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.
After the seminal work of finding high temperature superconductivity in a ceramic copper oxide material1, efforts to understand the cuprate have long been a central part of modern condensed matter physics. Although many intriguing aspects of its electronic behaviors have been unveiled and great progress has been made, an understanding of its superconducting mechanism, the novel interplay between competing phases, and its relationship to other correlated phenomena is still far from clear2,3,4,5,6,7,8,9,10,11,12. One simple, well-defined but still open question is what determines the superconducting transition temperature (Tc), or its material dependency. For example, the Tc of the single layer cuprates can be different by a factor of two; ~40 K for La2CuO4 and ~90 K for HgBa2CuO4. On the one hand, it may be too early to ask this question, while the superconducting mechanism itself still remains elusive. On the other, however, figuring out the detailed features behind the material dependency can provide the crucial hint for further understanding of superconductivity and other related properties. In fact, many directly or indirectly related theoretical studies have been performed on this issue13,14,15,16,17,18,19,20,21,22,23,24,25
In this regard, a notable suggestion has recently been made26,27,28,29. According to this theory, the Tc of the single-layer cuprate can be described with a two-orbital model that considers both the and Wannier orbitals, and the energy level offset between the two orbitals, ΔE, plays a key role in determining Tc. Whereas the larger value of this energy separation produces the higher Tc (e.g., the case of HgBa2CuO4) due to the better one band feature achieved, the smaller value results in the lower Tc (e.g., the case of La2CuO4) in spite of the better nested Fermi surface. The calculated Eliashberg parameter (λ) based on the many-body calculation using the fluctuating exchange (FLEX) approximation30,31 clearly exhibits a linear dependence on ΔE while the other parameters are shown to be less important. Further, this theory can be extended to the bilayer case29, which explains the correlation between the Fermi surface shape and Tc24.
A possible experimental test to verify this two-band theory is to examine the correlation between Tc and ΔE, the Fermi surface shape, or the partial density of states of the 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 ΔE or the Fermi surface shape was obtained from the LDA and used as the “inputs” for the many-body FLEX calculation in Refs 26, 27, 28, 29, the experimentally determined ΔE 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 Tc is higher for larger ΔE32, the actual experimental value of ΔE 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 ΔE 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 ΔE. It is problematic since a quantitative comparison is required in between the theory and experiment, while only the qualitative comparison was made regarding Tc in Refs 26, 27, 28, 29.
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 ΔE without introducing adjustable parameters.
In Ref. 17, a quantum chemical approach was adopted to evaluate the energy level offset between Cu- and 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 Refs 26, 27, 28, 29. In the sense mentioned above, the calculated ΔE 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 ΔE for La2CuO4 is too small to account for the maximum Tc of 40 K in the La2CuO433. 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 conceptually38. Here we first examine the electronic structure and the two-band theory for the material-dependent Tc 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 Tc 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 La2CuO4 whose noticeable Tc 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(a,b) show the band dispersion and projected density of states (PDOS) of La2CuO4 calculated by LDA and QSGW, respectively. The LDA result is in good agreement with the previous calculation (see, for example, Ref. 26 and 27). In the QSGW, several important differences are noted. First, the band width of both eg orbitals are significantly reduced, by about 1.30 and 0.65 eV for and , respectively, indicating that the band width overestimation (or effective mass underestimation) problem of LDA is somehow overcome by the QSGW procedure. Another key difference is that the separation between the and bands becomes larger in QSGW, as seen in Fig. 1(b). The energy level, , is shifted from −0.14 (LDA) to −0.03 (QSGW) while is from −0.98 (LDA) to −1.68 (QSGW) (indicated by the arrows in Fig. 1; see also Table 1). It is a factor of two difference in ; 0.84 eV in LDA and 1.66 in QSGW. The correct estimation of this quantity is important especially in the two-band theory for Tc26,27,28,29. The large value of is indeed consistent with the RIXS data as will be discussed further below.
The same features are also found in HgBa2CuO4, as presented in Fig. 1(c,d). The band width is reduced by ~0.75 eV in QSGW compared to LDA and its center position of PDOS moves slightly upward by 0.15 eV. While the dispersion in this material is already quite small due to the thicker blocking layer, its band width in QSGW is further reduced. is 2.25 in QSGW, again noticeably larger than the LDA value of 1.67 eV (see Table 1).
In QSGW, the of both La2CuO4 and HgBa2CuO4 is enhanced compared to LDA/GGA. How does this affect the theoretical estimation of Tc? 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’ problem39, the “best” that should be adopted in the FLEX evaluation of Tc may be lying somewhere in between the QSGW and LDA/GGA values. This can provide better quantitative agreement with the experiment, especially in La2CuO4, for which the LDA/GGA value of is found to be too small to account for Tc = 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 La2CuO4 (HgBa2CuO4). The same feature is found for the apical O-pz PDOS. As a result, the energy difference, ΔEp = Eapical − Einplane, is changed from 1.68 (1.00) in LDA to 0.55 (−0.86) in QSGW for the case of La2CuO4 (HgBa2CuO4), see Table 1. The correct estimation of ΔEp is also important for understanding Tc since it is an underlying quantity to determine (≈ ΔE ≈ ΔEd + ΔEp) in combination with other parameters. Note that our is different from ΔEd and ΔE in Ref. 26, 27, 28, 29 where ΔE ≈ ΔEd + ΔEp, , , and all of the parameters are calculated from the maximally localized Wannier orbital analysis. The ΔE contains the contribution from oxygen hybridization. We note, however, that our becomes effectively quite similar with ΔE in Ref. 26, 27, 28, 29 since we set our Emin,max to cover only the anti-bonding band complex. Actually one can make it almost equal, i.e., , by fine-tuning the Emin,max range.
Some other changes produced by QSGW are also noted. The 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 t2g complex is lowered (red color), the 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 systems40.
The shape of the Fermi surface is important for understanding cuprate superconductivity. For example, its nesting is crucial for the spin fluctuation pairing. Also, a notable correlation between the experimentally observed Tc at the optimal doping () 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 eg orbitals is reduced by 0.15e per unit cell. Notable features are found in the QSGW Fermi surface for La2CuO4. Contrary to the LDA result of Fig. 3(a), Fig. 3(b) has the pocket centered at (π, π) point as in HgBa2CuO4 Fermi surface (see Fig. 3(c,d)). This feature is in good agreement with ARPES data41 which also reports the pocket centered at (π, π) point. Further, the -orbital character (dark purple) is significantly reduced and the character (bright yellow) is dominant in the QSGW result, which is distinctive from the LDA in which the significant amount of components are observed near (π, 0) and (0, π).
In the case of HgBa2CuO4, 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 orbital character is dominant and band is well separated from Fermi level already in LDA due to the thicker blocking layers enhancing two-dimensional feature, the LDA result is quite similar to QSGW.
Comparison with RIXS
We now turn to the comparison with the RIXS data. Recently, Sala et al.32 successfully extracted the important model parameters for several different cuprate materials based on RIXS spectra. In this subsection, we examine the material dependent parameters by QSGW and compare them to the experimental values. With Ref. 32 as our main reference, we include two more compounds, namely, Sr2CuO2Cl2 and CaCuO2. Although Sr2CuO2Cl2 is also one of the single-layer cuprates, it has Cl—Cu—Cl bonding (instead of O—Cu—O) along c-axis and therefore the naive comparison of parameters such as and ΔEp in the line of Tc discussion may be misleading. This material is excluded in Ref. 26, 27, 28, 29 because of the same reason.
Our results are summarized in Table 1, Figs 4 and 5. The values of 10Dq, defined as the difference between two energy levels of and dxy (see Fig. 4(a)), are larger in the QSGW calculation by ~62–125% than the LDA values. While LDA underestimates 10Dq compared to the experiment, QSGW slightly overestimates, which is related to the tendency that Cu-t2g bands are pushed down relative to eg, as was also observed in the previous QSGW calculations for other transition-metal oxides40,42,43. 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 Tc is (or 4Ds + 5Dt in Ref. 32). Figures 4(b) and 5 clearly show that the calculated values of 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 Tc materials, CaCuO2 and HgBa2CuO4 (see Fig. 5). This can be taken as a strong support for the two-band theory in the sense that the LDA value of as an input for FLEX provides qualitative information of material dependence, while the 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 dxy and dyz,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 Tc trend of the cuprates is its significant enhancement in the thin film form. Locquet et al. reported44 that Tc can be controlled by epitaxial strain by about factor of two45. The underdoped La2CuO4 with its bulk Tc of 25 K exhibits a higher and lower Tc of ~49 K and 10 K when it is grown on SrLaAlO4 (SLAO) and SrTiO3 (STO) substrates, respectively44. 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, aSTO = 3.905 and aSLAO = 3.755 Å, for La2CuO4, which originally has a0 = 3.782 Å and c0 = 13.25 Å. As expected, the optimized out-of-plane parameters get smaller and larger under the tensile and compressive strain, respectively; and . As a result, the ratio between the out-of-plane and in-plane Cu—O distance, r = dapical/dinplane, is found to be 1.32, 1.28, and 1.24, for aSLAO, a0 and aSTO, respectively.
The calculated values of are plotted in Fig. 6. Both LDA and QSGW predict that gets enhanced and reduced under compressive and tensile strain, respectively, which is consistent with the experimental observation44. The reduction of at a = aSTO is about 0.16 eV in both LDA and QSGW, and the enhancement at a = aSLAO 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 La2CuO4, and they are in good agreement with the ARPES data41. Important model parameters including the key quantity for the two-band theory of Tc, , 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 Tc in La2CuO4 produced by the LDA input33.
Quasiparticle self-consistent GW
The QSGW42,43,46 calculates H0 (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 H0 (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 Vxc is generated aswhere εi and |ψi〉 refer to the eigenvalues and eigenfunctions of H0, respectively, and Re[Σ(ε)] is the Hermitian part of the self-energy42,43,46. With this Vxc, one can define a new static one-body Hamiltonian H0, 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 semiconductors42,43 to the various 3d transition metal oxides42,43,47 and 4f-electron systems48, have demonstrated its capability in the description of weakly and strongly correlated electron materials.
We used our new implementation of QSGW49 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 H2 through Kr2 with the significantly low APW energy cutoff of ~4 Ry, by including localized MTOs51. A key feature of this scheme for QSGW is that the expansion of Vxc can be made with MTOs, not APWs, which enables us to make the real space representation of Vxc at any k point.
We performed the calculations with the experimental crystal structures52,53,54,55, and used 10 × 10 × 10, 12 × 12 × 12, 12 × 12 × 8, and 14 × 14 × 14 k points for LDA calculations of Sr2CuO2Cl2, La2CuO4, HgBa2CuO4, and CaCuO2, respectively. As for QSGW calculations, in order to reduce the computation cost, the number of k points were reduced to be 5 × 5 × 5, 7 × 7 × 7, 8 × 8 × 4, and 8 × 8 × 8 for the first Brillouin zone of Sr2CuO2Cl2, La2CuO4, HgBa2CuO4, and CaCuO2, respectively. The MTO radii used in our calculations were as follows: (i) 1.58, 1.04, 0.89, and 1.38 Å for Sr, Cu, O, and Cl in Sr2CuO2Cl2, (ii) 1.43, 0.97, and 0.86 Å for La, Cu, and O in La2CuO4, (iii) 1.10, 1.59, 1.05, and 0.83 Å for Hg, Ba, Cu, and O in HgBa2CuO4, and (iv) 1.54, 1.01, and 0.86 for Ca, Cu, and O in CaCuO2.
Many of the key parameters in this study are defined in terms of the energy levels of each orbital, such as and . To quantify them we simply take the center of mass position of PDOS:where gα(E) is PDOS for a given orbital α. An ambiguity is inevitably introduced in determining Emin,max, and we set the range to cover the whole antibonding band complex for Cu-eg states. (, ) for La2CuO4 is (−1.95 eV, 2.05 eV) in LDA and (−2.20, 1.55) in QSGW. For HgBa2CuO4, the band dispersion changes and the values of and are redefined accordingly: (, ) = (−2.40, 2.50) in LDA and (−2.55, 1.65) in QSGW. If we choose two different ranges for two eg 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 method26,27. Even if the ranges are set to cover the whole window of Cu-eg bands including bonding parts, the trend reported in this work does not change. The same is true for O-2p and Cu-t2g 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 function26,27.
How to cite this article: Jang, S. W. et al. Quasiparticle self-consistent GW study of cuprates: electronic structure, model parameters, and the two-band theory for Tc. Sci. Rep. 5, 12050; doi: 10.1038/srep12050 (2015).
We thank Ryotaro Arita for helpful comment and Prof. Hiroshi Katayama-Yoshida for hosting the helpful discussion. S.W.J. and M.J.H. were supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2014R1A1A2057202). The computing resource is supported by National Institute of Supercomputing and Networking/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2014-C3-050) and by Computing System for Research in Kyushu University. T. K. was supported by the Advanced Low Carbon Technology Research and Development Program (ALCA), the “High-efficiency Energy Conversion by Spinodal Nano-decomposition” program of the Japan Science and Technology Agency (JST), and by the JSPS Core-to-Core Program (A) Advanced Research Networks (Computational Nano-materials Design on Green Energy).
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