In 2014, a review of 56 panels was published by the present author, with the aim to describe, using the concept of polarons, the properties of the superconductivity in the hole-doped cuprates [1]. From that review, here, I extract and rearrange those suitable for possibly giving an interested reader a shorter access to the understanding achieved. A decade earlier, ten relevant articles were published in a book [2] indicating the status, which, in the present effort, has become more concrete thanks to new experimental and theoretical progress.

1 Bipolarons

The discovery of superconductivity in hole-doped LaCuO 4 [3] resulted from the concept of Jahn–Teller (JT) polarons, as theoretically postulated in the group of Thomas at the University of Basilea [4]. In such a polaron, a hole carrier is trapped near a Cu2+ ion, which in octahedral oxygen symmetry is degenerate regarding its d orbitals. Locally, the oxygen ions are displaced according to the JT active conformations. Very early after the discovery of this superconductivity, it was recognized that such a polaron with effective spin S = 1 /2 would be nearly immobile in the antiferromagnetic (AFM) insulating lattice of the Cu2+ ions of a cuprate at low temperatures (Mott AFM). This is because if it moved, the spins of the Cu2+ would be forced to be turned around, which requires a large amount of energy. Maintaining the polaronic concept, but avoiding the S = 1 /2 difficulty, the bipolaron with S = 0, i.e., two polarons with antiparallel spin S, was proposed quite early [5]. In case, this would be an amenable way to understand the much investigated copper-oxide superconductor, the structure of such a possible quasiparticle nevertheless remained open. This situation lasted until Kabanov and Mihailovic [6, 7] proposed the intersite JT bipolaron in 2000, see Panel 1 (illustration on the right from [6, 7]).

Panel 1

figure a

In brief, this quasiparticle consists of two JT polarons next to each other in the lattice, having an O2− ion in common and the spins of the two JT polarons antiparallel, such that S = 0 results. Furthermore, the spins of the two JT polarons would also be oriented antiparallel to those on the Cu2+ ions of the bipolaron. The spins of the latter remain oriented antiferromagnetically to the Cu2+ spins in the AFM lattice, see Panel 2.

Panel 2

figure b

The two authors arrived at their proposal from three experiments: (1) The local structure determination with x-ray scattering in Bianconi’s group in Rome [8] (Panel 3, right; from [8]); (2) the structure as determined by inelastic neutron scattering in the group of Egami in Oak Ridge [9, 10] was also relevant (Panel 3, left; from [7]) and (3) especially the analysis of the electron paramagnetic resonance results of Sichelschmitt in Elschner’s group in Darmstadt by Kochelaev in terms of a three-spin JT polaron (Panel 3, center; from [6]) [11].

Panel 3

figure c

The three-spin JT polaron consists of one hole carrier trapped near two Cu2+ ions, with only an oxygen ion between them. Trapping a second hole carrier results in the intersite JT bipolaron, which has a lower energy than the three-spin JT polaron, despite the fact that the two carriers with the same charge repel each other. The bipolaron is basically an effective negative U center as proposed earlier by Anderson to occur in disordered semiconductors [12]. The reason for this negative U is the motion of the ions of the center, which overcomes the Coulomb repulsion between the two charges. The proof of the existence of such centers was a substantial success. With it, the absence of magnetic resonance in doped disordered semiconductors (spin S = 0) was understood at once. It could also explain the absence of conductivity in an interval with the presence of a mobility edge, etc. Anderson assumed in his paper that the mobility of the negative U center could be neglected because of the presence of the random electrical potential present in disordered materials. This is, however, not the case in the cuprates, which are basically regular crystals. Because of their mobility, bipolarons, as they are called in the high-temperature superconductors (HTS), can interact and as such clusters are charged, they repel each other by the Coulomb force. However, there is also the elastic force present, which overcomes the Coulombic one and leads to clustering and stripe formation as impressive simulations at the Josef Stefan Institute in Ljubljana have shown [13, 14]. Actually, the interaction is such that the bipolarons align along their {10} axis rather than perpendicularly to each other (Kochelaev, pers. comm.). The well-known stripes result, see Panel 4 (from [14]).

Panel 4

figure d

Panel 4 shows snapshots of simulations published in ref. [13, 14] and obtained for t = 0.04, n = 0.2, and v l (1,0), as a function of v l (1,1), where t is the reduced temperature (T/T c ), n is the density, and v l (1,0) and v l (1,1) stand for the short-range elastic nearest and next-nearest neighbor interactions. Clearly visible is the formation of elongated clusters/stripes, which, depending on the interaction of v l (1,1), are aligned along {1, 0} or {1, 1} in the plane. Interestingly, it was found that the hole-rich clusters with an even number of particles are more stable than those with an odd number.

2 The Phase Diagram

With the bipolaron model, it is possible to reproduce many features of the hole-doped cuprate phase diagram. It enables not only the quantitative prediction of the critical concentration at 6% hole doping for the occurrence of superconductivity on the underdoped side, but also an estimate of the T c from experimentally measured values of the pairing energy [15]. Furthermore, there exists a quasi-metallic region below a temperature T ∗(n) > T c (n) first detected via specific-heat experiments and showing a gap in tunneling data. T ∗(n) decreases nearly linearly with n, as was first reported by Deutscher [16] from Andreev reflections. A more recent diagram including data from many experiments is shown below in Panel 5 [17].

Panel 5

figure e

Panel 5 reproduces data from many structurally different cuprate systems showing the progression of T ∗ and T c , where the former curve approaches the latter tangentially above optimum doping. This excludes all magnetism-based theories, which yield a quantum critical point at T = 0 near optimum doping, because the T ∗ line would have to intersect that of T c near this doping level.

With the bipolaron model, a number of properties could be clarified. For instance, the nature of the pseudogap shown: At a given doping level n as a function of decreasing T above T ∗(n), single JT bipolarons exist, which below this temperature start to aggregate and form clusters or better stripes, see Panel 6 (from [1]).

Panel 6

figure f

In these cluster stripes, type 0D superconductivity is present as a number of tunneling experiments have shown gap-type signatures [18]. Upon further lowering the temperature, phase coherence and 3D superconductivity set in at T c (n). Important in this process is that the metallic-type stripes are by no means static entities. They emerge in the AFM lattice at a time, move, and disappear. This dynamic behavior has been documented in the group of Mihailovic in Ljubljana [19], i.e., these cuprates are intrinsically heterogeneous in a dynamic way. The pulse probe work at the Joseph Stefan Institute, more recent investigations by Oyanagi using EXAFS to record Cu–O in-plane bond lengths as a function of temperature, extensive EPR of Mn probes, and the oxygen isotope effect on the penetration depth have all been shown in a recent review by Shengelaya and Müller [20] to yield quite a clear picture of the dynamics of the heterogeneous state present in the layered hole-doped copper oxides.

3 Oxygen Isotope Effects

In the classical superconductors, the isotope effect was substantial in pointing to and supporting the Bardeen–Cooper–Schrieffer (BCS) theory. The key point is summarized in Panel 7: the shift in T c is proportional to the mass of an elemental superconductor to an exponent α, which is 1/2, as predicted by BCS and observed in most of the cases.

Panel 7

figure g

In cuprates, relevant oxygen isotope effects, i.e., substitution of the naturally abundant 16O in the oxide by the 18O isotope, have been reported on T c (n) and T ∗(n). We start with the former, in which two groups carried out nearly simultaneously experiments [21, 22], both done at optimal doping, i.e., with n yielding a maximal T c . A nearly vanishing effect was found. This indicated a purely electronic origin of the superconductivity observed in the cuprates, and supported the resonating valence bond (RVB) theory of Anderson [23] and the t-J model of Zhang and Rice [24]. These theories were since then followed also with similar models by a large part of the community. In contrast, using slightly underdoped samples, the group of Frank reported a clear isotope effect [25]. Since then, the group of Keller at the University of Zurich, with a substantial effort, not only confirmed the Canadian results, but even measured the oxygen isotope effect as a function of the doping for four different compounds [26, 27], see Panel 8 (from [28]).

Panel 8

figure h

This figure was obtained by Steven Weyeneth, whose picture appears on the right side, and the author [28]. It shows the vanishing effect at optimum doping plus its growing to a value of 1, twice as large as the one obtained by BCS theory! The curve shown follows the data very closely and was obtained by Kresin and Wolf in 1994. Its formula is quite simple, see Panel 9 [29].

Panel 9

figure i

Therein γ(n) is a slowly varying function of n, and has been taken as constant in obtaining the curve in Panel 8. The formula was obtained under the assumption that the polarons are aligned along the c-axis of the crystal, as shown in Panel 9. This theory has been overlooked because it considered polarons with their axis along c and because when it was published, experiments were scarce. Moreover, the derivation was mathematically not complete. Later the group of Keller, by selectively doping the 18O especially in and out of the CuO 2 planes—a real tour de force—showed that the main contribution to the oxygen isotope effect resulted from the oxygens in the C u O 2 planes [30]. Therefore, the relevance of the formula was not obvious. The present author then recognized that the formula should remain correct for polarons lying in-plane, and the excellent agreement with the data in Panel 8 supports this view. Furthermore, a mathematically amenable re-derivation of the formula was accomplished recently by Kochelaev, Müller, and Shengelaya [31]. This re-derivation in a reproducible manner is based on the original paper of Kresin and Wolf [29]. Therefore, Panel 8, in the opinion of the author, unambiguously supports the polaronic origin of the superconductivity in the hole-doped copper oxides.

Regarding the oxygen isotope effect present on T ∗(n), quite early two entirely different types of experiments were carried out: First, via x-ray near-edge absorption spectroscopy (XANES) and then with inelastic neutron scattering. We start with the results of XANES done by Lanzara et al. in Rome [32], see Panel 10 (from [32]).

Panel 10

figure j

Shown in Panel 10 is the ratio R of the fluorescence counts A 1 and B 1 near the absorption edge as a function of temperature. R depends on the nearest neighbors either out of plane by A 1 or in plane by B 1. One can therefore monitor the change in the Cu2+ to local O neighbor distance either in or out of plane on an extremely short time thanks to the x-rays. At T ∗, a substantial change is detected for the in-plane distance, with the largest oxygen isotope effect of 70 K ever published to the day. This effect was discussed away by the followers of the entirely electronic theories by proposing that, in fact, at this temperature, a structural phase transition (SPT) was present in the La 1.94Sr 0.06CuO 4. However no such SPT was found there.

To confirm the above-mentioned important finding, experiments with an entirely different method, namely, inelastic neutron scattering and the stoichiometric superconductor YBCO 1248 were carried out in the group of Furrer at the PSI in Würenlingen (Switzerland). In the compound YBCO, the rare-earth Y 3+ was replaced by the HO3+ with partially filled 4f shell, whose internal transitions could be detected by neutron scattering [33], see Panel 11.

Panel 11

figure k

Shown in Panel 11 is the transition of the ground state to the first excited Γ4 of HO3+ as a function of temperature. It is proportional to the \({A_{2}^{2}}\) crystal field. At T ∗, a jump occurs with a 16O tO18O isotope shift of 50 K comparable to that observed in LSCO with XANES (Panel 10). Also shown is a picture of D. Rubio Temprano, whose thesis work was performed under the direction of Prof. Albert Furrer at the ETH Zurich.

To substantiate the enormous oxygen isotope shift at T ∗ further, experiments in LSCO were carried out in which the La3+ was substituted in part by HO3+, whose Γ1 to Γ4 transition was recorded for both 16O tO18O and 63Cu tO65Cu substitution [34]. Whereas for the former, an isotope shift of 10 K occurs, there is none for the latter. The reason is that in LSCO, the Cu2+ nearest neighbors are oxygen ions on octahedral lattice sites. There exists an inversion symmetry in their motion that is not crystal-field-active. In contrast, in YBCO, an isotope shift is observed for 63Cu tO65Cu substitution, as expected, because here, the Cu2+ has five nearest neighbors, located on a pyramid, and the copper lacks inversion symmetry (the so-called Röhler mode is active). Note also that the isotope effect on T* is negative, i.e., opposite to the one observed on T c (n). This is what one expects from the Kresin–Wolf formula, in which the derivative of the transition temperature in question appears; see Panel 12 (adapted from [34]). Ignoring consistently and over many years the oxygen isotope effects presented above, which clearly support the polaronic basis of the superconductivity in hole-doped cuprates, is scientifically not acceptable.

Panel 12

figure l

4 The Vibronic Theory

In quantum physics, there are two basic ways to deal with these phenomena. The particle description of Heisenberg and the Schrödinger wave equation, which are, as Pauli has shown [35], equivalent. This appears to be also the case in the superconductivity of the copper oxides. The first, the quasi particle one, has been outlined in Section 1, the second is the vibronic one by Bussmann-Holder and Keller [26, 27]. In it, a lower d-band is coupled via phonon interaction to a higher-lying p-band. The former is at the basis of the single-band electronic models of RVB [23] and t-J [24]. However, it is just this coupling which yields the high transition temperature. The essence of this theory is given in Panel 13.

Panel 13

figure m

The Hamiltonian in Panel 13 consists of four terms: The first encompasses the electronic d-functions of the Cu2+ ions present, found in the t-J model [24]. The second is due to the oxygen p-band. The third is the vibronic interaction between the two bands: In it, the first is the linear interaction between the electronic densities and the respective ionic displacements, and the second is the exchange between the two bands. The latter is responsible for only one transition temperature and not for two, as predicted by the author almost two decades ago [36]. The last term describes the usual lattice dynamics. This model yields polarons with the binding energy amount shown. They are proportional to the square of the interaction constant γ and inversely proportional to the ionic mass and the phonon frequency squared, see [26, 27].

Actually, from this theory, with sufficiently large vibronic coupling, the effective interaction energy U eff becomes negative, i.e., bipolarons are formed [26, 27]. From the vibronic theory, the oxygen isotope effects on T c (n) could be well described with the local JT conformation t 2. Also, the oxygen isotope effect on T ∗(n) followed (Bussmann-Holder, pers. comm, 2015). The results obtained from the vibronic theory use the mean field approach. Thus, in the regions where the carrier doping is low and the charge distribution becomes more grainy, the experiments deviate from this theory. This was the case for the oxygen isotope effect on T c (n), but the earlier theory by Kresin and Wolf of 1994 yielded a spectacular agreement, as outlined in Section 3.

5 The Symmetry of the Superconducting Wave Function

The above overview constitutes the four first sections of this review. In the present section, attention is turned to the symmetry of the condensate and then in the next one to the superconducting behavior of the electron-doped copper oxides. The electronic single-band t-J theory and those related to it predicted a d-wave symmetry for the superfluid. Photoemission, tunneling, and especially the much hailed tricrystal experiments of Tsuei and Kirtley [37] apparently proved the reality of this prediction and contributed to the acceptance of these theories. What was overlooked and pointed out by the present author was that all these experiments probed the surface only [38] because of the extremely short coherence length. Experiments sensitive to the bulk, such as susceptibility, very early and recent NMR, photo-reflectivity, and especially muon spin rotation, yielded a different answer, see [1]. Actually, because the muons stop near the surface or in the bulk, depending on their incoming kinetic energy, the crossover from pure d-wave at the surface to s + d symmetry in the bulk obtained from group theory by Iachello [39] could be quantitatively demonstrated. Therefore and in contrast to the long followed electronic single-band theories, these experiments prove unequivocally the vibronic character in which a lower-lying d-band is coupled to a higher-lying oxygen-based p-band as described in Section 4.

We start by summarizing, in Panel 14 (from [1]), the essential observations obtained so far regarding the symmetry of the superconducting wave function in the hole-doped copper oxides [40]. They will then be shown in more detail in the subsequent panels.

Panel 14

figure n

In the CuO 2 plane, the symmetry is 100% d at the surface, and inside it is so far up to 75% d and 25% s, in agreement with the group-theoretical analysis of Iachello [39], and, surprisingly, 100% s along the tetragonal c-axis. The latter property could be derived from the vibronic theory discussed in Panel 13 (Bussmann-Holder, pers. comm, 2015).

The time span regarding the symmetry of the superconducting wave function started quite early in 1991, when Bulut and Scalapino [41] published their calculation on the inverse NMR T 2G relaxation times as a function of temperature down from the phase transition, as shown in Panel 15 (from [42]), i.e., for d- and s-wave symmetry.

Panel 15

figure o

Included in the figure are also the data on the T 2G relaxation times of LSCO of Brinkmann’s group at the University of Zürich from 1995 [42]. As seen, they lie in between the theoretically calculated two curves. Interpolating linearly between the two, one arrives at 20 to 25% s for the bulk. However, as noted later, this constitutes a lower limit for the s-wave component since, owing to the magnetic field, which is necessary for NMR, vortices are present, which yield an enhanced d-wave component at their border. It was in 1995 that the present author proposed the presence of s- and d-symmetry for the bulk superconducting wave function [36].

Since the experiments which yielded the above-described property of the bulk already in the mid-1990s, other experiments supported this and differences from the surface have been emphasized by the author in Phil. Mag. [38] as well in a conference review [43]. Both are in full agreement with a group theoretical effort by Iachello [39], in which he deduced a 100% d-wave symmetry on the surface towards a and s + d in the bulk a-b plane of the cuprate superconductor. To stay with the intention of a compact presentation, the reader interested historically or more deeply may look into the references cited, and we present here the results obtained later at the Swiss Muon Source at the PSI in Villigen by R. Khasanov and collaborators, initiated by Prof. H. Keller using muon spin rotation. Muon rotation is a local probe so that the muons, depending on their energy, stop either near the surface or in the bulk, and thus are ideally suited for investigating the wave function as a function of the distance from the surface. The relaxation rate σ sc(T) is inversely proportional to the square of the London penetration depth λ(T). At low magnetic field, the two s- and d-components are clearly visible in Panel 16 (from [44]), but less so at higher magnetic fields. This is due to the presence of more vortices at the surface and therefore has d-character [43], see also Panel 14. On the left, the decomposition of σ sc(T) into the two components is shown.

Panel 16

figure p

In the bulk, the symmetry was investigated at the PSI by the group of Khasanov: In Panel 17 (left from [45], right adapted from [46]), we show the square of the inverse of the London penetration depth as detected by muons stopped in the bulk of LaSCO along the crystallographic a-, b-, and c-direction. For a and b, because of the nearly tetragonal crystal, they are the same, as also seen in the left figure. But along the c-axis, it is pure s! This is borne out by the vibronic theory, see Panel 13.

Panel 17

figure q

In NMR, as well as in muon spin rotation experiments, a magnetic field has to be present which automatically manifests that also vortices are present. At their surface, for normal conductivity, the character is d-wave. Therefore, the average deduced symmetry is an enhanced d-wave one as compared to the bulk without magnetic field [43]. A more reliable s-to-d ratio can be deduced from the x-ray data of Oyanagi [47], obtained, of course, in the absence of a magnetic field. He recorded the local distance fluctuations between the in-plane Cu2+ and ligand O2− in YBCO as a function of temperature and found substantial enhancement, i.e., peaks, towards the temperatures T ∗ and T c . The analysis of Bussmann-Holder with her vibronic theory gave a 40% s- and 60% d-wave character of the superconducting wave function in the bulk [48]. These amounts may be regarded as the intrinsic ones (Panel 18).

Panel 18

figure r

To obtain superconductivity in electron-doped copper oxides, a special material treatment was required. In their paper, Dagan and Greene [49] summarized some of these efforts in which the importance of the presence of additional holes was mentioned. In this paper, they reported superconductivity in electron-doped Pr 2−x Ce x CuO 4, which was tested with resistivity and Hall angle measurements. Basically, they found that the mobility of the electrons is larger than that of the holes present and mask the latter in the normal conducting phase, but the holes present induce the superconductivity. They arrived at the conclusion that “the electrons have no (or a very small) contribution to superconductivity in the electron-doped cuprate superconductors” [49]. This important conclusion justifies the restriction of the present review to the hole-doped copper-oxide superconductors.

6 Summary and Conclusions

In Sections 1 and 3, the theories for the polaronic properties of hole-doped copper-oxide superconductors are described with their particle and Schrödinger wave function properties, due to Kabanov and Mihailovic [6, 7] and Bussmann-Holder [26, 27], respectively. These two theories have been shown to yield the following properties of the superconductors in question. (a) In the latter theory, the coupling of the lowest (Cu) d-type band (occurring also in the t-J model) to the next higher (O) p-band yields the high transition temperatures observed. (b) The phase diagram of all hole-doped cuprates results from the particle picture, see Section 2. (c) In Section 3, the observed, in part substantial oxygen isotope effects at both the crossover temperature T ∗(n) to the pseudogap region and from the latter at T c (n) to superconductivity is obtained. (d) In Section 4, an overview regarding the symmetry of the superconducting wave function is given. Here, the essential property is that the latter is different in the a-b plane at the surface from that in the bulk and nearly 100% s along the c-axis. (d) In a final note, the situation in the electron-doped cuprates is sketched, in that also there the presence of hole pockets is the key.

In conclusion, the two quite equivalent polaronic theories described deliver the main properties of the superconductivity in the copper oxides known. In these theories, the Born–Oppenheimer principle, in which the electronic degrees of freedom can be decoupled from the motion of atomic nuclei, mostly valid in condensed matter physics, is not valid, as evident from the essential presence of bipolarons. However, the single-band electronic theories, such as the RVB or the t-J model, assume at their outset the validity of this approximation; therefore, they do not apply, and are unable to explain most of the observations presented in Sections 3 and 4. However, the progression from the t-J single-band model to the vibronic theory by coupling the ground-state single d-band to the one above it, which is the oxygen p-band, yields the valid vibronic theory. This essential step is described in ref. [2] in a didactic manner in the last chapter by Bussmann-Holder et al.