Oxygen hole content, charge-transfer gap, covalency, and cuprate superconductivity

Significance

Modern theoretical methods solve a long-standing mystery of cuprate high-temperature superconductivity, identifying crucial quantities that optimize the transition temperature. Superconducting cuprates have very different transition temperatures, and even if the optimal value of the superconducting transition temperature is obtained for a given parent compound by varying doping, there is no correlation between optimal doping and transition temperature. Instead, it has been found experimentally that the optimal transition temperature is controlled by oxygen hole content or by the size of the charge-transfer gap. Our calculations show that these two quantities are correlated and that together with covalency they lead to an effective superexchange interaction between copper atoms that ultimately controls the optimal superconducting order parameter.

Abstract

Experiments have shown that the families of cuprate superconductors that have the largest transition temperature at optimal doping also have the largest oxygen hole content at that doping [D. Rybicki et al., Nat. Commun. 7, 1–6 (2016)]. They have also shown that a large charge-transfer gap [W. Ruan et al., Sci. Bull. (Beijing) 61, 1826–1832 (2016)], a quantity accessible in the normal state, is detrimental to superconductivity. We solve the three-band Hubbard model with cellular dynamical mean-field theory and show that both of these observations follow from the model. Cuprates play a special role among doped charge-transfer insulators of transition metal oxides because copper has the largest covalent bonding with oxygen. Experiments [L. Wang et al., arXiv [Preprint] (2020). https://arxiv.org/abs/2011.05029 (Accessed 10 November 2020)] also suggest that superexchange is at the origin of superconductivity in cuprates. Our results reveal the consistency of these experiments with the above two experimental findings. Indeed, we show that covalency and a charge-transfer gap lead to an effective short-range superexchange interaction between copper spins that ultimately explains pairing and superconductivity in the three-band Hubbard model of cuprates.

Although several classes of high-temperature superconductors have been discovered, including pnictides, sulfur hydrides, and rare earth hydrides, cuprate high-temperature superconductors are still particularly interesting from a fundamental point of view because of the strong quantum effects expected from their doped charge-transfer insulator nature and single-band spin-one-half Fermi surface (1, 2).

Among the most enduring mysteries of cuprate superconductivity is the experimental discovery, early on, that the hole content on oxygen plays a crucial role (2–5). Oxygen hole content ($2n_p$) is particularly relevant since NMR (5, 6) suggests a correlation between optimal $T_c$ and $2n_p$ on the ${\mathrm{C}\mathrm{u}\mathrm{O}}_2$ planes: A higher oxygen hole content at the optimal doping of a given family of cuprates leads to a higher critical temperature. This is summarized in figure 2 of ref. 6. The charge-transfer gap also seems to play a central role for the value of $T_c$, as suggested by scanning tunneling spectroscopy (7) and by theory (8). Many studies have shown that doped holes primarily occupy oxygen orbitals (3, 9–11). This long unexplained role of oxygen hole content and charge-transfer gap on the strength of superconductivity in cuprates is addressed in this paper.

The vast theoretical literature on the one-band Hubbard model in the strong-correlation limit shows that many of the qualitative experimental features of cuprate superconductors (12, 13) can be understood (14), but obviously not the above experimental facts regarding oxygen hole content. Furthermore, variational calculations (15) and various Monte Carlo approaches (16, 17) suggest that $d$-wave superconductivity in the one-band Hubbard model may not be the ground state, at least in certain parameter ranges (18, 19).

It is thus important to investigate more realistic models, such as the three-band Emery-VSA (Varma–Schmitt-Rink–Abrahams) model that accounts for copper–oxygen hybridization of the single band that crosses the Fermi surface (20, 21). A variety of theoretical methods (8, 22–27) revealed many similarities with the one-band Hubbard model, but also differences related to the role of oxygen (28, 29).

Investigating the causes for the variation of the transition temperature $T_c$ for various cuprates is a key scientific goal of the quantum materials roadmap (30).* We find and explain the above correlations found in NMR and in scanning tunnelling spectroscopy; highlight the importance of the difference between electron affinity of oxygen and ionization energy of copper (21, 31); and, finally, document how oxygen hole content, charge-transfer gap, and covalency conspire to create an effective superexchange interaction between copper spins that is ultimately responsible for superconductivity.

We do not address questions related to intraunit-cell order (32, 33).

Model

In second-quantized notation, the three-band Emery-VSA Hubbard model (2, 20, 21) on the square lattice is

$$H=\sum _{\mathbf{k}\sigma }{\mathrm{\Psi }}_{\mathbf{k}\sigma }^{†}{\mathbf{h}}_0\left(\mathbf{k}\right){\mathrm{\Psi }}_{\mathbf{k}\sigma }+U\sum _i{n}_{i\uparrow }^d{n}_{i\downarrow }^d,$$ [1]

where the multiplet ${\mathrm{\Psi }}_{\mathbf{k}\sigma }^{†}=$ ($d_{\mathbf{k},\sigma }^{†}$, $p_{\mathbf{k},\sigma }^{x†}$, $p_{\mathbf{k},\sigma }^{y†}$) contains the creation operators for electrons on the copper $d_{x^2-y^2}$ and the oxygen $p_x$ and $p_y$ orbitals ($\mathbf{k}$ is the wavevector and $\sigma$ the spin projection) and $n_{i\sigma }^d=d_{i\sigma }^{†}{d}_{i\sigma }$. Taking the distance between unit cells to be unity, the noninteracting Hamiltonian ${\mathbf{h}}_0\left(\mathbf{k}\right)$ is given in Eq. 2:

$$\begin{array}{ll}\hfill & {\mathbf{h}}_0\left(\mathbf{k}\right)=\hfill \\ \hfill & \left(\begin{array}{ccc}\hfill {ϵ}_d\hfill & \hfill t_{pd}\left(1-e^{-i{k}_x}\right)\hfill & \hfill t_{pd}\left(1-e^{-i{k}_y}\right)\hfill \\ \hfill t_{pd}\left(1-e^{i{k}_x}\right)\hfill & \hfill {ϵ}_p+2t_{pp}^{\prime }\cos k_x\hfill & \hfill t_{pp}\left(1-e^{i{k}_x}\right)\left(1-e^{-i{k}_y}\right)\hfill \\ \hfill t_{pd}\left(1-e^{i{k}_y}\right)\hspace{0.17em}\hfill & \hfill \hspace{0.17em}{t}_{pp}\left(1-e^{-i{k}_x}\right)\left(1-e^{i{k}_y}\right)\hfill & \hfill {ϵ}_p+2t_{pp}^{\prime }\cos k_y\hfill \end{array}\right).\hfill \end{array}$$ [2]

The on-site energies on the Cu and O orbitals are noted ${ϵ}_d$ and ${ϵ}_p$, respectively. We chose not to include the $-2t_{pp}$ contribution to the on-site oxygen energy, unlike ref. 2. The first-neighbor Cu-O hopping is $t_{pd}$, the first-neighbor (diagonal) O-O hopping is $t_{pp}$, and O-O hopping through the Cu site is $t_{pp}^{\prime }$. Finally, we consider only interactions on the copper atom ($U$). This is justified by density functional theory plus Hubbard interaction calculations showing the on-site oxygen and intersite interactions to be much smaller than the on-site copper interaction (34, 35). The Hamiltonian parameters and the unit cell are visualized in Fig. 1A.

Fig. 1.

(A) Schematic view of the three-band Hubbard model. The $d$-shaped orbitals sit on copper atoms whereas the two types of $p$-shaped orbitals ($p_x$ and $p_y$) sit on the surrounding oxygen atoms. (Inset) The $2\times 2$ cluster used for the CDMFT calculation. (B) Total and partial densities of states of the interacting model for parameter set Eq. 3 (ionic case) for 12% hole doping at $T=0$. (C) The same, for parameter set Eq. 4 (covalent case) for 13% hole doping at $T=0$. LHB stands for lower Hubbard band, UHB for upper Hubbard band, CTG for charge-transfer gap, CTB for charge-transfer band, and ZRSB for Zhang–Rice singlet band. Note that the isolated peak in the lower Hubbard band appears clearly only for the ionic case Eq. 3.

The three-band Hubbard model describes a charge-transfer insulator at large values of $U$ and at a filling of five electrons per unit cell (the undoped state). Such an insulating state is realized as $U$ increases and splits the Cu band into lower and upper Hubbard bands, such that the upper band is pushed beyond the oxygen-dominant band (around ${ϵ}_p$), leading to an insulating gap between the two. In the strong-coupling limit, excitations from the undoped state require a transfer of charge from the O orbital to the Cu orbital; hence the insulating gap is referred to as a charge-transfer gap (CTG) and the insulating state as a charge-transfer insulator (36). The central band is further called the charge-transfer band (CTB). This is different from a Mott insulator (e.g., in the one-band Hubbard model) where the insulating (or Mott) gap appears between the lower and upper Hubbard bands. On doping the charge-transfer insulator, the holes primarily go into the oxygen orbitals and another band appears at the Fermi level referred to as the Zhang–Rice singlet band (ZRSB) (27, 37, 38). These features of the three-band Hubbard model can be seen in Fig. 1 B and C, where we have shown the density of states from our cellular dynamical mean-field theory (CDMFT) computations. The mixed Cu-O character of the ZRSB, which crosses the Fermi level, indicates the signature of the Zhang–Rice singlet physics. Zhang–Rice singlets are characterized by singlet states formed between the Cu orbital and the adjacent O orbitals that participate in the low-energy physics of cuprates and form the basis for the one-band Hubbard model (39, 40).

Fig. 1 B and C shows the density of states for the following two sets of parameters (in units of $t_{pp}\sim 0.65\hspace{0.17em}\mathrm{e}\mathrm{V}$), using as a zero of energy the copper site energy ${ϵ}_d=0$:

$${ϵ}_p=7t_{pd}=1.5t_{pp}^{\prime }=1$$ [3]

$${ϵ}_p=2.3t_{pd}=2.1t_{pp}^{\prime }=0.2.$$ [4]

In these units, ${ϵ}_p$ measures the difference between the oxygen and copper site energies. We refer to Eq. 3 as the ionic case, since ${ϵ}_p$ is large, and we refer to Eq. 4 as the covalent case, since ${ϵ}_p$ is much smaller.

In Eq. 4 the values of the parameters have been obtained following the estimates for Bi2212 in Weber et al. (8); ${ϵ}_p$ has been further obtained by subtracting the double-counting contribution (8). Most cuprates have the same value for $t_{pp}\sim 0.65\hspace{0.17em}\mathrm{e}\mathrm{V}$. The range of microscopic parameters that we explore around Eq. 4, what we call the covalent case, covers most of the values typical of cuprates (8).

The less realistic parameters for the ionic case in Eq. 3 are obtained from ref. 24. As seen from the density of states (Fig. 1 B and C), the main difference between the two parameter sets is the absence of a clear lower Hubbard band (LHB) for the covalent case Eq. 4 compared to the ionic case Eq. 3. This is due to a much smaller value of ${ϵ}_p$ in the former, leading to better covalency and to a mixing of the lower Hubbard band and the charge-transfer band. This is a more realistic description of cuprates, whereas the ionic case Eq. 3 describes a scenario with well-separated lower Hubbard band and charge-transfer band. Also, the covalent case Eq. 4 has a well-formed Zhang–Rice singlet band compared to the ionic case Eq. 3. We performed CDMFT computations for various parameter sets around the ionic case Eq. 3 (at zero temperature using an exact diagonalization impurity solver and at finite temperatures using a quantum Monte Carlo solver) and the covalent case Eq. 4 (only at zero temperature). Finite temperature calculations with the quantum Monte Carlo solver were not possible for the covalent case because of the sign problem. For the ionic case, the average sign was small as seen in SI Appendix, Fig. S1. For more details on the impurity solvers see Materials and Methods.

Results

We define the order parameter $\psi =⟨\widehat{\mathrm{\Delta }}⟩/N$, where $N$ is the total number of unit cells in the lattice and $\widehat{\mathrm{\Delta }}$ is the pairing operator

$$2\widehat{\mathrm{\Delta }}=\sum _{\langle i{j\rangle }_x}\left(d_{i,\uparrow }{d}_{j,\downarrow }-d_{i,\downarrow }{d}_{j,\uparrow }\right)-\sum _{\langle i{j\rangle }_y}\left(d_{i,\uparrow }{d}_{j,\downarrow }-d_{i,\downarrow }{d}_{j,\uparrow }\right)+\mathrm{H}.\mathrm{c}.,$$ [5]

where ${⟨ij⟩}_x$ and ${⟨ij⟩}_y$ indicate nearest-neighbor copper atoms in the $\widehat{x}$ and $ŷ$ directions, respectively.

As we aim to relate a change in oxygen occupation to a change in optimal $T_c$, we need to obtain multiple superconducting domes, corresponding to different parameter sets, and compare them. This is shown in Fig. 2A as a function of oxygen and copper hole contents for parameters in the vicinity of the ionic case (41). The presentation follows that of the experimental paper of ref. 6. There is a correlation between the optimal $T_c$ (vertical lines) and $2n_p$. Increasing $2n_p$ leads in general to a higher $T_c$. However, this correlation is not absolute: For instance, the green and red domes at around $n_d\approx 0.76$ overlap without having the same maximum $T_c$.

Fig. 2.

Superconducting domes as a function of atomic hole contents. (A) dynamical mean-field critical temperature as a function of oxygen $2n_p$ and copper $n_d$ hole contents for various parameter sets obtained by varying one parameter at a time, starting from the ionic case Eq. 3 at $U=12$. The solid black line $n_d+2n_p=1$ corresponds to the parent compound. (B and C) Superconducting order parameter at $T=0$ as a function of oxygen hole content ($2n_p$), for various parameter sets that deviate slightly from, respectively, the ionic Eq. 3 and the covalent Eq. 4 cases with parameters indicated in the keys.

As a proxy for the superconducting strength (SI Appendix, Fig. S2), we also show the order parameter at zero temperature as a function of $2n_p$ in Fig. 2 B and C for variations of the parameters in the vicinity, respectively, of the ionic case Eq. 3 and of the covalent case Eq. 4. (SI Appendix, Fig. S3 shows a different proxy). For the covalent case, we studied a range of parameters that correspond to those seen in cuprates. The maximum superconducting order parameter decreases with $U$, as in the one-band case (42–45). We also see a positive correlation in the height of the superconducting domes (the maximum order parameter) with optimal $2n_p$, except beyond a certain $2n_p$ where the maximum order parameter decreases. As we now discuss, this corresponds to the closing of the charge-transfer gap, a case not encountered in the cuprates. When the CTG is closed, superconductivity mediated by long wave-length spin fluctuations may still occur (46–48), but long wave-length fluctuations are not taken into account here. As in the one-band case (43, 44), superconductivity is maximized (48) close to the metal–insulator transition, where the potential and kinetic energy terms in the Hamiltonian have comparable contributions.

To see this trend more clearly and to compare all the data presented thus far, we plot in Fig. 3A the maximum order parameter that can be obtained for a given set of microscopic parameters, and we group the results by color according to the model parameter that is varied. The domes are shown in faint colors for the covalent case. We do not show the domes for the ionic cases for clarity. The black arrows mark the reference parameters for the ionic Eq. 3 and covalent Eq. 4 cases, respectively, for $U=12$. The dark blue and green curves (solid squares) and the curves below them are for parameters in the vicinity of the ionic case and the other curves above are for parameters in the vicinity of the covalent case. The four lowest curves were obtained at finite temperature and are therefore lower than the ones obtained at $T=0$. If we set aside the purple curve that corresponds to the closing of the charge-transfer gap, the correlation between the maximum order parameter and $2n_p$ still applies locally across all parameter sets. The slopes of all curves are similar. It is quite striking that the covalent case, corresponding more closely to parameters actually encountered in cuprates, has maximal values of the order parameter at given oxygen hole content that are larger than those for the ionic case. This is explained below, in our discussion of pairing.

Fig. 3.

(A) Maximum order parameter for each of the superconducting domes in Fig. 2 as a function of the corresponding oxygen hole content ($2n_p$). The open symbols mark the top of the full superconducting domes, shown with faint lines, for parameters near the covalent case Eq. 4. The solid symbols correspond to changes in microscopic parameters from the ionic case Eq. 3. The four curves with the smallest order parameters and solid circles were obtained at $\beta =60$. For clarity, we show only the maximum order parameter and not the full domes for the ionic cases. We indicate the points corresponding to the reference ionic and covalent parameters with solid arrows. The values of the microscopic parameters that differ from the two basic models appear in Fig. 2. (Inset) Maximum order parameter as a function of total doping. There is clearly no correlation between maximum order parameter and total doping. (B) Oxygen hole content ($2n_p$) versus normalized CTG at optimal doping. We have normalized the CTG with the total bandwidth to compare across different parameter sets. (C) Maximum order parameter as a function of the normalized CTG at optimal doping. For each color, only one parameter is changed. The arrows on the colored segments of the plots show the effect of an increase of the respective parameters indicated in the key and in Fig. 2.

Fig. 3 A, Inset shows a very important result. There, we plot the same achievable maximum order parameter, but this time as a function of total doping, or hole content, as is usually done in plots of the phase diagram. We clearly note the absence of any correlation between the maximum order parameter and the total doping, in stark contrast with what we see in the main plot of Fig. 3A.

Finally, following experimental (7) and theoretical (8) work linking the CTG to the critical temperature, we inspect the relation of the CTG, accurately accessible for $T=0$ computations, to both oxygen occupation and order parameter. Fig. 3B shows the optimal oxygen hole content $2n_p$ as a function of the CTG, normalized to the bandwidth; the two quantities are clearly correlated. We can explain this trend very easily by turning to the density of states. Indeed, when increasing the CTG, we reduce the overlap of the oxygen spectrum with the upper Hubbard band. This reduces the oxygen weight in the upper Hubbard band and thus increases it in the occupied band. This means a smaller oxygen hole content $2n_p$.

Furthermore, we note from Fig. 3C that the order parameter decreases monotonously as the CTG increases, except for very low values of the CTG, as seen also in ref. 49; this is consistent with the experimental observation (7) that the maximum $T_c$ decreases as the CTG increases. Once the CTG has opened, increasing the CTG reduces both the order parameter and the oxygen hole content ($2n_p$), hence making them behave monotonously.

Fig. 3C clearly shows that the order parameter is larger for the model parameters near the covalent case Eq. 4 compared to the ionic case Eq. 3, demonstrating that a lower value of ${ϵ}_p-{ϵ}_d$ favors superconductivity, as also seen in ref. 8 and suggested early on in refs. 21 and 31.

Pairing.

Why are oxygen hole content and CTG related to the propensity to superconduct, as we just saw, and why is there a difference between the superconducting tendency of the ionic and the covalent cases when they both have the same CTG (equivalently, oxygen hole content)? In this section, we offer a physical interpretation that links with insights from the one-band Hubbard model where superconducting $T_c$ scales with the superexchange energy $J=4t^2/U$ (43, 44, 50, 51), suggesting that in the one-band model, pairing is driven by short-range antiferromagnetic spin fluctuations (52).^† This view is supported by recent experiments (55, 56) that demonstrate the importance of antiferromagnetic paramagnons for pairing in cuprates. In the three-band model, the suppression of the superconducting order parameter with the CTG (Fig. 3C) also suggests that an effective superexchange $J$ is at play, since here the CTG plays the role of the Mott gap.

This point of view is reinforced when, following ref. 52, we look at various components of the spin susceptibility, along with the cumulative value of the order parameter in Fig. 4A. The $\left(\pi ,\pi \right)$ component of the imaginary part of the cluster spin susceptibility ${\chi }^{″}$ is seen to dominate at low frequency, suggesting that spin fluctuations play an important role. The cumulative value of the order parameter $I_F\left(\omega \right)$ takes the form

$$I_F\left(\omega \right)={\int }_0^{\omega }\frac{d{\omega }^{\prime }}{\pi }\hspace{0.17em}\mathrm{I}\mathrm{m}\hspace{0.17em}{F}_{ij}^R\left({\omega }^{\prime }\right),$$ [6]

where $F^R\left(t\right)=-\iota \theta \left(t\right)⟨\left\{c_{i\uparrow }\left(t\right),c_{j\downarrow }\left(0\right)\right\}⟩$ is the retarded Gorkov function, with $i$ and $j$ nearest neighbors. $I_F\left(\omega \right)$ gives the contribution to the order parameter of frequencies up to $\omega$. Hence its infinite frequency limit ($\underset{\omega \to \infty }{lim}{I}_F\left(\omega \right)$) is the order parameter $⟨c_{i\uparrow }{c}_{j\downarrow }⟩$. As can be seen from Fig. 4A, $I_F\left(\omega \right)$ grows mostly at low frequencies and attains an almost constant value after decreasing to around $60\%$ of its maximum value. It grows up to half of its asymptotic value (horizontal dotted line in Fig. 4 A, Inset) as soon as $\omega \approx 0.1$ (black vertical dotted line), which is close to the peak in ${\chi }^{″}\left(\pi ,\pi \right)$ (blue vertical dotted line). We observe such a coincidence between the position of the peak in the $\left(\pi ,\pi \right)$ susceptibility and the frequency where $I_F\left(\omega \right)$ reaches to half of its asymptotic value for various other parameter sets near both ionic Eq. 3 and covalent Eq. 4 cases at $T=0$. This suggests that short-range antiferromagnetic spin fluctuations are behind pair formation for the three-band Hubbard model as well. The frequency dependence clearly illustrates the retarded nature of pairing.

Fig. 4.

Origin of Pairing. (A) Cumulative order parameter $I_F\left(\omega \right)$ (black line) as a function of $\omega$ and $\left(\pi ,\pi \right)$, $\left(\pi ,0\right)$, and $\left(\mathrm{0,0}\right)$ components of the imaginary part of the spin susceptibility ${\chi }^{″}$ on the cluster, for $U=12$ and the covalent case Eq. 4 at $T=0$. (Inset) The same, with a zoom-in on low values of $\omega$. The $\left(\pi ,\pi \right)$ component of ${\chi }^{″}$ is the strongest and its peak (blue dotted line) is correlated with the position of $\frac{1}{2}{I}_F\left(\omega \to \infty \right)$ (black dotted line). (B) The superexchange $J$ as a function of the CTG normalized to the total bandwidth. Superexchange is estimated, for different microscopic parameters, from the location in frequency of the pole with the highest residue in the imaginary part of the spin susceptibility ${\chi }^{″}\left(\omega ,\left(\pi ,\pi \right)\right)$ at half-filling in the normal state (no antiferromagnetism, no superconductivity). The decrease of $J$ with increasing CTG suggests that the CTG plays the role of $U$ in the usual expression $J=4t_{\mathrm{e}\mathrm{ff}}^2/U$. The fact that for a given CTG, $J$ is always smaller for the ionic case, suggests that the effective hopping $t_{\mathrm{e}\mathrm{ff}}$ is smaller in the ionic case, which is expected. Inset shows the optimal superconducting order parameter for various microscopic parameters as a function of $J$. This time, by contrast with Fig. 3 A and C where neither oxygen hole content nor CTG was sufficient to determine the optimal order parameter, here $J$ determines it for both covalent and ionic cases.

That an effective superexchange interaction $J$ between copper atoms is present in the three-band model can be surmised also from Fig. 4B. There we take the frequency at which the largest peak in ${\chi }^{″}\left(\pi ,\pi \right)$ occurs at half-filling in the normal state (no antiferromagnetism, no superconductivity) as a measure of $J$. The validity of this estimate was verified in ref. 57: Given the small size of the cluster, the long-wavelength spin fluctuations are not detected by ${\chi }^{″}\left(\pi ,\pi \right)$ so, instead, this quantity is sensitive to the energy it takes to flip neighboring spins. We see in Fig. 4B that $J$ decreases with increasing CTG normalized to the bandwidth for both ionic and covalent cases. In other words, the CTG in the three-band model plays the role of $U$ in the one-band model. The fact that for a given CTG, $J$ is always smaller for the ionic case, suggests that the effective hopping $t_{\mathrm{e}\mathrm{ff}}$ entering $J$ is smaller in the ionic case than in the covalent case, as expected.

Finally, Fig. 4 B, Inset makes another strong argument that this $J$ is at the origin of superconductivity. It shows the optimum order parameter as a function of $J$ for various microscopic parameter sets. For both the realistic parameter sets (8) for the covalent case Eq. 4 and those for the ionic case Eq. 3, the optimal order parameter is proportional to $J$ estimated at half-filling in the normal state. This time, ionic and covalent cases that have the same $J$ have essentially the same optimal order parameter, contrary to Fig. 3 A and C. This is a strong argument in favor of short-range spin fluctuations as the source of superconductivity in the three-band Hubbard model, as surmised in recent experiments (55, 56, 58).

Experiments (55, 56) have access to estimates of $J$ from paramagnons at the actual doping. They use the value of ${\chi }^{″}\left(\pi ,0\right)$ to access $J$. SI Appendix, Fig. S4 using the estimate of $J$ obtained from ${\chi }^{″}\left(\pi ,0\right)$ in the normal state gives essentially the same results as those of Fig. 4B.

Discussion

Given that our unit of energy $t_{pp}$ does not vary much among cuprates, the range of parameters that we have explored for the covalent case, while relatively small, is sufficient to cover the range of parameters that corresponds to existing families of cuprate high-temperature superconductors (8). The ionic case corresponds to a large change in model parameters from the covalent case Eq. 4 and it is not a realistic model for cuprate superconductivity. Different parameter sets correspond to different compounds or to the same compound in different physical situations. For example, ${ϵ}_p$ (${ϵ}_d=0$) and hence the CTG are strongly influenced by the presence and location of the apical oxygen (59) whose crucial role in determining the CTG and $T_c$ has also been studied in detail in ref. 48. Also, applying a positive pressure on compounds clearly increases the hopping parameters. There is thus room for increasing $T_c$.

The percentage change in each parameter that is necessary for a 1% relative change in the optimal superconducting order parameter is shown in Table 1 for the two cases studied here. The reference values are taken to be the calculations done with Eq. 3 (ionic case) and Eq. 4 (covalent case) at $U=12$. Clearly, the percentage change of parameters that is necessary to increase the optimal superconducting order parameter is larger for the covalent case than for the ionic case. Since the reference values of the optimal order parameter of both cases are close, this suggests that the parameters for the covalent case are closer to the maximum achievable value of the order parameter in the parameter space of the three-band model. Interaction on copper, $U$, has the largest impact on the superconducting order parameter while oxygen–copper hopping $t_{pd}$ is the second most important parameter.

Table 1.

Percentage change of each parameter that is necessary to cause a 1% relative increase in the optimal superconducting order parameter (SC) and in the optimal CTG

	Model
	Covalent		Ionic
	SC	CTG	SC
$\mathrm{\Delta }U/U$, %	−0.38	0.24	−0.14
$\mathrm{\Delta }{t}_{pd}/t_{pd}$, %	0.57	−0.68	0.23
$\mathrm{\Delta }{ϵ}_p/{ϵ}_p$, %	3.21	−1.65	0.25
$\mathrm{\Delta }{t}_{pp}/t_{pp}$, %	1.66	−0.48	−2.5
$\mathrm{\Delta }{t}_{pp}^{\prime }/t_{pp}^{\prime }$, %	−25.38	15.16	−1.4

Reference model parameters are Eq. 3 for the ionic case and Eq. 4 for the covalent case with $U=12$ for both cases.

Independent of the details of microscopic parameters, oxygen hole content increases the maximum order parameter while the CTG decreases it, as illustrated in Table 1 for the covalent case. This is consistent with our observation that the oxygen hole content at optimal doping and the CTG are almost perfectly anticorrelated (Fig. 3B) because, as could be argued from the densities of states in Fig. 1 B and C, small CTG means large oxygen character of available hole states and vice versa.

Ultimately, the link between oxygen hole content, CTG, covalency, and superconductivity is provided by the effective superexchange $J$ between copper atoms. The value of $J$ decreases with increasing CTG, which plays the role of $U$ in the one-band model. We also saw in Fig. 4B that, for a given CTG value $E_{ct}$, $J$ is larger in the covalent case. This is consistent with the fact that with $J=4t_{\mathrm{e}\mathrm{ff}}^2/E_{ct}$, the effective hopping $t_{\mathrm{e}\mathrm{ff}}$ is larger in the covalent case. In that case, we can attribute the larger value of the maximum superconducting order parameter for the covalent case at a particular $E_{ct}$ to a larger $t_{\mathrm{e}\mathrm{ff}}$; this along with the fact that the maximum superconducting order parameter decreases with increasing CTG suggests that the maximum order parameter is essentially proportional to $J$.

Covalency then, which is mostly controlled by the difference between the ionization energy of the transition metal and the oxygen affinity (21, 31), is important for high-temperature superconductivity. Among $3d$ transition metals, copper forms the most covalent bonds with oxygen, and hence other transition metal oxides are less likely to be high-temperature superconductors. Nickelates (60), for example, are more ionic and have a lower $T_c$. In the absence of precise estimates for the CTG that also controls the value of $J$, this suggests that combining transition metals with other chalcogens or pnictogens that might form strong covalent bonds could be a promising way to look for compounds that superconduct at room temperature and ambient pressure.

Summary and Conclusion

First, we have shown that the experimental correlation between oxygen hole content ($2n_p$) at optimal doping and optimal $T_c$ (or order parameter) is satisfied in both ionic and covalent limits of the three-band model, thus resolving a long-standing theoretical issue. In each of the two separate cases (covalent or ionic), changes in different model parameters that correspond to the same $2n_p$ lead, with few exceptions, to the same superconducting order parameter. By contrast, a given value of the total doping can lead to different maximum order parameters, depending on which model parameter is varied. Hence, keeping two axes for hole content, one for oxygen and one for copper, is a better way to draw the phase diagram of cuprates.

Second, we have also understood why $2n_p$ and the maximum superconducting order parameter are correlated; this is because $2n_p$ and the size of the CTG are almost perfectly anticorrelated. It is convenient then to focus on the CTG as one of the important parameters that control superconductivity. Since the CTG should be nearly independent of temperature starting from room temperature to around 100 K, measuring the CTG in the normal state should give an indication of whether or not a specific compound can be a high-temperature superconductor. A minimum CTG is needed for superconductivity to appear in the strong-correlation limit, but a large CTG is detrimental.

Third, as observed experimentally, we have shown that the value of the optimal superconducting order parameter is mostly controlled by the superexchange $J$, estimated from the spectrum of spin fluctuations. Thus, the inverse relation between the optimal superconducting order parameter and the CTG value $E_{ct}$ is natural since we expect that $J\sim 1/E_{ct}$. Comparing the results for the ionic and covalent cases, we have also shown that for a given CTG, larger covalency increases $J$, as expected from the scaling of $J$ with the square of the effective hopping between copper atoms.

In summary, the experimental demonstrations of correlations between optimal superconducting transition temperature and three apparently unrelated observables, namely 1) oxygen hole content, 2) CTG, and 3) the strength of spin fluctuations as measured by $J$, can be understood from a unified theoretical perspective if we also take into account the role of covalency in determining $J$. Interactions and covalency lead to an effective superexchange interaction between copper atoms that is ultimately at the origin of superconductivity in the three-band model, a realistic model for cuprates. This suggests additional avenues to discover compounds that superconduct at room temperature and ambient pressure.

Materials and Methods

CDMFT with Continuous-Time Quantum Monte Carlo Solver.

We use CDMFT (61) on a cluster of four unit cells (Fig. 1 A, Inset) to capture the local fluctuations and superexchange interaction, crucial for $d$-wave superconductivity. The environment is a bath of noninteracting electrons that hybridizes self-consistently with the cluster. Since the eight oxygen atoms on the cluster are uncorrelated, they may be integrated out and incorporated as a constant hybridization function in the lattice Green function that is needed for the CDFMT self-consistency relation. We can then concentrate on a four-site copper cluster. Because there is no direct hopping between copper atoms, at finite $T$ the cluster in an infinite bath problem can be solved using the continuous-time quantum Monte Carlo algorithm based on the segment algorithm (62) as in ref. 24. We use 96 processors with about 5 to 20 $\times 10^6$ Monte Carlo steps per processor. We need a typical 100 CDMFT iterations to reach self-consistency. Up to 400 iterations are necessary close to phase transitions. Once convergence is reached, all observables are obtained by computing their average over 15 iterations. For an iteration, the typical Monte Carlo sign is 0.1. We detail the average sign for various parameter sets and temperatures in SI Appendix, Fig. S1.

CDMFT with Exact Diagonalization Solver.

For computations at zero temperature, we use an exact diagonalization impurity solver (63) to solve the system of a four-site copper cluster hybridized to eight bath orbitals. The bath is parameterized based on the irreducible representations of the point group $C_{2v}$ (64, 65). To allow for superconductivity, we let the $U\left(1\right)$ gauge symmetry be broken in the bath; we achieve this by including anomalous hybridization terms between the cluster and the bath along with the regular hybridization terms. The bath parameters are adjusted in the CDMFT self-consistently so that the bath resembles as closely as possible to the real environment of the cluster. It takes around 50 iterations to achieve the self-consistency. After the CDMFT procedure has converged, we obtain various observables by computing the averages of the corresponding operators.

There are two ways to calculate the average of a one-body operator $Ô={\sum }_{\alpha ,\beta }{O}_{\alpha \beta }{c}_{\alpha }^{†}{c}_{\beta }$ in CDMFT: 1) lattice average gives the average value of the operator $Ô$ on the infinite lattice and 2) cluster average gives the average value of the operator $Ô$ on the cluster. The lattice average of $Ô$ is defined as

$$⟨Ô⟩=\oint \frac{d\omega }{2\pi }\frac{d^2\tilde{\mathbf{k}}}{{\left(2\pi \right)}^2}\mathrm{t}\mathrm{r}\left[\mathbf{O}\left(\tilde{\mathbf{k}}\right)\mathbf{G}\left(\tilde{\mathbf{k}},\omega \right)\right],$$ [7]

where $G\left(\tilde{\mathbf{k}},\omega \right)$ is the lattice Green function and $\tilde{\mathbf{k}}$, the reduced wave vector, belongs to the Brillouin zone of the superlattice. The cluster average of $Ô$, on the other hand, is defined as

$${⟨Ô⟩}_c=\oint \frac{d\omega }{2\pi }\mathrm{t}\mathrm{r}\left[\mathbf{O}{\mathbf{G}}_c\left(\omega \right)\right],$$ [8]

where ${\mathbf{G}}_c\left(\omega \right)$ is the cluster Green function. The Nambu formalism is used to include the anomalous counterpart of the Green function, the Gorkov function, as the off-diagonal block matrix of the full Green function matrix represented in the one-body Nambu basis. The pairing operator is expressed as a one-body operator in the Nambu basis, and the order parameter is calculated as its average, in a similar manner to the average of a one-body operator as discussed above. We have used the lattice average method Eq. 7 to obtain the order parameter and the hole doping in our computations.

We do not take into account the competition with antiferromagnetism so that results within about 5% of half-filling do not accurately reflect the phase diagram.

Data Availability

All relevant data are included in this article and SI Appendix. The datasets and the codes used for data analysis have been deposited in the Open Science Framework repository (66) (https://osf.io/jnbq9/).