Multi-Band Superconductivity, Polarons, Jahn-Teller Polarons, Heterogeneity and High-Temperature Superconductivity

1. Introduction

The first oxide which exhibits superconductivity was already discovered in 1964, namely reduced SrTiO₃ (STO), which becomes superconducting at $T_{\mathrm{c}}$ = 0.3 K [1]. At this time this was a remarkable observation which caused novel theoretical interpretations well beyond BCS theory [2]. Especially the soft transverse optic mode of STO was supposed to play a crucial role for the pairing mechanism. New interest in doped STO arose only recently when more attention was paid to the ultralow carrier density in this compound. Soon after STO superconductivity was reported in 1964 in the tungsten bronze Na_xWO₃ with $T_{\mathrm{c}}$ = 0.5 K [3]. In 1973 LiTi₂O₄ was discovered with $T_{\mathrm{c}}$ = 13.7 K followed by the BaPb_1−xBi_xO₃ in 1975 with $T_{\mathrm{c}}$ = 13 K [4]. In 1980 Nb doped SrTiO₃ with a maximum of $T_{\mathrm{c}}$ = 0.8 K was observed which is the first two-gap superconductor [5]. Like in the BaPb_1−xBi_xO₃ system, $T_{\mathrm{c}}$ adopts a dome like dependence on x, respectively Nb, now well known from the cuprate high-temperature superconductors (HTS’s). What is remarkable about these oxide is the fact that the electron (hole) concentration is only $n=2-4\times {10}^{21}/{\mathrm{cm}}^3$. This is almost one order of magnitude smaller than in a normal (superconducting) metal. While this fact has long been overlooked, recently it has attracted a lot of attention where many new interpretations have been offered [6].

Following, however, KAM’s strategy, he concluded that if the density of states in these oxides is so low, then the pairing interaction must be extremely large in order to achieve the observed superconducting transition temperatures. In conventional weak coupling BCS theory the formula for $T_{\mathrm{c}}$ is [7]:

$$T_{\mathrm{c}}=1.13{\Theta }_{\mathrm{D}}exp[-{\displaystyle \frac{1}{N\left(E_{\mathrm{F}}\right)V^{*}}}],$$

(1)

where $N\left(E_{\mathrm{F}}\right)$ is the electron concentration per elementary cell at the Fermi edge, $V^{*}$ is the electron-electron interaction, which must be attractive to obtain superconductivity. Then one notices that the product $N\left(E_{\mathrm{F}}\right)V^{*}$ must exhibit a certain order of magnitude to obtain a finite transition temperature $T_{\mathrm{c}}$. This requires for the oxides that if the electron concentration $N\left(E_{\mathrm{F}}\right)$ is small, the interaction $V^{*}$ must be very strong to achieve these large values of $T_{\mathrm{c}}$. This leads directly to the question:

What is so special about oxides? Why holes instead of electrons?

The oxygen ion and also the heavier chalcogenide ions in their doubly negatively charged state are peculiar as compared to almost all other ions, since they are unstable as free ions.

In simple oxides like MgO, CaO, SrO, the oxygen ion polarizability $\alpha \left({\mathrm{O}}^{2-}\right)$ depends linearly on the volume V of the oxygen ion, in tetrahedrally coordinated oxides this dependence is enhanced to a $V^2$ dependence. In anisotropic configurations as realized in spinels and ferroelectric perovskite oxides $\alpha \approx V^{3-4}$ [8]. In addition, it is a strong function of temperature and pressure. These properties are a consequence of the fact that the outer 2p-electrons of O₂₋ tend to delocalize and hybridize with nearest neighbor transition metal d-states. The degree of hybridization can be triggered through the dynamics thereby leading to a "dynamical covalency" [9]. The oxygen ion cannot be assigned to a fixed rigid ionic radius but instead space increments have to be introduced which can be related to $p-d$ hybridization degrees of freedom.

Quantum mechanical calculation of the oxygen ion polarizability within the Watson sphere model [10], where the free O₂₋ ion is surrounded by a homogeneously $2+$ charged sphere show that upon varying the radius $R_{\mathrm{W}}$ of this sphere, the oxygen ion polarizability can be obtained as a function of its volume (Figure 1) [11].

Apparently, the wave function does not shift rigidly with increasing $R_{\mathrm{W}}$, but forms increments which tend to overlap with the coordinating transition metal ions. This is best exemplified by looking at the difference charge density (Figure 2) where charge is depleted in the intermediate regions.

The consequences of this unusual behavior are: O₁₋ is a bound state, O₂₋ is resonant and unstable. Therefore, electron doping is impossible in oxides, hole doping is the only possibility to dope oxides, thus leading to hole superconductivity [13]!! Alternatives are that strong anharmonicity dominates oxides with tendencies to lattice and electronic instabilities! Besides of the above extraordinary role of the oxygen for hole doped superconductivity, KAM was inspired by his friend Harry Thomas to consider Jahn-Teller (JT) ions to play another special role [14]. While primarily starting with nickelates, the missing success in the search for HTSC lead to cuprates [15]. By assuming an electron to be located at a certain lattice site, two of the ligands move in and the other two move out: the energy is lowered. The same is true for the reversed case and a second minimum appears. The energy gain is the Jahn-Teller stabilization energy $E_{\mathrm{JT}}$ (Figure 3).

The ground-state energy for the Jahn-Teller polaron is then given by:

$$E_k^{\left(0\right)}=E_0^{\left(0\right)}+{\displaystyle \frac{{\hslash }^2{k}^2}{2m_{\mathrm{eff}}}}$$

(2)

with the effective mass $m_{\mathrm{eff}}$ being enlarged as compared to the bare electronic mass. Limiting cases of this simplified model yield the relevant physics of the problem as outlined above, namely $E_{\mathrm{JT}}\ll t$ where t is the hopping integral. The distortion due to the coupling to the electronic motion is small and is almost unaffected, i.e., free-electron-like. With increasing JT energy, the distortion increases, and for $E_{\mathrm{JT}}\gg t$ an isolated JT molecular complex results. The competition between localization and itineracy is thus an inherent property of the JT polaron problem. Combining both of the above, namely oxygen ion polarizability and instability and JT ions limits the search for HTSC to rather few systems, namely nickelates and cuprates. In addition, for both of these oxide compounds, very much alike to ferroelectric type perovskite oxides, it is known that it is almost impossible to grow them without any defects or in perfect stoichiometry. One of the prototypical perovskites, SrTiO₃ (STO) changes its properties completely upon replacing almost homoeopathic amounts of Sr by Ca. The same holds for introducing very small dopings of Nb. This implies that intrinsic heterogeneity governs the ground state character. Similarly, also cuprate HTS’s are non homogeneous and superconductivity is only observed in a limited doping region. Typically they are grown as ceramics and single crystals are difficult to achieve. This leads to the conclusion that perovskite oxides including cuprates are non-uniform and this structural diverseness is essential for their physical properties.

2. Isotope Effects

For BCS type superconductors the prediction of an isotope effect on $T_{\mathrm{c}}$ was a hallmark to highlight the role of the electron-phonon coupling. With a few exceptions an isotope effect has always been observed in conventional superconductors. That is why also cuprate HTS’s have been readily tested for the isotope effect. The first oxygen (₁₆O/₁₈O) isotope effect (OIE) experiments were carried out on Ba₂YCu₃O₇ and Ba₂EuCu₃O₇ samples at optimum doping and only a very tiny OIE on $T_{\mathrm{c}}$ ($\Delta T_{\mathrm{c}}/T_{\mathrm{c}}<0.2\%$) was found [16]. The conclusions from this experiment were drawn almost immediately after this report, namely that the missing effects must be interpreted in terms of a novel pairing mechanism beyond electron-phonon interaction. Since the undoped parent compounds of cuprate HTS’s are all antiferromagnets (AFM), the proximity between HTSC and antiferromagnetism was taken as evidence that magnetic fluctuations are the origin for the hole pairing in cuprates (see, e.g., Ref. [17]). Even though a number of further isotope experiments on various doping levels were carried out subsequently where isotope effects were observed [18,19,20], the first results [16] dominated the scene. However, a breakthrough result was achieved by the Stuttgart [21] and Zurich [22,23] groups almost simultaneously, when they concentrated on the question, which oxygen ions in optimally doped YBa₂Cu₃O_x (YBCO) systems are responsible to the OIE [site-selective oxygen isotope effect (SOIE)]. In the structure of YBCO there are three different oxygen lattice positions: plane (p), apex (a), and chain (c). The Zurich group applied a two-step oxygen-exchange process in order to prepare partially substituted ₁₆O/₁₈O YBa₂Cu₃O_6+x samples required for SOIE investigations [23]. As a result, quite opposite to expectations, namely the apex oxygen ions (a) should exhibit the largest OIE [22], it were the (p) oxygen ions being mainly responsible for the total OIE [23]. In order to demonstrate that this finding also holds in the underdoped region, additional SOIE studies were performed for Y_1−xPr_xBa₂Cu₃O_7−δ ($x=0,0.3,0.4$) [24,25], and the results are summarized in (Figure 4). It is obvious from the figure that at all doping levels the main contribution to the total OIE arises from the (p) oxygen ions.

This was verified in various experiments, but also through a rather unconventional strategy, where the OIE on the magnetic penetration depth $\lambda$ (superfluid density $\propto {\lambda }^{-2}$) was investigated using various experimental techniques [magnetization, magnetic torque, muon-spin rotation ($\mu$SR)] [26]. As an example of such a study, the results of the OIE on the in-plane superfluid density observed for single-crystal La_2−xSr_xCuO₄ using magnetic torque is depicted in Figure 5 [27], where a pronouned doping dependent OIE on the in-plane superfluid density [$\propto {\lambda }_{ab}^{-2}\left(0\right)$] is evident. Even though within BCS theory no OIE on the penetrations depth (superfluid density) is expected, these experiments were done under the assumption that a bi-polaronic mechanism would support an OIE on the penetration depth (superfluid density) with a site specific result. This was indeed observed in a SOIE study of the in-plane magnetic penetration depth ${\lambda }_{ab}$ in Y_0.6Pr_0.4Ba₂Cu₃O_7−δ by means of $\mu$SR [25]. Consistent with the SOIE on $T_{\mathrm{c}}$ (see Figure 4) also the one on ${\lambda }_{ab}$ was largest for oxygen atoms in the CuO₂ planes (p), whereas the one on the (ac) lattice positions was present, but much smaller. These unexpected and amazing results demonstrate clearly that two different structural components of the cuprate HTS systems play different roles in the respective physics supporting the viewpoint of two order parameters and heterogeneity as essential ingredients, which will be addressed in more detail below.

3. Essential Ingredients for High-Temperature Superconductivity: Heterogeneities and Mixed-Order Parameters in Cuprate Superconductors

Clear experimental evidence for intrinsic heterogeneity in the cuprates has been given by extended X-ray absorption fine structure (EXAFS) spectroscopy and pair distribution function analysis, where a stripe-like ordering of these regions has been detected [28]. Charge-rich areas and the fully symmetric undistorted charge-poor regions have been observed. Very intriguing demonstrations of their existence have been provided by the group of A. Bianconi where clear correlations between oxygen rich regions and charge density wave (CDW) puddles have been detected [29].

Opposite to a s-wave BCS superconductor and the conclusions obtained from the OIE experiments, the missing OIE at optimum doping readily led to the assumption that cuprate HTS’s are governed by a single d-wave order parameter. This was supported by in-plane phase-coherent tunneling experiments (see, e.g., Refs. [30,31,32]) and a number of other experimental studies. In contrast, Sun et al. [33] performed a tunneling experiment along the c-axis of optimally doped YBa₂Cu₃O_7−δ (YBCO) which is consistent with a pure s-wave order parameter. Furthermore, in a nuclear magnetic resonance (NMR) experiment the temperature dependence of the NMR spin relaxation time $T_2$ in different YBCO compounds was investigated [34]. The experimental results lie in between the theoretical calculations for a s- and a d-wave superconductor [35], however closer to the d-wave one. Based on these and other experimental findings as well as theoretical considerations, KAM proposed that cuprate superconductors must have a mixed $s+d$-wave order parameter that reflects the intrinsic inhomogeneities (hole-rich and hole-poor stripe-like regions) in these systems [36,37,38].

In order to test the proposed concept of KAM we performed detailed muon-spin rotation ($\mu$SR) experiments. $\mu$SR is extremely sensitive to detect any kind of local and global magnetic signatures in magnetic systems, including superconductors. In particular, $\mu$SR is a powerful tool to explore the pairing symmetry (s, d, $s+d$) of a cuprate HTS in the bulk of a single-crystal sample. This is exemplified by measuring the temperature dependence of the magnetic penetration depths ${\lambda }_a$, ${\lambda }_b$, and ${\lambda }_c$ along the principal crystallographic axes a, b, and c. The University of Zurich group has conducted such studies on three different cuprate HTS’s revealing consistent and generic results [39,40,41]. As an example, the results of such a $\mu$SR study for single-crystal YBa₂Cu₃O_7−δ are displayed in Figure 6, where the temperature dependences of the $\mu$SR relaxation rates ${\sigma }_a\propto {\lambda }_a^{-2}$, ${\sigma }_b\propto {\lambda }_b^{-2}$, and ${\sigma }_c\propto {\lambda }_c^{-2}$ are plotted [40]. Note that ${\sigma }_a\left(T\right)\propto {\lambda }_a^{-2}\left(T\right)$ as well as ${\sigma }_b\left(T\right)\propto {\lambda }_b^{-2}\left(T\right)$ exhibit a characteristic "up-turn" at low temperature. This is the signature of a coupled $s+d$-wave order parameter in the CuO₂ plane with a dominant d-wave order parameter ($\approx 75\%$) and a small s-wave contribution ($\approx 25\%$) [40,42]. However, ${\sigma }_c\left(T\right)\propto {\lambda }_c^{-2}\left(T\right)$ is in accordance with a pure s-wave order parameter. While this experiment does not only highlight the capabilities of $\mu$SR, it also provides clear evidence of the coexistence of two order parameters in cuprate HTS’s with major contributions of the d-wave order parameter in the a- and b-direction whereas along the c-direction the s-wave order parameter is realized in agreement with the c-axis tunneling results [33] mentioned above.

Of course also other experiments have confirmed the intrinsic heterogeneity of cuprate HTS’s and the two-component character of the condensate. Only few examples are given here: As already mentioned above early on after performing the first NMR experiments Bulut and Scalapino [35] analyzed these in terms of a purely electronic model based on spin fluctuations and concluded that a d-wave only approach was unable to reproduce the experimental data. Some years later it was shown that by using a mixed $s+d$-wave order parameter a very satisfactory description of the experiments could be achieved [43]. Furthermore, scanning tunneling microscope (STM) spectroscopy demonstrated that the electron tunneling process in STM is essentially incoherent. These observations are favored by the s-wave pairing mechanism. However, the conductance curves were found to be substantially smeared in comparison with the conventional excitation spectra predicted in BCS (isotropic s-wave) superconductors, suggesting the anisotropy of the gap functions in cuprate HTS’s [44]. NMR experiments were able to disclose that the observed NMR shifts stem from two components and provide clear evidence for intrinsic heterogeneity [45,46]. Studies of the nonlinear conductivity in cuprate HTS’s indentified that intrinsic and universal superconducting gap inhomogeneity is highly relevant to understanding the superconducting properties of the cuprates [47]. Theoretically, a similar conclusion for the coexistence of $s+d$-wave superconductivity was reached by using a quantum Monte Carlo approach [48]. Another ansatz was based on the slave boson approach for a two band model, where again coexistence regions of $s+d$ channels were considered [49]. Of course a number of related models exist, but this is beyond the scope of the article.

4. Theoretical Understanding

The simplest and most transparent model for understanding the above findings is provided by a two-band model where one band has mainly s-electron character whereas the second one constitutes the d-electron states. Interactions between both types of carriers are the important ingredient for two-band superconductivity. The Hamiltonian H reads [50]:

$$H=H_0+H_1+H_2+H_{12}$$

(3)

with

$$\begin{array}{ccc}\hfill H_0& =& \sum _{k_1,\sigma }{\xi }_{k_1}{c}_{k_1\sigma }^{+}{c}_{k_1\sigma }+\sum _{k_2,\sigma }{\xi }_{k_2}{d}_{k_2\sigma }^{+}{d}_{k_2\sigma }\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill H_1& =& -\sum _{k_1,k_1^{\prime },q}{V}_1(k_1,k_1^{\prime })c_{k_1+q/2\uparrow }^{+}{c}_{-k_1+q/2\downarrow }^{+}{c}_{-k_1^{\prime }+q/2\downarrow }{c}_{k_1^{\prime }+q/2\uparrow }\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill H_2& =& -\sum _{k_2,k_2^{\prime },q}{V}_2(k_2,k_2^{\prime })d_{k_2+q/2\uparrow }^{+}{d}_{-k_2+q/2\downarrow }^{+}{d}_{-k_2^{\prime }+q/2\downarrow }{d}_{k_2^{\prime }+q/2\uparrow }\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill H_{12}& =& -\sum _{k_1,k_2,q}{V}_{12}(k_1,k_2)[c_{k_1+q/2\uparrow }^{+}{c}_{-k_1+q/2\downarrow }^{+}{d}_{-k_2+q/2\downarrow }{d}_{k_2+q/2\uparrow }+h.c.]\hfill \end{array}$$

(7)

where $c^{+}$, c, $d^{+}$, d are electron creation and annihilation operators with spin $\sigma$ and momentum k. $H_0$ is the kinetic energy with hopping integrals $\xi$ and $V_i$ (i =1,2) are the intraband pairing interactions. The term $V_{12}$ represents the interband pairing interaction where electrons are pairwise exchanged between the two bands.

The gap equations are self-consistently derived and are given by:

$$\begin{array}{ccc}\hfill \langle c_{k_1\uparrow }^{+}{c}_{-k_1\downarrow }^{+}\rangle & =& {\displaystyle \frac{{\overline{\Delta }}_{k_1}^{*}}{2E_{k_1}}}tanh{\displaystyle \frac{\beta E_{k_1}}{2}}={\overline{\Delta }}_{k_1}^{*}{\Phi }_{k_1}\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill \langle d_{k_2\uparrow }^{+}{d}_{-k_2\downarrow }^{+}\rangle & =& {\displaystyle \frac{{\overline{\Delta }}_{k_2}^{*}}{2E_{k_2}}}tanh{\displaystyle \frac{\beta E_{k_2}}{2}}={\overline{\Delta }}_{k_2}^{*}{\Phi }_{k_2}\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill {\overline{\Delta }}_{k_1}& =& \sum _{k_1^{\prime }}{V}_1(k_1,k_1^{\prime }){\overline{\Delta }}_{k_1^{\prime }}{\Phi }_{k_1^{\prime }}+\sum _{k_2}{V}_{12}(k_1,k_2){\overline{\Delta }}_{k_2}{\Phi }_{k_2}\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill {\overline{\Delta }}_{k_2}& =& \sum _{k_2^{\prime }}{V}_2(k_2,k_2^{\prime }){\overline{\Delta }}_{k_2^{\prime }}{\Phi }_{k_2^{\prime }}+\sum _{k_2}{V}_{21}(k_1,k_2){\overline{\Delta }}_{k_1}{\Phi }_{k_1}\hfill \end{array}$$

(11)

Numerical solutions of the above equations yield the dependence of the gaps as a function of the chemical potential (Figure 7), i.e. doping and the interband interaction term $\gamma$. The dome like dependence of the coupled gaps reproduces very well the in the cuprate HTS’s observed dependence of $T_{\mathrm{c}}$ as a function of doping. In addition, the figure clarifies that $\gamma$ plays a central role for the magnitude of $T_{\mathrm{c}}$ and the gaps which increase nonlinearly with $\gamma$. In all cases the d-wave gap (squares) is the dominant one whereas the s-wave gap (circles) is almost 60% smaller than the d-gap.

In order to bring these results in contact with experimental data the average gap values have been plotted as functions of $T_{\mathrm{c}}$ and compared data for Y_1−xCa_xBa₂Cu₃O_7−δ in Figure 8 [51].

Apparently, good agreement between experiment and theory is given when taking two bands with inter- and intraband interactions into account. A special effect of using this approach is the fact, that $T_{\mathrm{c}}$ can significantly be enhanced by the interband coupling term even if the intraband couplings exhibit rather moderate values (Figure 9).

This has early on been highlighted in the papers by Suhl, Matthias, and Walker [52] and Moskalenko [53] that even independent of the sign of the interband coupling enhancements of $T_{\mathrm{c}}$ as compared to single band approaches a la BCS occur. As can be seen in Figure 9b, even in the case of both intraband interactions being too small to induce superconductivity, moderate values of $V_{12}$ of the order of 0.5 lead to superconductivity at $T_{\mathrm{c}}\simeq 50\mathrm{K}$ for mixed symmetry $s+d$ order parameters and $T_{\mathrm{c}}\simeq 28\mathrm{K}$ for $s+s$ symmetry order parameters.

5. Isotope Effects in a Multiband Polaron Approach

Polaron formation takes place, if the coupling between the electrons (holes) and the lattice is neither too strong nor too weak as has been outlined above. The mediate strength coupling between holes (electrons) and lattice vibrations (displacements) has two coupled and inseparable consequences, namely, the hole (electron) kinetic energy is slowed down, since it carries a lattice cloud, and the lattice energy is shifted caused by the hole (electron) drag. One without the other is not possible. The easiest ansatz is here the Lang-Firsov approximation [54], which takes both renormalizations simultaneously into account. For the renormalized electron creation and annihilation operators this yields [54]:

$${\tilde{c}}_{\sigma i}^{+}=c_{\sigma i}^{+}exp\left({\gamma }_i{\tilde{p}}_i\right),{\tilde{c}}_{\sigma i}=c_{\sigma i}exp(-{\gamma }_i{\tilde{p}}_i)$$

(12)

where ${\tilde{p}}_i$ is the normalized crystal momentum. For the lattice Hamiltonian the following renormalizations hold [54]:

$$\tilde{H}=\sum _{q,j}\hslash {\tilde{\omega }}_{q,j}({\tilde{b}}_{q,j}^{+}{\tilde{b}}_{q,j}+1/2)$$

(13)

with

$${\tilde{b}}_q^{+}=b_q^{+}+\sum _q{\gamma }_i\left(q\right)c_i^{+}{c}_i,{\tilde{b}}_q=b_q+\sum _q{\gamma }_i\left(q\right)c_i^{+}{c}_i,$$

(14)

and

$${\tilde{\omega }}_{q,j}^2={\omega }_{q,j}^2-{\displaystyle \frac{{\gamma }_{q,j}^2}{N\left(E_F\right)}}\sum _k{\displaystyle \frac{1}{\epsilon \left(k\right)}}tanh{\displaystyle \frac{\epsilon \left(k\right)}{k_{B}T}}$$

(15)

with

$$\epsilon \left(k\right)=-2t_1[cos\left(k_{x}a\right)+cos\left(k_{y}b\right)]+4t_{2}cos\left(k_{x}a\right)cos\left(k_{y}b\right)\mp t_4{[cos\left(k_{x}a\right)-cos\left(k_{y}b\right)]}^2-\mu .$$

(16)

The nearest $t_1$, second nearest neighbor $t_2$, and interplanar $t_4$ hopping integrals are modified through the interaction with the lattice. Within this approach an isotope effect on $T_{\mathrm{c}}$ may arise from either of the hopping integrals, or from different combinations of them. Interestingly, the OIE on $T_{\mathrm{c}}$ stemming from $t_1$ yields the wrong trend, whereas the combined action of $t_2$ and $t_4$ follow the experimental data (see Figure 10).

This result allows to draw conclusions on the involved lattice displacements, namely those where oxygen atoms displace towards the neighboring Cu ions are irrelevant (Figure 11, left), whereas a Jahn-Teller type mode (Figure 11, right) supports the observed OIE (see Figure 10) when combined with a c-axis displacement.

Very shortly also the lattice response (Eq. 8) is mentioned which shows some unconventional signatures caused by the polaronic coupling (Figure 12). This is best exemplified through the Debye-Waller factor which at high temperatures T appears to be conventional, i.e., decreasing steadily with decreasing T. However, around the so-called stripe temperature $T^{*}$ a rapid increase sets in signaling polaron coherence followed by a sudden drop near $T_{\mathrm{c}}$ and another increase below $T_{\mathrm{c}}$. The drop at $T_{\mathrm{c}}$ is a consequence of the zero line nodes of the d-wave component of the order parameter whereas the following increase stems from the s-wave component. This finding supports the two-component character and the coexistence of an s-wave and a d-wave order parameter. Upon inspecting the structure of a cuprate HTS, a possible assignment of them can be made in terms of the antiferromagnetic spin alignment in the planes and an out-of-plane almost ferroelectric type polar component (see Figure 13).

6. Conclusions and Outlook for Further Search for High-Temperature Superconductivity

A reasonable undertaking to discover novel superconductors would be to look or transition metal oxides or chalcogenides wherethe metal ion should be a JT active one with the tendency to valence instability, thus ensuringthat the ground state is intrinsically heterogeneous. In combination with the unconventional largepolarizability of oxygen, respectively chalcogen ions, fluctuating charge transfer is enabled, "dynamical covalency", which enhances the charge-lattice coupling locally and allows for unusual lattice anomalies in conjunction with possible pairing instabilities. The electron-lattice interaction is not the conventional BCS electron-phonon interaction, which is long-range, but a local one outside the Born-Oppenheimer approximation. This leads to unconventional isotope effects, which are typically overlooked or even ignored in purely electronic or spin-fluctuation-based pairing scenarios. Furthermore, combined order parameters are a consequence that not only leads to agreement with puzzling experiments, but also accounts for the complexity in the electronic structure of cuprates, often avoided in strongly simplified alternative procedures. As pairing glue we have identified (bi-) polaron formation, the starting point for the discovery of high-temperarure superconductivity.