Apical oxygen vibrations dominant role in d-wave cuprate superconductivity and its interplay with spin fluctuations
AApical oxygen vibrations dominant role in cuprate superconductivity and its interplaywith spin fluctuations.
Baruch Rosenstein ∗ and B. Ya. Shapiro † Microscopic theory of a high T c cuprate Bi Sr CaCu O x based on main pairing channel ofelectrons in CuO planes due to lateral vibrations of the apical oxygen atoms in adjacent the
SrO ionic insulator layer is proposed. Similar ionic substrate phonon model was used recently to explainvery high T c in novel one unit cell F eSe on perovskite STO. A microscopic vibration theory identifiesthe 40 mev phonon mode coupled to conducting
CuO planes with λ ∼ ,
5. It naturally explain thekink in dispersion relation observed by ARPES and the and effect of the isotope substitution.To describe the pseudogap physics by a single band fourfold symmetric t − t (cid:48) Hubbard model, thehopping parameters t (cid:48) ∼ − . t and the on side repulsion energy U ∼ . t are chosen. The electronicsystem is still strongly correlated, but U is weak enough to be effectively described by the meanfield model and its perturbative extensions. In particular the fragmentation of the Fermi surfacein underdoped samples and the non-circularity of the Fermi Surface are described well within the”symmetrized Hartree - Fock” approximation. The T ∗ transition line dividing the pseudogap (locallyantiferromagnetic) and paramagnetic phases and susceptibility (describing spin fluctuations couplingto 2DEG) are also obtained within this approximation. The superconducting gap was calculated inthe framework of the weak coupling approximation for both the phonon and the spin fluctuationschannels. The dominant ”glue” responsible for the d - wave pairing is the phonon mode rather thanspin fluctuations, although the later enhances superconductivity by 10-15%. The dependence ofthe superconducting gap and certain normal state properties, like the kink in dispertion relation,ondoping, temperature and effect of the O → O isotope substitution are obtained. PACS numbers: PACS: 74.20.Mn, 74.20.Rp,74.72.Hs a r X i v : . [ c ond - m a t . s up r- c on ] M a r INTRODUCTION.
For decades the only superconductors with critical temperature above 90 K under ambient conditions were cuprateslike Y Ba Cu O − δ ( T c = 93 K at optimal doping) and Bi Sr CaCu O x ( Bi K ). They are generally char-acterized 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 createdby ”charging” CuO planes: hole doping the anti - ferromagnetic (AF) parent material. Third, the conducting layersare separated by several insulating ionic oxide planes. Fourth, as doping decreases past optimal the pseudogap isopened and closed Fermi surface (FS) splits into four arcs[1] (a topological transition). Fifth is the d - wave symmetryof the order parameter below the ”superconducting dome” on the phase diagram. It is widely believed[2] that, al-though the insulating layers play a role in charging the
CuO planes, the (still not clearly identified) bosons responsiblefor 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 − K was fabricated by deposition of a single unit cell layer (1UC) of F eSe on insulating substrates like
SrT iO (STOboth[3] (001) and[4] (110)), T iO (rutile[5] (100) and anatase[6] (001)) and[7] BaT iO . Note that the first three of thecharacteristic cuprate features listed above are manifest in these systems as well. Indeed, the insulating substrates areagain the perovskite oxide planes. The electron gas residing in the F eSe layer[8] is charged (doped) by the perovskitesubstrate. The remaining two of the five cuprate features are clearly distinct in the new superconductor family. TheFermi surface is nearly round in sharp contrast to the rhomb - shaped one in cuprates. There are neither pseudogapnor the electron ”pockets”. Furthermore the symmetry of the order parameter is the nodeless s - wave[9]. Generallythe system is much simpler than the cuprates and much progress in understanding of its superconductivity mechanismwas achieved.The role of the insulating substrate in
F eSe/ST O seems to extend beyond the charging [8]. While the physicalnature of the pairing boson in cuprates is still under discussion, it became clear that superconductivity mechanismin 1UC
F eSe/ST O should at least include the substrate phonon exchange. Although there are theories based onan unconventional boson exchange within the pnictide plane (perhaps spin fluctuations exchange[10], as in pnictides’theories[11]), an alternative point of view was clearly formed[12, 13] based on idea that the pairing in the
F eSe planeis 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 superconductivityhas been the isotope effect. When the isotope O in surface layers of the ST O substrate was substituted[14] by O , the gap at low temperature (6 K ) decreased by about 10%. Detailed measurements of the phonon spectrumvia electron energy loss spectroscopy [15] demonstrated that the interface phonons are very energetic (the ”hard”longitudinal optical (LO) branch appears at Ω h = 100 mev ). The phonons couple to 2DEG with relatively smallcoupling constant[14] λ (cid:39) .
25, deduced from the intensity of the replica bands identified by ARPES [16]. Importantlythe 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[13]. Sincethe BCS scenario, T c ≈ Ω h e − /λ , is clearly out, one had to look for other ideas like the extreme, delta like, FSPmodel[17], for which T c ≈ λ λ Ω h . This lead[13] to sufficiently high T c for small λ . Unfortunately the EPI parametersto achieve such a strong FSP in ionic substrate are unrealistic. In a recent work[18] we developed a sufficiently precisemicroscopic model of phonons in adjacent insulating T iO layer of the STO substrate and found an additionalΩ s = 50 mev LO interface phonon. Since coupling of the Ω s to the electron gas in the F eSe layer is practicallythe same as that of the hard Ω h mode, it greatly enhances pairing. The momentum dependence of the EPI matrixelements has an exponential FSP, exp [ − pd a ], where d a is the distance between the ionic layer and 2DEG. Calculatedcoupling λ , critical temperature, replica band and other characteristics of the superconducting state are consistentwith experiments. It demonstrated that the perovskite ionic layer phonons constitute a sufficiently strong ”glue” tomediate high T c superconductivity.A question arises whether similar phononic pairing mechanism occurs in cuprates. Of course there is a structuraldifference between the cuprates and the 1UC F eSe/ST O in that the the bulk layered cuprates contain many
CuO planes, while there is a single
F eSe layer. The difference turns out to be insignificant, since it was demonstrated[19, 20]that even two unit cells of optimally doped Bi T c as the bulk material. Also recently a CuO monolayer on top of Bi Sr CaCu O δ film (1UC CuO ) wassynthesized[21] with surprisingly high the critical temperature of 100 K . The pairing is of a nodeless s-wave varietyas in 1UC F eSe/ST O in striking contrast with Bi circular [22] 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 theyears. In particular the CuO layer oxygen atoms breathing and buckling modes[23] and the apical oxygen enharmonic c axis vibrations have been considered[26][27][28]. 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 FSPphonons depend on the shape of the Fermi surface (FS, 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 FS and s - wavefor a more circular one like that of 1UC F eSe/ST O or CuO/BiSCCO . Early work in this direction was summarizedin ref. [17]. It was found that at weak coupling the Lorenzian FSP led to increase of T c , while at strong couplingthe phonon contribution was detrimental due to large renormalization parameter. Consensus emerged that the EPIalone is not strong enough to get such a high T c . EPI exchange can enhance, but cannot be the major cause of thed - wave pairing.In view of the experience with 1UC F eSe/ST O , is is natural to ask whether the lateral apical oxygen phononexchange that naturally has exponential FSP, due to distance d a between the conducting and insulating layers, canlead to the d-wave pairing in cuprates. It immediately reminds a high T c ”smoking gun” that was observed ofmore than a decade ago. It was discovered[28] that the superconducting gap is (locally) anti- correlated precisely tothe distance, d a , between the Cu atoms and the apical oxygen atoms just below/above. This is the first ”smokinggun” pointing at crucial role of the apical oxygen atoms. The second smoking gun is the tunneling experiment[29]that the authors describe best: ”We find intense disorder of electron - boson interaction energies at the nanometerscale, along with the expected modulations in d I/dV . Changing the density of holes has minimal effects on both theaverage mode energies and the modulations, indicating that the bosonic modes are unrelated to electronic or magneticstructure. Instead, the modes appear to be local lattice vibrations, as substitution of O for O throughout thematerial reduces the average mode energy by approximately 6% - the expected effect of this isotope substitutionon lattice vibration frequencies.” This is an indication that vibrating oxygen atoms are out of the CuO plane. Wetherefore 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 crystalswere thoroughly studied. Evidence consists of the ”kink” in quasiparticle dispersion relation in normal state[30–32]measured by ARPES, large isotope effect observed mainly in underdoped samples[34] and the statistics of the STMmeasurements[29]. The kinks should be attributed to EPI, since their locations (energies) change[32] by 6% uponsubstitution of the O isotope by O . The distribution of d I/dV is independent of doping in a wide range. Inparticular its average value is 40 mev and is too shifted by 6% upon the isotope substitution[29].In the present paper we construct a theory of a high T c cuprate that based on the idea of dominant pairing dueto apical phonon. To demonstrate the case, it is crucial to consider consistently a simple enough microscopic modelof cuprates that describes (at least qualitatively) most features of the material over the whole doping - temperaturephase diagram (including both normal (pseudogap, strange metal, Fermi liquid) and superconducting (underdopedto overdoped) states. To be more specific we apply the microscopic theory of the perovskite layer phonons and theircoupling to 2DEG to arguably the best studied cuprate superconductor Bi mev phonon mode detectedby ARPES and other experiments is identified, explains the dispersion relation kink and constitutes the main pairing”glue”. To describe the pseudogap physics of 2DEG in the CuO planes we adopt the fourfold symmetric t − t (cid:48) singleband Hubbard model[2] with on site repulsion energy U . It turns out that in order to describe faithfully the pseudogapphysics (AF Mott insulator at very small doping x , the locally AF state with nonzero pseudogap all the way to thepseudogap transition line [35][36, 37]), parameters of the model are restricted to a rather narrow ”window” around t (cid:48) ∼ . t and U ∼ t . This parameters range of Hubbard model without phonons has been repeatedly consideredtheoretically[51][38][39][40]. The electronic system is still can be classified as strongly correlated, but U is weak enoughto be effectively described by the mean field model and its perturbative extensions. In particular the fragmentationof the Fermi surface in underdoped samples and the non-circularity of FS are described well within the ”symmetrizedHartree - Fock” approximation[41]. Pseudogap and susceptibility (describing spin fluctuations coupling to 2DEG) canalso treated within this approximation. The superconducting gap is calculated in the framework of the weak couplingapproximation for both the phonon and the spin fluctuations channels.We find that the dominant ”glue” responsible for the d - wave pairing is the phonon mode rather than spinfluctuations, although the later enhances superconductivity by about 10-15%. As mentioned above two featuresturned out to be sufficient to trigger robust apical phonon d - wave pairing: the rhombic shape of the FS and theexponential FSP of the apical lateral phonon optical mode. The dependence of the superconducting gap on doping,temperature and effect of the O → O isotope substitution are obtained. In normal state the dimensionless EPIstrength is λ ∼ .
5, thus justifying the use of the weak coupling approach. The phonons naturally explain the kink indispersion relation and effect of the isotope substitution on it.The paper is organized as follows. In Section II a sufficiently precise phenomenological model of the lateral opticalphonons in ionic crystal is developed. In Section III a model of the correlated (due to the short range Coulombrepulsion) electron gas is presented. Experimental constraints leading to choice of parameters are discussed. SectionIV is devoted to normal state properties: the pseudogap phenomena (including the T ∗ line, topological transition in thequasi - particle spectrum) and renormalization of the electron Green’s function due to phonons. This allows calculationof the quasi - particle spectrum, the location of kink in dispersion relation (including the isotope dependence) andthe EPI coupling λ . In Section V superconductivity is studied in the framework of weak coupling dynamic Eliashbergapproach (beyond BCS approximation). Both phonon and spin fluctuation pairing are accounted for over the fulldoping range. The isotope effect exponent is determined. In the last Section results are summarized and discussed,their limitations and extensions commented. A simplified general picture of the d - wave pairing by apical phononsand its coexistence with spin fluctuations (or other pairing ”glue”) is presented. THE DOMINANT PHONON MODE COUPLED TO 2DEDWhat phonons are contributing most to the electron - electron pairings?
Although the prevailing hypothesis is that superconductivity in cuprate is ”unconventional”, namely not to bephonon - mediated, the phonon based mechanism has always been a natural option to explain extraordinary super-conductivity in cuprates. As mentioned in Introduction, the most studied phonon glue mode has been the oxygenvibrations within the
CuO plane[23][44][17]. As argued in ref.[18], in the context of high T c F eSe on perovskitesubstrates, lateral vibrations of the oxygen atoms in the adjacent ionic perovskite layer can couple sufficiently stronglyto 2DEG residing in the
CuO plane to be a viable option. Qualitatively one of the reasons is that the
SrO layerconstitutes a strongly coupled ionic insulator. Unlike the metallic layer where screening is strong, in an ionic layerscreening is practically absent and a simple microscopic theory of phonons and their coupling exists[45]. It was repeat-edly noticed[12] that vibrations in c directions contribute little to pairing. Let us start with a brief description of thestructure of the perhaps best studied high T c material Bi Sr CaCu O δ . Then the microscopic lateral vibrationsmodel is presented, while their coupling to the electron gas is considered in the next Section. FIG. 1. The profile 3D view of three layers comprising relevant part of the one unit cell :molecule” of Bi Cu (brown) O (orange), the apic phonon layer: Sr (cyan) O (red). The third layer: Bi (violet) O (dark red). Sizes ofatoms are inversely proportional to the values of the Born - Mayer inter - atomic potential parameter parameter b in Eq.(1). The structure of the quarter of the Bi Sr CaCu O δ unit cell near the conducting layer is schematically depictedin Figs. 1, 2. 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 Fig.1 (where Cu is drawn as a brown sphere, O -small orange spheres) and the left most chart in Fig. 2. Besides the single CuO layer only two insulating oxidelayers are assumed to be relevant. The closest layer at distance d a = 1 . A , see the second chart from left in Fig.2, consists of heavy Sr atoms (cyan rings) and light ”apical” oxygen (small red circle). The next layer is BiO , seethe third chart from left in Fig. 2 ( Bi - violet large ring, O - small dark red circles). Below this layer the patternis replicated in reverse order. Of course Bi Ca . In this paper we neglect the FIG. 2. Atomic lateral positions of the three layers a. the 2DEG layer consisting of Cu at R Cu = (cid:0) , , z Cu (cid:1) and two O atomsat R Ox = (cid:0) a, a/ , z Cu (cid:1) and R Oy = (cid:0) a/ , a, z Cu (cid:1) . b. the apical phonon layer containing the Sr at origin R Sr = (0 , ,
0) and R Oy = (cid:0) a/ , a, z Cu (cid:1) and O at R = ( a/ , a/ , Bi at R Bi = (cid:0) a/ , a/ , z Bi (cid:1) and O at R O = (cid:0) , , z Bi (cid:1) .d. The top view: all the three layer’s projections are superimposed. effects of tunneling between the CuO layers. Out of plane spacings counted from the SrO layer are specified in TableI. The chart on the right in Fig.2 is a view from above with sphere radii corresponding to the repulsive Born - Meyerpotential ranges given in Table I. Unit cell including both the metallic layer and the substrate is marked by the blackframe in Fig. 2.
TABLE I. Atomic parameters determining lateral apical oxygen vibrations.atom
Cu O Sr O Bi O mass (a.u.) 64 16 88 16 209 16 A ( kev ) 13 .
919 2 .
143 20 .
785 2 .
143 63 .
922 2 . b ( A − ) 3 .
561 3 .
788 3 .
541 3 .
788 3 . . Z . − . . − .
95 1 . − . z ( A ) 1 .
84 1 .
84 0 0 − . − . The square translational symmetry in the lateral ( x , y ) directions of the system has the lattice spacing of a = 3 . A and coincides with the distance between the Cu atoms. Distances between the layers are also given Table I neglectingsmall canting. The crystal has very rich spectrum of phonon modes. However very few have a strong coupling to 2DEGand even fewer can generate lateral (in plane) forces causing pairing. While phonons within the CuO planes havebeen extensively studied both theoretically[17, 23] and experimentally, the conclusion is that they do not constitutea strong enough ”glue”. It is reasonable to expect that the modes most relevant for the electron - phonon couplingare the vibrations of the atoms in the adjacent
SrO layer, see Fig.2. This is in conformity with the first and second”smoking gun” experiment findings[28][29]: the ”glue” is independent of the doping and anything else that happensin the 2DEG in the
CuO 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[45] andelectrostatic interaction of ionic charge, V XY ( r ) = (cid:112) A X A Y exp (cid:20) (cid:0) b X + b Y (cid:1) r (cid:21) + Z X Z Y e r , (1)with values of coefficients A and b listed in Table I. The ionic charges Z are estimated from the DFT calculatedMilliken charges[46]. 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 knownthat the acoustic phonons contribute little to the pairing[12, 47]. Atoms in neighboring layers can also be treated asstatic. Moreover one can neglect more distant layers. Even the influence of the lower BiO layer (below the last layershown in Fig.1) is insignificant due to the distance. Consequently the dominant lateral displacements, u α m , α = x, y ,are of the oxygen atoms directly beneath the Cu sites at R + r m , where the lattice sites r m and position within theunit cell R are: R = a (cid:18) , (cid:19) ; r m = a ( m , m ) . (2)The dynamic matrix D αβ q is calculated by expansion of the energy to second order in oxygen displacement (details inAppendix A), so that the phonon Hamiltonian in harmonic approximation is: H ph = 12 (cid:88) q (cid:26) M du α − q dt du α q dt + u α − q D αβ q u β q (cid:27) . (3)Here M is the oxygen mass. Summations over repeated components indices is implied. Now we turn to derivation ofthe phonon spectrum. TABLE II. Parameters describing the electron gas in
CuO layers.parameter t t (cid:48) /t U/t a value 0 . eV − .
184 1 . . A Two eigenvalues, the transversal (red) optical (TO) and the longitudinal (blue) optical (LO) modes are given inFig. 3. One observes that there are transversal modes are in the range Ω q ∼ − mev respectively. The energy ofLO modes is larger than that of the corresponding TO, although the sum Ω LO q + Ω T O q is nearly dispersionless. At Γthe splitting is small, while due to the long range Coulomb interaction there is hardening of LO and softening of TOat the BZ edges. The dispersion of the high frequency modes is small, while for the lower frequency mode it is morepronounced.
26 28 31 3336 3639 3941 41 k y ( π / a ) k y ( π / a )
22 25 27 30 32 k y ( π / a ) k y ( π / a ) FIG. 3. Spectrum of the lateral apical oxygen vibrations in the
SrO plane. a. longitudinal optical modes, b. transverseoptical modes. Note moderate dispersion of the longitudinal mode.
INTERACTING ELECTRON GAS AND THE ELECTRON - PHONON COUPLING
Our model consists of the 2DEG interacting with phonons of a polar insulator: H = H e + H ph + H e − ph . (4)Here the phonon part was given in Eq.(3) above. The electron part including strong short range Coulomb repulsion H e will be defined next, while the coupling between the electronic and vibrational degrees of freedom, H e − ph , issubject of the second Subsection. The t − t (cid:48) Hubbard model of the 2DEG in
CuO layers.
The electron gas of Bi Cu d x − y orbitals.Neglecting the inter - layer tunneling, the simplest t − t (cid:48) Hamiltonian is: K = (cid:88) x,y c σ † x,y (cid:0) − t (cid:0) c σx +1 ,y + c σx,y +1 (cid:1) + t (cid:48) (cid:0) c σx +1 ,y +1 + c σx +1 ,y − (cid:1)(cid:1) + h.c − µn x,y . (5)Here c † is the electron creation operator with σ = ↑ , ↓ being the spin projection. Only nearest and next to nearestneighbors hopping terms are included. In momentum space it reads: K = (cid:88) k c σ † k ( (cid:15) k + (cid:15) (cid:48) k − µ ) c σ k , (6)where the dispersion relation is modeled by just two contributions: (cid:15) k = − t (cos [ ak x ] + cos [ ak y ]) ; (7) (cid:15) (cid:48) k = − t (cid:48) cos [ ak x ] cos [ ak y ] .Summations are always over the 2D Brillouin zone, − π/a < k x , k y < π/a . The dispersion relation thus is simplifiedwith respect to a ”realistic” one[48], in which splitting due to tunneling is also taken into account and more distanthops are included. Values of the hopping parameters will be fixed at t = 300 meV , t (cid:48) = − . t , see Table II,independently of chemical potential µ determining the (hole) doping x . Reasons for such a choice will be given afterthe phase diagram will be presented in the next Section.Screened Coulomb interactions are described on the lattice model level by the on site Hubbard repulsion term[2] V = U (cid:88) m n ↑ m n ↓ m , (8)with n σ m = c σ † m c σ m being the spin σ occupation on the site m . The value of U will be fixed at U = 1 . t , see TableII and commented on in Subsection IVA. To conclude the 2DEG part of Hamiltonian is H e = K + V. Due to strongrepulsion, 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.
TABLE III. Parameters describing the electron gas in
CuO layers.parameter t t (cid:48) /t U/t a value 0 . eV − .
184 1 . . A Electron - phonon coupling
The lateral apical oxygen phonon’s interaction with the 2DEG on the adjacent
CuO layer d a = 1 . A above the SrO plane is determined by the electric potential created the charged apical oxygen vibration mode u m at arbitrarypoint r is:Φ ( r ) = (cid:88) m Ze (cid:113) ( r − R m − u m ) + d a , (9)Here the apical oxygen charge taken to be Z = − .
95, see Table I. This value is slightly below the charge at whichtransition to fourfold symmetry is spontaneously broken and the charge density wave emerges. It is important thatthe by vibrating charged oxygen atoms reside directly below Cu atoms. Influence on the electron - phonon couplingof vibrating Bi and O atoms of the next layer, see Figs.1-2, is further reduced since they are not situated directlybeneath the Cu sites.The Hamiltonian for interaction with electrons on the 3 d x − y Cu orbitals with wave functions ϕ l ( r , z ), H ei ,expanded to first order in the oxygen vibrations consequently is, H ei = − e (cid:90) r Φ ( r ) n r = − Ze (cid:88) l , m ( r l − R m ) · u m (cid:16) ( r l − R m ) + d a (cid:17) / n l . (10)The interaction electron-phonon Hamiltonian in momentum space has a standard form[49] H eph = Ze (cid:88) q n − q g α q u Aα q , (11)with EPI matrix element (see details and comments in ref.[18]), g q = (cid:88) m e ia q · m r m ( r m + d a ) / = 2 πe − qd a q q . (12)It is well known that only longitudinal phonons contribute to the effective electron - electron interaction, as is clearfrom the scalar product form of the Eq.(11). The last equality is only approximate (precision 2%, see Fig. 4). FIG. 4. Square of the matrix element of the electron - phonon coupling. Decreases exponentially as function of quasi -momentum momentum away from the Γ point. The forward scattering peak region occupies a significant portion of theBrillouin zone.
To conclude Eqs.(6,3,11) define our microscopic model. Now we turn to description of the normal state propertiesof 2DEG, including the influence of the EFI.
NORMAL STATE PROPERTIES: PSEUDOGAP, EPI COUPLING STRENGTH AND KINK INDISPERSION RELATION.
The so called normal state of cuprates (with an exception of highly overdoped regime where Landau liquid de-scription is sufficient) exhibits a host of ”abnormal” phenomena. These include pseudogap in underdoped regimeresulting in fracture of the Fermi surface, significant charge and spin susceptibility due to strong anti - ferromagneticcorrelations (leading to enhancement of the d - wave pairing) any many other ”strange” features. Many of thesephenomena will be described in the framework of the Hubbard model defined in the previous Section. Coupling tophonons also affects the normal properties such as the dispersion relation. The strength of EPI will be estimated andthe quasi - particle self energy calculated perturbatively. ”Symmetrized” mean field description of the pseudogap physics in the underdoped cuprate
One of the striking normal state phenomena in underdoped cuprates is pseudogap[1, 36]. In the present paper weadopt a point of view that links pseudogap to the short range anti - ferromagnetic order within each of the
CuO layers. There is no doubt that at very low doping the (AF) Mott insulator state is formed. However the long rangeAF order is lost at a relatively small doping. Moreover above this doping 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 asystem of AF domains with certain domain size L c .Generally local (STM) probes described in Introduction provide distribution of quantities like pseudogap withinthe domains. On the other hand ARPES, thermodynamic and transport experiments provide information on all thescales, namely after averaging over the domains. Theoretically this is achieved via renormalization group scheme (seefor example in the present context ref. [38]). In our work a simpler ”symmetrization” approach will be employed.It consists of averaging over the domains with different staggered magnetization directions performed on the semi -mean field level[41]. Choice of parameters
While the lattice spacing a is firmly determined by experiment (and is nearly independentof doping for small x ), the microscopic [25] or phenomenological[48] estimates for other electron gas parameters likethe energy scales U, t, t (cid:48) , µ vary considerably in different one band Hubbard approaches. Let us first express theparameters (energies) in units of the hopping amplitude t . The range of acceptable values of t (cid:48) /t is rather limited.It is negative and small. If one chooses | t (cid:48) | /t < .
15, the AF Mott state at very low doping does not appear. Atvalues larger than | t (cid:48) | /t > .
25 the shape of the Fermi surface in the underdoped regime is qualitatively different fromthe one observed by ARPES[50]. The available range therefore is, − . < t (cid:48) /t < .
15. The value of t (cid:48) = − . t ischosen to tune the Lifshitz (topological) transition from the full Fermi surface to the fractured one (four arcs) occursat experimentally observed[19] doping x opt = 0 . U . Generally ab initio calculations of the ”affinity” favour very large values of U of order of U = 1 − eV [2]. This was recently observed tobe consistent with experimentally determined value of the exchange spin - spin coupling J sufficient to support purelymagnonic mechanism of superconductivity[56]. However in most theoretical approaches that utilize Hubbard modelmuch lower values of U/t of order 2 < U/t < t − t (cid:48) Hubbard model within the HFapproximation values
U/t > . U/t is also detrimental to qualitatively understand the experimentally observed pseudogap values[36] and locationof the pseudogap disappearance line T ∗ [19, 37], since the energy values become an order of magnitude larger thanseveral tens of meV at lowest dopings. Therefore we are forced to consider smaller values. Since in our approach themechanism of superconductivity does not hinge on the spin fluctuations, this is feasible. Relatively low values of U were adopted in several approaches like renormalization group[38] or FLEX[39]. An advantage is that the mean fielddescription of the pseudogap physics is reliable[38] and even perturbative calculation of susceptibility and other AFfluctuations effects[40] is possible.The value of t = 0 . eV is chosen in accordance with ARPES[48] and the pseudogap[37] experiments. Pseudogap in Hartree - Fock approximation.
The Hartree - Fock theory of the t − t (cid:48) model, defined by Eq.(5), has been thoroughly investigated over theyears[51][52]. The spin rotation SU (2) symmetry in anti - ferromagnet is broken down to its U (1) subgroup. Theon site magnetization, M = (cid:0) n A ↓ − n A ↑ (cid:1) = (cid:0) n B ↑ − n B ↓ (cid:1) , is considered to be oriented along the spin space z axis.0The lattice translation symmetry consequently is reduced to a smaller one on two sublattices I = A, B . The sublattice A consists of odd ( x + y ) sites, while B contains even ( x + y ) sites. Position within the sublattices can be specifiedby integers i = 1 , ...N/ ≡ N (cid:48) and i = 1 , ..N , namely c Ai ,i = c i − i ,i and c Bi ,i = c i + i ,i .Hamiltonian in the magnetic quasi - momentum k space becomes (integer momenta) is, K = (cid:88) k k (cid:110) − (cid:16) c A † k h ∗ k a B k + h.c. (cid:17) + c I † k ( ε (cid:48) k − µ ) c I k (cid:111) , (13)where h k = t (cid:26) (cid:20) πiN (2 k − k ) (cid:21) + exp (cid:20) πiN (cid:48) k (cid:21) + exp (cid:20) πiN k (cid:21)(cid:27) ; (14) ε (cid:48) k = − t (cid:48) cos (cid:20) πN k (cid:21) cos (cid:20) πN ( k − k ) (cid:21) .The HF equations takes a form (using n A ↑ ≡ n , n A ↓ = n electron densities on each site, no charge density waveappear in the model considered), n = F [ n , n ] ; n = F [ n , n ] , (15)where the function F is defined by F [ n , n ] = 1 N N (cid:48) (cid:88) k (cid:26) f F (cid:2) E − k (cid:3) − ∆ pg + x k x k (cid:18) tanh (cid:20) E + k T (cid:21) − tanh (cid:20) E − k T (cid:21)(cid:19)(cid:27) . (16)Here ∆ pg ≡ U M is the pseudogap energy and f F ( ε ) ≡ (exp [ ε/T ] + 1) − is the Fermi - Dirac distribution. The newquasi - particle (hole in our case) spectrum consists of two branches E ± k = ε (cid:48) k − µ + U n + n ± x k . (17)and x k ≡ ∆ pg + | h k | . (18)The HF equations, Eq.(15) were solved numerically by iterations with N = 128 and periodic boundary conditions.The profile of the pseudogap as function of the hole doping, x = 1 − n , for a wide range of temperatures is given inFig. 5. The set of electron gas parameters is given in Table II and its choice was discussed above. The dependencechanges little at temperatures below 50 K .The values of pseudogap are qualitatively agree with measured[37] in Bi T ∗ , as function of doping x is shown in the phase diagram Fig. 6 (green line). It starts at the quantumcritical point x ∗ = 0 . , rapidly increases (almost vertically although a slight bending is visible) intersecting with thesuperconducting transition temperature T c at x opt = 0 . K . The factthat the transition is second order is verified by the fitting of the pseudogap curves near T ∗ in Fig. 5 by a power∆ pg ∝ ( x − x ∗ ) ν , with mean field critical exponent ν = 1 /
2. It simultaneously satisfied the criticality condition (where n = n ): 1 = 1 N N (cid:48) (cid:88) k U x k (cid:18) tanh (cid:20) E + k T (cid:21) − tanh (cid:20) E − k T (cid:21)(cid:19) . (19)There is no experimental consensus on the shape of this line at small temperatures[33][35], while order of magnitudeis consistent with tunneling experiments[37]. In our model the low temperature segment, T < T c , of the line exhibitsa weak first order transition with small latent heat.1 p se udog a p ( m ev ) FIG. 5. Pseudogap as function of doping for various temperatures.
Fragmentation of the Fermi surface
In 2D the Mermin - Wagner theorem [53] states that fluctuations for systems that have a continuous symmetryare strong enough to destroy long range order at any nonzero temperature. The order parameter locally exists, butaverages out due to incoherence of its “phase” over the sample. A more rigorous approach would be to divide thedegrees of freedom into two scales, large distance correlations, and short distance correlations. It can be performedfor certain bosonic models using renormalization group ideas, especially when the Berezinskii - Kosterlitz - Thoulesstype transition is involved. However such an approach is complicated in fermionic models in which order parameteris quadratic in fermionic operators[54]. A much simpler symmetrization approach that does not involve the explicitseparation of scales was proposed in ref.[41]. It was demonstrated by comparing with determinantal Monte Carlosimulations and for small sizes to exact diagonalization that he symmetrization therefore qualitatively takes intoaccount the largest available scale by “averaging over” the global symmetry group and agrees to within 5% with exactand MC results. We start with symmetrization of the HF Green function (GF). For (conserved) spin projection σ theGF on magnetic BZ is a 2 × G σmk k = 1 x k − ( − iω m + E (cid:48) k ) − iω m + E (cid:48) k − ( − σ ∆ pg h ∗ k h k − iω m + E (cid:48) k + ( − σ ∆ pg , (20)where E (cid:48) k = ε (cid:48) k + U n − µ and σ = 0 for ↑ and 1 for ↓ .The relation between the matrix on magnetic Brillouin zone and the symmetrized Matsubara Green’s function onthe whole BZ (nonmagnetic, since the symmetry is restored), − π/a < k x , k y ≤ π/a is[41], G symmk x k y = 14 (cid:88) σ (cid:16) G σAAm,k x ,k x + k y + e ik x a G σABm,k x ,k x + k y + e − ik x a G σBAm,k x ,k x + k y + G σBBm,k x ,k x + k y (cid:17) . (21)Here G IJ are elements of the matrix of Eq.(20). As a result the Green’s function (after analytic continuation) is, G sym ( ω, k ) = 12 (cid:18) Z + k ω + iη + E + k + Z − k iω + iη + E − k (cid:19) ; (22) Z ± k = (cid:15) k (cid:113) ∆ pg + | (cid:15) k | ± (cid:15) k was defined in Eq.(7) and η is the damping parameter. The dispersion relation in the nonmagnetic basistakes a form E ± k = (cid:15) (cid:48) k − µ + U n ± (cid:113) ∆ pg + | (cid:15) k | , (23)2 ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ Δ pg > Δ pg > Δ sc > Δ sc > Δ pg = Δ pg = Δ sc = Δ sc = t e m p e r a t u r e ( K ) FIG. 6. The doping - temperature phase diagram of a hole doped cuprate. The green curve marks the pseudogap transition T ∗ . Solid line represents the (mean field) second order transition, while the dashed segment represent weakly first order one,The parabolic curve is the experimental superconductor - normal critical temperature in Bi T c of our model, while the blue points are critical temperatures due to the apical phonon’s pairing only (that is whenthe spin fluctuations are ignored). where (cid:15) k was defined in Eq.(7). This is quite similar to one obtained in the slave boson approach to the t-J[55] andRVB[57] approaches. In particular they exhibit the fractured Fermi surface, see the upper row in Fig.7, qualitativelysimilar to ARPES observation[50, 58].The spectral weigh (imaginary part of the symmetrized Green function, Eq.(20), at Fermi surface, i.e. at zerofrequency) for five values of doping are shown. Two in the underdoped region , x = 0 . , .
15, optimal, x = x opt =0 . x = 0 .
18, 0 .
22. One observes that as the doping increases the length of the four Fermi arcsincreases until the topological (Lifshitz) transition to a single FS at x opt . Upon further hole doping the area of theenclosed region of BZ decreases. Note that the FS doesn’t extend to the BZ boundary as seen in early experiments[48],however more recent measurements[50] apparently are consistent with this picture. Phonon renormalization of the quasi - particle self energy and coupling constant λ ph Self energy due to phonons.
The quasiparticle (HF) self - energy is renormalized due to interaction with phonons. It generally leads to char-acteristic features of the spectrum like satellite bands[16][10], kinks in dispersion relation[30][32], etc. at energies ofthe order of the phonon frequency Ω above and below Fermi level. In our case (for details see a more general case3
FIG. 7. Upper row: the quasi - particle spectral weight in both the underdoped ( x = 0 . , . x = 0 . , and overdoped ( x = 0 . , .
22) systems. Four Fermi arcs in underdoped case coalesce into a closed Fermi surface at the(Lifshitz) topological transition at optimal doping. Lower row: Spin susceptibility distribution of (in meV − ) for the samedoping leveks. The distribution is continuous through the Lifshitz transition at optical doping. Note that Brillouin zone inthe upper row is centered at the chrystallographic Γ point, while in the lower row it is shifted to the M point. This allows aconvenent focus on the peak around the AF order Q = ( π/a, π/a ) point. considered in ref.[18] and references therein) the Matsubara self energy for x > x opt (and temperature above T c ) in(gaussian or renormalized) perturbation theory is:Σ n k = (cid:0) πZe (cid:1) TM N (cid:88) l ,m e − ld a ω b m + Ω iω n + m − E k + l , (24)where E p ≡ (cid:15) p + (cid:15) (cid:48) p − µ + U n/ . (25)The dispersion relations are given in Eq.(7) and M is the oxygen ion mass. Second order ”gaussian” perturbationtheory[59] is justified at weak coupling, so that it should be confirmed in the following subsection that the dimensionlesseffective electron - electron coupling λ ph is indeed small. Summing over the bosonic Matsubara frequencies, ω b m =2 πT m , one obtains (after analytic continuation to physical frequency),Σ ( ω, k ) = (cid:0) πZe (cid:1) M Ω N (cid:88) l e − d a l I ω, k + l ; (26) I ω p = f B [Ω] − f F [ − E p ] + 1 ω + iη + Ω − E p + f B [Ω] + f F [ − E p ] ω + iη − Ω − E p ,where f B [ ε ] = (exp [ ε/T ] − − is the Bose distribution.In the underdoped case ( x < x ∗ ) we make use of the symmetrized correlators of the previous Subsection. Thesymmetrization is justified for description of the ARPES data, since it is a nonlocal probe, presumably over areaslarger than the AF domain size. The results are similar in form to the underdoped case: I ω, k = Z + k (cid:32) f B [Ω] − f F (cid:2) − E + k (cid:3) + 1 ω + iη + Ω − E + k + f B [Ω] + f B (cid:2) − E + k (cid:3) ω + iη − Ω − E + k (cid:33) (27)+ (cid:8) Z + , E + → Z − , E − (cid:9) .Here energies E ± k and weights Z ± k are given in Eqs.(22,23). These expressions will be used for calculation of both theelectron phonon coupling constant and the dispersion relation of quasi - particles.4 x opt EP I s t r e ng t h λ FIG. 8. Dimensionless coupling λ ph at the nodal point on the Fermi surface as function of doping. Dimensionless electron - electron coupling λ Generally the dimensionless coupling constant is defined in terms of the self energy as λ k = − ddω Σ ( ω, k ) | ω =0 + . Inthe overdoped case (see Appendix B for details and expressions in a more cumbersome underdoped case) one obtains(at zero temperature): λ k = (cid:0) πZe (cid:1) M Ω N (cid:88) l e − d a l (cid:40) θ [ − E k + l ]( E k + l − Ω) + θ [ E k + l ]( E k + l + Ω) (cid:41) . (28)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 shown in Fig.8. At each doping the location of the FS point was given by an analytic solution (explicit,albeit cumbersome). As expected it has a maximum of λ ph = 0 .
62. Upon deviation from the angle 45 ◦ the couplingdecreases. This is consistent with the experimental value estimated recently[42] at 30 K to be λ ph = 0 .
41 at optimaldoping at k = (0 .π ). In the underdoped cases it vanishes at small angles due to finite extent of the Fermi arc,see Fig. 7. One observes that the decrease of λ ph in the overdoped case is rather slow (linear). This might berelated to inaccuracy of the simple chemical potential description at large doping, as will be discussed in SectionIV. Generally the averaged over the Fermi surface coupling constant belongs to an intermediate range[60]. Suchcoupling is sufficient (as will be shown also in the next Section) to provide high d-wave T c superconductivity of orderof 80 − 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 phononsthe 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 isotopesubstitution 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”) innormal state approximately 45 meV below Fermi level[30–32]. Although some other theories appeared, the large5 - - - - - - - - / Ω R e [ d Σ / d ω ] - - - - - - - - - - - -
101 binding energy / Ω R e [ d Σ / d ω ] - - - FIG. 9. Derivative of the self energy with respect to frequency at energies below the Fermi level. On the left: optimally dopedand overdoped systems: x=0.164 (red), x=0.166 (yellow), x=0.18 (green) ,x=0.22 (cyan). On the right: underdoped cases:x=0.12 (green), x=0.13 (blue), x=0.14 (violet), x=0.15 (pink).Insets show the dependence on the O → O substitution(dashed lines) isotope effect[34] (substitution of O isotope by O ), observed mainly in underdoped samples) provides evidencethat he kinks should be attributed to EPI.To determine the kink position observed directly, let us differentiate the self energy Eq.(26) with respect to frequency ω . The real part of the integrand is: ddω I ω, p = − f B [Ω] − f F [ − E p ] + 1( ω + iη + Ω − E p ) − f B [Ω] + f F [ − E p ]( ω + iη − Ω − E p ) , (29)where E p was defined in Eq.(25). In the underdoped regime one similarly obtains, ddω I ω, p = Z + (cid:32) f B [Ω] − f F (cid:2) − E + k + l (cid:3) + 1 (cid:0) ω + iη + Ω − E + k + l (cid:1) + f B [Ω] + f F (cid:2) − E + k + l (cid:3)(cid:0) ω + iη − Ω − E + k + l (cid:1) (cid:33) (30)+ Z − (cid:8) E + → E − (cid:9) ,where the energies E ± and Z ± were defined in Eq.(22,23).To characterize the kink in dispersion relation, we calculate the derivative in range of frequencies between − . − . . < x < .
22. The kink position (zero value of the derivative) is around ω = − Ω = − meV . In the inset thedashed lined are the same quantity but for a heavier isotope O , namely with the oxygen atom mass M replacedby αM , α = 18 /
16. The location is shifted by approximately 6%, as was indicated in the ARPES experiment[31].Similar picture in the underdoped region, 0 . < x < .
15 is presented in the right plot in Fig.9. Now we turn to themain objective of the present study - d - wave superconductivity.
SUPERCONDUCTIVITY.
Although the main emphasis of the paper is on the apical phonon mechanism of the d - wave superconductivityin 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 enhancesuperconductivity. The purpose of the present Section is to quantitatively compare the role of these two contributionsand show how they coexist (complement each other) in the d - wave superconducting state. We start from the6derivation of the phonon exchange d wave ”potential” (mainly near the Γ point of BZ) and then proceed to the spinfluctuation 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 offreedom to calculate the effective electron - electron interaction. We start with the phonons.
EPI
The Matsubara action for EPI, Eq.(11), and phonons are A eph [ ψ, u ] = Ze T (cid:88) m , q n − m, − q [ ψ ] g α q u αm, q ; (31) A ph = M T (cid:88) m, q u α − m, − q Π αβm, q u βm, q ; ,where n − n, − q [ ψ ] = (cid:80) k ,m ψ ∗ σ k − q ,m − n ψ σ k ,m and g was defined in Eq.(12). The polarization matrix is defined via thedynamic matrix of Eq.(3): Π αβn, q = (cid:0) ω bn (cid:1) δ αβ + M − D αβ q , α, β = x, y, calculated in Appendix A. Since the action isquadratic in the phonon field u , the partition function is gaussian and can be integrated out exactly, see details inref.[18]. As a result one obtains the effective density - density interaction term for of electrons A pheff = 12 T (cid:88) q ,n n n, q v phn q n − n, − q , (32)where the effective electron - electron frequency dependent ”potential” is: v phn, q = − (cid:0) πZe (cid:1) M e − d a | q | ω b n + Ω . (33)The expression is ”exact” for harmonic phonons (we have neglected the transversal mode and small dispersion of thelongitudinal mode spectrum[18], see Fig.3). An approximate expression for the effective interaction due to the electroncorrelations effects will be derived next. The potential exhibits the central ”inverted” (that is negative) ”peak” thatwe will call the apical phonon dip due to the exponential form of the matrix element shown in Fig.4. The secondbosonic ”glue” is generated by the correlation effects. Susceptibility of 2DEG and the effective electron - electron interaction due to Hubbard correlation.
Since, as explained in Subsection IIIA, the on site Coulomb repulsion constant in our scheme, U = 1 . t , is not verylarge (can be classified as an range with short range order and moderate fluctuations), the gaussian expansion[41]is applicable One starts with the mean field GF and considers the rest of the action as a perturbation. In theoverdoped case, it is just a ”renormalized” Kohn-Luttinger perturbation theory[61]. We therefore calculate theeffective interaction due to correlations in the second order in U . Generally, utilizing the inversion symmetry, theeffective interaction in the spin singlet channel has a form: A coreff = 12 T (cid:88) q .n n n q v corn q n − n, − q ; (34) v corm q = U + U χ m q ,where χ m q is the electronic susceptibility. The positive constant U in Eq.(34) is just the direct first order Coulombrepulsion suppressing the s-wave pairing, but having no impact on the d - wave pairing.The well known Kohn-Luttinger diagrams[40, 61] give in the overdoped case, x > x ∗ , the following dynamicMatsubara susceptibility: χ m q = − TN (cid:88) n p iω m + n − E p + q iω n − E q (35)= 1 N (cid:88) p f F [ E p + q ] − f F [ E p ] iω m + E p − E p + q ,7where E p was defined in Eq.(25). This is calculated numerically for sufficiently large values of N = 128 and harmonics | m | (cid:54)
32. For m = 0 and q = we use the L’hopitail’s limit f F [ E p + q ] − f F [ E p ] E p − E p + q ≈ T (1 + cosh [ E p /T ]) . (36)In the lower row of Fig.7 the static part, namely zero frequency is given x = x opt = 0 . x = 0 .
18 and x = 0 . x < x ∗ , one calculates the same two diagrams on the magnetic BZ, 0 < k <π, − π < k < π , namely using the GF of Eq.(20). Since we are interested in the dynamic susceptibility on the scaleof the Cooper pairs, the full sublattice matrix should be used. This is derived in Appendix B, where a rather bulkyexpression, Eq.(53) is given. It turns out that after symmetrization it is not much different from the overdoped casesusceptibility as is shown in Fig.7. The symmetrization of the susceptibility matrix is made as in ref.[41]). The zerofrequency χ sym ,k x ,k y at T = 50 K is plotted for x = 0 .
12 and x = 0 .
15. The dependence on temperature in the relevantrange (
T < 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. Notealso the fine structure of the susceptibility: there are two characteristic local maxima near point M , while the pointitself is a local minimum. The splitting is very small. In this paper we do not consider possible fourfold symmetrybreaking (or nematicity). This effective electron - electron couplings will be used in the gap equation. Superconducting gap in overdoped system
To complete the electronic effective action, one adds to Eqs.(32) and (34) the electronic part, A eff = 1 T (cid:88) n k (cid:26) ψ ∗ σn k G − n k ψ σn k + 12 n n k v n k n − n, − k (cid:27) ; (37) v n q = v phn q + v corn q ,where G is the (HF) Green’s function and v ph and v cor are given by Eq.(33) and Eq.(34) respectively.The standard superconducting gap equation is,∆ m k = − TN (cid:88) n p v m − n, k − p G − ∗ n p ∆ − n p G − n p + ∆ ∗ n p . (38)Here the (Matsubara) gap function is related to the anomalous GF, (cid:10) ψ σm k ψ ρn p (cid:11) = δ n + m δ k + p ε σρ F m k ( ε σρ - the anti-symmetric tensor), by ∆ m k = TN (cid:88) n p v m − n, k − p F n, p . (39)The gap equation was solved numerically by iteration for N = 128 and 64 frequencies. It converges to the d - wavesolution. An example of the gap distribution over the BZ (for the optimal doping at T = 50 K ) is given in Fig. 10.The absolute value of the Matsubara gap function has a maximum near the crystallographic X point (0 , π ). Thisvalue as function of doping and temperature is given in Fig. 11. The blue part of the surface corresponds to x ≥ x opt .The line of vanishing gap determines the critical temperature values on the phase diagram in Fig.6 (red squares). Inthe optimal and overdoped domains it agrees well with the parabolic experimental dependence (dashed curve) takenfrom ref.[19]. If one neglects the magnon contribution, namely takes v = v ph , the temperatures are lower by 10-15%(red circles).One observes that the decrease of T c is rather slow (linear) at large doping compared to the experiment. Whendoping becomes of order 30% it is expected to significantly impacts the effective mesoscopic lattice model parameters( µ, U, t, t (cid:48) ). In underdoped cases the pseudogap should be taken into account. The results are the yellow part of thesurface in Fig.11 for the gap and critical temperatures shown on the left hand side of the phase diagram, Fig.6.8 - - - - k x ( π / a ) k y ( π / a ) FIG. 10. The d - wave solution of the gap equation for optimal doping, x = 0 .
166 at 50 K . Superconducting gap in underdoped system
In an AF domain (considered to be larger than the Cooper pair) the fourfold symmetry is broken. As a consequenceone uses basis consisting of two sublattices I = A, B and the magnetic BZ defined in Subsection IVA. The electroniceffective action, in this basis takes a form A eff = 1 T (cid:88) n k (cid:26) ψ ∗ σIn k (cid:2) G − σn k (cid:3) IJ ψ σJn k + 12 n σIn k v σρIJn k n ρJ − n, − k (cid:27) , (40)where G is the (HF) Green’s function is given in Eq.(20) and v = v ph + v cor in Appendix B. The symmetrizedsusceptibility in the underdoped cases of x = 0 . , .
15 are given in Fig.7. One observes that the distribution iscontinuously crosses over to the overdoped one via the (Lifshitz) topological transition at optical doping.The anomalous Green’s function is also a 2 × (cid:68) ψ σIn, k ψ ρJ − n, − k (cid:69) = ε σρ F IJn k . (41)Assuming the up-down (singlet) pairing[18], see Appendix C,[∆ n k ] = 0 ∆ ↑↓ n k ∆ ↓↑ n k ↑↓ IJn k = (cid:88) m p v ↑↓ IJn − m, k − p F ↑↓ IJ , (42)9 FIG. 11. Superconducting (maximal) d - wave Matsubara gap as function of dopings and temperatures. Underdoped partsare in brown, while the overdoped in blue. the gap equation in matrix form becomes, (cid:104) ∆ ↑↓ n k (cid:105) = − (cid:88) m p [ v n − m, k − p ] ∗ (cid:110)(cid:2) G − ↓ m p (cid:3) † (cid:2) ∆ ↑↓ m p (cid:3) − (cid:2) G − ↑ m p (cid:3) + (cid:2) ∆ ↑↓ m p (cid:3) † (cid:111) − , (43)and the same for ∆ ↓↑ n k . The star product denotes the matrix element multiplication.The iteration solution for the same system size, as in the overdoped case, converges to the d - wave solution forwide range of initial conditions. The maximum gap as function of doping and temperature is given in Fig. 11 (theyellow part of the surface). The line of vanishing gap determines the T c values on the phase diagram in Fig.6 (redsquares). In the underdoped domain it comes short of the parabolic experimental dependence[19] (dashed curve). Ifone neglects the magnon contribution, namely takes v = v ph , the temperatures are lower by 10 −
15% (red circles).
Isotope effect
The influence of the oxygen isotope substitution, O → O on superconductivity can be gauged by calculation ofthe change of the (Matsubara) gap at a temperature below T c . In Fig. 12 we plot the The deduced exponent, α = 1816 log ∆ (cid:0) O (cid:1) ∆ ( O ) , (44)0at temperature T = 15 K . x opt α FIG. 12. The doping dependence of the isotope effect exponent α , Eq.(44).. The same exponent was estimated by measuring the T c isotope effect in various hole doped cuprates[62], mostlyin Y Ba Cu O − x and La − x Sr x CuO . Qualitatively the exponent is very small in overdoped and optimally dopedmaterials, but becomes significant at strongly overdoped case. In Ba Sr CaCu O the experimental results arescarce, but order of magnitude is the same as in Fig.12. The isotope effect exponential is small, α = 0 .
05, and nearlyindependent of doping at optimal and overdoped systems, however it fast increases when the doping is reduced belowoptimal (reaches α = 0 . x = 0 . DISCUSSION AND CONCLUSIONS.
Theory of superconductivity of high T c perovskite cuprates based on the dominant lateral apical phonon pairingmechanism was proposed. It is comprehensive in a sense that the whole range dopings. is considered includinganomalous normal state properties of the perovskites. To demonstrate the basic principles we limited ourselves inthis paper to a simplest sufficiently generic model. To describe the pseudogap physics of 2DEG in the CuO planes weadopt the fourfold symmetric t − t (cid:48) single band Hubbard model with on site repulsion energy U of moderate strength.Doping is controlled by the chemical potential.The results are following. Considering a typical cuprate superconductor Bi CuO planes. The approach isjustified in ionic crystals. The most important for the pairing mode is found to be the optical lateral (within the
SrO plane) mode at 40 meV , see Fig.3. The dimensionless electron - electron attraction exhibits an exponential forwardscattering peak, see Fig.4, and is estimated to have the strength of λ ∼ .
5, see Fig.8.When parameters of the model were fixed at t (cid:48) ∼ − . t and U ∼ . t , t − . eV , the mean field T ∗ line, greencurve in phase diagram, fig.6, become a crossover between short range correlated anti - ferromagnetic pseudogap phaseand the paramagnetic one. Within the mean field approximation the phase transition is second order (although longrange averaging over domains makes it a crossover, see [54] for the RG approach not attempted in this paper) withpseudogap given in Fig.5. The T ∗ becomes first order at a small segment below the superconducting dome in phasediagram, Fig.6. The quasi - particle spectrum undergoes a topological (Lifshitz) transition. The closed Fermi surfaceabove the T ∗ line disintegrates into four Fermi arcs below it, see Fig.7.1 k x ( π / a ) k y ( π / a ) - - - - FIG. 13. The overall potential including both the phonon central dip at Γ and the correlation peak at M .. Note that the dipis larger than the peak leading to dominance of the phonon channel. 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 ( O → O ) dependence, see Fig.9. Since theelectron - phonon coupling λ is moderate, weak coupling dynamic Eliashberg approach is applicable to calculate thegap function and critical temperature T c . One has to go beyond the BCS approximation due to important dependenceof the phonon mediated pairing on frequency. Both phonon and spin fluctuation pairing are accounted for over the fulldoping range. It is found that the critical temperatures above 90 K at optimal doping can be reached, see Fig. 6. Thedominant ”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 10-15%. Comparison of the doping dependence of T c withexperimental[19]is qualitatively fair, although . underdoped are slightly underestimated, while strongly overdopedoverestimated. The isotope effect (of the O → O substitution) has a very small nearly doping independent nearand critical doping, but grows fast when doping is reduced (reaches α = 0 . x = 0 . M point (corner of Brillouin zone) since theinteraction is repulsive. Both regions of the BZ contribute to superconductivity and fortunately do no interfere witheach other. Indeed the phonon peak decreases exponentially to just 10% at distance k ph = 1 /d a = π a , where d a is thevertical distance of the CuO layer from the
SrO layer, see Fig. 4. The susceptibility becomes negligible at distance π a from M , see Fig.7. Hence the BZ is effectively utilized.To summarize, two features turned out to be sufficient for robust apical phonon d - wave pairing. The first is therhombic shape of the Fermi surface. The second is the exponential FSP of the apical lateral phonon optical modeand, to a lesser degree, constructive cooperation with the spin fluctuation channel.Restriction of the description of the electron gas to one band Hubbard model with just two parameters t, t (cid:48) fornearest neighbor and next to nearest neighbor hopping obviously makes the model less realistic to quantitativelydescribe real materials like Bi t (cid:48)(cid:48) . In addition the tunneling between the conducting CuO planes via a metallic layer and the nematicity (deviations from the fourfold symmetry) should be added. Theselead to a characteristic splitting of the spectrum[48]. This is left for future work. Of course the phenomena broadlytermed ” unusual normal and superconducting properties of high T c cuprates ” contains many more features. In thispaper we have emphasized ones that are directly linked to the phonon exchange. Other like the ”strange metal”behavior in a similar model with relatively low U were recently address[39].Experimentally the main claim of the paper, namely that the ”glue” that creates d - wave pairing is the phononexchange of a very specific nature, the apical oxygen’s (that is one belonging to an insulating layer, SrO , adjacentto the conducting
CuO layer) lateral vibrations, can be further directly strengthened or falsified by suppression suchvibrations as in refs[28][29] or actively focus on these modes and their coupling. Since one or to unit cell perovskiteswere recently fabricated[21][19][20] perhaps apical oxygen atoms can be distinguished from the rest. An alternativeroute 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 (not addressed here) that presumably originates from correlation effects [39]. Themodification can be isolate by isotope substitution. Superconducting properties due to this particular mechanisms inaddition to T c and order parameter studied, are also sensitive to the isotope substitution. An example is magnetizationcurves[63] that simply depend on T c (via Ginzburg - Landau description[59]). Acknowledgements.
We are grateful Prof. D. Li, L. L.Wang, Y. Guo, J.Y. Lin and Y. Yeshurun for helpful discussions. Work of B.R.was supported by NSC of R.O.C. Grants No. 101-2112-M-009-014-MY3.
APPENDIX A. THE APICAL OXYGEN LATERAL VIBRATION MODES
The approximate method of determining the relevant vibration modes is the same as previously used for the
F eSe on STO superconductor, see details in Appendix A of ref. [18]. Dynamic degrees of freedom are the O atoms in the SrO layer, see Fig.2. Hamiltonian for these degrees of freedom is H ph = K ph + W , (45)where kinetic energy is K ph = M (cid:88) n (cid:18) ddt u n (cid:19) , (46)and the potential energy part consists of interatomic Born - Meyer potentials defined in Eq.(1) and Table I. Onlyinteractions of the ”dynamic” oxygen atoms in the SrO with neighboring
BiO below and
CuO above are taken intoaccount: W = 12 (cid:88) n , m v SrO (cid:2) R Sr − R + r n − r m − u m (cid:3) + v CuO (cid:2) R Cu − R + r n − r m − u m (cid:3) + v OO (cid:2) R Ox − R + r n − r m − u m (cid:3) + v OO (cid:2) R Oy − R + r n − r m − u m (cid:3) + v BiO (cid:2) R Bi − R + r n − r m − u m (cid:3) + v OO (cid:2) R O − R + r n − r m − u m (cid:3) (47)+ 12 (cid:88) n (cid:54) = m v OO [ r n − r m + u n − u m ] .3Here the apical oxygen position R and the lattice vectors r n were defined in Eq.(2), while positions of the heavy Sr, Cu, Bi , see Figs. 1-2, are R Sr = ; (48) R Cu = R + z Cu (cid:98) z ; R Bi = R + z Bi (cid:98) z .The interlayer spacings are given in Table I and (cid:98) z ≡ (0 , , CuO layer and that in the BiO layer are: R Ox = ( a, a/ , z Cu ) ; R Oy = ( a/ , a, z Cu ) ; (49) R O = (0 , , z Bi ) . Vibrations of heavy atoms and even oxygen in other planes are not expected to be significant due to their massor distance from the
SrO layer oxygen atoms. Some effects of those vibrations can be accounted for by the effectiveoxygen mass, while more remote layers above and below the important layer were checked to be negligible. Harmonicapproximation consists of expansion around a stable minimum of the energy. Expressions for the derivatives are givenin ref.[18]. This leads to the following expression for the dynamic matrix D αβ k = (cid:88) n (cid:2) v CuO (cid:3) (cid:48)(cid:48) αβ [ R Cu − R + r n ] + (cid:2) v OO (cid:3) (cid:48)(cid:48) αβ [ R Ox − R + r n ]+ (cid:2) v OO (cid:3) (cid:48)(cid:48) αβ [ R Oy − R + r n ] + (cid:2) v SrO (cid:3) (cid:48)(cid:48) αβ [ R Sr − R + r n ] + (cid:2) v BiO (cid:3) (cid:48)(cid:48) αβ [ R Bi − R + r n ]+ (cid:2) v BiO (cid:3) (cid:48)(cid:48) αβ [ R O − R + r n ] + (1 − exp [ − i k · r n ]) (cid:2) v OO (cid:3) (cid:48)(cid:48) αβ [ r n ] . (50)These matrix elements determine the frequencies (eigenvalues) for the two polarizations presented in Figs. 3. APPENDIX B. NORMAL STATE PROPERTIES IN AF PHASEEPI in the magnetic Brillouin zone
The connection between the electron - electron attraction due to phonons given in Eq.(33) in the usual ”param-agnetic” basis, that is full BZ (marked by v k x ,k y here) in the underdoped cases should be represented as a matrixelements in the sublattice space defined on a smaller magnetic BZ. The matrix, v phk k = 12 v k ,k − k + v k + π,k − k + π ( v k ,k − k − v k + π,k − k + π ) exp [ − iak ]( v k ,k − k − v k + π,k − k + π ) exp [ iak ] v k ,k − k + v k + π,k − k + π , (51)was used to calculate the phonon effects in both normal and superconducting state. Susceptibility
The susceptibility matrix that enters the effective electron - electron interaction strength due to (the Hubbardrepulsion induced) correlations is calculated in the postgaussian approximation as Lindhard type diagrams given inFig.B1. They are similar to the paramagnetic case[61][40]. The propagators of the diagrams however, Eq.(20), aredefined on magnetic BZ and have two sublattice indices. The spin singlet pairing contribution to elements comes fromthe left and center diagrams: χ IIm, q = TN N (cid:48) (cid:88) n p (cid:16) − G ↓ IIm + n, q + p G ↓ IIn p + G ↓ IIm + n, q + p G ↑ IIn p (cid:17) ; (52) χ ABl,q = − TN N (cid:48) (cid:88) n p (cid:16) − G ↓ ABm + n, q + p G ↓ BAn p + G ↓ ABm + n, q + p G ↑ BAn p (cid:17) ; χ BAl, q = χ AB ∗− l, − q , N (cid:48) = N/ I = A, B . The third diagram vanishes.Summing up over integers n , one obtains χ AAm q = P m q − Q m q ; (53) χ ABm q = R m q ; χ BBm q = − P m q − Q m q ,where P m q = 12 N N (cid:48) (cid:88) p (cid:0) L m (cid:2) E − p , E − q + p (cid:3) + L m (cid:2) E − p , E + q + p (cid:3) + L m (cid:2) E + p , E − q + p (cid:3) + L m (cid:2) E + p , E + q + p (cid:3)(cid:1) ; (54) Q m q = ∆ pg N N (cid:48) (cid:88) p x p (cid:0) L m (cid:2) E + p , E − q + p (cid:3) − L m (cid:2) E − p , E + q + p (cid:3) + L m (cid:2) E + p , E + q + p (cid:3) − L m (cid:2) E − p , E − q + p (cid:3)(cid:1) ; R m q = 12 N N (cid:48) (cid:88) p h ∗ q + p h p x q + p x p (cid:0) L m (cid:2) E − p , E + q + p (cid:3) + L m (cid:2) E + p , E − q + p (cid:3) − L m (cid:2) E + p , E + q + p (cid:3) − L m (cid:2) E − p , E − q + p (cid:3)(cid:1) .Here L m [ E , E ] = f F [ E ] − f F [ E ]2 iπT m + E − E , (55)∆ pg is the pseudogap energy, x p is defined in Eq.(18), E ± p in Eq.(17) and h p in Eq.(14). n, p n-m, p - q I J -n,- p m-n, q - p n-m, p - q m-n, q - p n, p -n,- p I I n-m, p - q m-n, q - p n, p -n, - p FIG. 14. Three second order diagrams determining the effective electron - electron interaction due to spin fluctuations. Botthspin and sublattice indices are indicated. While the diagram on left and center give nonzero contributions of Eq.(), the thirdvanishes due to conflict in assigning spin indices to propagators in the loop.
APPENDIX C. DERIVATION OF THE GAP EQUATION IN UNDERDOPED SYSTEM
We derive the Gorkov’s equations within the functional integral approach[64, 65] starting from the effective electronaction for grassmanian fields ψ ∗ σ k ,n and ψ σ k ,n . To simplify the presentation it is useful to lump the quasi - momentum andthe Matsubara frequency into a single subscript, { n, k , k } → α, and the spin and sublattice into the four component5spinor { σ, I } → a . The action of Eq.(40) takes a standard multicomponent four - Fermi form studied for example inref.[18]: A [ ψ ] = ψ ∗ aα T abα ψ bα + 12 ψ ∗ aβ ψ aχ + β v ab − χ ψ ∗ bγ ψ bγ − χ . (56)The hopping 4 × a = { σ, I } , b = { ρ, J } in the following form, T ab { n,k ,k } = δ σρ (cid:0) − iω n + ε (cid:48) k − µ + Un (cid:1) + σ σρz ∆ pg − δ σρ h ∗ k − δ σρ h k δ σρ (cid:0) − iω n + ε (cid:48) k − µ + Un (cid:1) − σ σρz ∆ pg , (57)with I and J being the row and the column indices.Gorkov equations in matrix form are: − G α T α − F α ∆ ∗ tα = I ; (58) G α ∆ α − F α T t − α = 0,where F α is the anomalous GF and the matrix gap function is defined [∆ α ] in components as∆ bcα = (cid:88) χ v bcα − χ F bcχ . (59)The corresponding gap equation is∆ bcα = − (cid:88) χ v bcα − χ (cid:20)(cid:16) T t − χ [∆ χ ] − T χ + ∆ † χ (cid:17) − (cid:21) bc . (60)The singlet Ansatz Eq.(42) leads to Eq.(43).6 ∗ [email protected] † [email protected][1] T. Timusk and B. Statt, Rep. Prog. Phys.
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