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The vibronic approach to high-Tc superconductivity
Physica C 175 ( 1991) 644-650 North-Holland
The vibronic approach to high-Tc superconductivity Mladen Georgiev Faculty of Science, Myilly Point Campus, Northern Territory University, PO Box 40146, Casuarina, NT 0811, Australia Dipartimento di Fisica, II Universiti* degli Studi di Roma, Via Emanuele Carnevale, I-O0173 Roma, Italy
Received 18 January 1991
Recent experimental and theoretical studies have been stressing the relationship between the pairing mechanism and the motion of apical oxygensin high-Tomaterials. Earlier we had proposeda similar model of vibronicaUycoupledchargetransfer from planes to chains producing asymmetricallydistorted Cu-O polyhedra to move and pair as small polarons. Numerical data for LaSCO proved to be in optimistic agreement with the experimental To. This model is now revisited both as a theoretical background and as a backing up of experimental evidence. Its particular relationship with the inferred ferroelectricity of high-To materials will be analyzed.
Apologia Ever since the discovery of a high-To superconductivity in layered perovskites by Bednorz and Miiller, a tremendous effort has been in progress to unravel the pairing mechanism [ 1 ]. It is the purpose of this paper to emphasize certain experimental facts that seem to be lending support to a fairly unconventional view. By this we mean the vibronic approach, a term originally borrowed from the quantum chemistry of molecules. In what follows we shall once more reevaluate our vibronic model against the background of new data accumulating rapidly in the literature.
1. Introduction Earlier we proposed a pairing model appropriate to the Cu-O system [2,3]. The essentials are: opposite parity Cu and O states (or their hybrids) mix vibronically, which renders a Cu-O bond unstable against uneven modes via the pseudo-Jahn-Teller effect. The vibronic mixing materializes through diVisiting Fellow from Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, 72 Lenin Boulevard, Bulgaria.
polar coupling of the vibrational mode to the electron density, c-axis dipoles will have the best chance to couple to and survive screening from the 2D charge carriers in CuO2 planes. Examples are provided by Cu ( 3dz2 ) and O (2pz) orbitals both spread along the c-axis, while the relevant modes are A2u in tetragonal La2_xSrxCuO 4 and Blu in orthorhombic YBa2Cu3OT. The C u ( I I ) - O ( I V ) bonds in YBCO (Argonne nomenclature) are of particular interest. Now the static electronic states hybridize only weakly due to the large bond length. By vibronic mixing, double-well potentials are produced for the apical O (IV) ions and transitions incited across them transferring charge from in-plane Cu ( I I ) / O ( I, II ) to localized O ( I V ) levels. As a result, O ( I V ) ions will tend to either occupy stable off-center sites or delocalize by interwell tunneling. These O(IV) ions will display anomalous vibrational amplitudes, as observed [ 4 ]. Distorted bipyramids will result, asymmetrical with respect to the Cu (I) ion in between. The associated parity violation gives rise to an inversion dipole and a high dynamic polarizability of the Cu ( I I ) - O (IV) moiety. This opens a polarization-induced pairing channel [2 ] for in-plane holes which may be responsible for the high-T¢ superconductivity [5]. Physically this is equivalent to a travelling off-centered (ferroelectric) instability coupled to the in-
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M. Georgiev/ Vibronic approach to high-Tosuperconductivity
plane charge carders which promotes the pairing [2 ]. We shall now revisit both the premises of this model and the supporting experimental evidence. Earlier work has been surveyed elsewhere [ 3 ]. Related local vibronic models have also been proposed by others but we are constrained to mention only a few [6-8].
2. Electronic transitions
The infrared (IR) optical spectral of high-To materials are believed to hold the key to understanding the pairing mechanism. On increasing the photon energy, they appear to be composed to phonon lines in the far infrared (FIR) superimposed on a background electronic activity that peaks in the mid infrared (MIR). The MIR feature is found in the IR absorption (or optical conductivity) peaked at about 0.25 eV in all the high-To superconductors [9]. This is followed by another electronic band in the near infrared (NIR) [10]. Both bands are clearly associated with intragap states: they appear e.g. at 0.13 eV and 1 eV in semiconducting YBa2Cu307_x ( x ~ 0 . 8 ) , while the bandgap of YBa2Cu306 is at about 1.7 eV (even though shifting to lower energies on increasing x). Corresponding bands in La2CuO4 appear at 0.5 eV and 1.4 eV, while the gap is at 2 eV. Raman-scattering probing of FIR and MIR reveals that the intragap states ofYBa2fu307 (YBCO) couple strongly to phonons, for they exhibit a Fano antiresonance [ 11 ]. The Raman spectrum below Tc displays polarization and orientation dependent structures resolving it into two components of A~8 and B~8 symmetries with respect to the x - y in-plane interchange. (In concomitance, the IR optical conductivity of YBCO also resolves into plane and chain components [ 12 ].) While Als originates from intraband transitions in CuO2 planes, B~s is attributed to charge transfer transitions (CTT) from a nonbonding (NB) py O(IV) band to the Cu-O hybridized inplane band. The chain-to-plain charge transfer has been computed implying that the MIR gap state may correspond to CTT perpendicular to the CuO2 planes [ 13,5 ]. The excited-state configuration where an extra hole is added to a nearby O or Cu site is found to have a sharply reduced energy relative to the undoped ground state. Low-lying Cu 3+ 0 2 - Cu 2+ O 1 -
-
-
645
CTT occur because of the proximity of the Cu 2+ and 0 2 - ionization potentials meaning a near degeneracy in energy of these two Cu-O complexes, a remarkable intrinsic property of the Cu-O system. Assigning the relevant electronic orbitals is of chief importance. "Ab initio" quantum-chemistry calculations on cluster models suggest that the extra holes enter the antibonding (AB) combination of Cu(dx2_y2) and O(2p,x,y) orbitals. In the Cu2+O l state the hole resides on the NB O(p~,y) orbital. The AB-NB CTT is a transverse mode of electron transfer from the O(p~) band to the Cu(dy~_z2) component with a transition dipole normal to the chain, large oscillator strength and energy from 0.1 to 0.5 eV. It amounts to a high dynamic polarizability along the O ( I V ) - C u ( I ) - O ( I V ) axis. Now a specific view on high-To emerges [ 5 ] which suits quite well with the crystalline structure and chemistry of layered perovskites: pairing mediated by CTT across the O (IV)-Cu ( I ) - O (IV) moiety along the c-axis between CuO2 planes. CTT create a retarded attraction between oxygen holes leading to an s-wave BCS-like superconductive state. Holes pair effectively via the monopole-to-induced-dipole coupling by virtue of the high doping-made polarizability of Cu-O bonds. Although the supercurrents themselves run in the CuO2 planes, the attractive pairing mechanism involves the CuO3 chains and is not two-dimensional for that matter! Several experiments lend strong support to the CTT high-To model [ 5 ]: its efficiency depending on the C u ( I I ) - O ( I V ) separation, it has responded to external pressure or to Co substitution in the expected manner. Another experiment using polarized Cu-edge X-ray absorption spectroscopy reveals that extra holes in YBCO reside mainly on a Cu-O orbital oriented along the c-axis which involves the O(IV) p~ and Cu(II) dz2 states. It is inferred that a second hole band different from the in-plane Cu(3dx2_y~)-O(2px, 2py, ~) hybrid should be present. In conformity with this nuclear quadrupole moment measurements suggest that the bridging O (IV) oxygens should be regarded as O 1- rather than 0 2 - . There being a clear correlation between Tc and the O ( I V ) - C u ( I I ) separation [14], the relevance of these observations to superconductivity is implied.
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M. Georgiev / Vibronic approach to high-To superconductivity
3. Phonon coupling
In so far as the O ( I V ) - C u ( I I ) CTT are phonon coupled, they may incite considerable anharmonic motion along the c-axis. Indeed, there is convincing evidence of O(IV) anharmonicity in YBCO [ 4,15,16 ]. However, vibrational modes coupled to the superconductivity should also exhibit anomalies near To. Indeed, anomalous Cu-O modes near Tc in ErBa2Cu307_y have been evidenced by ion channeling along the c-axis [ 17 ]. A 30% increase across Tc is observed in the amplitudes of Cu and O ions indicating a strong coupling. Neutron powder diffraction reveals abnormally large amplitudes of O(IV) ions at Cu-O chains, concomitant with these ions occupying energy minima at about 0.1 A off-side the chain line [ 18 ]. Off-side displacements result from the coupling to odd-parity (uneven) modes. Polarized FIR reflection studies have sorted out uneven modes, A2~ parallel and Eu normal to the c-axis [ 19 ]. While IR modes in semiconducting YBa2Cu306 exhibit normal oscillator strenghts, those in superconducting YBa2Cu307 have very high strengths. The C u ( I ) - O ( I V ) stretching mode is reported to be 645 cm- 1. Raman and FIR experiments on YBa2Cu307_x clearly show lattice coupling to the superconducting systems [ 20 ], as evidenced by a Fano effect on two Raman modes, at 120 and 335 cm -~, and by an anomalous softening at T~ of several IR peaks at 275, 311 and 569 c m - 1. However, the closest relationship to Tc is shown by a B~s tetragonal symmetry phonon mode at 340 cm -1 of a vertical O ( I I ) - O ( I I I ) vibration in the CuO2 planes [ 21 ]. This mode softens at and displays a behavior with T~ in an external magnetic field. Raman modes in resonance with the MIR gap state of YBa2Cu3OT_x are found by exciting with subgap IR light [22]. A line at about 500 cm-1 assigned to the C u ( I ) - O ( I V ) axial stretching mode appeared on both intragap and intergap excitation, its frequency rising with increasing x. This hardening suggests a reduction of the bond length related to a resonance interaction with the localized electronic excitation. The mode polarization implies that the electronic CTT is also along the c-axis. It is remarkable that the 340 cm-~ in-plane mode, coupled beyond doubt to the superconducting transition, is only excited by visible light, possibly being
in resonance with higher energy states. On the other hand the dual coupling of the stretching mode to both higher and lower energy excitations may prove an essential feature of the chains-to-planes CTT. It is also remarkable that virtually the same 500 cm -~ Raman frequency appears at about 510 cm-~ in the photoinduced optical absorption of YBa2Cu307_x and at 579 c m - t in YBCO. Another even frequency at 436 cm-~ also displays a correlated IR appearance. Such dual Raman and IR activity may signify breaking up of the inversion symmetry at the Cu (I) site as a result of coupling to the charge carriers. An axial distortion of the CuO5 pyramids has been suggested [ 3,22 ]. For tetragonal La2CuO4 the A2u modes are along the c-axis, while the Eu modes are a-b-planar vibrations. The A2u stretching of vertex O (3) oxygens is reported to be at 520 cm-1, whereas the AEu stretching of a-b-plane O ( l , 2) is at 683 cm -~ [23]. Oxygen isotope constants and transverse-optic AEu modes at 240 and 500 cm-~ in LaE_xSr~CuO4 display anomalies for x ~ 0.15 where T¢ is at maximum, possibly due to coupled lattice fluctuations [ 24 ].
4. The vibronic CT model
There have been a few attempts to ascribe vibronic features to high-To superconductivity. One early work addresses the peculiarities of the Cu-O system as of relevance to the electronic mid-infrared band [25 ]. This band is assigned to "internal electronic and vibrational excitations of a spin-zero charge-transfer complex (P+) formed between a hole and an essentially single paramagnetic square planar CuO4 unit (po)-. Non-vanishing dipole moments occur for electronic transitions from NB O to AB CuO orbitals, only in-plane Cu(3dx2_y2) and O(2px.y) states being considered. The antisymmetric vibrational Eu mode modulates the AB mixture which incites CTT across P+ producing the IR band. The derived frequency-dependence conductivity agrees with the experimental spectrum of LaSCO by fitting a set of electronic and vibrational parameters. The polarizability ol(to) varies between low frequency a ( 0 ) and high frequency c~e limits with fitting values 700 A 3 and 180 A 3, respectively. The higher value of the former (vibronic) limit is due to
647
M. Georgiev / Vibronic approach to high-T~superconductivity
phonon renormalization which elevates the pairing [2,3 ]. Superconducting are either the highly correlated electrons of po or the P+'s themselves delocalizing as polarons with a fluctuating electric dipole moment and high polarizability. Attractive monopole-induced dipole interactions with P+ or between pairs of P+ lead to enhanced carrier pairing and high-T¢ superconductivity: the Rice-Wang model contains all the essentials of a vibronic theory. The Hamiltonian of interband vibronic CTT reads [2,3]
/~=
E
(ij)mn
+l ~ ijmn
~m.+X
im
Eim ~ a im + aim
l~(ijl m n )a imajn + + ajmain ,
with [ 3 ] t~l~ = ½E, u [ 1 - ( E,u/ 4EjTgu ) Xllg u2 =
]/2sinh (x~,o),
( 2EsT~u/htottsu) [ 1 _ ( Egu/ 4EjTgu)2]
iF'in =Eim --EjTgu [ 1 "{- (Esu/4EjTgu) 2 ] + ½hdgiigu ,
~ii,u = oJi~ [ 1 - ( E ~ d 4 E j T , J H=
~
+ tijmnaimajn + ~ Elm aim + aim
+½E
[KijmnQijmn ~ + Pijmn/mijmn]
(ij)mn ijmn
+E
ijmn
ijmn
W ( q"" l m n ) a i , ,+a j , + ajmai, .
(3)
Ptg~ =Ptsu [ 1 - (EgJ4EjTgu) 2 ] ,
( 1)
Here m, n=g,u are the band labels, while i , j , l label sites in the Cu-O manifold, a;+s (a;s) creates (annihilates) an in-plane hole at site i, while a~ (a~) does so for a (side) chain hole. The bands are assumed narrow, nearly degenerate, and of opposite parities. (ij) label neighbouring sites and tom, is the hopping integral between these. Qom~, Po,,, and Morn ~ are coordinates, momenta, and reduced masses of intra (i = j ) or inter (i ~ j ) site nuclear oscillators coupled to the electronic system through the mixing constants Gore~. While the latter (even) mode couples to the itinerant in-plane g-holes, such as the breathing Cu-O vibration, the former (uneven) promotes gu CT from planes to chains which leads to efficient g-hole pairing. W(ijl mn ) are hole interaction constants. In a particular case a m-hole at site j couples to the electric dipole p,,~ induced through m - n charge transfer at another hole site l: W ( ljl m n ) = UoS,~ -ptm~ • (ejRo/kR~)(1-~m~).
] 1/2,
rP(ljl g u ) = -~tgu" ( G R o / k R ~ ) ,
im
G ijmn Qo,,,,, a ~,,, ÷ ay,,
+½~
3/2 ,
(2)
Here ptm~ is the m - n mixing dipole, Uo=etej/kR ~ is the intersite Coulomb repulsion. Excluding the lattice in the adiabatic approximation leads to a renormalized electronic Hamiltonian
locally along the mixing-mode coordinate Qttg~ at Qomm= 0. Similar equations hold good for quantities renormalized through coupling to the intersite modes Qom~ at Qttg~=O through now the Coulomb energy enters in stead of the dipole-dipole term. Here 2 EjTm~= GomJ2Kom~ are Jahn-TeUer energies, Es~= IE g - E u l . Because of the local mixing the renormalized hopping term/o-,~ splits into two components: an onsite tunneling part t~,-~and an intersite hopping part t0mm. We get a pairing energy vM,~ = 0ol, o
= ½ 6,,u(eiRu/kR~t+esRjt/kR~) 2 ,
(4)
which is of the monopole-induced dipole type, with A2
otl,u = P t,J E, u =o~,~u(EMP~,~) [ 1 - ( E M 4 E ~ )
~] ,
(5)
the vibronic polarizability, where/~8~ = 2tttsu, and 2 at,~ =Pt,~/Egu ,
(6)
the electronic polarizability, t~m~ and otm~ correspond to the low and high frequency limits of the Rice-Wang model. Because of the polaron band narrowing ~mn/Olmn > 1 for 4En-,~>Em~. Another choice of interaction constant leads directly to hole dimers, whether bipolarons or Cooper pairs W ( i j l m n ) = U o s . , . + (2pm~ " P n m / k R 3 i j ) ( 1 --¢~mn),
which renormalizes to
(7)
M. Georgiev / Vibronic approach to high- T~ superconductivity
648
with a second-order polarisation operator
IYV(ijl mn )= UijOm, + ( 2~m, •~ , , , / k R ~ ) ( 1 - J m , ) .
(8)
This gives rise to a pairing energy of the Van der Waals type U v d w = ~JImn = ( h t m n / 2 )
( &tmn/kR3) 2.
(9)
The normal-state charge carriers are off-centered polarons, viz. holes coupled to B~,-distorted CuOs bipyramids [ 2 ]. These polarons may pair to form rigid bipolarons with a translational mass
mt=h2/2tbd2 =h 2 Ub/4tvdb 2 2
(10)
where tv= tom,, Ub = Uvdw, while db is the bipolaron hopping distance. These bipolarons will Bose-condense at [ 3 ] Tcb = 3.3 lfi2n z/3/kBrn~/3m~-D/3,
( 11 )
where D is the dimensionality of conduction, e.g. D = 2 for in-plane conductivity, m~ is the local mass of off-center tunneling perpendicular to the planes:
ml=h2/{llgud~l.
(12)
However, if the VdW energy is much too low, Cooper pairing may precede with a condensation temperature [6 ]
Tcp= 1.14( hw/kB) Xexp{ -N/2EjT[Nu(O) ] ~/2},
(13)
where N is the number of unit cells, while Ng(0) and N~(0) are the densities of stable at the Fermi level for g and u. The Hamiltonian eq. ( l ) is closely related to ones describing ferroelectricity in crystals regarded as a cooperative pseudo-Jahn-Teller phenomenon [26]. Expanding eq. ( 1 ) into band waves we get
H= ~ Em(k)a+mkamk+ ~ (hto~+l/2)b+b~ k,m
+ Y" Ga~(k) [hw~(q)/2K~(q)N)'/2(ba~+b+~) k,q,v
(14)
where v is the phonon-mode polarization. A renormalised phonon frequency is obtained from eq. (14) at the central Q = 0 configuration,
t~ 2 =tO 2 [ 1 + P¢( ~q) ] ,
× [ng(k-q) - n,(k) ]/{ [Eg(k) - E u ( k - q ) ]2_ (fioJ¢)2},
(16)
where ~2 is the unit-cell volume, n,, (k) is the number of particles in the m band. Pq being always negative, it may become large enough along a wavevector q to turn thq imaginary rendering the central configuration unstable against q. There will be a phase transition at t3¢ = 0. In the limit of narrow electronic bands
nm(k) = e x p ( - E , , ( k ) / 2 k B T ) × 2cosh (E,, (k)/2kB T ) ,
P¢(O)=-(4EjT/Egu)tanh(Eg,/4kBT)
.
(17)
From eq. (16) we see that only the g-holes will exert a destabilizing effect on the paraphase, while u-holes will tend to stabilize it. The narrow-band model is physically equivalent to a system of noninteracting Cu-O molecules, each with a double-well potential
E + _ ( Q ) = ½ {KQ2+,-[(4G2QZ+EZu]~/2}
(18)
along Q = Qugu. The free energy of weakly-interacting vibronic dipoles will therefore be (c.f. ref. [2] )
F= - 2kB T log{cosh [ (4G2Q 2+E2,)l/2/4kB T] } + ½ kQ 2 - , + (2p2/kR 3) [4G2Q2/(4G2Q2+E2p) ] , (i9) where the signs ( - , + ) are for antiferroelectric or ferromagnetic alignments, respectively. The average equilibrium configurations ( Q ) of the lower-symmetry phase will be found on solving 1 =tanh [ (4G2QZ+E2p)l/Z/4kB T]
× [4EjT/(4GEQ2+E2p) ~/z]
qv
+ + + a,kam~_k ) X (amkanq-k
Pq (~q) =4EjTQ(2x)-3 J dk[Eg (k) - E u ( k - q ) ]
(15)
+ , - ( 2p2 / kR3EjT) [4EjTEsp/ ( 4G2Q2 + E2p) 12 . (2o) Now the Curie temperature is obtained from ( Q ) = 0 resulting in
Tcf = ( EgJ 4kB ) /tanh -1 [ ( EgJ 4EjT ) -,+4(2ae/kR3)l.
(21)
M. Georgiev/ Vibronic approach to high-Tosuperconductivity
Because of the tan - ~z term in the denominator which effectively exceeds 1 only at z close to 1, eqs. ( 11 ) and (21) could yield transition temperatures, matching each other at any reasonable value of the entering parameters, only for the ferroelectric alignment. For this reason the MID pairing channel, c.f. eq. (4), may have to be preferred over the VdW bipolaron mechanism, eq. (9), should there be any correlation between the superconductivity and the ferroelectricity in high-To materials. The interrelation between ferroelectricity and highT~ superconductivity is also addressed elsewhere [ 27,28 ], while evidence for the near coincidence of superconducting and ferroelectric phase transitions in YBCO is obtained from ultrasonic measurements [29 ]. A high dielectric constant along the c-axis has also been reported resulting from the enlarged polarizability [ 5 ].
5. Discussion We have described the essentials of a pairing theory based on vibronically-coupled charge transfer from planes to chains and its relationship to the observed ferroelectricity of high-T¢ materials. In the strong coupling case, the experimental NIR and MIR electronic peaks may be attributed to the optical absorption of off-center hole polarons and bipolarons, respectively [ 3 ]: this would set the peak energies at PNIR~ 4EjT and Pr4MR~ Ub- Quantitative estimates have already been presented elsewhere [3] to show that realistic T~ values of a Van der Waals bipolaronic superconductivity result from the vibronic interpretation of the IR electronic spectra of LaSCO. Further extensions should clearly describe the optical conductivity associated with vibronically coupled CTT across C u - O bonds, as well as the effect of possible (anti) ferroelectric and antiferromagnetic correlations. Finally, it would be nice to see just where this theory leads to numerically. From the optical data of section 2 we now set Eja-=0.25 eV, Ub=0.25 eV, ho9 = 80 meV (645 c m - 1) and also Eg~ = 0.8 eV from the Hubbard data [ 30,3 ] to characterise a "vibronic superconductor". A compromise dielectric constant k = 5 and db= 4 A are used too. From eqs. (3) we find x2gu = 1.35 eV and Eg~ = 2/tlgu =0.04447 eV. We
649
next compute K~ = 24.63565 eV/• 2 and K2 = 12.31782 eV/A 2 for one- and two-oxygen mass oscillators, respectively, yielding the interwell halfseparation in eq. (18) qoi=2(hto/Ki)x#s u for i = 1, 2 to be qot=0.06621 A and q2o=0.09364 A, in good agreement with the experimental data of section 2. We next insert this into eqs. (10) and (12) to calculate the corresponding bihole masses: translational mass m t = [0. l 1906 (eV)2/(2t,-j~) 2] me and off-centered mass roll = 19544.746 rne = 0.66 Mo, m12= 9772.374 me= 0.33 Mo. Now we insert into eq. ( 11 ) to compute condensation temperatures Tel = 1 0 8 . 6 6 / ( m t / m e ) 2/3 K and Tc2= 1 3 6 . 9 0 / ( m t / me) 2/3 K, at nb= 1021 cm -3. Clearly, the hopping term tugg of the in-plane carriers is essential for obtaining the experimental condensation temperature of about 100 K. From eq. (4) complemented by the Coulomb term we compute &tsu =76.42 flk 3 and ot~u= 11.80 /~k3 at Rtt=Rj~= 3.12 A for two holes residing at C u ( I I ) atoms in neighboring unit cells. From eq. (5) the polarizability ratio is 6.5 against 4 from the Rice-Wang data. Our calculations show that sensible vibronic parameters can be arranged favourably so as to produce realistic high-T~ values of a bipolaron superconductivity. Under these conditions the Curie temperature, eq. (21), is obtained for 8ae/kR3=0.2 which implies that the correlations between neighbouring ferroelectric dipoles are to be accounted for: this is to be expected for c-axis aligned local off-center O (IV) polarons leading to arrays of ferroelectric arrangements. It also seems likely that the higher Tc values in YBCO may be boosted by both two-oxygen Blu coupling and a lower bihole binding energy Ub, as seen from the optical spectra. To summarize finally, the following scenario seems emerging for a high-Te mechanism in YBCO: as a hole is transferred from a CuO2 plane to an apical oxygen ion, it activates and/or couples to the local B~u mode which shifts the O ( I V ) off-center. This creates a dynamically-polarizable entity at the displaced-oxygen site which mediates the pairing of other in-plane holes. The essential point here is the hole-phonon coupling and indeed local B2u modes are predicted to be very sensitive to the charge on the in-plane Cu or O ions [31 ].
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M. Georgiev / Vibronic approach to high-To superconductivity
References [ 1 ] J.C. Phillips, Physics of High-To Superconductors (Academic Press, San Diego, 1989). [2] M. Georgiev and M. Borissov, Phys. Rev. B 39 (1989) 11624-11631. [3] M. Borissov and M. Georgiev, M. Phil. Mag. B 60 (1989) 293-306. [4] A.W. Hewat, Europhys. Lett. 19 (1988) 73-76. [5] A.R. Bishop, R.L. Martin, K.A. Miiller and Z. Tesanovic, Z. Phys. B 76 (1989) 17-24. [6] B.Ya. Shapiro, Phys. Lett. A 127 (1988) 239-241. [7]Yu.A. Izyumov and M.I. Katsnel'son, Fiz. Met. & Metalloved. 66 (1988) 1083-1088 (in Russian). [8 ] D.P. Clougherty, K.H. Johnson and M.E. McHenry, Physica C 162-164 (1989) 1475-1476. [9] M. Reedyk, D.A. Bonn, J.D. Garrett, J.E. Greedan, C.V. Stager, T. Timusk, K. Kamaras and D.B. Tanner, Phys. Rev. B 38 (1988) 11981-11984. [ 10 ] C. Taliani, R. Zamboni and F. Licci, Solid State Commun. 64 (1987) 911; C. Taliani, R. Zamboni, G. Ruani, F.C. Matacotta and K.I. Pokhodnya, Solid State Commun. 66 (1988) 487; C. Taliani, R. Zamboni, G. Ruani, A.J. Pal, F.C. Matacotta, Z. Vardeny and X. Wea, in: High-To Superconductors, Electronic Structure, eds. A. Bianconi and A. Marcelli (Pergamon, Oxford, 1989) pp. 95-101. [ 11 ] H. Monien and A. Zawadowski, Phys. Rev. Lett. 63 ( 1989 ) 911. [12] Z. Schlesinger, R.T. Collins, F. Holtzberg, C. Feild., S.H. Blanton, U. Welp, G.W. Crabtree, Y. Fang and J.Z. Liu, Phys. Rev. Lett. 65 (1990) 801. [ 13] Z. Tesanovic, A.R. Bishop and R.L. Martin, Solid State Commun. 68 (1988) 337. [14] R.J. Cava, B. Batlogg, S.A. Sunshine, T. Siegrist, R.M. Fleming, K. Rabe, L.F. Schneemeyer, D.W. Murphy, P.K. van Dover, P.K. Gallagher, S.H. Glarum, S. Nakahara, R.C. Farrow, J.J. Krajewski, S.M. Zahurak, J.V. Waszczak, J.H. Marshall, P. Marsh, L.W. Rupp Jr., W.F. Peck and E.A. Rietman, Physica C 153-155 (1988) 560.
[ 15 ] S.D. Conradson and I.D. Raistrick, Science 243 (1989) 1340. [ 16 ] S.D. Conradson, I.D. Raistrick and A.R. Bishop, Science 248 (1990) 1394. [17] R.P. Sharma, L.E. Rehn, P.M. Baldo and J.Z. Liu, Phys. Rev. Lett. 62 (1989) 2869. [ 18 ] M. Francois, A. Junod, K. Yvon, P. Fischer, J.J. Capponi, P. Strobel, M. Marezio and A.W. Hewat, Physica C 153155 (1988) 962. [ 19 ] M. Bauer, I.B. Ferreira, L. Genzel, M. Cardona, P. Muragaraj and J. Maier, Solid State Commun. 72 ( 1989 ) 551. [20] C. Thomsen, M. Cardona, R. Liu, T. Ruf, B. Gegenheimer and E.T. Heyen, Physica C 162-164 (1989) 1079. [21 ] T. Ruf, C. Thomsen, R. Liu and M. Cardona, Phys. Rev. B 38 (1988) 11985. [22] R. Zamboni, G. Ruani, A.J. Pal and C. Taliani, Solid State Commun. 70 (1989) 813. [23 ] A.V. Bazhenov, T.N. Fursova, V.B. Timofeev, A.S. Cooper, J.P. Remeika and Z. Fisk, Phys. Rev. B 40 ( 1989 ) 4413. [24]M.K. Crawford, M.N. Kunchur, W.E. Farneth, E.M. McCarron III and S.J. Poon, Phys. Rev. B 41 (1990) 282. [25] M.J. Rice and Y.R. Wang, Phys. Rev. B 36 (1987) 8794. [26] I.B. Bersuker and V.Z. Polinger, Vibronic Interactions in Molecules and Crystals (Springer, Berlin, 1989 ). [ 27 ] A. Bussmann-Holder, A. Simon and H. Buttner, Phys. Rev. B 39 (1989) 207. [28] D.A. Cardwell and N.M. Shorrocks, Phys. Rev. B 40 (1989 ) 6564. [29] V. Muller, C. Hucho, K. de Groot, D. Winau, D. Maurer and K.H. Rieder, Solid State Commun. 72 (1989) 997. [30] F. Mila, Phys. Rev. B 38 (1988) 11358. [ 31 ] S. Onari, T. Mori, K. Nakaoka, T. Matsuzawa, T. Hiroaki, A. Ono, S. Hidaka and T. Arai, Physica C 162-164 (1989 ) 1343.
The vibronic approach to high-Tc superconductivity Mladen Georgiev Faculty of Science, Myilly Point Campus, Northern Territory University, PO Box 40146, Casuarina, NT 0811, Australia Dipartimento di Fisica, II Universiti* degli Studi di Roma, Via Emanuele Carnevale, I-O0173 Roma, Italy
Received 18 January 1991
Recent experimental and theoretical studies have been stressing the relationship between the pairing mechanism and the motion of apical oxygensin high-Tomaterials. Earlier we had proposeda similar model of vibronicaUycoupledchargetransfer from planes to chains producing asymmetricallydistorted Cu-O polyhedra to move and pair as small polarons. Numerical data for LaSCO proved to be in optimistic agreement with the experimental To. This model is now revisited both as a theoretical background and as a backing up of experimental evidence. Its particular relationship with the inferred ferroelectricity of high-To materials will be analyzed.
Apologia Ever since the discovery of a high-To superconductivity in layered perovskites by Bednorz and Miiller, a tremendous effort has been in progress to unravel the pairing mechanism [ 1 ]. It is the purpose of this paper to emphasize certain experimental facts that seem to be lending support to a fairly unconventional view. By this we mean the vibronic approach, a term originally borrowed from the quantum chemistry of molecules. In what follows we shall once more reevaluate our vibronic model against the background of new data accumulating rapidly in the literature.
1. Introduction Earlier we proposed a pairing model appropriate to the Cu-O system [2,3]. The essentials are: opposite parity Cu and O states (or their hybrids) mix vibronically, which renders a Cu-O bond unstable against uneven modes via the pseudo-Jahn-Teller effect. The vibronic mixing materializes through diVisiting Fellow from Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, 72 Lenin Boulevard, Bulgaria.
polar coupling of the vibrational mode to the electron density, c-axis dipoles will have the best chance to couple to and survive screening from the 2D charge carriers in CuO2 planes. Examples are provided by Cu ( 3dz2 ) and O (2pz) orbitals both spread along the c-axis, while the relevant modes are A2u in tetragonal La2_xSrxCuO 4 and Blu in orthorhombic YBa2Cu3OT. The C u ( I I ) - O ( I V ) bonds in YBCO (Argonne nomenclature) are of particular interest. Now the static electronic states hybridize only weakly due to the large bond length. By vibronic mixing, double-well potentials are produced for the apical O (IV) ions and transitions incited across them transferring charge from in-plane Cu ( I I ) / O ( I, II ) to localized O ( I V ) levels. As a result, O ( I V ) ions will tend to either occupy stable off-center sites or delocalize by interwell tunneling. These O(IV) ions will display anomalous vibrational amplitudes, as observed [ 4 ]. Distorted bipyramids will result, asymmetrical with respect to the Cu (I) ion in between. The associated parity violation gives rise to an inversion dipole and a high dynamic polarizability of the Cu ( I I ) - O (IV) moiety. This opens a polarization-induced pairing channel [2 ] for in-plane holes which may be responsible for the high-T¢ superconductivity [5]. Physically this is equivalent to a travelling off-centered (ferroelectric) instability coupled to the in-
0921-4534/91/$03.50 © 1991 - Elsevier SciencePublishers B.V. (North-Holland)
M. Georgiev/ Vibronic approach to high-Tosuperconductivity
plane charge carders which promotes the pairing [2 ]. We shall now revisit both the premises of this model and the supporting experimental evidence. Earlier work has been surveyed elsewhere [ 3 ]. Related local vibronic models have also been proposed by others but we are constrained to mention only a few [6-8].
2. Electronic transitions
The infrared (IR) optical spectral of high-To materials are believed to hold the key to understanding the pairing mechanism. On increasing the photon energy, they appear to be composed to phonon lines in the far infrared (FIR) superimposed on a background electronic activity that peaks in the mid infrared (MIR). The MIR feature is found in the IR absorption (or optical conductivity) peaked at about 0.25 eV in all the high-To superconductors [9]. This is followed by another electronic band in the near infrared (NIR) [10]. Both bands are clearly associated with intragap states: they appear e.g. at 0.13 eV and 1 eV in semiconducting YBa2Cu307_x ( x ~ 0 . 8 ) , while the bandgap of YBa2Cu306 is at about 1.7 eV (even though shifting to lower energies on increasing x). Corresponding bands in La2CuO4 appear at 0.5 eV and 1.4 eV, while the gap is at 2 eV. Raman-scattering probing of FIR and MIR reveals that the intragap states ofYBa2fu307 (YBCO) couple strongly to phonons, for they exhibit a Fano antiresonance [ 11 ]. The Raman spectrum below Tc displays polarization and orientation dependent structures resolving it into two components of A~8 and B~8 symmetries with respect to the x - y in-plane interchange. (In concomitance, the IR optical conductivity of YBCO also resolves into plane and chain components [ 12 ].) While Als originates from intraband transitions in CuO2 planes, B~s is attributed to charge transfer transitions (CTT) from a nonbonding (NB) py O(IV) band to the Cu-O hybridized inplane band. The chain-to-plain charge transfer has been computed implying that the MIR gap state may correspond to CTT perpendicular to the CuO2 planes [ 13,5 ]. The excited-state configuration where an extra hole is added to a nearby O or Cu site is found to have a sharply reduced energy relative to the undoped ground state. Low-lying Cu 3+ 0 2 - Cu 2+ O 1 -
-
-
645
CTT occur because of the proximity of the Cu 2+ and 0 2 - ionization potentials meaning a near degeneracy in energy of these two Cu-O complexes, a remarkable intrinsic property of the Cu-O system. Assigning the relevant electronic orbitals is of chief importance. "Ab initio" quantum-chemistry calculations on cluster models suggest that the extra holes enter the antibonding (AB) combination of Cu(dx2_y2) and O(2p,x,y) orbitals. In the Cu2+O l state the hole resides on the NB O(p~,y) orbital. The AB-NB CTT is a transverse mode of electron transfer from the O(p~) band to the Cu(dy~_z2) component with a transition dipole normal to the chain, large oscillator strength and energy from 0.1 to 0.5 eV. It amounts to a high dynamic polarizability along the O ( I V ) - C u ( I ) - O ( I V ) axis. Now a specific view on high-To emerges [ 5 ] which suits quite well with the crystalline structure and chemistry of layered perovskites: pairing mediated by CTT across the O (IV)-Cu ( I ) - O (IV) moiety along the c-axis between CuO2 planes. CTT create a retarded attraction between oxygen holes leading to an s-wave BCS-like superconductive state. Holes pair effectively via the monopole-to-induced-dipole coupling by virtue of the high doping-made polarizability of Cu-O bonds. Although the supercurrents themselves run in the CuO2 planes, the attractive pairing mechanism involves the CuO3 chains and is not two-dimensional for that matter! Several experiments lend strong support to the CTT high-To model [ 5 ]: its efficiency depending on the C u ( I I ) - O ( I V ) separation, it has responded to external pressure or to Co substitution in the expected manner. Another experiment using polarized Cu-edge X-ray absorption spectroscopy reveals that extra holes in YBCO reside mainly on a Cu-O orbital oriented along the c-axis which involves the O(IV) p~ and Cu(II) dz2 states. It is inferred that a second hole band different from the in-plane Cu(3dx2_y~)-O(2px, 2py, ~) hybrid should be present. In conformity with this nuclear quadrupole moment measurements suggest that the bridging O (IV) oxygens should be regarded as O 1- rather than 0 2 - . There being a clear correlation between Tc and the O ( I V ) - C u ( I I ) separation [14], the relevance of these observations to superconductivity is implied.
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M. Georgiev / Vibronic approach to high-To superconductivity
3. Phonon coupling
In so far as the O ( I V ) - C u ( I I ) CTT are phonon coupled, they may incite considerable anharmonic motion along the c-axis. Indeed, there is convincing evidence of O(IV) anharmonicity in YBCO [ 4,15,16 ]. However, vibrational modes coupled to the superconductivity should also exhibit anomalies near To. Indeed, anomalous Cu-O modes near Tc in ErBa2Cu307_y have been evidenced by ion channeling along the c-axis [ 17 ]. A 30% increase across Tc is observed in the amplitudes of Cu and O ions indicating a strong coupling. Neutron powder diffraction reveals abnormally large amplitudes of O(IV) ions at Cu-O chains, concomitant with these ions occupying energy minima at about 0.1 A off-side the chain line [ 18 ]. Off-side displacements result from the coupling to odd-parity (uneven) modes. Polarized FIR reflection studies have sorted out uneven modes, A2~ parallel and Eu normal to the c-axis [ 19 ]. While IR modes in semiconducting YBa2Cu306 exhibit normal oscillator strenghts, those in superconducting YBa2Cu307 have very high strengths. The C u ( I ) - O ( I V ) stretching mode is reported to be 645 cm- 1. Raman and FIR experiments on YBa2Cu307_x clearly show lattice coupling to the superconducting systems [ 20 ], as evidenced by a Fano effect on two Raman modes, at 120 and 335 cm -~, and by an anomalous softening at T~ of several IR peaks at 275, 311 and 569 c m - 1. However, the closest relationship to Tc is shown by a B~s tetragonal symmetry phonon mode at 340 cm -1 of a vertical O ( I I ) - O ( I I I ) vibration in the CuO2 planes [ 21 ]. This mode softens at and displays a behavior with T~ in an external magnetic field. Raman modes in resonance with the MIR gap state of YBa2Cu3OT_x are found by exciting with subgap IR light [22]. A line at about 500 cm-1 assigned to the C u ( I ) - O ( I V ) axial stretching mode appeared on both intragap and intergap excitation, its frequency rising with increasing x. This hardening suggests a reduction of the bond length related to a resonance interaction with the localized electronic excitation. The mode polarization implies that the electronic CTT is also along the c-axis. It is remarkable that the 340 cm-~ in-plane mode, coupled beyond doubt to the superconducting transition, is only excited by visible light, possibly being
in resonance with higher energy states. On the other hand the dual coupling of the stretching mode to both higher and lower energy excitations may prove an essential feature of the chains-to-planes CTT. It is also remarkable that virtually the same 500 cm -~ Raman frequency appears at about 510 cm-~ in the photoinduced optical absorption of YBa2Cu307_x and at 579 c m - t in YBCO. Another even frequency at 436 cm-~ also displays a correlated IR appearance. Such dual Raman and IR activity may signify breaking up of the inversion symmetry at the Cu (I) site as a result of coupling to the charge carriers. An axial distortion of the CuO5 pyramids has been suggested [ 3,22 ]. For tetragonal La2CuO4 the A2u modes are along the c-axis, while the Eu modes are a-b-planar vibrations. The A2u stretching of vertex O (3) oxygens is reported to be at 520 cm-1, whereas the AEu stretching of a-b-plane O ( l , 2) is at 683 cm -~ [23]. Oxygen isotope constants and transverse-optic AEu modes at 240 and 500 cm-~ in LaE_xSr~CuO4 display anomalies for x ~ 0.15 where T¢ is at maximum, possibly due to coupled lattice fluctuations [ 24 ].
4. The vibronic CT model
There have been a few attempts to ascribe vibronic features to high-To superconductivity. One early work addresses the peculiarities of the Cu-O system as of relevance to the electronic mid-infrared band [25 ]. This band is assigned to "internal electronic and vibrational excitations of a spin-zero charge-transfer complex (P+) formed between a hole and an essentially single paramagnetic square planar CuO4 unit (po)-. Non-vanishing dipole moments occur for electronic transitions from NB O to AB CuO orbitals, only in-plane Cu(3dx2_y2) and O(2px.y) states being considered. The antisymmetric vibrational Eu mode modulates the AB mixture which incites CTT across P+ producing the IR band. The derived frequency-dependence conductivity agrees with the experimental spectrum of LaSCO by fitting a set of electronic and vibrational parameters. The polarizability ol(to) varies between low frequency a ( 0 ) and high frequency c~e limits with fitting values 700 A 3 and 180 A 3, respectively. The higher value of the former (vibronic) limit is due to
647
M. Georgiev / Vibronic approach to high-T~superconductivity
phonon renormalization which elevates the pairing [2,3 ]. Superconducting are either the highly correlated electrons of po or the P+'s themselves delocalizing as polarons with a fluctuating electric dipole moment and high polarizability. Attractive monopole-induced dipole interactions with P+ or between pairs of P+ lead to enhanced carrier pairing and high-T¢ superconductivity: the Rice-Wang model contains all the essentials of a vibronic theory. The Hamiltonian of interband vibronic CTT reads [2,3]
/~=
E
(ij)mn
+l ~ ijmn
~m.+X
im
Eim ~ a im + aim
l~(ijl m n )a imajn + + ajmain ,
with [ 3 ] t~l~ = ½E, u [ 1 - ( E,u/ 4EjTgu ) Xllg u2 =
]/2sinh (x~,o),
( 2EsT~u/htottsu) [ 1 _ ( Egu/ 4EjTgu)2]
iF'in =Eim --EjTgu [ 1 "{- (Esu/4EjTgu) 2 ] + ½hdgiigu ,
~ii,u = oJi~ [ 1 - ( E ~ d 4 E j T , J H=
~
+ tijmnaimajn + ~ Elm aim + aim
+½E
[KijmnQijmn ~ + Pijmn/mijmn]
(ij)mn ijmn
+E
ijmn
ijmn
W ( q"" l m n ) a i , ,+a j , + ajmai, .
(3)
Ptg~ =Ptsu [ 1 - (EgJ4EjTgu) 2 ] ,
( 1)
Here m, n=g,u are the band labels, while i , j , l label sites in the Cu-O manifold, a;+s (a;s) creates (annihilates) an in-plane hole at site i, while a~ (a~) does so for a (side) chain hole. The bands are assumed narrow, nearly degenerate, and of opposite parities. (ij) label neighbouring sites and tom, is the hopping integral between these. Qom~, Po,,, and Morn ~ are coordinates, momenta, and reduced masses of intra (i = j ) or inter (i ~ j ) site nuclear oscillators coupled to the electronic system through the mixing constants Gore~. While the latter (even) mode couples to the itinerant in-plane g-holes, such as the breathing Cu-O vibration, the former (uneven) promotes gu CT from planes to chains which leads to efficient g-hole pairing. W(ijl mn ) are hole interaction constants. In a particular case a m-hole at site j couples to the electric dipole p,,~ induced through m - n charge transfer at another hole site l: W ( ljl m n ) = UoS,~ -ptm~ • (ejRo/kR~)(1-~m~).
] 1/2,
rP(ljl g u ) = -~tgu" ( G R o / k R ~ ) ,
im
G ijmn Qo,,,,, a ~,,, ÷ ay,,
+½~
3/2 ,
(2)
Here ptm~ is the m - n mixing dipole, Uo=etej/kR ~ is the intersite Coulomb repulsion. Excluding the lattice in the adiabatic approximation leads to a renormalized electronic Hamiltonian
locally along the mixing-mode coordinate Qttg~ at Qomm= 0. Similar equations hold good for quantities renormalized through coupling to the intersite modes Qom~ at Qttg~=O through now the Coulomb energy enters in stead of the dipole-dipole term. Here 2 EjTm~= GomJ2Kom~ are Jahn-TeUer energies, Es~= IE g - E u l . Because of the local mixing the renormalized hopping term/o-,~ splits into two components: an onsite tunneling part t~,-~and an intersite hopping part t0mm. We get a pairing energy vM,~ = 0ol, o
= ½ 6,,u(eiRu/kR~t+esRjt/kR~) 2 ,
(4)
which is of the monopole-induced dipole type, with A2
otl,u = P t,J E, u =o~,~u(EMP~,~) [ 1 - ( E M 4 E ~ )
~] ,
(5)
the vibronic polarizability, where/~8~ = 2tttsu, and 2 at,~ =Pt,~/Egu ,
(6)
the electronic polarizability, t~m~ and otm~ correspond to the low and high frequency limits of the Rice-Wang model. Because of the polaron band narrowing ~mn/Olmn > 1 for 4En-,~>Em~. Another choice of interaction constant leads directly to hole dimers, whether bipolarons or Cooper pairs W ( i j l m n ) = U o s . , . + (2pm~ " P n m / k R 3 i j ) ( 1 --¢~mn),
which renormalizes to
(7)
M. Georgiev / Vibronic approach to high- T~ superconductivity
648
with a second-order polarisation operator
IYV(ijl mn )= UijOm, + ( 2~m, •~ , , , / k R ~ ) ( 1 - J m , ) .
(8)
This gives rise to a pairing energy of the Van der Waals type U v d w = ~JImn = ( h t m n / 2 )
( &tmn/kR3) 2.
(9)
The normal-state charge carriers are off-centered polarons, viz. holes coupled to B~,-distorted CuOs bipyramids [ 2 ]. These polarons may pair to form rigid bipolarons with a translational mass
mt=h2/2tbd2 =h 2 Ub/4tvdb 2 2
(10)
where tv= tom,, Ub = Uvdw, while db is the bipolaron hopping distance. These bipolarons will Bose-condense at [ 3 ] Tcb = 3.3 lfi2n z/3/kBrn~/3m~-D/3,
( 11 )
where D is the dimensionality of conduction, e.g. D = 2 for in-plane conductivity, m~ is the local mass of off-center tunneling perpendicular to the planes:
ml=h2/{llgud~l.
(12)
However, if the VdW energy is much too low, Cooper pairing may precede with a condensation temperature [6 ]
Tcp= 1.14( hw/kB) Xexp{ -N/2EjT[Nu(O) ] ~/2},
(13)
where N is the number of unit cells, while Ng(0) and N~(0) are the densities of stable at the Fermi level for g and u. The Hamiltonian eq. ( l ) is closely related to ones describing ferroelectricity in crystals regarded as a cooperative pseudo-Jahn-Teller phenomenon [26]. Expanding eq. ( 1 ) into band waves we get
H= ~ Em(k)a+mkamk+ ~ (hto~+l/2)b+b~ k,m
+ Y" Ga~(k) [hw~(q)/2K~(q)N)'/2(ba~+b+~) k,q,v
(14)
where v is the phonon-mode polarization. A renormalised phonon frequency is obtained from eq. (14) at the central Q = 0 configuration,
t~ 2 =tO 2 [ 1 + P¢( ~q) ] ,
× [ng(k-q) - n,(k) ]/{ [Eg(k) - E u ( k - q ) ]2_ (fioJ¢)2},
(16)
where ~2 is the unit-cell volume, n,, (k) is the number of particles in the m band. Pq being always negative, it may become large enough along a wavevector q to turn thq imaginary rendering the central configuration unstable against q. There will be a phase transition at t3¢ = 0. In the limit of narrow electronic bands
nm(k) = e x p ( - E , , ( k ) / 2 k B T ) × 2cosh (E,, (k)/2kB T ) ,
P¢(O)=-(4EjT/Egu)tanh(Eg,/4kBT)
.
(17)
From eq. (16) we see that only the g-holes will exert a destabilizing effect on the paraphase, while u-holes will tend to stabilize it. The narrow-band model is physically equivalent to a system of noninteracting Cu-O molecules, each with a double-well potential
E + _ ( Q ) = ½ {KQ2+,-[(4G2QZ+EZu]~/2}
(18)
along Q = Qugu. The free energy of weakly-interacting vibronic dipoles will therefore be (c.f. ref. [2] )
F= - 2kB T log{cosh [ (4G2Q 2+E2,)l/2/4kB T] } + ½ kQ 2 - , + (2p2/kR 3) [4G2Q2/(4G2Q2+E2p) ] , (i9) where the signs ( - , + ) are for antiferroelectric or ferromagnetic alignments, respectively. The average equilibrium configurations ( Q ) of the lower-symmetry phase will be found on solving 1 =tanh [ (4G2QZ+E2p)l/Z/4kB T]
× [4EjT/(4GEQ2+E2p) ~/z]
qv
+ + + a,kam~_k ) X (amkanq-k
Pq (~q) =4EjTQ(2x)-3 J dk[Eg (k) - E u ( k - q ) ]
(15)
+ , - ( 2p2 / kR3EjT) [4EjTEsp/ ( 4G2Q2 + E2p) 12 . (2o) Now the Curie temperature is obtained from ( Q ) = 0 resulting in
Tcf = ( EgJ 4kB ) /tanh -1 [ ( EgJ 4EjT ) -,+4(2ae/kR3)l.
(21)
M. Georgiev/ Vibronic approach to high-Tosuperconductivity
Because of the tan - ~z term in the denominator which effectively exceeds 1 only at z close to 1, eqs. ( 11 ) and (21) could yield transition temperatures, matching each other at any reasonable value of the entering parameters, only for the ferroelectric alignment. For this reason the MID pairing channel, c.f. eq. (4), may have to be preferred over the VdW bipolaron mechanism, eq. (9), should there be any correlation between the superconductivity and the ferroelectricity in high-To materials. The interrelation between ferroelectricity and highT~ superconductivity is also addressed elsewhere [ 27,28 ], while evidence for the near coincidence of superconducting and ferroelectric phase transitions in YBCO is obtained from ultrasonic measurements [29 ]. A high dielectric constant along the c-axis has also been reported resulting from the enlarged polarizability [ 5 ].
5. Discussion We have described the essentials of a pairing theory based on vibronically-coupled charge transfer from planes to chains and its relationship to the observed ferroelectricity of high-T¢ materials. In the strong coupling case, the experimental NIR and MIR electronic peaks may be attributed to the optical absorption of off-center hole polarons and bipolarons, respectively [ 3 ]: this would set the peak energies at PNIR~ 4EjT and Pr4MR~ Ub- Quantitative estimates have already been presented elsewhere [3] to show that realistic T~ values of a Van der Waals bipolaronic superconductivity result from the vibronic interpretation of the IR electronic spectra of LaSCO. Further extensions should clearly describe the optical conductivity associated with vibronically coupled CTT across C u - O bonds, as well as the effect of possible (anti) ferroelectric and antiferromagnetic correlations. Finally, it would be nice to see just where this theory leads to numerically. From the optical data of section 2 we now set Eja-=0.25 eV, Ub=0.25 eV, ho9 = 80 meV (645 c m - 1) and also Eg~ = 0.8 eV from the Hubbard data [ 30,3 ] to characterise a "vibronic superconductor". A compromise dielectric constant k = 5 and db= 4 A are used too. From eqs. (3) we find x2gu = 1.35 eV and Eg~ = 2/tlgu =0.04447 eV. We
649
next compute K~ = 24.63565 eV/• 2 and K2 = 12.31782 eV/A 2 for one- and two-oxygen mass oscillators, respectively, yielding the interwell halfseparation in eq. (18) qoi=2(hto/Ki)x#s u for i = 1, 2 to be qot=0.06621 A and q2o=0.09364 A, in good agreement with the experimental data of section 2. We next insert this into eqs. (10) and (12) to calculate the corresponding bihole masses: translational mass m t = [0. l 1906 (eV)2/(2t,-j~) 2] me and off-centered mass roll = 19544.746 rne = 0.66 Mo, m12= 9772.374 me= 0.33 Mo. Now we insert into eq. ( 11 ) to compute condensation temperatures Tel = 1 0 8 . 6 6 / ( m t / m e ) 2/3 K and Tc2= 1 3 6 . 9 0 / ( m t / me) 2/3 K, at nb= 1021 cm -3. Clearly, the hopping term tugg of the in-plane carriers is essential for obtaining the experimental condensation temperature of about 100 K. From eq. (4) complemented by the Coulomb term we compute &tsu =76.42 flk 3 and ot~u= 11.80 /~k3 at Rtt=Rj~= 3.12 A for two holes residing at C u ( I I ) atoms in neighboring unit cells. From eq. (5) the polarizability ratio is 6.5 against 4 from the Rice-Wang data. Our calculations show that sensible vibronic parameters can be arranged favourably so as to produce realistic high-T~ values of a bipolaron superconductivity. Under these conditions the Curie temperature, eq. (21), is obtained for 8ae/kR3=0.2 which implies that the correlations between neighbouring ferroelectric dipoles are to be accounted for: this is to be expected for c-axis aligned local off-center O (IV) polarons leading to arrays of ferroelectric arrangements. It also seems likely that the higher Tc values in YBCO may be boosted by both two-oxygen Blu coupling and a lower bihole binding energy Ub, as seen from the optical spectra. To summarize finally, the following scenario seems emerging for a high-Te mechanism in YBCO: as a hole is transferred from a CuO2 plane to an apical oxygen ion, it activates and/or couples to the local B~u mode which shifts the O ( I V ) off-center. This creates a dynamically-polarizable entity at the displaced-oxygen site which mediates the pairing of other in-plane holes. The essential point here is the hole-phonon coupling and indeed local B2u modes are predicted to be very sensitive to the charge on the in-plane Cu or O ions [31 ].
650
M. Georgiev / Vibronic approach to high-To superconductivity
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