Journal of Physics and Chemistry of Solids 61 (2000) 467–471 www.elsevier.nl/locate/jpcs
Charge inhomogeneity and high temperature superconductivity V.J. Emery a,*, S.A. Kivelson b a
Department of Physics, Brookhaven National Laboratory, Upton, NY 11973-5000, USA Department of Physics, University of California at Los Angeles, Los Angeles, CA 90095, USA
Abstract When charge is doped into a correlated insulator, it forms an inhomogeneous state which may be fluctuating, ordered or glassy. The outstandingly successful BCS theory of superconductivity was not designed for doped insulators, and an appropriate theory must incorporate the local electronic structure on the scale of the superconducting coherence length. Unlike conventional materials, superconductivity in the cuprates is driven by lowering the in-plane zero-point kinetic energy. There is extensive experimental evidence in support of this picture. Inelastic X-ray scattering can give information about the existence and energy scale of the charge structure, especially in optimally doped materials. q 2000 Published by Elsevier Science Ltd. Keywords: D. Superconductivity; Charge inhomogeneity
1. Introduction The high temperature superconductors  are quasi-two dimensional doped insulators, obtained by chemically introducing charge carriers into a highly correlated antiferromagnetic insulating state. Despite the fact that there is a large “Fermi surface” that contains all the pre-existing holes and the doped holes , it is impossible to understand the behavior of the system and, in particular, the origin of high temperature superconductivity unless the nature of the doped insulating state is incorporated into any theory. For example, whereas in conventional superconductors the superfluid density ns is given by all of the holes in the Fermi sea, for the high temperature superconductors ns vanishes as the dopant concentration tends to zero [3,4]. In our view, the essential feature of the doped correlated insulating state is charge inhomogeneity [5–8], or “topological doping” . Specifically, in d dimensions, the charge forms “stripes” or one-dimensional arrays of
d 2 1-dimensional structures that are also antiphase domain walls for the background spins. In d 1 there is an array of charge solitons [10,11], in d 2 there are linear “rivers of charge” threading through the antiferromagnetic background [5–7] and in d 3 there are arrays of charged planes [7,8], as * Corresponding author. Tel.: 11-516-344-3765; fax: 11-516344-2918. E-mail address: [email protected]
observed in the manganates . These self-organized structures may be fluctuating or they may form ordered or glass phases. They are a consequence of the tendency of the correlated antiferromagnet to expel the doped holes, and they lead to a lowering of the zero-point kinetic energy. The theoretical arguments that lead to this picture are summarized in Section 2. The substantial experimental support for this picture of the doped-insulating state of the high temperature superconductors is described in Section 3. The primary evidence comes from neutron and X-ray scattering experiments which observe the influence of the charge structure on the spin or lattice structures. What does this have to do with high temperature conductivity? It has been shown that, in La22xSrxCuO4, the superconducting transition temperature Tc is inversely proportional to the stripe spacing for underdoped and optimally doped materials . Also, the coexistence of superconductivity with a cluster spin glass phase [14,15] is consistent with the idea of superconducting rivers of charge that separate the background spins into loosely coupled regions. Moreover, rather general and phenomenological arguments indicate that the BCS many-body theory, which is so successful for conventional superconductors, must be revised for the high temperature superconductors. (Section 4) Once this is accepted, it is clear that any new many-body theory must be based on the local electronic structure of the doped insulator at distances of the order of
0022-3697/00/$ - see front matter q 2000 Published by Elsevier Science Ltd. PII: S0022-369 7(99)00338-8
V.J. Emery, S.A. Kivelson / Journal of Physics and Chemistry of Solids 61 (2000) 467–471
the superconducting coherence length. A new mechanism and many-body theory of superconductivity, based on local charge inhomogeneity has been developed  and the essential ideas will be described in Section 4. 2. Topological doping It has been known for a long time that the motion of a hole doped into an antiferromagnet is inhibited by the creation of strings of broken bonds [17,18]. Consequently, when there is a finite density of holes, the system will strive to lower its kinetic energy. This is accomplished first by establishing a locally charge inhomogeneous state, and then by a transition to a high-Tc superconducting state. Neutral holes would follow a rather different route: the system would separate into a hole-free antiferromagnetic phase and a hole-rich (magnetically disordered or possibly ferromagnetic) phase, in which the holes are mobile and the cost exchange energy is less than the gain in kinetic energy [19–22]. On the other hand, charged holes can do no more than phase separate locally, by forming arrays of linear metallic stripes [5–7] which are “topological” in nature, since they are antiphase domain walls for the antiferromagnetic background spins . This structure lowers the kinetic energy along the stripe but makes it more difficult, if anything, for a single hole to move perpendicular to the stripe direction. A hop transverse to a stripe takes the hole far above Fermi energy . However, as we shall see, pairs of holes can move more easily transverse to a stripe, and they lower their kinetic energy by making the system a high temperature superconductor. For a perspective on the circumstances in which physics is driven by the kinetic energy see Ref. .
theory of the phase transition  the spin order parameter S~q~ and the charge order parameter r2Q~ first couple in third order
S~q~ ·S~q~ r2Q~ , so it is necessary that the ordering ~ 2~q; or in other words, the wavevectors satisfy Q length of the spin modulation is twice that of the charge modulation. This relation is found to be satisfied experimentally [26,27,29], and it implies that the charge is effectively concentrated along antiphase domain walls in the magnetic order, which gives the precise meaning of the concept of topological doping . The existence of charge stripes in this material gained further support from hard X-ray scattering experiments  that detected the associated lattice displacements. The observation of essentially ordered stripes allowed a study of the evolution of the spin and charge order parameters, which not only provided input into the mechanism of stripe formation by showing that they are charge driven, but also established that inelastic incommensurate magnetic peaks observed previously [32– 35] in La22xSrxCuO4 were produced by fluctuating stripes. Recently, inelastic incommensurate magnetic peaks have been observed in underdoped YBa2Cu3O72d by neutron scattering experiments , which established that YBa2Cu3O72d and the La22xSrxCuO4 family have a common spin structure. These are the only families of materials in which extensive neutron scattering experiments have been performed, and the data strongly suggest that the concept of topological doping will be established in other cuprate superconductors, once samples of sufficient size for neutron scattering experiments become available. Indeed there is growing evidence of similar behavior in Bi2Sr2CaCu2O81d : preliminary neutron scattering experiments show incommensurate magnetic peaks, and a calculation of the effects of stripes in ARPES experiments  produced regions of degenerate states and a flat section of the “Fermi surface”, as observed experimentally [38,39].
3. Experimental support The idea that the kinetic energy of a single doped hole is frustrated by the antiferromagnetic structure of the background spins is supported by angular resolved photoemission spectroscopy (ARPES), which found that the bandwidth of the doped hole is controlled by the exchange integral J, rather than the hopping amplitude t . Macroscopic phase separation can take place whenever the dopants are mobile, as for example in oxygen-doped and photo-doped materials. We have reviewed the experimental evidence for this behavior elsewhere . The existence of charge and spin stripes in the La22xSrxCuO4 family was established in an elegant series of experiments by Tranquada and co-workers [26,27]. In particular, the Nd-doped material La1.62xNd0.4SrxCuO4 has spin and charge stripes that would be ordered, if it were not for the inevitable structural disorder of the material . The Nd does not dope the CuO2 planes but rather causes a transition to the low-temperature tetragonal structure, which provides an orienting potential for the stripes [26,27,29]. In a Landau
4. Superconductivity There are strong indications that this local electronic structure is closely related to high temperature superconductivity. First of all, in La22xSrxCuO4, the value of Tc is inversely proportional to the spacing between stripes in underdoped and optimally doped materials . Secondly, m SR experiments [14,15] have found evidence for a phase in which superconductivity coexists with a cluster spin glass. In YBa2Cu3O72d , the spin freezing temperature goes to zero when the superconducting Tc is more than 50 K. It is difficult to see how these two phases could coexist unless there is a glass of metallic stripes dividing the CuO2 planes into antiferromagnetic regions that are coupled into a cluster spin glass. 4.1. BCS many-body theory It has been argued that the quasiparticle concept does not
V.J. Emery, S.A. Kivelson / Journal of Physics and Chemistry of Solids 61 (2000) 467–471
apply to many synthetic metals, including the high temperature superconductor . This idea is supported by ARPES experiments on the high temperature superconductors, which show no sign of a normal-state quasiparticle peak near the points
0; ^p and
^p; 0 where high temperature superconductivity originates . If there are no quasiparticles, there is no Fermi surface in the usual sense of a discontinuity in the occupation number nk~ at zero temperature. This undermines the very foundation of the BCS meanfield theory, which is a Fermi surface instability that relies on the existence of quasiparticles. A major problem for any mechanism of high temperature superconductivity is how to achieve a high pairing scale in the presence of the repulsive Coulomb interaction, especially in a doped Mott insulator in which there is poor screening. In the high temperature superconductors, the coherence length is no more than a few lattice spacings, so neither retardation nor a long-range attractive interaction is effective in overcoming the bare Coulomb repulsion. Nevertheless ARPES experiments [42,43] show that the major component of the gap function is proportional to cos kx 2 cos ky : It follows that, in real space, the gap function and hence, in BCS theory, the net pairing force, is a maximum for holes separated by one lattice spacing, where the bare Coulomb interaction is very large (,0.5 eV, allowing for atomic polarization). It is not easy to find a source of an attraction that is strong enough to overcome the Coulomb force at short distances and achieve high temperature superconductivity in a natural way by the usual Cooper pairing. There is phenomenological support for this point of view. In BCS theory, an estimate of Tc is given by Tc , D0 =2 where D 0 is the energy gap measured at zero temperature. This is a good approximation for conventional superconductors because the classical phase ordering temperature Tu is very high. An approximate upper bound on Tc is obtained by considering the disordering effects of only the classical phase fluctuations as Tc , Tu AV0 ; where V0 is the zero temperature value of the “phase stiffness” (which sets the energy scale for the spatial variationof the superconducting phase) and A is a number of order unity . V0 may be expressed in terms of the superfluid density ns(T) or, equivalently, the experimentally measured penetration depth l (T) at T 0: V0
Éc2 a p 4m 16p
where a is a length scale that depends on the dimensionality of the material. For a conventional superconductor such as Pb, Tu is about 10 6 K, which implies that phase ordering occurs very close to the temperature at which pairing is established . For the high temperature superconductors, especially underdoped materials, D0 =2Tc varies with doping and takes values much greater than one. On the other hand, Tu provides a quite good estimate of Tc for the high temperature
superconductors , and it can be improved by making a plausible generalization of the classical phase Hamiltonian . A major reason for this discrepancy is that BCS theory was not designed for doped insulators. It gives a large superfluid density that involves all of the holes enclosed by the large Fermi surface, and is not proportional to the density of doped holes. 4.2. Mechanism and many-body theory of superconductivity The frustration of the motion of holes doped into an antiferromagnetic insulator suggests that high temperature superconductivity is not driven by the potential energy (i.e. by an attractive interaction, possibly mediated by the exchange of some kind of boson) but by the system’s drive to lower its zero-point kinetic energy. This is precisely what emerges from the stripe picture . The existence of a spin glass state for a substantial range of doping in the high temperature superconductors [14,15] implies that the stripe dynamics is slow and that the motion of holes along the stripe is much faster than the fluctuation dynamics of the stripe itself. Thus an individual stripe may be regarded as a finite piece of one-dimensional electron gas (1DEG) that is located in an active environment of the undoped spin regions between the stripes. Then it is appropriate to start out with a discussion of an extended 1DEG in which the singlet pair operator P † may be written P† c†1" c†2# 2 c†1# c†2" ;
creates a right-going
i 1 or left-going
i 2 where fermion with spin s . In one dimension, the fermion operators of a 1DEG may be expressed in terms of Bose fields and their conjugate momenta (f c(x),p c(x)) and (f s(x),p s(x)) corresponding to the charge and spin collective modes, respectively. In particular, the pair operator P † becomes p p P† , ei 2puc cos
3 where 2x uc ; pc [10,11]. In other words, there is an operator relation in which the amplitude of the pairing operator depends on the spin fields only and the (superconducting) phase is a property of the charge degrees of freedom.p Now, if the system acquires a spin gap, the amplitude cos
2pfs acquires a finite expectation value, and superconductivity will appear when the charge degrees of freedom become phase coherent. The spin gap temperature can be quite high, even in a 1DEG, and it is generically distinct from the phase ordering temperature [10,11,46–50] because phase order is destroyed by quantum fluctuations, even at zero temperature. For a 1DEG, a spin gap occurs only if there is an attractive interaction for the spin degrees of freedom. However, this is no longer true if the 1DEG is in contact with an active (spin) environment, as in the stripe phases. We have shown that pair hopping between the 1DEG and the environment will generate a spin gap in both the stripe and the environment, even for purely repulsive interactions . At a lower temperature the motion of
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a pair from stripe to stripe establishes global phase coherence and high temperature superconductivity . This last step determines the value of Tc, especially in underdoped and optimally doped materials . The phase diagram itself is consistent with this picture. The physics evolves in three stages. Above the superconducting transition temperature there are two crossovers, which are quite well separated in at least some underdoped materials. The upper crossover is indicated by the onset of short-range magnetic correlations and by the appearance of a pseudogap  in the c-axis optical conductivity (perpendicular to the CuO2 planes), which might possibly indicate the establishment of a charge glass phase. The lower crossover is where a spin gap or pseudogap (which is essentially the amplitude of the superconducting order parameter) is formed. Finally, superconducting phase order is established at Tc, and in fact determines the value of Tc .
Acknowledgements We would like to acknowledge frequent discussions with J. Tranquada. This work was supported at UCLA by the National Science Foundation grant number DMR93-12606 and, at Brookhaven, by the Division of Materials Sciences, US Department of Energy under contract No. DE-AC0298CH10886.
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5. The role of inelastic X-ray scattering Inelastic X-ray scattering could contribute to this subject in several ways. First of all, extensive neutron scattering experiments have been carried out on only two sets of materials (the La22xSrxCuO4 family and YBa2Cu3O72d ) because large single crystals of other materials have been unavailable. For the most part, there is neutron scattering evidence of incommensurate peaks in the spin structure factor, but rather little evidence of the charge behavior. Also, no clear evidence of incommensurate spin peaks has been obtained in optimally doped YBa2Cu3O72d . In view of the information from La1.62xNd0.4SrxCuO4, the presence of low-energy spin peaks also provides evidence for charge stripes. However, the absence of incommensurate spin peaks does not provide evidence for the absence of charge stripes: an ordered array of charge stripes could cut the localized spins in the CuO2 planes into regions that are fully spin gapped, which would be very favorable for superconductivity, but unfavorable for establishing the existence of stripes. Thus it is desirable to obtain more direct evidence for the charge stripes themselves, and this is where inelastic X-ray scattering may be able to help. Stripes are relevant for high temperature superconductivity provided their energy scale is much lower than the superconducting gap, which is of the order of 30 meV in YBa2Cu3O72d and Bi2Sr2CaCu2O81D . However, the m SR evidence for a cluster spin glass suggests that, in some regions of the phase diagram, the charges are static for the lifetime of a muon, 2 ms. Evidence for such a charge glass would require elastic scattering measurements that find an ordering temperature that decreases as the resolution improves, and they may well require a resolution of 1 meV or better.
   
        
         
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