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Dopant induced valence transition and fermion pairing in a model of superconducting copper oxides
Solid State Communications,Vol. 69, No. 5, pp.527-530, 1989. Printed in Great Britain.
oo38-1098189 $3.00 + .OO Pergamon Press plc
DOPANT INDUCED VALENCE TRANSITION AND FERMION PAIRING IN A MODEL OF SUPERCONDUCTING COPPER OXIDES S. Mazumdar Department
of Physics, University
of Arizona,
Tucson, AZ 85721. USA
(Received 22 October 1988 by C.N.R. Rao)
We propose opposed to a the high T, several other
that doping of the semiconducting copper oxides leads to a valence transition, simple valence fluctuation. The model explains seemingly conflicting experiments oxides, and also provides a unified theoretical approach for superconductivity systems with strong Coulomb correlation.
as in in
thesemodels, the proximity of the superconducting state to the structural distortion is therefore puzzling. The optical measurements,3 now confirmed in single crystals, find a strong electronic absorption at -0.4 eV that tracks T, and the volume fraction of the Meissner effect in La,,Sr,CuO,: the oscillator strength of the absorption peaks at x _ 0.15-O. I7 and is very weak for x < 0.1 and x > 0.3. Similarly, the absorption is seen in YBa$&O,_s for 6 = 0.2 but not for 6 = 0.8. Furthermore, this electronic absorption is coupled7 to vibrational modes. The intensity of the simple chargetransfer (CT) process Cu2+01- ++ Cus+Oz- is expected to increase monotonically with doping, in contrast to what is seen experimentally, indicating a different origin of the dopant-induced absorption. A consistent description of all these conflicting normal-state data seems a prerequisite to a theory of the SC. The theoretical model proposed here is a modified two-band extended Hubbard model that takes into account explicitly the Madelung energy of the CuOs plane. F-simplicity. we focus on LaSrCuO (hereafter 214). but we emphasize that the same basic mechanism applies to YBaCuO (hereafter 123). Indeed, a very precise description of the role of the chain oxygen in 123 can be given. Within the model, doping leads to a valence transition. as opposed to the valence fluctuationa that occursin standard two-band description. At a critical dopant concentration. the true Cu valence jumps from 2 to 1 (Cus+ * Cu’+). so that the holes migrate to the oxygens, leaving an effectively quarter-filled O-band, with the number of holes p per 0 site nearly one-half. The common features between the copper oxides, the organics and LiTi,O, are Coulomb correlations and p _ l/2, while the common feature between the copper oxides and doped BaBiOS is a highly correlated O-band with many more holes than expected from formal valences. We discuss experimental signatures of the valence transition. along with results of numerical calculations. Coulomb correlations at p _ l/2 drive a CDW and give rise to new optical absorptions. The pairing mechanism for SC involves bipolarons in a correlated CDW. We briefly discuss numerical results establishing this. We observe that (a) many other experiments indicate Cui+ in the superconductors (though not in the semiconductors), and (b) the photoemission data are not in contradiction to these ideas. Finally. we make several experimental predictions that can establish the validity of the model.
The possibility that superconductivity (SC) in the is driven by repulsive Coulomb copper oxides interactions has recently been widely discussed. Motivated by the distinctive antiferromagnetism (AF) observed in the semiconducting compositions.’ models based on half-filled Cu bands with strong on-site repulsion have generated the greatest interest. But these models provide no obvious explanation for the equally distinctive anomalous structural distortions’ and optical properties.$ In the present work, we demonstrate that a Coulomb-driven 4k, charge density wave (CDW) in an approximately quarter-filled oxygen band can explain the structural distortions and th-cal absorption and is consistent with the observed AF behavior as a function Further, our model provides natural of doping. explanations for the jump in the number of carriers in Hall measurements and for the (otherwise confusing) behavior of the magnetic susceptibility as a function of doping. In addition. our theory makes a number of explicit, testable predictions. Finally, our approach provides a unified framework for Coulomb-driven SC, within which it may be possible to explain SC in the high T, oxides, the “low” T, oxides LiTi,O, and doped BaBiO,, and the organic superconductors. The structural distortions and the optical properties are, like the AF. intrinsic features of the materials. The structural distortions in the copper oxides are seen in both LaSrCuO and YBaCuO at the superconducting compositions, at temperatures T, > T,. but within the orthorhombic phase (i.e., we are not referringto the tetragonal-orthorhombic transitio;iiT These distortions and lattice softening have been seen in velocity,z sound ultrasound,z x-ray” and neutron scattering5 measurements. A specific heat anomaly is seen6 in superconducting YBaCuO at T,, while high resolution recent x-ray scattering’ and ultrasound measurement? indicate that the structural anomalies are limited to the CuO, planes. The latter, along with the behavior of sound velocity2 at and below T,, as well as the unusually large oscillator strengths of certain infrared modes.’ very strongly indicate CDW. even though the very weak isotope effect suggests a minimal role of electron-phonon coupling. Similarly, neither the oneband nor the two-band Hubbard models explains CDW formation. The CDW here would involve the breathing mode vibrations of the 0.1-0 bonds. Such a CDW is destroyed by the large Hubbard U necessary to explain Within the antiferromagnetism in the semiconductor. 527
DOPANT INDUCED VALENCE TRANSITION
528
Vol. 69, No. 5
The proposed Hamiltonian is H = II, c
nit
nt’
+ UP c
i
+
Idp
njtnj’ + Id c
c
(a$cjo
nia + An c
i,u
j
+
C&Q) + tpp
(ij).u
c
c$c~,~ + E,
(1)
(jj7.u
Here the vacuum consists of Cu’+(d”‘) and 02-(pa), and a$ and cf create holes on Cu and O-sites, respectively. Charge counting in La.$uO, (assuming Las+) shows that only one of the two oxygens in a CuO, unit is definitely Oa-. while the charges on the Cu and the second 0 depend on the parameters in Eq. (1) as well as doping. Here Ud,U, are the usual Hubbard repulsions and tdn and tpp are the hopping integrals between neighboring Cu and 0 and between neighboring O-sites. For large Ud, U, and in the limit td,,t,, * 0. the ground state of the semiconductor is determined by the inequality,’ I,+A,-IAEM,I
njo
j.u
$0
.
(2)
qu
where I, is the second ionization energy of Cu. A, the second electron affinity of 0 and AE, the difference m the Madelung energies between the configuration having the coppers as Cu2+ and the one in which the coppers Here E, = are Cu’+ and half the oxygens are O’-. ZVii,(l + ni)(l + ni,) + ZVj.,(2 - nj )(2 - n.,) + ZVij (1 + ni)(2 - nj) < 0, with i,i’ c b; j.j’ e 0. A smaller lefthand side (LHS) favors Cu*+. while a larger LHS favors Cu’+. Importantly, distinct “Cu*+” and “Cu’+” phases are possible even with finite bandwidth, as long as the bandwidths are small compared to AEM. which is usually the case for ionic solids. The LHS in Eq. (2) as a whole is written as a site energy difference c in the usual two-band description; the consequence of explicit described inclusion of EM is the valence transition below. All band and cluster calculations’s indicate that CuO, is very close to the borderline in Eq. (2) (i.e., E - 0). This is primarily due to the large Id of Cu. which is related to the stability of the d”’ configuration, and the positive A’,. Only the large AE,,,, in a solid drives the system towards Cua+. Thus a recent molecular calculation with extensive configuration interaction finds” a d-count of 9.95. The very strong dependence of the Neel temperature on y in La,CuO,_,, is indirect evidence that the system is close to the a border: for y + 0 and the slightest self-doping, valence transition Cu2+ + Cu’+ becomes possible. Undoped La,CuO,_x is therefore exactly analogous to the mixed-stack organic CT solids,‘a except that in the latter the ionicities vary between 0 and 1 instead of between 1 and 2. In many of these CT solids, pressure or temperature-induced first-order valence transitions transition) have been knownI for (“neutral-to-ionic” years and are explained” within Eq. (2): increased pressure or reduced temperature increases Et,,, and drives the system from the neutral to the ionic side. Calculations’s indicate that for systems close to the border, even a minute change in E, can drive the transition. Doping in the oxides causes an intrinsic variation as opposed to the temperature or pressureof E,, induced extrinsic variation. Each replacement of Oz- by
O’- reduces EM severely: Indeed, the dopant-induced effect on EM is larger than what is expected from pressure or temperature-induced lattice contraction. For a system close to the border, this decrease in EM can drive a sharp Cua+ + Cu’+ transition. As in one dimension (lD),‘a a sharp transition is For a model CuO, plane close to found numerically.” the border of Eq. (2). the slightest doping does indeed This is because even with cause a valence transition. finite tp& the relative magnitudes of Id and AE, are very. large. These results will be published elsewhere. Beyond the transition, the empty Cu’+ band is no longer is described by relevant, and the p - l/2 0-sublattice the one-band Hamiltonian, c:~cj’o + H(t3 + H,_,
.
(3)
Here tj - (t&,/A0 + tPf. with AE the energy of the C.T. Cu’+O’- Cu Oa-. H(t,) a second-neighbor hopping among half the oxygens that is irrelevant for what follows, and He_, an electron-phonon coupling discussed below. The Coulomb repulsion Vj between nearest-neighbor O’- holes originates in Ehl and is extremely important for the description of this p - l/2 system. Signatures of the valence transition are seen in the x-dependence of the Hall coefficient Ru and the high temperature magnetic susceptibility x in Laa_,SrxCuO,. Note that the number of carriers is much larger than x after the transition. as instead of having only dopantinduced O’-, now we have half the oxygens O’-, and nearly all of them are mobile above T,. Such a precipitous jump in the Hall number has actually been seen in R, measurements at x > 0.15. This jump in Ru has not been explained before and is not expected within the existing models. In measurements of x, three different regions are found:‘a x < 0.06, 0.06 < x < 0.25 and x > 0.25. Y(X) ,.. decreases in region 1, rises rapidly in region 2 and exhibits a peak at x - 0.25 (where the actual x is larger Within the simple than at x = 0) and decreases again. two-band model with fixed 6, only the behavior seen-in region 1 is expected: the decrease is due to17 the repulsion between holes occupying neighboring Cu and 0 that competes with Ud and promotes double occupancy of Cu-site by holes. This is also seen’l numerically. The behavior of regions 2 and 3 cannot be explained within this picture. Within the present valence transition model. regions 2 and 3 are described by the one-band model of Eq. (3), with region 2 being p - l/2. Near p - l/2, both U, and Vj reduce double occupancy of sites by holes and x is very large, while bevond this. p Y l/2 and x is expected to decrease again.‘r The sharp transition that occurs between region I and region 2 is not expected in x measurements. For large U,. V. drives a 4kF CDW in the p - l/2 system in which eack O’- has only Oa- neighbors and vice versa, leading to a structure -as shown -in Fig. 1. Since the Cu’+ - O’- bond is stronger than the Cu’+ 02- bond, the hopping integral tdp L of the form tdP -
DOPANT INDUCED VALENCE TRANSITION
Vol. 69, No. 5
Fig. I. The oxygen-based and shaded circles represent occur where the lines cross.
4kp 02-
CDW. Here and O’-, and
the open the Cui+
t$,[l + o(nj - l/2)], where 01 is an electron-phonon coupling constant. This modulation of t,,s gives the H,_, in Eq. (3). and gives rise to the observed lattice distortion at T,. Note that the actual CDW is driven by and the lattice distortion is a the Coulomb repulsion, consequence. Most importantly, we emphasize that because of this electron-phonon coupling, we would predict nonzero isotope effect. even though the pairing is finally electronically driven. The 0.4 eV energy gap is due to V.. For V. # 0, a nearest-neighbor CT 02-Oi++ O1-OJ- has a ‘finite energy barrier within Eq. (3) that gives rise to an electronic absorption.18 Clearly, the oscillator strength of the absorption peaks at p y l/2. where nearly every We have nearest-neighbor hop has this energy barrier. argued that the Hall data suggest that x .. 0.15 in La,_,Sr,CuO, is closest to p = l/2 0-sublattice; therefore, this is where the largest oscillator strength is expected. For x < 0.15, the system ‘is still in the Cu2+ phase. while for x much larger than 0.15. p is much greater than one-half and the oscillator strength is weak. confirmed experimentally.3 This is other Any mechanism would require that the reduced oscillator strength for x > 0.15 is due to increased oxygen deficiency. but this is clearly against the increase in Hall number. Only a limited number of experiments can be discussed here, but several groups have pointed out the absence of local moments due to Cu*+ in the superconductors from epr, nmr, nqr, and magnetic scattering experiments. and have even suggested Cu’+. From Raman scattering studies, spin fluctuations have been claimedI in both the semiconducting and the superconducting 123, but the strength of the Raman peak in the superconductor is vanishingly weak compared to of SC in ReportsZO that in the semiconductor. support the idea of Cu’+ in the CuYBa,Ag,O, _ -The photoemission data are in compound. -not The Cu XPS spectrum contradiction to this either. indicates the absence of Cu3+. but the peaks are too broad to distinguish between Cu’+.Cu*+ or true mixed From the relative intensities of the multiple valence.8 XPS peaks, several recent authors have claimed” that the bulk of the Cu is Cu’+. This is in agreement with the split peaks in O(ls) XPS that suggest both O*- and More detailed discussions will be presented OZ-. elsewhere. important feature of the present The most description is that it provides a unified starting point for understanding SC in many other correlated systems. Within Eq. (3). which explains” the normal state -I
529
properties of the complete family of organic conductors. all physical properties are strongly p-dependent. It is significant that the few known organic superconductors (probably only two dozen, out of literally hundreds of conductors) are all quasi two-dimensional (2D) and p l/2. In LiTi,O,, the relevant Ti-band is again p - l/2 (one d-electron per two Ti atoms), while the Verwey transition in magnetite (Fe,O,) with the same spine1 structure and p = l/2 (as only two out of the three Fe atoms, one Fez+ and the other Fes+, OCCUDY eauivalent octahedral sites, and charge separation occurs between these two) has been explained22 within Eq. (3). Elsewhere we will argue that doped BaBiO, has a threedimensional (3D) O-based p - l/3 4k, CDW. It is significant that the electronegativities of Pb, Bi and Cu. which determine the O-band occupancy, are virtually the Very similar optical featuresZS and O(ls) XPS same. spectraa are seen in the superconducting oxides and BaPbBiO. The most probable pairing mechanism involves mobile bipolarons in a 4k, CDW. analogous somewhat to the weak coupling bipolaron model.*s Physically, the pairing is easiest to see in the strong coupling U, - co, tj + 0 limit. An extra hole added to the p - l/2 system occupies a cage, as shown in Fig. 2(a). Unlike in 1D. this hole can move over the unoccupied sites in the CDW% -with an effective hopping integral t” - t3/2V, where V is smaller than V; for finite U. Thus the extra hole behaves as a mdbile polaron in the CDW. Nearest-neighbor hops by the outer_ holes of the cage in Fig. 2(a) have energy barriers 2V when the case is occupied. If a second hole is now added ‘io a neighboring cage as in Fig. I(b), it is easily seen that the energy barrier to nearest-neighbor hops by the outer hoJe common to both cages is substantially lowered - to V. This configuration interaction binds the polarons into Note that the binding here is in second bipolarons. However, the binding energy is comparable to t”. order. Numerically, this binding has been confirmedz6 in a 4 x 4 lattice, in which it is found that E(l0) + E(8) < 2E(9) for nonzero V;, where E(N) is the enersv of the system with N fer’mions. The mobile bipolaron is more therefore stable than two isolated polarons. Condensation of these bipolarons can then lead to SC, as has been suggestedz5 in the case of the 2k, CDW. From the numerical results,26 a weak second-neighbor repulsion can destroy clusters of three polarons while leaving the bipolaron stable. The mechanism of the binding is the same in all the correlated superconductors mentioned above, but the binding is proportional to t2, and copper oxides with the largest tj have the highest +,.
(a) x
n
(b)
.
x
.
x
.
x
.
l
x
x
x
l
x
.
x
x
l
x
.
x
.
)(
.
x
l
x
.
x
.
x
.
.
x
x
x
x
x
.
x
x
l
x
.
x
.
x
.
Fig. 2. (a) Single polaron and (b) bipolaron CDW (see text). Here the crosses are holes are empty sites.
in a 2D 4k, and the dots
530
DOPANTINDUCEDVALENCETRANSITION
To conclude, we suggest that Coulombdriven CDW is the key to understanding SC in narrow band systems. Several experimental predictions that can test the validity of the model are made. First, more thorough investigations of the Ag-based oxides should be made: the present model predicts SC in such systems. Second, structural investigations2*‘~5 have been limited to the superconducting compositions. These should be extended both to the semiconducting regions, as well as to highly doped 214. We predict that lattice softening should be absent in both regions. Superlattice peaks, or at least precursors in scattering measurements, are predicted in superconducting 214, 123 as well as the more recent Bi and Ti-based compounds. A specific heat anomaly in superconducting 214 at T, is predicted. The 0.4 eV
1. 2. 3.
8. 9. 10. 11. 12. 13.
D. Vaknin et al., Phys. Rev. Lett. 58. 2802 (1987); J. M. Tranquada et al., Phys. Rev. Lett. 60, 156 (1988). S. Bhattacharya et al., Phys. Rev. Lett. 60. 1181 (1988). and references therm. S. Etemad et al.. Phvs. Rev. B 37. 3396 (1988): K. Kamaras et .al.. _Phys. Rev. Lett. 59, 919 (1987). P. M. Horn et al., Phys. Rev. Lett. 59. 2772 (1987). D. McK. Paul et al., Phys. Rev. Lett. 58. 1976 (1987). T. Laegrid et al., Nature 330, 637 (1987). S. L. Herr et al., Phys. Rev. B 36, 733 (1987). See also. S. Etemad et al., Proceedings of the Materials Research Society Meeting, 1987. J. Zaanen and A. M. Oh%. Phys. Rev. B 37 (1988). H. M. McConnell et al.. Proc. Natl. Acad. Sci. USA 53. 46 (1965). Mark S. Hybertsen and L. F. Mattheiss. Phys. Rev. Lett. 60. 1661 (1988). and references therein. A. Redondo, S. Mazumdar. and D. K. Campbell, to be published. J. B. Torrance et al., Phys. Rev. Lett. 47, 1747 (1981). S. Kuwajima and Z. G. Soos. Synth. Met. 19, 489 (1987).
Vol.
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absorption should exist in all the superconductors. Further analysis of O(ls) XPS should look for bulk amounts of Oi-. 0”-nmr may be an alternate method to detemine true oxygen valences. Acknowledgments - The author acknowledges illuminating discussions with S. Bhattacharya (Exxon Research and Engineering Company). Part of this work was done while the author was at National Chemical Laboratory. Pune, India. Bhattacharya has informed Note added in proof: us that preliminary measurements find that softening is absent in the undoped La,CuO,. in agreement with the above prediction.
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
26.
S. Mazumdar and S. N. Dixit. to be published. N. P. Ona et al.. Phvs. Rev. B 35. 8421 (1987). L. F. Scl-meemeyer et al., Phys. Rev. B 35, 8421 (1987). S. Mazumdar and A. N. Bloch, Phys. Rev. Lett. 50, 207 (1983). J. Hubbard, Phys. Rev. B 17. 494 (1978): S. Mazumdar and Z. G. Soos. Phys. Rev. B 23, 2810 (1981). K. B. Lvons et al.. Phvs. Rev. Lett. 60. 732 (1988). K. K. Pan et al., Phys. Lett. A125. 147 (1987); Y. D. Yao et al., preprint. D. D. Sarma, Phys. Rev. B 37. 7984 (1988). and references therein; D. D. Sarma and C. N. R. Rao, J. Phys. C 20. L659 (1987). J. R. Cullen and E. Callen. J. Appl. Phys. 41, 879 (1970). S. Uchida et al., Phase Transitions 8. 95 (1987). P. Ganguly and M. S. Hegde, Phys. Rev. B 37. 5106 (1988). P. Prelovsek et al., J. Phys. C20, L229 (1987); D. J. Scalapino et al., in Novel Mechanisms of Superconductivity. edited by V. Krezin and S. Wolf, Plenum, N.Y. (1987). S. Mazumdar and S. Ramasesha. Proceedings of ICSM 88, Synth. Met., in press.
oo38-1098189 $3.00 + .OO Pergamon Press plc
DOPANT INDUCED VALENCE TRANSITION AND FERMION PAIRING IN A MODEL OF SUPERCONDUCTING COPPER OXIDES S. Mazumdar Department
of Physics, University
of Arizona,
Tucson, AZ 85721. USA
(Received 22 October 1988 by C.N.R. Rao)
We propose opposed to a the high T, several other
that doping of the semiconducting copper oxides leads to a valence transition, simple valence fluctuation. The model explains seemingly conflicting experiments oxides, and also provides a unified theoretical approach for superconductivity systems with strong Coulomb correlation.
as in in
thesemodels, the proximity of the superconducting state to the structural distortion is therefore puzzling. The optical measurements,3 now confirmed in single crystals, find a strong electronic absorption at -0.4 eV that tracks T, and the volume fraction of the Meissner effect in La,,Sr,CuO,: the oscillator strength of the absorption peaks at x _ 0.15-O. I7 and is very weak for x < 0.1 and x > 0.3. Similarly, the absorption is seen in YBa$&O,_s for 6 = 0.2 but not for 6 = 0.8. Furthermore, this electronic absorption is coupled7 to vibrational modes. The intensity of the simple chargetransfer (CT) process Cu2+01- ++ Cus+Oz- is expected to increase monotonically with doping, in contrast to what is seen experimentally, indicating a different origin of the dopant-induced absorption. A consistent description of all these conflicting normal-state data seems a prerequisite to a theory of the SC. The theoretical model proposed here is a modified two-band extended Hubbard model that takes into account explicitly the Madelung energy of the CuOs plane. F-simplicity. we focus on LaSrCuO (hereafter 214). but we emphasize that the same basic mechanism applies to YBaCuO (hereafter 123). Indeed, a very precise description of the role of the chain oxygen in 123 can be given. Within the model, doping leads to a valence transition. as opposed to the valence fluctuationa that occursin standard two-band description. At a critical dopant concentration. the true Cu valence jumps from 2 to 1 (Cus+ * Cu’+). so that the holes migrate to the oxygens, leaving an effectively quarter-filled O-band, with the number of holes p per 0 site nearly one-half. The common features between the copper oxides, the organics and LiTi,O, are Coulomb correlations and p _ l/2, while the common feature between the copper oxides and doped BaBiOS is a highly correlated O-band with many more holes than expected from formal valences. We discuss experimental signatures of the valence transition. along with results of numerical calculations. Coulomb correlations at p _ l/2 drive a CDW and give rise to new optical absorptions. The pairing mechanism for SC involves bipolarons in a correlated CDW. We briefly discuss numerical results establishing this. We observe that (a) many other experiments indicate Cui+ in the superconductors (though not in the semiconductors), and (b) the photoemission data are not in contradiction to these ideas. Finally. we make several experimental predictions that can establish the validity of the model.
The possibility that superconductivity (SC) in the is driven by repulsive Coulomb copper oxides interactions has recently been widely discussed. Motivated by the distinctive antiferromagnetism (AF) observed in the semiconducting compositions.’ models based on half-filled Cu bands with strong on-site repulsion have generated the greatest interest. But these models provide no obvious explanation for the equally distinctive anomalous structural distortions’ and optical properties.$ In the present work, we demonstrate that a Coulomb-driven 4k, charge density wave (CDW) in an approximately quarter-filled oxygen band can explain the structural distortions and th-cal absorption and is consistent with the observed AF behavior as a function Further, our model provides natural of doping. explanations for the jump in the number of carriers in Hall measurements and for the (otherwise confusing) behavior of the magnetic susceptibility as a function of doping. In addition. our theory makes a number of explicit, testable predictions. Finally, our approach provides a unified framework for Coulomb-driven SC, within which it may be possible to explain SC in the high T, oxides, the “low” T, oxides LiTi,O, and doped BaBiO,, and the organic superconductors. The structural distortions and the optical properties are, like the AF. intrinsic features of the materials. The structural distortions in the copper oxides are seen in both LaSrCuO and YBaCuO at the superconducting compositions, at temperatures T, > T,. but within the orthorhombic phase (i.e., we are not referringto the tetragonal-orthorhombic transitio;iiT These distortions and lattice softening have been seen in velocity,z sound ultrasound,z x-ray” and neutron scattering5 measurements. A specific heat anomaly is seen6 in superconducting YBaCuO at T,, while high resolution recent x-ray scattering’ and ultrasound measurement? indicate that the structural anomalies are limited to the CuO, planes. The latter, along with the behavior of sound velocity2 at and below T,, as well as the unusually large oscillator strengths of certain infrared modes.’ very strongly indicate CDW. even though the very weak isotope effect suggests a minimal role of electron-phonon coupling. Similarly, neither the oneband nor the two-band Hubbard models explains CDW formation. The CDW here would involve the breathing mode vibrations of the 0.1-0 bonds. Such a CDW is destroyed by the large Hubbard U necessary to explain Within the antiferromagnetism in the semiconductor. 527
DOPANT INDUCED VALENCE TRANSITION
528
Vol. 69, No. 5
The proposed Hamiltonian is H = II, c
nit
nt’
+ UP c
i
+
Idp
njtnj’ + Id c
c
(a$cjo
nia + An c
i,u
j
+
C&Q) + tpp
(ij).u
c
c$c~,~ + E,
(1)
(jj7.u
Here the vacuum consists of Cu’+(d”‘) and 02-(pa), and a$ and cf create holes on Cu and O-sites, respectively. Charge counting in La.$uO, (assuming Las+) shows that only one of the two oxygens in a CuO, unit is definitely Oa-. while the charges on the Cu and the second 0 depend on the parameters in Eq. (1) as well as doping. Here Ud,U, are the usual Hubbard repulsions and tdn and tpp are the hopping integrals between neighboring Cu and 0 and between neighboring O-sites. For large Ud, U, and in the limit td,,t,, * 0. the ground state of the semiconductor is determined by the inequality,’ I,+A,-IAEM,I
njo
j.u
$0
.
(2)
qu
where I, is the second ionization energy of Cu. A, the second electron affinity of 0 and AE, the difference m the Madelung energies between the configuration having the coppers as Cu2+ and the one in which the coppers Here E, = are Cu’+ and half the oxygens are O’-. ZVii,(l + ni)(l + ni,) + ZVj.,(2 - nj )(2 - n.,) + ZVij (1 + ni)(2 - nj) < 0, with i,i’ c b; j.j’ e 0. A smaller lefthand side (LHS) favors Cu*+. while a larger LHS favors Cu’+. Importantly, distinct “Cu*+” and “Cu’+” phases are possible even with finite bandwidth, as long as the bandwidths are small compared to AEM. which is usually the case for ionic solids. The LHS in Eq. (2) as a whole is written as a site energy difference c in the usual two-band description; the consequence of explicit described inclusion of EM is the valence transition below. All band and cluster calculations’s indicate that CuO, is very close to the borderline in Eq. (2) (i.e., E - 0). This is primarily due to the large Id of Cu. which is related to the stability of the d”’ configuration, and the positive A’,. Only the large AE,,,, in a solid drives the system towards Cua+. Thus a recent molecular calculation with extensive configuration interaction finds” a d-count of 9.95. The very strong dependence of the Neel temperature on y in La,CuO,_,, is indirect evidence that the system is close to the a border: for y + 0 and the slightest self-doping, valence transition Cu2+ + Cu’+ becomes possible. Undoped La,CuO,_x is therefore exactly analogous to the mixed-stack organic CT solids,‘a except that in the latter the ionicities vary between 0 and 1 instead of between 1 and 2. In many of these CT solids, pressure or temperature-induced first-order valence transitions transition) have been knownI for (“neutral-to-ionic” years and are explained” within Eq. (2): increased pressure or reduced temperature increases Et,,, and drives the system from the neutral to the ionic side. Calculations’s indicate that for systems close to the border, even a minute change in E, can drive the transition. Doping in the oxides causes an intrinsic variation as opposed to the temperature or pressureof E,, induced extrinsic variation. Each replacement of Oz- by
O’- reduces EM severely: Indeed, the dopant-induced effect on EM is larger than what is expected from pressure or temperature-induced lattice contraction. For a system close to the border, this decrease in EM can drive a sharp Cua+ + Cu’+ transition. As in one dimension (lD),‘a a sharp transition is For a model CuO, plane close to found numerically.” the border of Eq. (2). the slightest doping does indeed This is because even with cause a valence transition. finite tp& the relative magnitudes of Id and AE, are very. large. These results will be published elsewhere. Beyond the transition, the empty Cu’+ band is no longer is described by relevant, and the p - l/2 0-sublattice the one-band Hamiltonian, c:~cj’o + H(t3 + H,_,
.
(3)
Here tj - (t&,/A0 + tPf. with AE the energy of the C.T. Cu’+O’- Cu Oa-. H(t,) a second-neighbor hopping among half the oxygens that is irrelevant for what follows, and He_, an electron-phonon coupling discussed below. The Coulomb repulsion Vj between nearest-neighbor O’- holes originates in Ehl and is extremely important for the description of this p - l/2 system. Signatures of the valence transition are seen in the x-dependence of the Hall coefficient Ru and the high temperature magnetic susceptibility x in Laa_,SrxCuO,. Note that the number of carriers is much larger than x after the transition. as instead of having only dopantinduced O’-, now we have half the oxygens O’-, and nearly all of them are mobile above T,. Such a precipitous jump in the Hall number has actually been seen in R, measurements at x > 0.15. This jump in Ru has not been explained before and is not expected within the existing models. In measurements of x, three different regions are found:‘a x < 0.06, 0.06 < x < 0.25 and x > 0.25. Y(X) ,.. decreases in region 1, rises rapidly in region 2 and exhibits a peak at x - 0.25 (where the actual x is larger Within the simple than at x = 0) and decreases again. two-band model with fixed 6, only the behavior seen-in region 1 is expected: the decrease is due to17 the repulsion between holes occupying neighboring Cu and 0 that competes with Ud and promotes double occupancy of Cu-site by holes. This is also seen’l numerically. The behavior of regions 2 and 3 cannot be explained within this picture. Within the present valence transition model. regions 2 and 3 are described by the one-band model of Eq. (3), with region 2 being p - l/2. Near p - l/2, both U, and Vj reduce double occupancy of sites by holes and x is very large, while bevond this. p Y l/2 and x is expected to decrease again.‘r The sharp transition that occurs between region I and region 2 is not expected in x measurements. For large U,. V. drives a 4kF CDW in the p - l/2 system in which eack O’- has only Oa- neighbors and vice versa, leading to a structure -as shown -in Fig. 1. Since the Cu’+ - O’- bond is stronger than the Cu’+ 02- bond, the hopping integral tdp L of the form tdP -
DOPANT INDUCED VALENCE TRANSITION
Vol. 69, No. 5
Fig. I. The oxygen-based and shaded circles represent occur where the lines cross.
4kp 02-
CDW. Here and O’-, and
the open the Cui+
t$,[l + o(nj - l/2)], where 01 is an electron-phonon coupling constant. This modulation of t,,s gives the H,_, in Eq. (3). and gives rise to the observed lattice distortion at T,. Note that the actual CDW is driven by and the lattice distortion is a the Coulomb repulsion, consequence. Most importantly, we emphasize that because of this electron-phonon coupling, we would predict nonzero isotope effect. even though the pairing is finally electronically driven. The 0.4 eV energy gap is due to V.. For V. # 0, a nearest-neighbor CT 02-Oi++ O1-OJ- has a ‘finite energy barrier within Eq. (3) that gives rise to an electronic absorption.18 Clearly, the oscillator strength of the absorption peaks at p y l/2. where nearly every We have nearest-neighbor hop has this energy barrier. argued that the Hall data suggest that x .. 0.15 in La,_,Sr,CuO, is closest to p = l/2 0-sublattice; therefore, this is where the largest oscillator strength is expected. For x < 0.15, the system ‘is still in the Cu2+ phase. while for x much larger than 0.15. p is much greater than one-half and the oscillator strength is weak. confirmed experimentally.3 This is other Any mechanism would require that the reduced oscillator strength for x > 0.15 is due to increased oxygen deficiency. but this is clearly against the increase in Hall number. Only a limited number of experiments can be discussed here, but several groups have pointed out the absence of local moments due to Cu*+ in the superconductors from epr, nmr, nqr, and magnetic scattering experiments. and have even suggested Cu’+. From Raman scattering studies, spin fluctuations have been claimedI in both the semiconducting and the superconducting 123, but the strength of the Raman peak in the superconductor is vanishingly weak compared to of SC in ReportsZO that in the semiconductor. support the idea of Cu’+ in the CuYBa,Ag,O, _ -The photoemission data are in compound. -not The Cu XPS spectrum contradiction to this either. indicates the absence of Cu3+. but the peaks are too broad to distinguish between Cu’+.Cu*+ or true mixed From the relative intensities of the multiple valence.8 XPS peaks, several recent authors have claimed” that the bulk of the Cu is Cu’+. This is in agreement with the split peaks in O(ls) XPS that suggest both O*- and More detailed discussions will be presented OZ-. elsewhere. important feature of the present The most description is that it provides a unified starting point for understanding SC in many other correlated systems. Within Eq. (3). which explains” the normal state -I
529
properties of the complete family of organic conductors. all physical properties are strongly p-dependent. It is significant that the few known organic superconductors (probably only two dozen, out of literally hundreds of conductors) are all quasi two-dimensional (2D) and p l/2. In LiTi,O,, the relevant Ti-band is again p - l/2 (one d-electron per two Ti atoms), while the Verwey transition in magnetite (Fe,O,) with the same spine1 structure and p = l/2 (as only two out of the three Fe atoms, one Fez+ and the other Fes+, OCCUDY eauivalent octahedral sites, and charge separation occurs between these two) has been explained22 within Eq. (3). Elsewhere we will argue that doped BaBiO, has a threedimensional (3D) O-based p - l/3 4k, CDW. It is significant that the electronegativities of Pb, Bi and Cu. which determine the O-band occupancy, are virtually the Very similar optical featuresZS and O(ls) XPS same. spectraa are seen in the superconducting oxides and BaPbBiO. The most probable pairing mechanism involves mobile bipolarons in a 4k, CDW. analogous somewhat to the weak coupling bipolaron model.*s Physically, the pairing is easiest to see in the strong coupling U, - co, tj + 0 limit. An extra hole added to the p - l/2 system occupies a cage, as shown in Fig. 2(a). Unlike in 1D. this hole can move over the unoccupied sites in the CDW% -with an effective hopping integral t” - t3/2V, where V is smaller than V; for finite U. Thus the extra hole behaves as a mdbile polaron in the CDW. Nearest-neighbor hops by the outer_ holes of the cage in Fig. 2(a) have energy barriers 2V when the case is occupied. If a second hole is now added ‘io a neighboring cage as in Fig. I(b), it is easily seen that the energy barrier to nearest-neighbor hops by the outer hoJe common to both cages is substantially lowered - to V. This configuration interaction binds the polarons into Note that the binding here is in second bipolarons. However, the binding energy is comparable to t”. order. Numerically, this binding has been confirmedz6 in a 4 x 4 lattice, in which it is found that E(l0) + E(8) < 2E(9) for nonzero V;, where E(N) is the enersv of the system with N fer’mions. The mobile bipolaron is more therefore stable than two isolated polarons. Condensation of these bipolarons can then lead to SC, as has been suggestedz5 in the case of the 2k, CDW. From the numerical results,26 a weak second-neighbor repulsion can destroy clusters of three polarons while leaving the bipolaron stable. The mechanism of the binding is the same in all the correlated superconductors mentioned above, but the binding is proportional to t2, and copper oxides with the largest tj have the highest +,.
(a) x
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Fig. 2. (a) Single polaron and (b) bipolaron CDW (see text). Here the crosses are holes are empty sites.
in a 2D 4k, and the dots
530
DOPANTINDUCEDVALENCETRANSITION
To conclude, we suggest that Coulombdriven CDW is the key to understanding SC in narrow band systems. Several experimental predictions that can test the validity of the model are made. First, more thorough investigations of the Ag-based oxides should be made: the present model predicts SC in such systems. Second, structural investigations2*‘~5 have been limited to the superconducting compositions. These should be extended both to the semiconducting regions, as well as to highly doped 214. We predict that lattice softening should be absent in both regions. Superlattice peaks, or at least precursors in scattering measurements, are predicted in superconducting 214, 123 as well as the more recent Bi and Ti-based compounds. A specific heat anomaly in superconducting 214 at T, is predicted. The 0.4 eV
1. 2. 3.
8. 9. 10. 11. 12. 13.
D. Vaknin et al., Phys. Rev. Lett. 58. 2802 (1987); J. M. Tranquada et al., Phys. Rev. Lett. 60, 156 (1988). S. Bhattacharya et al., Phys. Rev. Lett. 60. 1181 (1988). and references therm. S. Etemad et al.. Phvs. Rev. B 37. 3396 (1988): K. Kamaras et .al.. _Phys. Rev. Lett. 59, 919 (1987). P. M. Horn et al., Phys. Rev. Lett. 59. 2772 (1987). D. McK. Paul et al., Phys. Rev. Lett. 58. 1976 (1987). T. Laegrid et al., Nature 330, 637 (1987). S. L. Herr et al., Phys. Rev. B 36, 733 (1987). See also. S. Etemad et al., Proceedings of the Materials Research Society Meeting, 1987. J. Zaanen and A. M. Oh%. Phys. Rev. B 37 (1988). H. M. McConnell et al.. Proc. Natl. Acad. Sci. USA 53. 46 (1965). Mark S. Hybertsen and L. F. Mattheiss. Phys. Rev. Lett. 60. 1661 (1988). and references therein. A. Redondo, S. Mazumdar. and D. K. Campbell, to be published. J. B. Torrance et al., Phys. Rev. Lett. 47, 1747 (1981). S. Kuwajima and Z. G. Soos. Synth. Met. 19, 489 (1987).
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absorption should exist in all the superconductors. Further analysis of O(ls) XPS should look for bulk amounts of Oi-. 0”-nmr may be an alternate method to detemine true oxygen valences. Acknowledgments - The author acknowledges illuminating discussions with S. Bhattacharya (Exxon Research and Engineering Company). Part of this work was done while the author was at National Chemical Laboratory. Pune, India. Bhattacharya has informed Note added in proof: us that preliminary measurements find that softening is absent in the undoped La,CuO,. in agreement with the above prediction.
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
26.
S. Mazumdar and S. N. Dixit. to be published. N. P. Ona et al.. Phvs. Rev. B 35. 8421 (1987). L. F. Scl-meemeyer et al., Phys. Rev. B 35, 8421 (1987). S. Mazumdar and A. N. Bloch, Phys. Rev. Lett. 50, 207 (1983). J. Hubbard, Phys. Rev. B 17. 494 (1978): S. Mazumdar and Z. G. Soos. Phys. Rev. B 23, 2810 (1981). K. B. Lvons et al.. Phvs. Rev. Lett. 60. 732 (1988). K. K. Pan et al., Phys. Lett. A125. 147 (1987); Y. D. Yao et al., preprint. D. D. Sarma, Phys. Rev. B 37. 7984 (1988). and references therein; D. D. Sarma and C. N. R. Rao, J. Phys. C 20. L659 (1987). J. R. Cullen and E. Callen. J. Appl. Phys. 41, 879 (1970). S. Uchida et al., Phase Transitions 8. 95 (1987). P. Ganguly and M. S. Hegde, Phys. Rev. B 37. 5106 (1988). P. Prelovsek et al., J. Phys. C20, L229 (1987); D. J. Scalapino et al., in Novel Mechanisms of Superconductivity. edited by V. Krezin and S. Wolf, Plenum, N.Y. (1987). S. Mazumdar and S. Ramasesha. Proceedings of ICSM 88, Synth. Met., in press.