Atomic Stripe Formation in Infinite-Layer Cuprates

Abstract

High-temperature superconductivity appears in cuprate materials that have been tuned in a way where the copper–oxygen bond configuration and coordination is in a state of minimal energy. In competition with the Jahn–Teller effect, which impedes the formation of infinitely connected CuO₂ planes, the state of minimal energy persists for planar copper–oxygen bond length variations of up to 10%. We have synthesized the infinite-layer phases of CaCuO₂ and SrCuO₂ as single-crystalline films using molecular beam epitaxy and performed in-plane scanning transmission electron microscopy mapping. For the infinite-layer phase of CaCuO₂ with a short Cu–O bond length, the CuO₂ planes maintain their minimal energy by forming distinguished atomic stripes. In contrast, atomic stripe formation does not occur in the infinite-layer phase of SrCuO₂, which has a larger Cu–O bond length. The polar field provided by the charge reservoir layer in cuprates with infinitely connected CuO₂ planes holds the key over the emergence of superconductivity and is vital to maintain infinitely connected CuO₂ planes themselves.

Introduction

Cuprate superconductors have not lost their charm even after decades of research. This attention is borne out of the uniqueness that a certain set of spin, charge, and orbital degrees of freedom allow for superconducting transition temperatures under ambient pressure well in excess of the boiling temperature of liquid nitrogen. In transition-metal oxide (TMO) interfaces, the diverging lattice, electronic, orbital, and magnetic behaviors of the two adjacent materials lead to properties that are commonly diverse from those of the two individual bulk materials and this is not different for high-temperature superconducting cuprates. This would imply that one were at the brink to tailor and pitch the net electronic response behavior of interfaces. Much effort has been focused on the investigation of single individual interfaces while cuprate superconductors are built up from at least two or more interfaces. Cuprate superconductors are natural superlattices, composed of copper-oxide planes and planes of oxides, so-called charge reservoir layers (CRLs), with the sole function to shift the instability of tenorite^1,2 toward CuO₂ planes. In other words, the CRLs have the power to transform the monoclinic angle β = 99.467° of the tenorite crystallographic unit cell toward 90° in cuprate superconductors. The fact that the crystal structure and consequently its electronic response does not follow the trend set by other simple 3d TMOs is the consequence of a not yet well-understood interplay of electronic, spin, lattice, and orbital degrees of freedom.

In cuprate superconductors, the CRLs are made of rare-earth, alkaline earth, or post-transition-metal oxide strata that sandwich the CuO₂ planes. Nonetheless, this building principle finds its exception in cuprates with the so-called infinite-layer (IL) structure and simple cations, e.g., Ca²⁺ or Sr²⁺ ions, separate adjacent CuO₂ planes. Within a unit cell, the charge-neutrality criteria must be fulfilled and this is the task of the CRL. Furthermore, charge-carrier doping into the CuO₂ planes from the CRL by heterovalent cation substitution is considered as the fall-back explanation for the induction of superconductivity in cuprates. Regrettably, this seems not to function as expected for IL cuprates. In IL-SrCuO₂, trivalent rare-earth substitution (R³⁺) induces superconductivity^3,4 though the opposite is true for IL-CaCuO₂^1,5 (Figure S1). The fact that Sr_1–xR_x³⁺CuO₂ becomes superconducting and Ca_1–xR_x³⁺CuO₂ does not is puzzling particularly under the light that both materials are of equally high crystalline quality (Figures S2 and S6) when prepared by molecular beam epitaxy (see the Supporting Information). Apparently, a yet to be quantified mechanism is responsible for this difference and motivated us to link the superconducting transition temperature T_c of all cuprate superconductors to their in-plane copper–oxygen bond lengths d_Cu–O (Figure 1) in a phase diagram. In this context, two singularities emerge for d_Cu–O ≈ 1.895 and 1.950 Å, where superconductivity is effectively suppressed no matter the circumstances. In detail, if d_Cu–O ≈ 1.950 Å, at the boundaries between zones I and II (Figure 1), it is not just the vanishing superconducting state; upon shortening d_Cu–O, the coordination of copper changes from square planar into a state where the copper coordination is octahedral (O), pyramidal (P), or pyramidal (P) and square planar (S). At d_Cu–O ≈ 1.950 Å, the criteria of a quantum critical point⁶ are sufficiently fulfilled and we label it as the first quantum critical point (QCP 1) in the cuprate phase diagram. In terms of real cuprate materials, PrBa₂Cu₃O_7−δa⁹ (P) and Gd_2–xCe_xCuO₄¹⁰ (S) represent QCP 1 under equilibrium conditions. Indeed, these two materials are not superconducting. Yet, this does not come as a surprise and is known for long. With further decreasing d_Cu–O, superconductivity disappears yet again, this time for d_Cu–O ≈ 1.895 Å, which is between zones II and III in the phase diagram (Figure 1). For d_Cu–O ≈ 1.895 Å, another transition in the coordination of copper takes place. This time it is from O or P, or P and S (zone II)¹¹ to exclusively O (zone III).¹² Again, also this point suffices the requirements set forth to qualify as a QCP, labeled QCP 2 in the cuprate phase diagram (Figure 1). The phase diagram shown here has been assembled from unstrained cuprate bulk materials. For thin materials, where epitaxial strain controls the polarization field of the CRL, the absolute position of QCP 1 and 2 may be freely manipulated and was shown for La₂CuO₄ thin films.¹³ We note that cuprate superconductors located in zone II have the highest T_c values, and discussions about the absolute magnitude of the superconducting transition temperature are beyond the scope of this manuscript. It is, however, quite more important to note that IL-CaCuO₂ has a d_Cu–O that puts it right in the center of zone II, while being neither a metal nor a superconductor (Figure S1). We conjecture two conclusions that can be drawn from the cuprate phase diagram. First, the chemistry of strata above and/or below the CuO₂ planes influences the equilibrium bond length of the Cu–O bond. In other words, as CRLs are made up of various ions having different effective polarization strengths, the Cu–O bond lengths are subjected to the competition between the intrinsic Jahn–Teller effect in Cu²⁺ that impedes the formation of infinitely connected CuO₂ planes and the extrinsically applied polarization strength. Second, there are critical copper–oxygen bond lengths where the superconducting transition temperature has vanished (T_c = 0 K). Independently, in the absence of such CRL or polarizing strata, the copper–oxygen bonding scheme succumbs entirely to what is dictated by the Jahn–Teller effect as seen in tenorite—the symmetry is lowered from tetragonal to monoclinic. To approach a solution as to why IL-CaCuO₂ is not superconducting, one needs to consider the model of an undistorted (CuO₂)^2– plane without having to deal with influences stemming from polarization or strain, namely, bulk-like samples without CRLs. Using energy minimizing principles, Bianconi et al.¹⁴ reported d_Cu–O ≈ 1.965 Å, for a free-standing CuO₂ plane, where Cu is square-planar coordinated and we define this length as “Bianconi threshold”, d_B. Certainly, experimental verification of this value is a tough task although some groups report on tetragonal CuO ultrathin films²⁰⁻²² with d_Cu–O slightly (≈2 pm) shorter than the Bianconi threshold. More importantly is, however, the fact that cuprates with Cu–O bond lengths shorter than the Bianconi threshold¹⁴ are susceptible to stripe order, quite in contrast to cuprates with longer Cu–O bond lengths.²³⁻²⁷ Therefore, does IL-CaCuO₂ opt for stripes, too? Answering this question requires atomically resolved imagery of an ultrahigh-quality crystal.

Figure 1.

Phase diagram of cuprate superconductors where the superconducting transition temperature T_c is plotted as a function of d_Cu–O length for several cuprates. All compounds plotted here have infinitely connected CuO₂ planes and the listed cuprates do show superconductivity, except IL-CaCuO₂. In zones I and III, the coordination number of copper is unique, whereas all three coordinations may coexist in zone II. The Bianconi threshold is the length in the unstrained and undistorted CuO₂ plane.¹⁴ For every Cu–O bond length between 1.850 and 2.000 Å, there exist a chemistry (typically called charge reservoir layer¹⁵) that induces superconductivity, and this superconductivity appears throughout the phase diagram except for two distinct points at 1.950 and 1.895 Å, quantum critical points (QCPs) I and II, respectively. Copper–oxygen bond length values have been taken from refs¹⁶⁻¹⁹ and for CaCuO₂, Sr_1–xLa_xCuO₂, R₂CuO₄, La_2–xSr_xCuO₄, and Bi₂Sr₂CaCu₂O_8+δ are from our experiments. Cuprate superconductors containing copper in a square-planar coordination environment have vacant apical sites and such cuprates are therefore not susceptible to Jahn–Teller distortions (JT free in zone I). In contrast, the Jahn–Teller effect governs the entire formation of the crystal structure if copper is sixfold-coordinated (JT driven in zone III). Subtle deviations from an exact square-planar copper coordination cause a Jahn–Teller response that depends on the polarization field of the CRL above or below the CuO₂ planes, and cuprates in zone II are called “Jahn–Teller-compensated”.

Results and Discussion

To achieve such IL-CaCuO₂ crystals, we used molecular beam epitaxy.¹ For comparability, we synthesized IL cuprates CaCuO₂, SrCuO₂, and their siblings, where Ca has been substituted by 4% of Nd and Sr by 10% of La, respectively. Cross-sectional scanning transmission electron spectroscopy (STEM) images (Figure S6) of IL-Ca_1–xR_xCuO₂ and IL-Sr_1–xR_xCuO₂ confirm that these materials are, indeed, free of impurity phases or stacking faults. While those images are on the cation arrangements and, in principle, insensitive to possible defects related to oxygen sublattice imperfection, our specimens are reasonably free from such defects. It is widely known that some different phases are readily formed and contaminated the IL cuprate phase. Such impurity phases are also subject to the applied annealing conditions. Consequently, we applied the criteria of phase purity, which is a sufficient measure for oxygen sublattice perfection.⁵ Unfortunately, information from cross-sectional STEM alone is insufficient as it presents only a marginal snapshot of the copper–oxygen plane. Instead of cross-sectional STEM images, in-plane STEM images (see the Supporting Information) may unearth hidden distortions that are the result of the competition between the Jahn–Teller effect and the polarization fields which extend their influence on the CuO₂ planes. Accordingly, and we have taken high-resolution in-plane images (Figure 2) of the bare CuO₂ planes. These images reveal principally the CuO₂ plane, though there is more. For IL-CaCuO₂ and IL-Ca_0.96Nd_0.04CuO₂, the CuO₂ plane is blotched with rabbets, i.e., antisite boundaries, linear in shape with 90° corners. It is plausible that these CuO₂ planes ease their internal stress by forming rabbet structures and therefore the rabbet structures do form in IL-Ca_1–xR_xCuO₂ while they are absent in IL-Sr_1–xR_xCuO₂. In other words, cationic stripes formed for d_Cu–O < d_B but not for d_Cu–O > d_B. Independently, atomically resolved, element-specific line scans using energy-dispersive spectroscopy (Figure 2) verify that the Ca–Cu order has been upended (Figure 2, line scan (ii)) while forming rabbets and these rabbets have therefore a cationic character. In contrast, the oxygen position remains intact (Figure 3D). While anionic stripe formation is not seen for the IL-Ca_1–xR_xCuO₂ system, it is throughout possible and has been reported for doped oxygen anions in HgBa₂CuO_4+δ.²⁸ More importantly here is, however, that these cationic stripes are strong enough to sustain even with doping of 4%, as exemplified by atomically resolved, element-specific STEM images of IL-Ca_0.96Nd_0.04CuO₂ in Figure 3. The mechanism of cationic stripe formation is schematically visualized in Figure 3G, where a half-unit cell shift of the CuO₂ planes along the boundary gives rise to cationic stripes linked to structural dimorphism in the form of partially disordered nanoscale stripes.^14,29

Figure 2.

Cationic stripe formation in the CuO₂ planes of the infinite-layer CaCuO₂ system revealed by in-plane STEM mapping. (A) Perspective view on the CuO₂ plane of the infinite-layer cuprate with the tetragonal crystal structure (P4/mmm, S.G.: 123) of CaCuO₂. The dashed line is the scanning position and direction shown in (B) and the STEM intensity profiles are plotted in (C). The STEM scanning path in between two adjacent Cu sites and the detected element would be only Ca. (B) High-resolution cross-sectional transmission electron microscopy images taken of CaCuO₂ and Ca_0.96Nd_0.04CuO₂ grown on (001) LSAT substrates, SrCuO₂ on (001) SrTiO₃, and Sr_0.90La_0.10CuO₂ thin film grown on (110) DyScO₃ substrates. In (C), STEM intensity line scans taken at locations (i), (ii), and (iii) positions shown in (B) are plotted as a function of distance. Cationic stripes are formed in CaCuO₂ but not in SrCuO₂. In superconducting cuprates with a well Jahn–Teller-compensated CRL, e.g., Bi₂Sr₂CaCu₂O_8+δ, stripes within the CuO₂ planes are also not observed (Figures S2 and S3). Cross-sectional scanning transmission microscopy images of IL-CaCuO₂ and IL-SrCuO₂ are shown in Figure S6.

Figure 3.

(A) High-resolution STEM and energy-dispersive spectrometry (EDS)-mapped CuO₂ planes found in Ca_0.96Nd_0.04CuO₂. (B–E) Elemental distributions of Cu, Ca, O, and Nd, respectively. (F) Combined elemental map of Cu and Ca. Dashed lines represent irregularities and demonstrate the distortions of the CuO₂ planes in Ca_0.96Nd_0.04CuO₂. This is schematically shown in (G), where the CuO₂ planes are distorted by the presence of Ca at the Cu site. In Bi₂Sr₂Ca₂Cu₃O_10+δ, the position of Ca between the CuO₂ planes is slightly off its centric position^51,52 and the equilibrium position is different from what is observed in the infinite-layer CaCuO₂ phase.

After all, why do these cationic stripes form in IL-Ca_1–xR_xCuO₂ and are not seen in cuprates with comparable d_Cu–O, e.g., Bi₂Sr₂CaCu₂O_8+δ (Figures S2 and S3)? To answer this question, the polar character of the cuprates, where doubly negatively charged CuO₂ planes (CuO₂)^2– are sandwiched by doubly positively charged CRL (A_xO_y)²⁺ (A are chemical elements), has to be considered. Furthermore, real crystalline materials are inevitably subject to crystallographic imperfections³⁰⁻³³ and the resultant charge imbalance must be compensated within the smallest possible volume.³⁴ However, these considerations must incorporate also the impact of the Jahn–Teller principle in these cuprates³⁵ as local geometric arrangements of the O^2– ligands dictate them. Due to octahedral coordinated copper, cuprates in zone III (Figure 1) are subject to Jahn–Teller distortions and are built up from infinitely connected CuO₆ strata with massively elongated c-axis lengths; in zone III, the Jahn–Teller distortions are therefore explicit. Nonetheless, despite strong Jahn–Teller interventions in zone III, additional distortions—tilted CuO₆ octahedrons—are well known, for example, for La_2–xSr_xCuO₄ and La_2–xBa_xCuO₄.³⁶ Accordingly, cuprates in zone III have sufficient means to compensate for crystal imperfections without harming the CuO₂ planes. A prominent example where these mechanisms result in total loss of superconductivity while maintaining a static stripe structure is the “1/8-anomaly” in bulk La_2–xBa_xCuO₄.³⁷⁻⁴¹ As IL-CaCuO₂ is a cuprate with square-planar coordinated copper, zone I is of higher importance. Notably, the apical positions of these cuprates are vacant and their absence gives rise that the copper ions are not susceptible to Jahn–Teller distortions per se (JT free). Commonly, these cuprates crystallize in the T′ structure with d_Cu–O > d_B. The large d_Cu–O keeps cuprates with T′ structure away from cationic stripe formation, while being still perceptible to electronic stripes.^42,43 Instead, the T′ structure is known to be prone to point defects, so-called interstitial oxygen atoms, preventing superconductivity, which may be also the case with IL cuprates. Oxygen vacancies, if any, also matter as exemplified by the work by Leca et al.,⁴⁴ who grew superconducting IL cuprate films in a low oxidizing atmosphere and oxidized them after the growth. Solutions are, nonetheless, in place to eliminate such defects and instigate superconductivity. In IL-SrCuO₂, d_Cu–O almost coincides with d_B, and cationic replacement of Sr by La places d_Cu–O comfortably above d_B and gives rise to superconductivity. Note that any substitution of Ca in IL-Ca_1–xR_xCuO₂ (R = La, Nd, Sr) does not sufficiently expand d_Cu–O so that d_Cu–O >d_B.

In strict terms, four- and five-coordinated copper is not Jahn–Teller-active. Nonetheless, weak Jahn–Teller activities for four- and five-coordinated copper are known^11,45,46 and IL-Ca_1–xR_xCuO₂ has a d_Cu–O, where such influences are common. One might be inclined to raise the question as to why such a structure does even exist. The synthesis conditions of IL cuprates^1,3,5 are significantly off from thermodynamic equilibrium conditions. Moreover, the square-planar configuration of Cu²⁺ is rare among copper complexes^47,48 and infinitely connected CuO₄ strata are only known for IL and T′ structure cuprates. The famous compound, Egyptian Blue (CaCuSi₄O₁₀), does have square CuO₄ plaquettes with d_Cu–O ≈ 1.928 Å, (equal to IL-CaCuO₂) though they are not infinitely connected.^49,50 After all, the IL-Ca_1–xR_xCuO₂ system must be considered as metastable and highly susceptible to structural distortions driven by the Jahn–Teller effect to lower the energy of this crystallographic system. These effects cumulate in the formation of cationic stripes within the CuO₂ planes and this is irreconcilable with superconductivity.

Superconducting cuprates located in zone II, particularly those with very high T_c (HgBa₂Ca₂Cu₃O_8+δ, Tl₂Ba₂Ca₂Cu₃O_10+δ, and Bi₂Sr₂Ca₂Cu₃O_10+δ), accommodate a modified IL-CaCuO₂, where the calcium ions are now shifted further away from the CuO₂ plane (Figure 3H), and this modification is sufficient to lower the local symmetry of the system. For example, in Bi₂Sr₂Ca₂Cu₃O_10+δ, the infinite layer is sandwiched between BiO_1.5+δ layers that are sufficiently potent to outweigh local charge imbalances nascent from crystallographic imperfections, a function unachievable in the plain CaCuO₂ system. Actually, the distance between two adjacent CuO₂ planes that is coerced by adamant Ca²⁺ ions in IL-CaCuO₂ (c = 3.1766 Å) is, by far, shorter than what is seen in Bi₂Sr₂Ca₂Cu₃O_10+δ (c = 3.3077 Å).^53,54 Additionally, the bonds of BiO_1.5+δ are very adaptive,⁵⁴ in contrast to unmalleable calcium ions.

In cooperation, these two buffering mechanisms (the variable stoichiometry in BiO_1.5+δ and the tunability of the Ca equilibrium position) format the environment of copper in a way that tolerates the IL-CaCuO₂ strata to be stable in cuprates of zone II. In other words, for a given Jahn–Teller potential there exists a chemistry that maintains this crystal structure without violation of the CuO₂ plane stability and it is exactly this part that is missing for the infinite-layer phase of CaCuO₂.⁵⁵ In contrast to the general assumption where the Jahn–Teller effect is being considered as a consequence of the electronic degeneracy owed to an unconventional bonding scheme specific to cuprates to form infinitely connected CuO₂ planes,⁵⁶ it is the polarization of the chemicals above and below the CuO₂ planes that promote or quench the implicit Jahn–Teller instability. The fact that the electronic polarization counteracts Jahn–Teller distortions in cuprate superconductors which are, de facto, superlattices, results in structural rearrangements, i.e., stipes which is a commonly accepted mechanism in multiferroic systems.⁵⁷ Several local distortions are possible and one of them is suitable to stabilize an overall superstructure, i.e., stripe formation,³⁷ as observed here in IL-Ca_1–xR_xCuO₂. Those stripes take away the stabilizing criteria of the infinite-layer phase, thus resulting in a progressive self-distractive inroad of the CuO₂ planes. Regrettably, these aspects are not covered by methods commonly used to derive electronic response of materials, e.g., density functional theory.⁵⁸ However, Jahn–Teller effects are of sufficient strength to amplify related electronic properties by several orders of magnitude, as exemplified in Dy(Fe,Mn)O₃ perovskites.^35,58

The remaining point to be clarified is how the presence or absence of cationic stripes is reflected in the electronic response functions of the IL cuprate system. We characterized T_c, the Hall coefficient R_H, and the mean-free path length (see the Supporting Information) and combined those results with STEM electron energy loss spectroscopy data taken at the Cu L₃ and O K-edges (see the Supporting Information). For superconducting IL-Sr_0.90La_0.10CuO₂ there exist a finite density of states (DOS) at the Cu L₃ as well as the O K-edges (Figure 4A,B) and we assign a Cu 3d–O 2p hybridized state at the Fermi level, as is a common procedure for transition-metal oxides.^61,62 Knowing that cationic stripes are blotching the CuO₂ planes of IL-Ca_0.96Nd_0.04CuO₂, this is expected to change. Indeed, the DOS at Cu L₃ as well as the O K-edges is strongly depleted. The consequence of such a low DOS should become apparent in the electronic transport behavior. For all IL-Ca_0.96Nd_0.04CuO₂ samples investigated here, is at most 5 unit cells (Figure 4C) and this length is consistent with the cationic stripe density seen by in-plane STEM measurements (Figure 2). As expected, this situation becomes more dramatic for pristine IL-CaCuO₂. At the Fermi level, an electronic excitation gap appears while ≪ 1 u.c. Effectively, in IL-CaCuO₂, the polar environment around its CuO₂ planes prevents the formation of a metallic state, and so does IL-SrCuO₂. At the same time, the positive R_H seen for IL-Sr_0.90La_0.10CuO₂ turns negative. Interpolating between these extreme points by IL-Ca_0.85Sr_0.15CuO₂ shows a population of DOS at the Fermi level though electronic conduction is steadfastly insulating (Figure S1). This may also arise from a very minute difference in local oxygen sublattice perfection between IL-Ca_0.85Sr_0.15CuO₂ and two end members (IL-CaCuO₂ and IL-SrCuO₂), requiring more detailed investigations. If is small, then R_H is negative and vice versa, and this correlation suggests that reconstruction of the CuO₂ planes, i.e., stripe formation, is the very reason for the absence of superconductivity in cuprate superconductors. It seems that this might not be different for IL-SrCuO₂. Its electron energy loss spectroscopy (EELS) spectra are almost identical to those of IL-CaCuO₂ or IL-Ca_0.96Nd_0.04CuO₂, whereas rabbets are not apparent from the in-plane STEM imagery. As d_Cu–O is somewhat smaller than d_B, IL-SrCuO₂ appears to be at the brink of cationic stripe formation while the density of rabbets might be low enough to be considered as point defects. Hence, IL-SrCuO₂ is an antiferromagnetic insulator, both locally and globally (Figure S1).

Figure 4.

(A) Electron energy loss spectra of Cu and O embedded in CuO₂ planes with Cu in a square-planar coordination environment. Here, we used CaCuO₂, SrCuO₂ Ca_0.85Sr_0.15CuO₂, Sr_0.90La_0.10CuO₂, and Ca_0.96Nd_0.04CuO₂. The Cu L edge is affected by the effective polarization brought onto the CuO₂ plane by varying the effective electrostatic potential from Ca²⁺ through Ca_1–x²⁺Sr_x (x = 0.15) to Sr_1–x²⁺La_x (x = 0.10). For Sr_0.90La_0.10CuO₂, high-temperature superconductivity appears at T_c = 41 K. The insulating states of CaCuO₂ and SrCuO₂ are distinct. For IL-CaCuO₂ and IL-Ca_0.96Nd_0.04CuO₂, one-dimensional cationic stripes are present and interrupting the CuO₂ planes, whereas in SrCuO₂, such stripes are not observed. For Ca_0.85Sr_0.15CuO₂, the electronic response is at the brink of a metallic state. The electronic response function seen at the Cu L edge (A) corresponds exactly to the O K edge (B), where the distribution of spectral weight is identical. (C) Plot of the superconducting transition temperature T_c as a function of the mean-free path length in CaCuO₂, Sr_0.90La_0.10CuO₂, and Ca_0.96Nd_0.04CuO₂. is limited due to cationic stripes in the CuO₂ planes, and this relates to the appearance of superconductivity. For the induction of superconductivity, is required to be at least 4 Å, which corresponds to the Cu–Cu distance (a₀) of the CuO₂ planes. For a small , the Hall coefficient R_H is negative (<−1×10^–8 Ωm/T), and for R_H ≈ 0 Ωm/T, ≈ 30 nm. For Bi₂Sr₂CaCu₂O_8+δ and Bi₂Sr₂Ca₂Cu₃O_10+δ, R_H is positive⁵⁹ with > 10 nm.⁶⁰

Conclusions

We have proposed a concept that combines all cuprate superconductors into a singular phase diagram. The control parameters of this superconducting cuprate phase diagram are d_Cu–O, the polar character of the CuO₂ planes, and the Jahn–Teller instability. We argue that this concept has important predictions that are essential for the comprehension of cuprate superconductors, which are de facto superlattices.

First, the local crystal structure in the vicinity of the CuO₂ planes is determined by the Jahn–Teller instability and Pauling’s second rule.³⁴ Notably, the CuO₂ planes are the playground of high-T_c superconductivity.

Second, doping aka substitution or addition of chemical elements modify the effective electronic polarization of the CRL and this change in polarization affects d_Cu–O. Heterovalent substitution, where additional electrons populate antibonding Cu 3d–O 2p orbitals, further modifies the effective polarization. The c-axis response of the cuprate is in no way influenced by substitution and simply follows Vegard’s law.

Third, the Cu–O bond lengths of cuprates with CRLs appear in a way where the Jahn–Teller effect responded to the imposed polarization strength of the CRLs.

Forth, one could say that, within certain boundary conditions, there exists for every d_Cu–O a chemistry, which is sufficiently potent to balance the Jahn–Teller influences in a way that allows for infinitely connected CuO₂ planes, thus superconductivity.

Fifth, if this balance cannot be maintained, the entirety of the cuprate will surrender to the Jahn–Teller principle and form cationic stripes. In the case of IL-CaCuO₂ and IL-Ca_0.96Nd_0.04CuO₂ presented here, cationic stripes disrupt the CuO₂ planes and are consequently fatal to superconductivity.

Sixth, the commonly observed disorder of the CRL for cuprates with very high T_c is helpful⁶³ to effectively counteract charge imbalances stemming from point defects in the CuO₂ planes. These imperfections must be, however, limited to the CRL, not the CuO₂ planes.⁶⁴

Finally, other order parameters, e.g., charge- and spin-density waves, the pseudogap, and long-range antiferromagnetic order (Table S1) are in no contradiction, though might be considered as a dressing of the superconducting state.

In this context, higher superconducting transition temperatures in cuprate material systems are reachable, and such an approach will require thin-film materials synthesis methods. To date, the common cuprate superconductor is a materials system, built by thermodynamic constraints that are defined by bulk synthesis methods. An elegant way to bypass those limits, while enabling novel cuprate material systems, are thin-film synthesis methods that cater to the need of a highly diversified stratification approach. A first step in this direction was done by Castro et al.,⁶⁵ using SrTiO₃ as a CRL. This new material, in its entirety inaccessible to bulk synthesis methods, is superconducting and more materials combinations should be considered to synthesize cuprates with higher T_c.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.1c01720.

• Growth of infinite-layer cuprates by molecular beam epitaxy; electronic transport measurements; superconducting Bi₂Sr₂CaCu₂O_8+δ crystals as well as related high-resolution in-plane scanning transmission electron microscope imagery; and energy scale calibration of electron energy loss spectroscopy at relevant oxygen energy levels (PDF)

The authors declare no competing financial interest.