGW results were in excellent agreement with RIXS data.
The present study shows that the first-principles band calculation can quantitatively reproduce the experimental observation by taking into account the correlation effects beyond LDA. We emphasize that it is not inconsistent with the previous study by Sakakibara et al. which takes the LDA result as an input for the many-body calculation of superconductivity. While the QSGW result cannot be used as a direct input for the FLEX-type calculation because of the partial double-counting of the many-body correlation, the "best" non-interacting Hamiltonian, that can serve as an input, may lie somewhere in between the LDA and QSGW. Obtaining a well-defined non-interacting Hamiltonian is, therefore, an important future direction for the first-principles-based description of high-temperature superconductivity, and it may quantitatively resolve the problem of low T c in La 2 CuO 4 produced by the LDA input 33 .

Methods
Quasiparticle self-consistent GW. The QSGW 42,43,46 calculates H 0 (non-interacting Hamiltonian describing quasiparticles or band structures) and W (dynamically-screened Coulomb interactions between the quasiparticles within the random phase approximation) in a self-consistent manner. While the 'one-shot' GW is a perturbative calculation starting from a given H 0 (usually from LDA/GGA), QSGW is a self-consistent perturbation method that can determine the one-body Hamiltonian within itself. The GW approximation gives the one-particle effective Hamiltonian whose energy dependence comes from the self-energy term Σ (ω) (here we omit index of space and spin for simplicity), and in QSGW, the static one-particle potential V xc is generated as where ε i and |ψ i 〉 refer to the eigenvalues and eigenfunctions of H 0 , respectively, and Re[Σ (ε)] is the Hermitian part of the self-energy 42,43,46 . With this V xc , one can define a new static one-body Hamiltonian H 0 , and continue to apply GW approximation until converged. In principle, the final result of QSGW does not depend on the initial conditions. Previous QSGW studies, ranging from semiconductors 42,43 to the various 3d transition metal oxides 42,43,47 and 4f-electron systems 48 , have demonstrated its capability in the description of weakly and strongly correlated electron materials.
Computation details. We used our new implementation of QSGW 49 by adopting the 'augmented plane wave (APW) + muffin-tin orbital (MTO)' , designated by 'PMT' 50,51 , for the one-body solver. The accuracy of this full potential PMT method is proven to be satisfactory in the supercell calculations of homo-nuclear dimers from H 2 through Kr 2 with the significantly low APW energy cutoff of ~4 Ry, by including localized MTOs 51 . A key feature of this scheme for QSGW is that the expansion of V xc can be made with MTOs, not APWs, which enables us to make the real space representation of V xc at any k point.
We performed the calculations with the experimental crystal structures [52][53][54][55] , and used 10 × 10 × 10, 12 × 12 × 12, 12 × 12 × 8, and 14 × 14 × 14 k points for LDA calculations of Sr 2 CuO 2 Cl 2 , La 2 CuO 4 , HgBa 2 CuO 4 , and CaCuO 2 , respectively. As for QSGW calculations, in order to reduce the computation cost, the number of k points were reduced to be 5 Many of the key parameters in this study are defined in terms of the energy levels of each orbital, such as − E x y If we choose two different ranges for two e g orbitals to include only the main peak of each orbital PDOS, we can actually produce the better agreement with the numbers in the previous study by Sakakibara et al. where the levels are defined using maximally localized Wannier function method 26,27 . Even if the ranges are set to cover the whole window of Cu-e g bands including bonding parts, the trend reported in this work does not change. The same is true for O-2p and Cu-t 2g levels. In other words, none of the reasonably defined energy ranges change our conclusion, and the values are well compared with those reported in the previous study using a maximally localized Wannier function 26,27 .