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Apical oxygen vibrations dominant role in d-wave cuprate superconductivity and its interplay with spin fluctuations

Microscopic theory of a high T c cuprate Bi 2 Sr 2 CaCu 2 O 8+x based on main pairing channel of electrons in CuO planes due to 40mev lateral vibrations of the apical oxygen atoms in adjacent the SrO ionic insulator layer is proposed. The separation between the vibrating charged atoms and the 2D electron gas creates the forward scattering peak leading in turn to the d -wave pairing within Eliashberg formalism. The phonon mode naturally explain the kink in dispersion relation observed by ARPES and the and effect of the O 16 → O 18 isotope substitution in the normal state. To describe the pseudogap physics a single band fourfold symmetric t−t′ Hubbard model, with the hopping parameters t′∼−0.17t and the on site repulsion e U ∼ 6t. It described the Mott insulator at low doping, while at higher dopping the pseudogap physics (still strongly correlated) can be be approximated by the symmetrized mean field model and with renormalized U incorporating screening. The location of the transition line T *between the locally antiferromagnetic pseudogap and the paramagnetic overdoped phases and susceptibility (describing spin fluctuations coupling to 2DEG) are also obtained within this approximation. The superconducting d - wave gap mainly due to the phonon channel but is assisted by the spin fluctuations (15%–20%). The dependence of the gap and T c on doping and effect of the isotope substitution are obtained and is consistent with experiments.


Introduction
For decades the only superconductors with critical temperature above 90 K under ambient conditions were cuprates like Bi 2 Sr 2 CaCu 2 O 8+x (Bi2212). They are generally characterized by the following five structural/ chemical/electronic peculiarities. First, they are all quasi-two dimensional (2D) perovskite layered oxides. Second, the 2D electron gas (2DEG) in which the superconductivity resides is created by 'charging' CuO planes: hole doping the anti-ferromagnetic (AF) parent material. Third, the conducting layers are separated by several insulating ionic oxide planes. Fourth, as doping decreases past optimal the pseudogap is opened and closed Fermi surface splits into four arcs [1] (a topological transition). Fifth is the d-wave symmetry of the order parameter below the 'superconducting dome' on the phase diagram. It is widely believed [2] that, although the insulating layers play a role in charging the CuO planes, the (still not clearly identified) bosons responsible for the pairing (so called 'glue') are confined to the CuO layer.
Several years ago another group of superconducting materials with critical temperature as high as T c = 60 − 106 K was fabricated by deposition of a single unit cell layer (1UC) of FeSe on insulating substrates like SrTiO 3 (STO both [3] ( ) 001 and [4] ( ) 110 ), TiO 2 and [5] BaTiO 3 . Note that the first three of the characteristic cuprate features listed above are manifest in these systems as well. Indeed, the insulating substrates are again the perovskite oxide planes. The electron gas residing in the FeSe layer [6] is charged (doped) by the perovskite substrate. The remaining two of the five cuprate features are clearly distinct in the new superconductor family. The Fermi surface is nearly round in sharp contrast to the rhomb-shaped one in cuprates. There are neither pseudogap nor the electron 'pockets'. Furthermore the symmetry of the order parameter is the noddles s-wave [7]. Generally the system is much simpler than the cuprates and much progress in understanding of its superconductivity mechanism was achieved. The role of the insulating substrate in FeSe/STO seems to extend beyond the charging [6]. While the physical nature of the pairing boson in cuprates is still under discussion, it became clear that superconductivity mechanism in 1UC FeSe/STO should at least include the substrate phonon exchange. Although there are theories based on an unconventional boson exchange within the pnictides plane (perhaps spin fluctuations exchange [8], as in pnictides theories [9]), an alternative point of view was clearly formed [10,11] based on idea that the pairing in the FeSe plane is largely due to vibration of oxygen atoms in a substrate oxide layer near the interface.
Historically a smoking gun for the relevance of the electron-phonons interactions (EPI) to superconductivity has been the isotope effect. When the isotope 16 O in surface layers of the STO substrate was substituted [12] by 18 O, the gap at low temperature (6 K) decreased by about 10%. Detailed measurements of the phonon spectrum via electron energy loss spectroscopy [13] demonstrated that the interface phonons are very energetic (the 'hard' longitudinal optical (LO) branch appears at Ω h = 100mev). The phonons couple to 2DEG with relatively small coupling constant [12] λ ; 0.25, deduced from the intensity of the replica bands identified by ARPES [14]. Importantly the interpretation of the replica bands was based on the forward peak in the electron -phonon scattering (FSP). Initially this inspired an idea that the surface phonons alone could provide a sufficiently strong pairing [11]. Since the BCS scenario, T c ≈ Ω h e −1/ λ , is clearly out, one had to look for other ideas like the extreme, delta like, FSP model [15,16] for which » W l l + T c h 2 3 . This lead [11] to sufficiently high T c for small λ. Unfortunately the EPI parameters to achieve such a strong FSP in ionic substrate are unrealistic. In a recent work [17] we developed a sufficiently precise microscopic model of phonons in adjacent insulating TiO 2 layer of the STO substrate and found an additional Ω s = 50mev LO interface phonon. Since coupling of the Ω s to the electron gas in the FeSe layer is practically the same as that of the hard Ω h mode, it greatly enhances pairing. The momentum dependence of the EPI matrix elements has an exponential FSP, [ ] pd exp 2 a , where d a is the distance between the ionic layer and 2DEG. Calculated coupling λ, critical temperature, replica band and other characteristics of the superconducting state are consistent with experiments. It demonstrated that the perovskite ionic layer phonons constitute a sufficiently strong 'glue' to mediate high T c superconductivity.
A question arises whether similar phononic pairing mechanism occurs in cuprates. Of course there is a structural difference between the cuprates and the 1UC FeSe/STO in that the the bulk layered cuprates contain many CuO planes, while there is a single FeSe layer. The difference turns out to be insignificant, since it was demonstrated [18,19] that even two unit cells of optimally doped Bi2212 sandwiched between insulating materials exhibits practically as high T c as the bulk material. Also recently a CuO monolayer on top of Bi2212 film was synthesized [20] with surprisingly high the critical temperature of 100 K. The pairing is of a noddles s-wave variety as in 1UC FeSe/STO in striking contrast with Bi2212 and other hole doped cuprates. The s-wave symmetry was explained by extremely strong charging [20,21]. In particular it was noticed that the Fermi surface becomes nearly circular [21] also in sharp contrast to the rhombic shape of hole doped cuprates.
The idea that phonons are at least partially responsible for the d-wave pairing has been contemplated over the years. In particular the CuO layer oxygen atoms breathing and buckling modes [22] and the apical oxygen c axis vibrations [23][24][25] have been considered. It is well established that phonons cause s-wave pairing in low T c materials, d-wave pairing is possible when FSP is present. It turns out that the nature of pairing for the FSP phonons depend on the shape of the Fermi surface, assumed to be fourfold symmetric throughout this paper. Our experience can be summarizes as follows. The pairing tends to be d-wave a for rhomb-like Fermi surface and s-wave for a more circular one like that of 1UC FeSe/STO or CuO/Bi2212. Early work in this direction was summarized in [15]. It was found that at weak coupling the Lorentzian FSP led to increase of T c , while at strong coupling the phonon contribution was detrimental due to large renormalization parameter. Consensus emerged that the EPI of CuO plane phonons alone is not strong enough to get such a high T c . EPI exchange can somewhat enhance, but cannot be the major cause of the d-wave pairing.
In view of the experience with 1UC FeSe/STO, is is natural to ask whether the lateral apical oxygen phonon exchange that naturally has exponential FSP, due to distance d a between the conducting and insulating layers, can lead to the d-wave pairing in cuprates. It immediately reminds a high T c 'smoking gun' that was observed of more than a decade ago. It was discovered [25] that the superconducting gap in Bi2212 is (locally) anticorrelated precisely to the distance, d a , between the Cu atoms and the apical oxygen atoms just below/above. This is the first 'smoking gun' pointing at crucial role of the apical oxygen atoms. The evidence of the anticorrelation is not conclusive since recently correlation single-layer Bi 2 Sr 0.9 La 1.1 CuO 6 , double-layer Nd 1.2 Ba 1.8 Cu 3 O 6 and infinite-layer CaCuO 2 was observed [26].
The second smoking gun is the tunneling experiment [27] that the authors describe best: 'We find intense disorder of electron-boson interaction energies at the nanometer scale, along with the expected modulations in d 2 I/dV 2 . Changing the density of holes has minimal effects on both the average mode energies and the modulations, indicating that the bosonic modes are unrelated to electronic or magnetic structure. Instead, the modes appear to be local lattice vibrations, as substitution of 18 O for 16 O throughout the material reduces the average mode energy by approximately 6% -the expected effect of this isotope substitution on lattice vibration frequencies.' This is an indication that vibrating oxygen atoms are out of the CuO plane. We therefore revisit this clear evidence in light of the lateral apical vibration superconductivity theory.
Unlike 1UC CuO, where no measurements of the phonon excitations were made to date, the bulk BSCCO crystals were thoroughly studied. Evidence consists of the 'kink' in quasiparticle dispersion relation in normal state [28][29][30] measured by ARPES, large isotope effect observed mainly in underdoped samples [31] and the statistics of the STM measurements [27]. The kinks should be attributed to EPI, since their locations (energies) change [30] by 6% upon substitution of the 16 O isotope by 18 O. The distribution of d 2 I/dV 2 is independent of doping in a wide range. In particular its average value is 40mev and is shifted by 6% upon the isotope substitution [27]. This indicates that if the phonon pairing mechanism is dominant the relevant phonons do not belong to the CuO planes. Phonons in cuprates were extensively studied within the microscopic (DFT) approach including the oxygen vibration mode [32].
In the present paper we construct a theory of a high T c cuprate that based on the idea of dominant pairing due to apical lateral longitudinal phonons (ALLP) along with minor AF fluctuations contribution. This 40mev phonon mode and its coupling including the matrix elements are described sufficiently well by the Born-Meyer approximation [33,34] that has been applied to cuprates [35]. To support the pairing scenario, it is crucial to present a simple enough microscopic model of cuprates that comprehensively describes (at least qualitatively) various features of the material over the whole doping-temperature phase diagram (underdoped to overdoped) including both normal and d -wave superconducting states. To be more specific we consider the effect of the ALLP pairing in the arguably best studied cuprate superconductor Bi2212. To describe the pseudogap physics of 2DEG in the CuO planes we limit ourselves to the fourfold symmetric -¢ t t single band Hubbard model [2] with on site repulsion energy U. In the absence of direct experimental determinations of U, one resorts to the first principle calculations. Most of the microscopic (DFT) determinations of U [36] are in the 'strong coupling Mott insulator' range U/t = 5 − 10, so that U is comparable to the bandwidth W. Recently however in a similar type of the first principle calculations [37] resulted in smaller values of U. It turns out within our approach that in order to describe the pseudogap physics, parameters of the model are restricted to a rather narrow 'window' around ¢~t t 0.2 , U ∼ 6t. Since the ALLP exchange is effective enough to be the dominant 'glue' responsible for the d-wave pairing, a simple description of the Hubbard model combining the RPA type coupling renormalization due to screening [38] and the symmetrized HF approach [39] is sufficiently accurate. The spin fluctuations exchange enhances superconductivity by 15%-20%.
Two conditions turned out to be sufficient to trigger robust apical phonon d-wave pairing: the rhombic shape of the Fermi surface and the exponential FSP of the ALLP mode. The dependence of the superconducting gap on doping, temperature and effect of the 16 O → 18 O isotope substitution are obtained. In normal state the dimensionless EPI strength is λ ∼ 0.6, thus justifying the use of the weak coupling approach [40]. The phonons naturally explain the effect of the isotope substitution on the kink in dispersion relation.
The paper is organized as follows. In section II a sufficiently precise phenomenological model of the lateral optical phonons in ionic crystal is developed. In section III an effective model of the correlated electron gas is presented. Section IV is devoted to normal state properties: the pseudogap phenomena (including the T * line, fragmentation of the quasi-particle spectrum) and renormalization of the electron Green's function due to phonons. This allows location of kink in dispersion relation (including the isotope dependence) and the EPI coupling λ. In section V superconductivity is studied in the framework of dynamic Eliashberg approach. Both the phonon and the spin fluctuation channels are accounted for over the full doping range. The isotope effect exponent is determined. In the last section results are summarized and discussed. A simplified general picture of the d-wave pairing by apical phonons and its coexistence with spin fluctuations is presented.

The model
Our model consists of the 2DEG interacting with phonons of a polar insulator: Hub ph e ph We start with the phonon. The electron part is the Hubbard model, while the coupling between the electronic and vibrational degrees of freedom, H e−ph , is subject of the last Subsection.
2.1. What phonons are contributing most to the electron-electron pairings?
Although the prevailing hypothesis is that superconductivity in cuprate is 'unconventional', namely not to be phonon-mediated, the phonon based mechanism has always been a natural option to explain extraordinary superconductivity in cuprates. As mentioned in Introduction, the most studied phonon glue mode has been the oxygen vibrations within the CuO plane [15,16,22,41]. As argued in [17], in the context of high T c 1UC FeSe on perovskite substrates, lateral vibrations of the oxygen atoms in the adjacent ionic perovskite layer can couple sufficiently strongly to 2DEG residing in the CuO plane to be a viable option. Qualitatively one of the reasons is that the SrO layer constitutes a strongly coupled ionic insulator. Unlike the metallic layer where screening is strong, in an ionic layer screening is practically absent and a simple microscopic theory of phonons and their coupling exists [33]. It was repeatedly noticed [10] that vibrations in c directions contribute little to pairing. Let us start with a brief description of the structure of the perhaps best studied high T c material Bi2212. Then the microscopic lateral vibrations model is presented, while their coupling to the electron gas is considered in the next section. The structure of the quarter of the Bi 2 Sr 2 CaCu 2 O 8+δ unit cell near the conducting layer is schematically depicted in figure 1. Electronic properties in both normal and superconducting states of cuprates are determined by holes (created by doping) in conducting CuO layers, see top layer in figure 1 (where Cu is drawn as a brown sphere, O -small orange spheres) and the left most chart in figures 8 (appendix A). Besides the single CuO 2 layer only two insulating oxide layers are assumed to be relevant. The closest layer at distance d a = 1.84A, see the second chart from left in figure 8a, consists of heavy Sr atoms (cyan rings) and light 'apical' oxygen (small red circle). The next layer is BiO, see the third chart from left in figure in figure 8(b) (Bi -violet large ring, O -small dark red circles). Below this layer the pattern is replicated in reverse order. Of course Bi2212 has metallic bilayers separated by Ca. In this paper we neglect the effects of tunneling between the CuO 2 layers. Out of plane spacings counted from the SrO layer are specified in table 1.
The translational symmetry in the lateral (x,y) directions of the system has the lattice spacing of a = 3.9A and coincides with the distance between the Cu atoms. Distances between the layers are also given table 1 neglecting small canting. The crystal has very rich spectrum of phonon modes. However very few have a strong coupling to 2DEG and even fewer can generate lateral (in plane) forces causing pairing. While phonons within the CuO planes have been extensively studied both theoretically [15,22] and experimentally, the conclusion is that they do not constitute a strong enough 'glue'. It is reasonable to expect that the modes most relevant for the electron -phonon coupling are the vibrations of the atoms in the adjacent SrO layer, see figure 1. This is in conformity with the first and second 'smoking gun' experiment findings [25,27]: the 'glue' is independent of the doping and  anything else that happens in the 2DEG in the CuO 2 layer simply because the phonons are originating in different layer.

Lateral apical oxygen optical phonon modes in the SrO layer
Phonons in ionic crystals are described by the Born-Meyer potential due to electron's shells repulsion [33] and electrostatic interaction of ionic charge, with values of coefficients A and b listed in table 1. The ionic charges Z are estimated from the DFT calculated Milliken charges [34]. In the SrO layer the charges are constrained by neutrality. Since oxygen is much lighter than Sr, the heavy atoms' vibrations are negligible. Obviously that way we lose the acoustic branch, however it is known that the acoustic phonons contribute little to the pairing [10,42]. Atoms in neighboring layers can also be treated as static. Moreover one can neglect more distant layers. Even the influence of the lower BiO layer (below the last layer shown in figure 1) is insignificant due to the distance. Consequently the dominant lateral displacements, a u m , α = x, y, are of the oxygen atoms directly beneath the Cu sites at ( ) = a m m r , m 1 2 . The dynamic matrix ab D q is calculated by expansion of the energy to second order in oxygen displacement (details in appendix A), so that the phonon Hamiltonian in harmonic approximation is: Here M is the oxygen mass. Summations over repeated components indices is implied. Now we turn to derivation of the phonon spectrum. Two eigenvalues, the transversal (red) optical (TO) and the longitudinal (blue) optical (LO) modes are given in figure 2. One observes that there are longitudinal modes are in the range Ω q ∼ 26 − 41mev and 22 − 32mev respectively. The energy of LO modes is larger than that of the corresponding TO, although the sum W + W LO TO q q is nearly dispersionless. At Γ the splitting is small, while due to the long range Coulomb interaction there is hardening of LO and softening of TO at the BZ edges. The dispersion of the high frequency modes is small, while for the lower frequency mode it is more pronounced.

The -¢ t t Hubbard model of the 2DEG in CuO layers
The electron gas of Bi2212 consists of two identical layers with tunneling between them. The effective single band model. Neglecting the inter-layer tunneling, the simplest -¢ t t Hamiltonian in momentum space is: Here † s c k is the electron creation operator with spin projection σ = ↑ , ↓ . Only nearest and next to nearest neighbors hopping terms are included:  4 cos cos . 5 x y x y k k Summations are always over the 2D Brillouin zone, − π/a < k x , k y < π/a. The dispersion relation thus is simplified with respect to a 'realistic' one [43,44] in which splitting due to tunneling is also taken into account and more distant hops are included. Values of the hopping parameters, see table 2 will be fixed independently of chemical potential μ determining the (hole) doping x. Reasons for such a choice will be given after the phase diagram will be presented in the next section. The on site repulsion is described by the on site Hubbard repulsion term [2] ( ) i being the spin σ occupation on the site {i x , i y }. Due to strong repulsion, even the model without phonons is highly nontrivial and will be treated approximately in the next section. Now we turn to the electron-phonon coupling.
While the lattice spacing a is firmly determined by experiment (and is nearly independent of doping for small x), the microscopic [45] or phenomenological [43] estimates for other electron gas parameters like the energy scales m ¢ U t t , , , vary considerably in different one band Hubbard approaches. The values of t = 0.3eV, U = 6 at zero doping will be used throughout the paper to fit numerous experimental quantities like the ARPES [43], the pseudogap characteristics [44]. The range of acceptable values of ¢ t t is rather limited. If one chooses | | ¢ < t t 0.12, the Mott state at very low doping does not appear [46]. At values larger than | | ¢ > t t 0.25 the shape of the Fermi surface in the underdoped regime is qualitatively different from the one observed by ARPES [47]. The value of ¢ =t t 0.17 is chosen to tune the Lifshitz (topological) transition from the full Fermi surface to the fractured one (four arcs) occurs at experimentally observed [18] doping x opt = 0.16.

Electron-phonon coupling
The lateral apical oxygen phonon's interaction with the 2DEG on the adjacent CuO layer d a = 1.84A above the SrO plane is determined by the electric potential created the charged apical oxygen vibration mode u m at arbitrary point r is: Here the apical oxygen charge taken to be Z = − 0.95, see table 1. This value is slightly below the charge at which transition to charge density wave occurs. The interaction electron-phonon Hamiltonian that accounts for the hole charge distribution in the CuO plane is derived in appendix A. The result in momentum space has a density -displacement form It is well known that only longitudinal phonons contribute to the effective electron-electron interaction, as is clear from the scalar product form of the equation (8). The precision of the last equality is 2%, see figures 9 and 10 in appendix A.
To conclude equations (4), (3), (8) define our microscopic model. Now we turn to description of the normal state properties of 2DEG, including the influence of the EPI.

Normal state properties: pseudogap, EPI coupling strength and kink in dispersion relation
The normal state of cuprates exhibits a host of phenomena including pseudogap in underdoped regime resulting in fracture of the Fermi surface, significant charge and spin susceptibility due to strong anti-ferromagnetic correlations (leading to enhancement of the d-wave pairing). These phenomena are described in the framework of the strongly coupled Hubbard model defined in the previous section. Unfortunately the theoretical description of the Hubbard model away from half filling (Monte Carlo [48], diagrammatic [49,50]) is either uncertain or extremely complicated. We use a much simpler approximation scheme including the RPA type coupling U renormalization [51] and symmetrized HF [39]. It provides a good agreement with the more sophisticated methods. Coupling to phonons also affects the normal properties such as the dispersion relation. The strength of EPI will be estimated and the quasi-particle self energy calculated perturbatively.

Renormalized mean field description of the Hubbard model
Hubbard model at moderate value U = 6 in the doping range x = 0.05-0.25 range is a strongly correlated fermion system that does not allow the Landau liquid description (except at high doping). Generally it is also out of applicability range of the HF approximation due to large vertex corrections [49,50]. However it is well known that the overdoped system has a well defined Fermi surface and can be very well described by the HF type twobody correlator [43]. In the underdoped phase one obtains an effective description in terms of 'RVB' correlators [52] that have recently been cast as a symmetrized HF [39]. Such an approach is consistent if the vertex corrections effectively lead to reduction of the coupling to a smaller value U . It turns out that MC and diagrammatic results can be approximated by such a scheme when the renormalized U , where χ 0 is the (Matsubara) charge susceptibility. This should be solved consistently with the HF equations and is described in both the overdoped and the underdoped phases in appendix B. The coupling U = 6 is reduced by screening to the renormalized values given in table 2. The HF equations were solved numerically by iterations on lattice N = 128 × 128 with periodic boundary conditions.
One of the striking normal state phenomena in underdoped cuprates is pseudogap [1,53]. In the present paper we adopt a point of view that pseudogap to the short range anti-ferromagnetic order within each of the CuO layers. The long range AF order is lost at a relatively small doping and the system becomes quasi two dimensional. In 2D one can model the short range order and the fluctuations effects [2] by considering the macroscopic sample as a system of AF domains with certain domain size. Generally local (STM) probes described in Introduction provide distribution of quantities like pseudogap within the domains. On the other hand ARPES, thermodynamic and transport experiments provide information on all the scales, namely after averaging over the domains. It is found that the value of the pseudogap is qualitatively agree with somewhat similar calculations [54] (improved by the renormalization group), the MC simulations and experiments [1,53].
The transition temperature T * as function of the hole doping, x = 1 − n, is given in figure 3 as the green line. It starts at the quantum critical point x * = 0.16, rapidly increases (almost vertically although a slight bending is visible) intersecting with the superconducting transition temperature T c . Then it curves towards the AF phase at small doping. The mean field transition happens to be second order with an exception of the small section below the 'superconducting dome' in figure 3 (marked by a phenomenological parabolic fit to experiment, see [18]).
In appendix B the expressions for the electron correlators in both phases is given. The spectral weight namely the imaginary part of the symmetrized Green function, equation (B7), at zero frequency, exhibits the fractured Fermi surface qualitatively similar to ARPES observation [47,55].
The spectral weigh for five values of doping are shown in the upper row in figure 4. Two are in the non-superconducting state, x = 0.02, 0.05, one in the underdoped region, x = 0.13, optimal, x = x opt = 0.16, and overdoped x = 0.2 regions. One observes that as the doping increases the length of the four Fermi arcs increases until the topological (Lifshitz) transition to a single Fermi surface at x opt . Upon further hole doping the area of the enclosed region of BZ decreases. Note that the Fermi surface does not extend to the BZ boundary as seen in early experiments [43] , however more recent measurements [47] apparently are consistent with this picture.
3.2. Phonon renormalization of the quasi-particle self energy and coupling constant λ ph 3.2.1. Self energy due to phonons The quasiparticle (HF) self-energy is renormalized due to interaction with phonons. It generally leads to characteristic features of the spectrum like satellite bands [8,14], kinks in dispersion relation [28,30], etc at energies of the order of the phonon frequency Ω above and below Fermi level. In our case (for details see a more general case considered in [17] and references therein) the Matsubara self energy for x > x opt (and temperature above T c ) in (gaussian or renormalized) perturbation theory is: The dispersion relations are given in equation (5) and M is the oxygen ion mass. Second order 'gaussian' perturbation theory [56] is justified at weak coupling, so that it should be confirmed in the following subsection that the dimensionless effective electron-electron coupling λ ph is indeed small. Summing over the bosonic Matsubara frequencies, w p = Tm 2 m b2 , one obtains (after analytic continuation to physical frequency), Here energies  E k and weights  Z k are given in equations (B9), (B10). These expressions will be used for calculation of both the electron phonon coupling constant and the dispersion relation of quasi-particles.

Dimensionless electron-electron coupling λ
Generally the dimensionless coupling constant is defined in terms of the self energy as In the overdoped case (see appendix B for details and expressions in a more cumbersome underdoped case) one obtains at zero temperature: Results of numerical computation at the nodal point on the Fermi surface in the doping range from x = 0.08 to x = 0.28 is performed. At each doping the location of the Fermi surface point was given by an analytic solution. As expected it has a maximum of λ ph = 0.62. Upon deviation from the angle 45°the coupling decreases. This is consistent with the experimental value estimated recently [57,58] at 30 K to be λ ph = 0.41 at optimal doping at ( ) p = k 0. . In the underdoped cases it vanishes at small angles due to finite extent of the Fermi arc, Generally the averaged over the Fermi surface coupling constant belongs to an intermediate range [40]. Such coupling is sufficient (as will be shown also in the next section) to provide high d-wave superconductivity T c ∼ 80 − 90 K at optimal doping, yet does not require the use of a rather problematic strong coupling Eliashberg theory. The coupling constitutes the bulk of the mechanism of superconductivity in the present paper (in addition to phonons the spin fluctuations also contribute to the overall effective coupling λ, see below).
The EPI renormalizes the quasiparticle spectrum and dynamics leading to several observations of the isotope substitution effect on the normal state properties. One of them is the 'kink' in dispersion relation.

The 'kink' function and the effect of the isotope substitution
It was established by ARPES early on that the hole dispersion relation abruptly changes derivative ('kink') in normal state approximately 45meV below Fermi level [28][29][30]. Although some other theories appeared, the large isotope effect [31] (substitution of 16 O isotope by 18 O), observed mainly in underdoped samples) provides evidence that he kinks should be attributed to EPI. To determine the kink position observed directly, let us differentiate the self energy equation (13) with respect to frequency ω. The real part of the integrand is: where E p was defined in equation (12). In the underdoped regime the expression is given in appendix B.
To characterize the kink in dispersion relation, we calculate the derivative in range of frequencies between − 1.3Ω to − 0.9Ω for three dopings, 0.13 (green), 0.15 (red) and 0.17 (violate) in figure 5. The kink position (zero value of the derivative) is around ω = − Ω = − 45meV . The dashed lined are the same quantity but for a heavier isotope 18 O, namely with the oxygen atom mass M replaced by αM, α = 18/16. The location is shifted by approximately 6%, as was indicated in the ARPES experiment [29]. Now we turn to the main objective of the present study: d-wave superconductivity.

Superconductivity
Although the main emphasis of the paper is on the ALLP mechanism of the d-wave superconductivity in the hole doped cuprates, in the present section we take into account also the magnetic fluctuation contribution. The reason is that the AF fluctuations were widely observed and in certain cases were shown to at least enhance superconductivity. The purpose of the present section is to quantitatively compare the role of these two contributions and show how they coexist (complement each other) in the d-wave superconducting state. We start from the derivation of the phonon exchange d wave 'potential' (mainly near the Γ point of BZ) and then proceed to the spin fluctuation one (mainly near the M point of the BZ).

Effective phonon and the spin fluctuation generated electron-electron interactions in spin singlet channel
In order to describe superconductivity, one should 'integrate out' the phonon and the spin fluctuations degrees of freedom to calculate the effective electron-electron interaction. We start with the phonons. The Matsubara action for EPI, equation * and g was defined in equation (9). The polarization matrix is defined via the dynamic matrix of equation (3): , α, β = x, y, calculated in appendix A. Since the action is quadratic in the phonon field u, the partition function is gaussian and can be integrated out exactly, see details in [17] . As a result one obtains the effective density-density interaction term for of electrons The expression is 'exact' for harmonic phonons (we have neglected the transversal mode and small dispersion of the longitudinal mode spectrum [17], see figure 2). An approximate expression for the effective interaction due to the electron correlations effects will be derived next. The potential exhibits the central 'inverted' (that is negative) 'peak' that we will call the apical phonon dip due to the exponential form of the matrix element. The second bosonic 'glue' is generated by the correlation effects. Since, as explained in Subsection IIIA, the renormalized on site repulsion constant in our scheme, U is not very large (see table 3), the gaussian expansion [39,59] is applicable. One starts with the mean field GF and considers the rest of the action as a perturbation. In the overdoped case, it is just a 'renormalized' Kohn-Luttinger perturbation theory [60]. We therefore calculate the effective interaction due to correlations in the second order in U r . Generally, utilizing the inversion symmetry, the effective interaction in the spin singlet channel has a form: where χ mq is the electronic susceptibility. The positive constant U r in equation (19) is just the direct first order Coulomb repulsion suppressing the s-wave pairing, but having no impact on the d-wave pairing. The well known Kohn-Luttinger diagrams [38,60] give in the overdoped case, x > x * , the following dynamic Matsubara susceptibility: where E p was defined in equation (12). This is calculated numerically for sufficiently large values of N = 256 and harmonics | |  m 32. In the lower row of figure 5 the static part, namely zero frequency is given x = x opt = 0.16 and x = 0.2. Similarly in the underdoped case, x < x * , one calculates the same two diagrams on the magnetic BZ, 0 < k 1 < π, − π < k 2 < π, namely using the GF of equation (B7). Since we are interested in the dynamic susceptibility on the scale of the Cooper pairs, the full sublattice matrix should be used. This is derived in appendix B, where a rather bulky expression, equation (B13) is given. It turns out that after symmetrization it is not much different from the overdoped case susceptibility as is shown in figure 5. The symmetrization of the susceptibility matrix is made as in [39]). The zero frequency c k k sym 0, , x y at T = 50 K is plotted for x = 0.13. The dependence on temperature in the relevant range (T < 300 K) is very weak. One observes that the evolution is smooth through the Lifshitz point x opt .
The general feature of the Matsubara susceptibility distribution over the BZ is that near the crystallographic M point the susceptibility is large, while near the Γ point it is small. This is crucial for the d-wave pairing. Note also the fine structure of the susceptibility: there are two characteristic local maxima near point M, while the point itself is a local minimum. The splitting is very small. In this paper we do not consider possible fourfold symmetry breaking (or nematicity). This effective electron-electron couplings will be used in the gap equation.

Superconducting gap
To complete the electronic effective action, one adds to equations (18) and ( figure 4 (red squares). In the underdoped domain it comes short of the parabolic experimental dependence [18] (dashed curve). If one neglects the magnon contribution, namely takes v = v ph , the temperatures are lower by 15%-20% (red circles).

Isotope effect
The influence of the oxygen isotope substitution, 16 O → 18 O on superconductivity can be gauged by calculation of the change of the (Matsubara) gap at a temperature below T c . In figure 6 we plot the The deduced exponent, 16 log , 27 16 18 at temperature T = 20 K. The same exponent was estimated by measuring the T c isotope effect in various hole doped cuprates [61], mostly in YBa 2 Cu 3 O 7−x and La 2−x Sr x CuO 4 . Qualitatively the exponent is small in overdoped and optimally doped materials, but increases at strongly overdoped case. In Ba 2 Sr 2 CaCu 2 O 7 the experimental results are scarce, but order of magnitude is the same as in figure 6.
The isotope effect exponential is small, at optimal and overdoped systems, however it slightly increases when the doping is reduced below optimal (reaches α = 0.08 at x = 0.1).

Thermal fluctuations and the inter-layer tunneling
In the bi-layer Bi2212 there are two types of tunneling. The first is a rather strong tunneling between adjacent layers within the bi-layer is estimated [43] to be ¢ = t meV 30 80 . It leads to appearance of the secondary band mentioned in section 2. The second tunneling amplitude is between the bi-layers in different cells. The 3D dispersion relation is k k are given in equation (4) and s -the inter bilayers separation. The order of magnitude is much smaller than ¢ t : t ⊥ = 1 − 2 meV. Superconductivity in a single CuO bi-layer is of the 2D Kosterlitz -Thouless type. The mean field critical temperature calculated above slightly overestimates T KT , where the modulus of the order parameter is established: T c − T KT ≈ T c Gi 2D . Here Gi 2D is the 2D Ginzburg number, Gi 2D = a 2 /dξ P , d is the thickness of the CuO bi-layer and ξ P is the lateral coherence length [56]. Due to the tunnelings t ⊥ between bi-layers makes the system 3D and the KT feature disappears.
The 2D/3D crossover occurs when ξ ⊥ ∼ s where ξ ⊥ are the coherence length in z -direction. Close to the critical temperature x~-v T T T

Summary
Theory of superconductivity of high T c cuprates based on the dominant ALLP pairing mechanism was proposed. It is comprehensive in a sense that the whole doping range is considered including anomalous normal state properties of cuprates like Bi 2 Sr 2 CaCu 2 O 8+x . To demonstrate the basic principles we limited ourselves in this paper to a simple sufficiently generic model: the pseudogap physics of 2DEG in the CuO planes is described by the fourfold symmetric -¢ t t single band Hubbard model with on site repulsion energy U of moderate strength. Doping is controlled by the chemical potential.
The results are following. The most important for the pairing mode for Bi2212 is found to be the optical longitudinal lateral (within the SrO plane) mode at 45meV, mostly due to vibration of apical oxygen atoms. The dimensionless electron-electron attraction exhibits an exponential forward scattering peak and is estimated to have the strength of λ ∼ 0.6. When parameters of the effective one band -¢ t t model of 2DEG were fixed at ¢~t t 0.17 and U ∼ 6t, t = 0.3eV, the mean field T * line, green curve in phase diagram, figure 3, become a crossover between short range AF pseudogap phase and the paramagnetic one. The quasi-particle spectrum undergoes a topological (Lifshitz) transition. The closed Fermi surface above the T * line disintegrates into four Fermi arcs below it, see figure 4.
Renormalization of the electron Green's function due to phonons allows calculation of the quasi-particle properties. Location of kink in dispersion relation including the observed isotope ( 16 O → 18 O) dependence, see figure 5. Since the electron-phonon coupling λ is moderate, weak coupling dynamic Eliashberg approach is applicable to calculate the gap function and critical temperature T c . One has to go beyond the BCS Figure 6. Isotopic effect critical exponent versus doping. approximation due to important dependence of the phonon mediated pairing on frequency. Both phonon and spin fluctuation pairing are accounted for over the full doping range. It is found that the critical temperatures above 90 K at optimal doping can be reached, see figure 4. The dominant 'glue' responsible for the d-wave pairing turns out to be the phonon mode rather than spin fluctuations, although the later enhances superconductivity by about 15%-20%. Comparison of the doping dependence of T c with experimental [18] is qualitatively fair, although . underdoped are slightly underestimated, while strongly overdoped overestimated. The isotope ( 16 O → 18 O substitution) effect is small at optimal doping but increases towards both the underdoped and the overdoped regions, see figure 6. This is consistent with observations [31]. To summarize, two features turned out to be sufficient for robust apical phonon d-wave pairing. The first is the rhombic shape of the Fermi surface. The second is the exponential FSP of the apical lateral phonon optical mode and, to a lesser degree, constructive cooperation with the spin fluctuation channel. The s-wave solution of the gap equation sometimes competes with the d-wave that appears only when the fourfold anisotropy of the Fermi surface is sufficiently pronounced. In these cases the central peak favors d-wave over the s-wave due to two reasons. First, the s-wave pairing due to the apical phonons is generally weaker than the CuO plane phonons since unlike in BCS large momentum q contributions are suppressed. Second, while the s-wave channel is suppressed by direct Coulomb repulsion, the d-wave is not (the quasi-local Coulomb repulsion drops out of the gap equation for the d-wave). We have explicitly compared energies and found that the s-wave loses in the range presented.

Concluding remarks
Restriction of the description of the electron gas to one band Hubbard model with just two parameters ¢ t t , for nearest neighbor and next to nearest neighbor hopping obviously makes the model less realistic to quantitatively describe real materials like Bi2212. These typically require either a three band much more complicated model or an effective one band model with more distant hopping terms like t″. In addition the tunneling between the conducting CuO planes via a metallic layer and the nematicity (deviations from the fourfold symmetry) should be added. These lead to a characteristic splitting of the spectrum [43] . This is left for future work. Of course the phenomena broadly termed 'unusual normal and superconducting properties of high T c cuprates' contains many more features. In this paper we have emphasized ones that are directly linked to the phonon exchange.
Experimentally the main claim of the paper, namely that the 'glue' that creates d-wave pairing is the phonon exchange of a very specific nature, the apical oxygen's (that is one belonging to an insulating layer, SrO, adjacent to the conducting CuO layer) lateral vibrations, can be further directly strengthened or falsified by suppression such vibrations as in [25,27] or actively focus on these modes and their coupling. Since one or to unit cell perovskite were recently fabricated [18][19][20] perhaps apical oxygen atoms can be distinguished from the rest. An alternative route is to look for secondary effects of this coupling on normal state properties, some calculated in the present paper. The phonons induce modifications in normal state like modification of dispersion relation on transport beyond the 'strange metal' resistivity behavior. The modification can be isolated by isotope substitution. Superconducting properties due to this particular mechanisms in addition to T c and order parameter studied, are also sensitive to the isotope substitution. An example is magnetization curves [62] that simply depend on T c (via Ginzburg-Landau description [56]).
while the potential energy part W consists of interatomic Born-Meyer potentials defined in equation (2). Only interactions of the 'dynamic' oxygen atoms in the SrO with neighboring BiO below and CuO 2 above are taken into account:   Consequently the dominant lateral displacements, a u m , α = x, y, are of the apical oxygen atoms. Vibrations of heavy atoms and even oxygen in other planes are not expected to be significant due to their mass or distance from the SrO layer oxygen atoms. Some effects of those vibrations can be accounted for by the    effective oxygen mass, while more remote layers above and below the important layer were checked to be negligible.
Harmonic approximation consists of expansion around a stable minimum of the energy. Expressions for the derivatives are given in [17]. This leads to the following expression for the dynamic matrix These matrix elements determine the frequencies (eigenvalues) for the two polarizations presented in figure 2.

A.2. The electron-phonon matrix elements
The microscopic derivation of the electron-phonon coupling of the holes residing in the CuO plane should in principle start at least from an effective three band (Emery) model of cuprate [63]. It is often reduced to the two band [64] model consisting of the Zhang-Rice singlet state, a symmetric combination of the (in plane) 2p x and 2p y O orbitals, and the d 3 x y 2 2Cu orbitals. Let us assume a simplified picture case that the hole's wave function is concentrated on two oxygen positions within the unit cell. Concentrating on one unit cell, drawn in figure 8(a) (left) as a black square, the 2D density is: Extent of the density distribution in the z direction is neglected.
The electron-ion electrostatic energy is, where the potential was given by equation (7)  , is considered to be oriented along the spin space z axis. The lattice translation symmetry consequently is reduced to a smaller one on two sublattices I = A, B. The sublattice A The HF equations takes a form (using n A↑ ≡ n 1 , n A↓ = n 2 electron densities on each site, no charge density wave appear in the model considered) , = n F n n n F n n , ; The fact that the transition is second order is verified by the fitting of the pseudogap curves near T * in figure 3 by a power ( ) D µ - There is no experimental consensus on the shape of this line at small temperatures [68,69], while order of magnitude is consistent with tunneling experiments [44]. In our model the low temperature segment, T < T c , of the line exhibits a weak first order transition with small latent heat.

B.2. Underdoped
In 2D the Mermin-Wagner theorem [70] states that fluctuations for systems that have a continuous symmetry are strong enough to destroy long range order at any nonzero temperature. The order parameter locally exists, but averages out due to incoherence of its 'phase' over the sample. A more rigorous approach would be to divide the degrees of freedom into two scales, large distance correlations, and short distance correlations. It can be performed for certain bosonic models using renormalization group ideas, especially when the Berezinskii-Kosterlitz -Thouless type transition is involved. However such an approach is complicated in fermionic models in which order parameter is quadratic in fermionic operators [71]. A much simpler symmetrization approach that does not involve the explicit separation of scales was proposed in [39]. It was demonstrated by comparing with determinantal Monte Carlo simulations and for small sizes to exact diagonalization that he symmetrization therefore qualitatively takes into account the largest available scale by 'averaging over' the global symmetry group and agrees to within 5% with exact and MC results. We start with symmetrization of the HF Green function (GF). For (conserved) spin projection σ the GF on magnetic BZ is a 2 × 2 sublattice matrix,

B.3. Symmetrization
The relation between the matrix on magnetic Brillouin zone and the symmetrized Matsubara Green's function on the whole BZ (nonmagnetic, since the symmetry is restored), − π/a < k x , k y π/a is [39],   figure 4 (red squares). In the optimal and overdoped domains it agrees well with the parabolic experimental dependence (dashed curve) taken from [18]. If one neglects the magnon contribution, namely takes v = v ph , the temperatures are lower by 15%-20% (red circles). One observes that the decrease of T c is rather slow (linear) at large doping compared to the experiment. When doping becomes of order 30% it is expected to significantly impacts the effective mesoscopic lattice model parameters (m ¢ U t t , , , ). In underdoped cases the pseudogap should be taken into account. The results are the yellow part of the surface in figure 13 for the gap and critical temperatures shown on the left hand side of the phase diagram, figure 3. The maximum gap as function of doping and temperature is given in figure 13 (the yellow part of the surface).