Dynamically screened Coulomb interaction in the parent compounds of hole-doped cuprates, trends and exceptions

TL;DR

This study explores how the strength of Coulomb interactions in parent cuprate compounds influences their maximum superconducting transition temperatures, revealing that stronger onsite repulsion correlates with higher $T_c$.

Contribution

It provides a detailed analysis of the relationship between Coulomb interaction strength and superconductivity in cuprates using constrained RPA, highlighting the importance of screening effects.

Findings

Superconductivity is favored by larger onsite Coulomb repulsion.

A clear trend links static Coulomb interaction $U$ to $T_{c\text{max}}$ in parent compounds.

Frequency dependence of $U$ offers insights into pairing mechanisms.

Abstract

Although the cuprate high-temperature superconductors were discovered already 1986 the origin of the pairing mechanism remains elusive. While the doped compounds are superconducting with high transition temperatures $T_{c}$ the undoped compounds are insulating due to the strong effective Coulomb interaction between the Cu $3d$ holes. We investigate the dependence of the maximum superconducting transition temperature, $T_{c\text{ max}}$, on the onsite effective Coulomb interaction $U$ using the constrained random-phase approximation. We focus on the commonly used one-band model of the cuprates, including only the antibonding combination of the Cu $d_{x^2-y^2}$ and O $p_x$ and $p_y$ orbitals, and find a clear screening dependent trend between the static value of $U$ and $T_{c\text{ max}}$ for the parent compounds of a large number of hole-doped cuprates. Our results suggest that superconductivity is favored by a large onsite Coulomb repulsion. We analyze both the trend in the static value of $U$ and its frequency dependence in detail and, by comparing to other works, speculate on the mechanisms behind the trend.

Figures

Tables

Table 1. Table 1: Energy windows used in the Wannier function construction (eV).

LaCuO4	-2.5$\to$2
YSr2Cu3O6	-2$\to$2.2
TlBa2CuO6	-2.2$\to$3
YBa2Cu3O6	-2$\to$3
YbBa2Cu3O6	-3$\to$2
HgBa2CuO4	-2.2$\to$2
NdBa2Cu3O6	-3$\to$2
LaBa2Cu3O6	-3$\to$2

Table 2. Table 2: Effect of external pressure on U 𝑈 U (eV) for YbBa 2 Cu 3 O 6 . External pressure was simulated by reducing the lattice parameters. For in-plane pressure the a 𝑎 a lattice parameter was reduced by 1 % percent \% and for hydrostatic pressure a 𝑎 a was reduced by 1 % percent \% and c 𝑐 c by 2 % percent 2 2\% .

	$U$	$U/t$
Normal	3.1	6.9
Hydrostatic	3.2	6.9
In-plane	3.2	6.6

Equations

$$P=P_d+P_r.$$

$$U(\omega )=[1-v{P}_r(\omega ){]}^{-1}v.$$

$$W(\omega )=[1-vP(\omega ){]}^{-1}v=[1-U(\omega )P_d(\omega ){]}^{-1}U(\omega ).$$

$$P({\mathbf{r},\mathbf{r}}^{\prime };\omega )$$

$$-\frac{{\psi }_{\mathbf{k}n}(\mathbf{r}){\psi }_{{\mathbf{k}}^{\prime }{n}^{\prime }}^{\ast }(\mathbf{r}){\psi }_{{\mathbf{k}}^{\prime }{n}^{\prime }}({\mathbf{r}}^{\prime }){\psi }_{\mathbf{k}n}^{\ast }({\mathbf{r}}^{\prime })}{\omega +{ϵ}_{{\mathbf{k}}^{\prime }{n}^{\prime }}-{ϵ}_{\mathbf{k}n}-i\delta }.$$

$$\tilde{H}=\left(\begin{array}[]{cc}H_{dd}&0\\ 0&H_{rr}\end{array}\right).$$

$$\displaystyle P_{d}(\mathbf{r,r}^{\prime};\omega)=\sum_{\mathbf{k}n}^{\text{occ}}\sum_{\mathbf{k}^{\prime}n^{\prime}}^{\text{unocc}}\Bigl{(}\frac{\phi_{\mathbf{k}n}^{\ast}(\mathbf{r})\phi_{\mathbf{k}^{\prime}n^{\prime}}(\mathbf{r})\phi_{\mathbf{k}^{\prime}n^{\prime}}^{\ast}(\mathbf{r}^{\prime})\phi_{\mathbf{k}n}(\mathbf{r}^{\prime})}{\omega-\epsilon_{\mathbf{k}^{\prime}n^{\prime}}+\epsilon_{\mathbf{k}n}+i\delta}$$

$$\displaystyle-\frac{\phi_{\mathbf{k}n}(\mathbf{r})\phi_{\mathbf{k}^{\prime}n^{\prime}}^{\ast}(\mathbf{r})\phi_{\mathbf{k}^{\prime}n^{\prime}}(\mathbf{r}^{\prime})\phi_{\mathbf{k}n}^{\ast}(\mathbf{r}^{\prime})}{\omega+\epsilon_{\mathbf{k}^{\prime}n^{\prime}}-\epsilon_{\mathbf{k}n}-i\delta}\Bigr{)}P_{\mathbf{k}n}P_{\mathbf{k}^{\prime}n^{\prime}}.$$

$${\Omega }_{n{n}^{\prime }}=\sqrt{\Delta {ϵ}_{n{n}^{\prime }}^2+2J_{n{n}^{\prime }}\Delta {ϵ}_{n{n}^{\prime }}},$$

Full text

I Introduction

One of the most important experimental insights about the high-temperature copper oxide superconductors is how $T_{c}$ is correlated with the materials structure expressed as functions of doping, pressure and compositions. The phase diagrams of the high $T_{c}$ cuprates as a function of doping concentration reveal a generic feature common to all compounds, showing the characteristic parabolic curve separating the superconducting and the normal phases. The crystal structure of the cuprates exhibit the generic copper-oxide planes where the dominant low-energy physics is beleived to be constrained. It is known for a long time that $T_{c\text{ max}}$ increases with the number of CuO2 layers and for a given number of layers there is a strong dependence of $T_{c\text{ max}}$ on the cuprate family. It was, however, not known at the microcsopic level on which quantum mechanical parameters $T_{c\text{ max}}$ depended. This puzzle was investigated and analyzed in detail by Pavarini et al. more than a decade ago and they found an interesting and important trend showing a correlation between $T_{c\text{ max}}$ and the hopping parameters Pavarini et al. (2001). Thorough investigation of the phase diagrams of the high $T_{c}$ cuprates on the other hand has revealed that the macroscopic properties of the copper oxides are decisively influenced by strong electron-electron interaction (large Hubbard $U$) between the copper $3d$ holes (see, e.g. Ref. Lee et al., 2006). The large Coulomb repulsion also profoundly influences other fundamental properties which do not follow the standard Fermi-liquid theory which is exhibited already in the case of zero doping in which the material becomes an antiferromagnetic Mott insulator. While a large Coulomb repulsion is at first thought not conducive for the formation of Cooper pairs leading to superconductivity, theoretical studies of the two dimensional single-band Hubbard model indicate that superconductivity can be favored by a large $U$Fukuyama and Yosida (1987); Yokoyama et al. (2012); Kaczmarczyk et al. (2013); Yanagisawa (2015); Yanagisawa et al. (2016).

The Heisenberg nearest neighbor exchange parameter $J$ is a quantity that is intimately related to the Hubbard $U$. In the large $U$ limit the quantities are directly related as $J=-4t^{2}/U$, where $t$ is the nearest neighbor hopping. In a recent experimental studyMallett et al. (2013) Mallet et al. investigate the dependence of $J$ on $T_{c\text{ max}}$ for the systems R(Ba,Sr)2Cu3Oy. It was shown that $J$ had a strong correlation with $T_{c\text{ max}}$ for the considered compounds. However, it was also shown that changing internal pressure by ion-substitution and varying the external pressure have identical effects on $J$ but opposite effects on $T_{c\text{ max}}$. On the other hand the refractivity sum was shown to have a strong correlation with $T_{c\text{ max}}$ which lead the authors to suggest a dielectric rather than a magnetic pairing mechanism.

The purpose of the present work is to delve deeper into the microscopic origin of the trend in $T_{c\text{ max}}$ by studying its dependence on the strength of the Coulomb repulsion, or Hubbard $U$. Although $U$ and $J$ are directly related in the limit where $U$ is much larger than the bandwidth, for the cuprate compounds $U$ is of the same order as the bandwidthWerner et al. (2015); Jang et al. (2016) and hence this relation is not guaranteed to hold. Further on, the value of $U$ is directly influenced by the dielectric screening and an investigation of the material dependence of $U$ could therefore be a route to understand the correlation between $T_{c\text{ max}}$ and the refractivity sum reported in Ref. Mallett et al., 2013.

We compute the Hubbard $U$ using the constrained random-phase approximation (cRPA)Aryasetiawan et al. (2004); Miyake and Aryasetiawan (2008) as implemented in the FLAPW codes FLEUR and SPEX Friedrich et al. (2010); The FLEUR group . The cRPA yields both the static (time-averaged) value and the full frequency dependence of $U$ and allows for a detailed analysis of the screening channels responsible for renormalizing the bare Coulomb interaction $v$.

We consider a wide range of hole-doped cuprate compounds starting from the well-studied La2CuO4 as well as TlBa2CuO6 and HgBa2CuO4 and continuing with the the compounds R(Ba,Sr)2Cu3O6 (R=Y, Yb, Nd, La) that were also studied experimentally in Ref. Mallett et al., 2013. In the latter compounds, changing the ion size yields a change of size of the unit cell and can therefore be considered as a change of the ”internal pressure” of the compoundMallett et al. (2013). We also explore the effects of external pressure on $U$ by systematically changing the lattice parameters.

With the exception of La2CuO4, we find a screening-dependent correlation between $T_{c\text{ max}}$ and both the static value of $U$ and the ratio $U/t$, suggesting that superconductivity in the cuprates is favored by a large onsite Coulomb repulsion. Contrary to $J$ we also find that external and internal pressures have the same effect on $U$, that is, $U$ increases with both internal and external pressure. However, the increase is not sufficiently large to account for the observed increase in $T_{c\text{ max}}$. In the numerical studies in Refs. Weber et al., 2012 and Acharya et al., 2018 it was found that superconductivity was favored by a small charge-transfer energy ($\epsilon_{d}-\epsilon_{p}$). Together with the trend in $U$ in the present paper this suggests that superconductivity may be favored by having a large $U$ and small charge-transfer energy, which would lead to a charge-transfer insulating parent compound with the lower Hubbard band below the $O$ $p$-states. Since La2CuO4 both has a large $U$ and a large charge-transfer energy this could offer an explanation of why La2CuO4 does not follow the trend in $U$.

We also consider the full frequency dependent $U(\omega)$ for these compounds. We analyze the different screening channels and show that the $p$-$d$ screening channel, that gives rise to peaks around 8-9 eV in all cuprate compounds, is much stronger in La2CuO4 than in the other compounds. In that sense La2CuO4 is an unusual case, and may not be a good representative prototype for a general cuprate compound. Furthermore we show that $U$ is highly material dependent, suggesting that the common assumption of using the same value of $U$ for all cuprate compounds can yield misleading conclusions.

II Method

II.1 cRPA

To study the materials dependence of the Hubbard $U$, we use the constrained random-phase approximation (cRPA) method Aryasetiawan et al. (2004); Miyake and Aryasetiawan (2008). In the cRPA method, the screening channels expressed in terms of the polarizations are decomposed into those within the model ($P_{d}$) and the rest ($P_{r}$):

[TABLE]

It can then be shown that the effective Coulomb interaction among the electrons residing in the model subspace (the $d$ subspace) is given by

[TABLE]

This effective interaction is physically interpreted as the Hubbard $U$, which is now a function of frequency. This interpretation is based on the fact that when $U$ is screened by the polarization $P_{d}$ of the model one obtains the fully screened interaction:

[TABLE]

In practice the polarization is computed from the LDA bandstructure Kohn and Sham (1965) within the random phase approximation (RPA) , which for a given spin is given by

[TABLE]

In the LDA the conduction band in the cuprates originates from the antibonding combination of the Cu $d_{x^{2}-y^{2}}$ with the Oxygen $p_{x}$/$p_{y}$ orbitals and has $d_{x^{2}-y^{2}}$ symmetry. The bonding and nonbonding bands, commonly referred to as the Opx/y bands are located around 6 eV below the Fermi energy. Commonly used models for the cuprates include

• •

the one-band model derived from the antibonding conduction band.

• •

the two-band model that apart from the conduction band include the narrow band just below the Fermi energy originating from the hybridization between the Cu 3$d_{z^{2}}$ and the apex Oxygen $p_{z}$ orbital.

• •

the three-band model that includes the antibonding conduction band as well as the bonding and nonbonding combinations.

• •

the four-band model that include all the above-mentioned bands.

In this work we focus on the one-band model since this provides the minimal low-energy model of the cuprates.

To define the $d_{x^{2}-y^{2}}$ model subspace we use Maximally Localised Wannier functions (MLWF:s) Marzari and Vanderbilt (1997); Mostofi et al. (2008); Sakuma (2013) that are derived from the LDA band structure. Hence, $P_{d}$ in Eq. 1 is the polarization within the $d_{x^{2}-y^{2}}$ conduction band. Since the $d_{x^{2}-y^{2}}$-band is not isolated we use the disentanglement approach Miyake et al. (2009) to get a well defined one-particle bandstructure and model polarization. In this method the hybridization between the model and the rest is cut in the Hamiltonian

[TABLE]

The $r$ subspace polarization is then calculated as $P_{r}=P-P_{d}$, where the full polarization $P$ and the $d$ subspace polarization $P_{d}$ are calculated for the disentangled bandstructure according to Eq. (4) . In Fig. 1 we show the Wannier interpolated bandstructures for the two compounds YSr2Cu3O6 and LaBa2Cu3O6.

We only compute $U$ for the undoped parent compounds. However, since the metallic screening from within the $d_{x^{2}-y^{2}}$ conduction band is removed within the cRPA, $U$ is expected to be very weakly dependent on the doping. Hence, the values of $U$ for the parent compounds can be used also in the doped cases.

II.2 Simulation of external pressure

We consider the effect of external pressure for the compound YbBa2Cu3O6 by scaling the lattice parameters. With the exception of La2CuO4, where the effect of pressure is approximately isotropicAkhtar et al. (1988); Schilling (2007) (i.e. the $a$ and $c$ lattice parameters are scaled by the same factor) a generic feature of the cuprates seems to be that the effect of hydrostatic pressure on the $c$ lattice parameter is around twice as big as that of the in-plane lattice parameter $a$Schilling (2007). Therefore we approximate hydrostatic pressure by scaling $a$ with $x\%$ and $c$ with $2x\%$. Experimentally $T_{c\text{ max}}$ increases with moderately applied pressure and decreases again at high pressure (larger than 5-7 GPa) Schilling (2007); Schilling and Klotz (1992). In Ref. Armstrong et al., 1995 $c$ was shown to decrease about $4\%$ and $a$ about $2\%$ with a pressure of 6 GPa for HgBa2Ca2Cu308+δ. Therefore we consider scalings of $a$ and $c$ below these numbers in this work.

In Ref. Hardy et al., 2010 both the uniaxial and hydrostatic pressure derivatives of $T_{c}$ were determined for HgBa2CuO4+δ close to the optimal doping level. It was found that $T_{c}$ increases with a decreasing unit cell area of the Cu-O planes as well as with an increasing separation of the planes. To investigate the effect of uniaxial pressure on $U$ we also consider scaling only $a$, which corresponds to applying only in-plane pressure.

III Computational details

We use the LDA bandstructure calculated with the full-potential linearized augmented plane-wave (FLAPW) code FLEUR The FLEUR group as a starting point. The MLWF:s were computed using Wannier90 library Marzari and Vanderbilt (1997); Mostofi et al. (2008); Sakuma (2013) and $U$ was computed within the cRPA as implemented in the SPEX-code Friedrich et al. (2010); The FLEUR group . We only considered spin-polarization for the compounds YbBa2Cu3O6 and NdBa2Cu3O6, since the other compounds are not spin-polarized within the LDA. All calculations were converged with respect to the FLAPW basis set, the number of bands used to compute the polarization function, the number of $\mathbf{k}$-points used in the LDA as well as cRPA calculation. For example this required the use of between 300-400 bands in the computation of the polarization function for the different compounds.

The bands used to construct the Wannier functions were defined using an energy window, where for each $\mathbf{k}$-point all states with an energy inside the energy window were used in the Wannier function construction. In Tab. 1 we present the energy windows for the different compounds. For the spin-polarized calculations $U$ was defined as the average matrix element over the two spin channels. However, both the value of $U$ and the nearest neighbor hopping $t$ were very similar for the two spin-channels.

For La2CuO4 and TlBa2CuO6 we used the reduced structures in Refs. McMahan et al., 1988; Singh and Pickett, 1992 while for the remaining materials we use the experimental structures. The crystal structure for YSr2Cu3O6 was taken from from Ref. YSr, , YBa2Cu3O6 from Ref. Swinnea and Steinfink, 1987, HgBa2CuO4 from Ref. Putilin et al., 1993, YbBa2Cu3O6, NdBa2Cu3O6 and LaBa2Cu3O6 from Ref. Chu, 1987.

IV Results and Discussion

IV.1 Static Interaction

In the one-band Hubbard model with a static Coulomb repulsion $U$ and a nearest neighbor hopping $t$ the only free parameter is the ratio $U/t$. Therefore this ratio provides a good measure of the degree of local correlations.

In Fig. 2 we show the static value of $U$ as well as the ratio $U/t$ for all compounds considered in this work. $U/t$ follows an increasing trend with increasing $T_{c\text{ max}}$. The smallest value of $U/t$ is approximately 6 and the largest approximately 9, which implies a substantial difference of the degree of local correlations in the compounds. Since the bandwidths in most of the compounds are similar, $U$ follows the same increasing trend as $U/t$, albeit not as clear. Hence, the trend in $U/t$ can mainly be attributed to the trend in $U$ and is not an effect of a trend in the hopping parameters. The only exception to the trend is La2CuO4 which has a remarkably large $U$ compared to the relatively low Tc. This suggests that La2CuO4, which is typically considered as a prototype of a cuprate high Tc superconductor, actually is an exceptional case. Furthermore, assuming that the trend in $U$ implies that high Tc superconductivity is favored by a large onsite Coulomb repulsion, the fact that La2CuO4 alludes the trend implies that there are other mechanisms that hamper superconductivity in this compound. While the compounds R(Ba,Sr)2Cu3O6 all have similar structures with two CuO layers it is interesting to note that both TlBa2CuO4 and HgBa2CuO4, which are single layer compounds, also follow the trend. This implies that $U/t$ indeed is an important parameter to obtain large Tc and at least as important as other parameters, such as the number of Cu-O layers.

The value of $U$ depends both on the screening properties and on the shape and extent of the Wannier basis functions. The value of the bare interaction $v$, on the other hand, only depends on the shape and extent of the Wannier basis functions. More localized Wannier functions yield larger values of $v$. Since the Wannier functions are derived from the bandstructure, trends in $v$ can be considered as bandstructure effects while trends in $U$ can depend both on the bandstructure and the screening process. By comparing the values of $v$ (lower left panel) and $U$ (upper left panel) in Fig. 2 one can conclude that the trend in $U$ is intimately related to the screening in the compounds. For example TlBa2CuO4 has a larger bare interaction but a smaller value of $U$ than HgBa2CuO4 due to the larger screening in the former compound. Also in YbBa2Cu3O6, which has the largest value of $v$, the screening is large compared to the other compounds. In $v/t$ (lower right panel of Fig. 2) the main exceptions to the trend are HgBa2CuO4 and TlBa2CuO4.

For the compound YbBa2Cu3O6 we simulated the external pressure by scaling the lattice parameters as discussed in Section II.2. Our results are summarized in Tab. 2. The effect of external pressure is small ($<2\%$ for reasonable pressures) both on $U$ and $U/t$. Thus $U$ cannot be used to understand the increase of $T_{c\text{ max}}$ upon applied external pressure. It is interesting to note that, contrary to the Heisenberg exchange parameter $J$Mallett et al. (2013), $U$ follows the same trend upon applied internal and external pressure. However, since also the hopping amplitude increases with a decreasing in-plane Cu distance, $U/t$ is unchanged upon applied hydrostatic pressure and decreases with in-plane pressure.

Taken together our results indicate that $U$ is an important parameter to get high $T_{c\text{ max}}$, and superconductivity is favored by a large onsite Coulomb repulsion. However, there are exceptions to this trend; La2CuO4 has a relatively large $U$ but low $T_{c\text{ max}}$ and upon applied external pressure the change in $U$ is not sufficient to account for the observed increase in $T_{c\text{ max}}$. This indicates that $U$ or $U/t$ are not the only important parameter for high $T_{c}$ superconductivity in the cuprates, rather there seem to be many competing mechanisms that taken together determine whether a material has a high $T_{c}$ or not. The shape of the Fermi surface, as indicated by the trend in $T_{c\text{ max}}$ with $t^{\prime}/t$ in Ref. Pavarini et al., 2001, is an example of one such important parameter.

IV.2 Comparison to other calculations

In this section we compare our results to other similar studies. The two main studies we will focus on are the ones by Jang et al. Jang et al. (2016) that compared $U$ for a number electron and hole-doped cuprates as well as Hirayama et al.Hirayama et al. (2018) who derived the low-energy Hamiltonian for La2CuO4 as well as HgBa2CuO4 in the one, two and three-band models using the MACE-schemeImada and Miyake (2010); Hirayama et al. (2013, 2017). We also briefly discuss the numerical studies by Weber et al.Weber et al. (2012) and Acharya et al.Acharya et al. (2018), using cluster DMFT and $GW$+DMFT based schemes respectively, showing a correlation between the charge-transfer energy and $T_{c}$, which indicates that superconductivity is favored by a small charge-transfer gap.

IV.2.1 Comparison to Jang et al.

In Ref. Jang et al., 2016 Jang et al. computed the static value of $U$ using the cRPA for the parent compounds of a number of both hole- and electron-doped cuprates. Of specific interest to this work they computed $U$ for La2CuO4 and HgBa2CuO4. For both these compounds they obtained a value of $U$ which is substantially smaller than the ones in this work (3.15 eV for La2CuO4 compared to 3.65 eV in this work and 2.15 eV for HgBa2CuO4 compared to 3.42 eV in this work). The origin of this difference is the difference in methods when computing the model polarization. We used the disentanglement approachMiyake et al. (2009) described in Section II.1 while Ref. Jang et al., 2016 used a weighting approachŞaşıoğlu et al. (2011) where the polarization is computed for the original LDA bandstructure and $P_{d}$ is defined as

[TABLE]

$P_{\mathbf{k}n}$ is the probability that the electron in state $|\phi_{\mathbf{k}n}\rangle$ resides in the $d$-subspace. This method generally yields smaller values of $U$ since not all metallic screening from the correlated (disentangled) band is removed. Furthermore, the final aim of our calculations is to use the $U$ values together with a Hamiltonian or hopping parameters in e.g. LDA+DMFT or $GW$+EDMFT calculations. The Hamiltonian for this type of calculation would exactly correspond to the $d$-block of the disentangled Hamiltonian in Eq. 5. Hence, in the disentanglement approach both the $U$-matrix and the hopping parameters are derived from the disentangled bandstructure, which is not the case in the weighting approach. We therefore consider the disentanglement approach to be a more appropriate method in this case.

In Ref. Jang et al., 2016 the focus was on the comparison between electron-doped and hole-doped cuprates and a general tendency of electron-doped cuprates to have a smaller $U$ was found. However, also some of the hole-doped cuprates had similarly small values of $U$ which lead the authors to conclude that the strong correlation enough to induce Mott gap may not be a prerequisite for the high-$T_{c}$ superconductivity. In this work we consider a wider range of hole-doped cuprates. Even though $U$ is generally larger using the disentanglement approach ($U$ for HgBa2CuO4 using the disentanglement approach is still larger than the value of $U$ computed for La2CuO4 with the weighting approach in Ref. Jang et al., 2016) at first sight our results seems to strengthen the conclusions in Ref. Jang et al., 2016. $U$ for YSr2Cu3O6 for example is only 2.33 eV which is approximately 0.6 times the bandwidth, and hence cannot be considered to be deep in the Mott-insulating regime. However, as discussed in e.g. Ref. Werner et al., 2015, cRPA for a pure one-band model could potentially underestimate $U$ due to the large spread of the Wannier basis states in this model. Hence, one should mainly focus on the trend rather than the absolute values of $U$ in a one-band model. The trend in the static value implies that, even though strong correlation enough to induce Mott gap may or may not be a prerequisite for the high-$T_{c}$ superconductivity, superconductivity is favored by strong local Coulomb repulsions.

Jang et al. found that the electron-doped cuprates, which have smaller $T_{c\text{ max}}$ than their hole-doped counterparts, also had smaller values of $U$. This observation fits with the trend reported in this paper. However, for Hg-based compounds with different number of CuO-layers, they also found that $U$ for the triple layer compound HgBa2Ca2Cu3O8 is smaller than the corresponding value of the double and single layer compounds, even though $T_{c\text{ max}}$ increases with the number of layers. This result contradicts the trend and therefore further illuminates the complexity of the problem with many competing mechanisms. In this particular case it points to two competing mechanisms to achieve high Tc superconductivity, namely a large $U$ on one hand and many CuO layers on the other hand.

Another interesting observation in Ref. Jang et al., 2016 was a correlation between $U$ and the average inverse apical Oxygen distance $1/h_{O}=\frac{1}{\sqrt{2}}(1/h_{O_{1}}+1/h_{O_{2}})$, where $h_{O_{1/2}}$ is the distance to the apical Oxygen above/below the Cu ion in the CuO plane. From Fig. 3 it is clear that, while we reproduce this correlation for YSr2Cu3O6, TlBa2CuO6, HgBa2CuO4 and La2CuO4, the remaining compounds considered in this work do not show any correlation between $U$ and $1/h_{O}$. For these materials the screening from the charge reservoir layers are important as will be discussed more in detail below.

IV.2.2 Comparison to Hirayama et al.

Hirayama et al.Hirayama et al. (2018) calculated the effective low-energy Hamiltonians for La2CuO4 as well as HgBa2CuO4 in the one, two and three band models using the MACE schemeImada and Miyake (2010); Hirayama et al. (2013, 2017). Of interest to this work are their values of $U$ in the one-band model (5.00 eV for La2CuO4 and 4.37 eV for HgBa2CuO4) which are substantially larger than the ones in this work. The MACE-scheme involves a cRPA calculation using the disentanglement approach, but for the 17-bands closest to the Fermi-energy (which includes the Cu $d$-bands and Oxygen $p$-bands) they computed the polarization from the $GW$ quasiparticle bandstructure rather than the LDA. This yielded smaller screening and hence larger values of $U$.

The reason that La2CuO4 has a relatively large $U$ was also analyzed. It was concluded that La2CuO4 has a larger value of $U$ than HgBa2CuO4 because the Oxygen $p$ orbitals are farther below the Cu $d_{x^{2}-y^{2}}$ orbitals in La2CuO4 which yields a different (more localized) character of the antibonding conduction band. This analysis would imply that both $U$ and the bare Coulomb interaction $v$ should be larger for La2CuO4, which also agrees with our results in Fig. 2.

IV.2.3 Comparison to Weber et al. and Acharya et al.

In Ref. Weber et al., 2012 Weber et al. studied Tc as a function of the charge-transfer energy $(\epsilon_{d}-\epsilon_{p})$ as well as the hopping parameters using cluster DMFT for the three-band model with fixed value of $U_{dd}=$ 8 eV. It was found that the charge-transfer energy shows an antilinear correlation with the static order parameter, i.e. decreasing $(\epsilon_{d}-\epsilon_{p})$ yielded a larger superconducting order parameter. It was also shown that the charge-transfer energy, computed from the LDA bandstructure, displayed an antilinear correlation with the experimental T${}_{c\text{ max}}$ for a large number of cuprates.

In Ref. Acharya et al., 2018 Acharya et al. studied how the displacement of the apical Oxygen in La2CuO4 affects the superconducting order parameter, optical gap as well as spin and charge susceptibilities using a one-shot combination of quasiparticle self-consistent $GW$ (QPSC$GW$) and DMFT. They used a static $U$ of 10 eV which is substantially larger than the static cRPA valueWerner et al. (2015), but ignored the frequency dependence. This large $U$ value was motivated by comparing full and restricted QPSC$GW$ calculations in Ref. Choi et al., 2016. It was found that pristine LaCuO4 was Mott-insulating but increasing the distance between the apical Oxygen and the Cu-O plane ($\delta$) yields a cross-over to a charge-transfer insulator (CTI). Increasing $\delta$ further shrinks the CTI gap and the gap collapses at the critical value $\delta_{c}=0.045$Å. They estimated $T_{c\text{ max}}$ from their calculated values of the superconducting order parameter and found that T${}_{c\text{ max}}$ increased with increasing $\delta$ until it reached its maximum value at $\delta=\delta_{c}$. These results support the conclusions by Weber et al. that superconductivity is favored by a small charge-transfer gap.

Since an estimation of the charge-transfer gap requires the use of a three-band model, straightforward comparisons between these two works and our results are difficult. However, as discussed in detail in Ref. Werner et al., 2015, in practical calculations for the three-band model using e.g. LDA+DMFT, $U_{pp}$ and $U_{pd}$ are typically ignored. In such, so called $d$-$dp$ model calculations, the $p$-$d$ screening should be included in the effective $U_{dd}$. Hence, the only difference between the $U_{dd}$ in the one-band model and the effective $U_{dd}$ in the $d$-$dp$ three-band model comes from the Wannier basis functions, which are more localized in the latter case. Therefore it is reasonable to assume that the effective $U_{dd}$ in the three-band model will follow the same trend as in the one-band model but with larger overall values. Both Weber et al. and Acharya et al. effectively decreased the charge-transfer energy while keeping $U_{dd}$ fixed. This yields a transition from a Mott insulator to a pure charge-transfer insulator with the lower Hubbard band below the Oxygen $p$-states. The same effect can be reached by keeping the charge-transfer gap fixed and increasing the $U$. The results in this work combined with the studies above therefore suggest that superconductivity in the hole-doped cuprates is favored by having a CTI parent compound.This can be achieved by having a large $U$ and/or a small charge-transfer energy. Due to the large charge-transfer energy in La2CuO4 the lower Hubbard band is in the same energy range as the Oxygen $p$ bandsWerner et al. (2015) in spite of the large $U$-value, which can explain why La2CuO4 does not follow the trend in Fig. 2.

IV.3 Frequency dependence

In Fig. 4 we show the imaginary part of the frequency dependent $U$ for a selected number of compounds. The main features that can be observed in all materials is a subplasmonic peak around 8-9 eV, as well as the main bulk plasmon around 30 eV. The 8-9 eV peak originates from screening from the Oxygen $p$ bands below the Fermi energyWerner et al. (2015). To provide a rough estimation of the position of the peak we will consider a two-level system. The poles of the response function for a two-level system is given by

[TABLE]

where $\Delta\epsilon_{nn^{\prime}}$ is the energy difference between the states and $J_{nn^{\prime}}$ the exchange interaction between the states. For La2CuO4 the Oxygen $p$ bands are relatively far below the Fermi energy which implies that $\Delta\epsilon_{nn^{\prime}}$ is large, and therefore the $p$-$d$ peak appears at relatively high energy in $U$.

It is also interesting to note that the $p$-$d$ peak is much more pronounced in La2CuO4 than in the other compounds. If the $p$-$d$ screening acts destructively for superconductivity this could offer an alternative explanation why La2CuO4 eludes the trend in the static $U$ in Fig. 2. However, it is possible that the large onsite Coulomb interaction, which is not accounted for when computing the cRPA $U$, suppresses the $p$-$d$ screening channel in the real material. The tendency of the cRPA to overestimate the low-energy screening channels between narrow bands close to the Fermi energy has also been indicated in model studies in Refs. Shinaoka et al., 2015; Han et al., 2018; Honerkamp et al., 2018.

In addition to these features the compounds with a rare-earth element RBa2Cu3O6 (R=Yb,Nd,La) exhibit a well pronounced low frequency structure around 0.5-1eV. In Fig. 5 we compare the DOS of YSr2Cu3O6 where this peak is absent and YbBa2Cu3O6 which displays the low-energy peak in Im$U$. From this comparison it is tempting to conclude that the metallic screening originates from the Yb spectral weight close to the Fermi energy, which originates from the narrow $4f$ band in the LDA bandstructure. However, these states are highly localized on the rare-earth ion and do not contribute much to the screening on the Cu ion. This is also apparent since LaBa2Cu3O6 does not have these 4$f$ bands but yet display a strong metallic screening. Rather, the origin of the low-energy screening channel is apparent in the bandstructure. In Fig. 6 the bandstructure of LaBa2Cu3O6 and YSr2Cu3O6 are compared. LaBa2Cu3O6 exhibits a narrow band close to the Fermi energy, which can be deduced to originate from the $d_{xz}$ and $d_{yz}$ orbitals from the out of plane Cu ion. The closeness to the Fermi-energy and the strong hybridization gives rise to the strong screening channel in Im$U$. In YSr2Cu3O6, this band is at much lower energy and not as strongly hybridized with the Cu$d_{x^{2}-y^{2}}$ conduction band.

IV.4 Screening analysis

By keeping the basis functions fixed and making use of an energy window to selectively remove different screening channels it is possible to dissect which screening channels that contribute to the trend. Screening from all states within the energy window as well as screening due to transitions from states within the energy window to the model subspace ($d_{x^{2}-y^{2}}$ conduction band) is removed. The bare interaction then corresponds to the case with an infinite energy window. We consider the following windows:

-8$\rightarrow$2 eV 2. 2.

-8$\rightarrow$12 eV

Window 1 excludes the $p$-$d$ screening as well as the additional low-energy screening channels while Window 2 also excludes the screening to higher lying bands.

Since we are not interested in the absolute values but only the relative trend we show the value of $U/t$ scaled by its maximum value for each energy window in Fig. 7. For the large energy window (-8$\rightarrow$12 eV) the picture is almost identical to the case with the bare interaction. This shows that the high energy screening affects all compounds in the same way and that the material specific screening is related to the low-energy screening channels within the energy window. This can be understood from the DOS in Fig. 5. For energies higher than 12 eV the majority of the spectral weight comes from the interstitial region of the FLAPW basis set and therefore corresponds to broad bands that are not expected to yield a very material specific screening.

The inclusion of the screening to states between 2 to 12 eV (see window -8$\rightarrow$2 eV) is dramatic and highly material specific. This is expected since this energy region contains a large spectral weight on the atoms in the charge reservoir layers (Sr/Ba, Y etc.), which is highly material specific. However, it is only upon the inclusion of the additional low energy screening channels in $U$ that a clear trend can be observed. Hence the trend cannot be attributed to any specific screening channel but all low energy screening channels, within the -8 $\rightarrow$ 12 eV energy window collectively contribute to the trend.

V Conclusions

We have computed the effective Coulomb interaction $U$ for the one-band model for the parent compounds of a number of hole-doped cuprate superconductors using the constrained random-phase approximation (cRPA). We find a screening dependent trend between the maximum superconducting transition temperature ($T_{c\text{ max}}$) and the static screened interaction $U$, suggesting that superconductivity is favored by a large onsite effective Coulomb interaction. The only exception to the trend is La2CuO4 which has a relatively large value of $U$ but the smallest $T_{c\text{ max}}$. From our data we suggest that $U$ is one out of many competing parameters to achieve high Tc superconductivity and that there are other mechanisms that hamper superconductivity in La2CuO4, such as the large charge-transfer energyWeber et al. (2012); Acharya et al. (2018). We also study the frequency dependence of $U$ and explain the different features. One of the most dominant features, present in all the studied compounds, is a peak in Im$U(\omega)$ at around 8-9 eV, which originates from screening from the O$p_{x/y}$ bands to the $d_{x^{2}-y^{2}}$ conduction band ($p$-$d$ screening). This peak is much more pronounced in La2CuO4 than in the other compounds which leads us to suggest that, apart from the large charge-transfer energy, the strong $p$-$d$ screening in La2CuO4 could be one possible mechanism that hamper superconductivity in this compound. For the compounds RBa2Cu3O6, R=La,Nd,Yb, we find an additional low-energy screening channel due to screening from the band derived from the out of plane Cu $d_{xz}$ and $d_{yz}$ states. This band is close to the Fermi energy and strongly hybridized with the $d_{x^{2}-y^{2}}$ conduction band for these compounds which yields an unusually strong screening mode.

Acknowledgements.

This work was supported by the Swedish Research Council. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at LUNARC We would like to thank O.K. Andersen for useful inputs and discussions. We would also like to thank C. Friedrich and S. Blügel for providing us with their FLAPW code.