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The van Hove singularity and high-Tc superconductivity: The role of nanoscale disorder and interlayer coupling
J. Ph\\r.
Chrm.
Solids
Vol.
52.
No.
11112.
pp.
1363-1370.
1991
0022.3697/91
Printed in Great Britain.
$3.00 + 0.00 Pergamon Press plc
THEVANHOVESINGULARITY AND HIGH-T, SUPERCONDUCTIVITY: THEROLEOFNANOSCALEDISORDER ANDINTERLAYERCOUPLING R.S. Markiewicz Physics Department and Barn&t
Abstract
Institute, Northeastern University, Boston, MA 02115 USA
- The van Hove singularity (vHs) model provides a unifying picture for understanding many
anomalous features of high-temperature
superconductors.
calculated Fermi levels in most high-Tc
cuprates.
The vHs is found to occur very close to the
The large doa peak (the logarithmic
singularity is
only modestly broadened by interlayer coupling) provides a simple explanation of enhanced superconductivity,
but also leads to striking intrinsic disorder, both as the material is doped away from the
vHs (potentially
Keywords:
nanoscale phase separation),
superconductivity,
and at the vHs (short-range
charge density wave, phase separation,
density wave order).
interlayer. Fermi surface
INTRODUCTION All two-dimensional band structures have a saddle-point van-Hove singularity (vHs), at which the density-of-states (dos) diverges logarithmically. Prior to the discovery of high-temperature superconductivity, a vHs model was introduced to study the competition between superconductivity and charge-density wave (CDW) formation (Balseiro and Falicov, 1979). If CDW formation could be suppressed, Hirsch and Scalapino (1986) showed that the dos peak provided a simple means of attaining high Tc’s , and a vHs model was very soon applied to the high-T, cuprates (Labbe and Bok, 1987, Dzyaloshinskii, 1987). The earlier theories (Labbe and Bok, 1987, Dzyaloshinskii, 1987, Friedel, 1989) used a simple tight-binding model, in which the vHs falls exactly at half-filling of the band; this leads to complications associated with the Mott-like insulating state found at half-filling. More recent theories (Markiewicz, 1988-1991e, Tsuei, 1990, Tsuei, et al. 1990, Newns, et al. 1991, Levin, et al. 1991, Si and Levin, 1991, Toby, et al., 1991) assume that the vHs occurs away from half-filling, as found in band-structure calculations, and corresponds to the hole doping of optimum T,. INTERLAYER COUPLING Because the BCS equation averages the dos over an interval of 4kr~T,, the vHs singularity can be smeared out over the same interval without significantly degrading T,. Thus, pairbreaking due to real phonon scattering is unlikely to produce significant T, reduction, since the electronphonon scattering rate deduced from dc conductivity is TV&’ N 2k~T. Thus, the factor most harmful to high Te is interlayer coupling, which broadens the vHs dos peak over an energy interval 4t,, where t, is the interlayer hopping parameter. It can be shown (Markiewicz, 1991b) that as long as t, < T,, there is no T, reduction, although there is a substantial reduction for larger t, values. Moreover, if the hopping involves interband interlayer hopping, the effective value of t, involved in broadening the dos is considerably reduced from the full t, value. For example, in BizSrzCaCuzOs (Bi-2212), interlayer hopping may involve hopping from a CuOz band on one layer to a BiO band on the next. In this case, the effective value t: involved in broadening the vHs peak is 2: N t i/AE, where AE is the separation of the bare CuOz and BiO bands at the vHs. This is illustrated in Fig. la (Markiewicz, 1991b): The single vHs peak associated with a CuOz plane is split by coupling with a BiO plane into three components. Since a value t, = 1OOmeV was chosen, the splitting of the dos peaks is of order SOOmeV, but the 1363
I364
R. S. MARKIEWICZ
broadening of the individual peaks is only of order t: N 10meV. Fig. lb illustrates another interesting aspect of interlayer coupling: while the dos peak C is derived from the CuOz plane vHs, the interlayer coupling has shifted the peak from the a-point of the Brillouin zone to a point along the T - it? line.
Fig. 1 (a) Normalized density of states for a simplified model of Bi-2212, showing the splitting and broadening of the vHs peak produced by interlayer coupling. (b) Fermi surface of model Bi-2212, near the energy of feature C in the dos. Solid lines show Fermi surface at k, = 0, dashed lines at k, = 7r/2c, with c the c-axis lattice constant. Letter C denotes the (incipient) topological change which gives rise to feature C in Fig. la. [After Markiewicz, 1991b] LOCATION OF THE vHs It has long been known that the vHs falls at the hole doping of optimum T, in local density approximation calculations for La2 _,Sr,CuOs (LSCO) (Mattheis, 1987, Xu, et al. 1987). For most other high-T, cuprates and bismuthates, the vHs also falls close to the calculated Fermi level (Markiewicz, 1991a). A conspicuous exception had been YBazCusOr_6 (YBCO), for which early band structure calculations did not place the vHs at the Fermi level of the stoichiometric material (5 = 0). However, the CuOz plane hole concentration can be systematically varied by doping or by deviations from oxygen stoichiometry (6 > 0). If this is accounted for by rigidly shifting the CuOz band with respect to the Fermi level, it is found that the vHs falls at the Fermi level at the hole doping which experimentally corresponds to optimum T, (Markiewicz, 1991a). Moreover, the most recent band structure calculations (Pickett, et al., 1990) do place the vHs at the Fermi level of the 07 compound, as illustrated in Fig. 2. The solid lines are CuOzplane like bands, the dashed line is a CuO chain band, which crosses a CuO:! plane band and strongly hybridizes with it near the X-point. The arrow points to the X-point vHs, which falls precisely at the Fermi level, near the center of the X-U-line, just as in LSCO. The feature along the r - Y line, labelled with an asterisk is also generated by plane-chain interaction, and is strikingly reminiscent of the vHs feature of Fig. lb. It should be noted that the band structure calculations predict a c-axis dispersion, due to interlayer coupling, of m 150meV, which would wash out the sharp dos peaks. However, interlayer coupling should be very sensitive to electron correlation effects and to strong electron-phonon coupling, both of which can act to sharpen the dos peak.
I365
The van Hove singularity model
Y
r Z
S
R
xv
r,t
Fig. 2 Fermi surface of YBCO, from Pickett, et al., 1990. Solid lines = predominantly CuOa planelike sections; dashed lines = predominantly chain-like sections. Arrows indicate position of the vHs, along the X-U-line. Asterisk indicates possible second vHs feature; compare point C of Fig. lb. INTRINSIC DISORDER At half filling of the CuOa plane band, LSCO is an antiferromagnetic insulator. This can be explained as a Mott transition (more appropriately, a transition to a charge-transfer insulator), and the transition can be described within the slave boson formulation (Sa de Melo and Doniach, 1990, Grilli, et al., 1990, Markiewicz, 1990a), using realistic band parameters (Hybertsen, et al., 1989, Martin, et al., 1991). Doping away from half filling leads to an instability against phase separation (Nagaev, 1974, Visscher, 1974, Zaanen and Gunnarson, 1989, Trugman, 1989, Emery, et al., 1990, Schulz, 1990, Markiewicz, 1990a). This transition can be understood as an intrinsic feature of the Mott transition. Mott’s picture is that the transition occurs at exactly half filling, to avoid the strong on-site Coulomb repulsion of two holes, independently of the band structure. In this case, many different normal states would converge to the same antiferromagnetic insulator at half filling, which is only possible if the transition is first order. In the presence of such a transition, it is impossible to infer the nature of the doped phase from the properties of the antiferromagnetic phase at half filling. At best, a Hubbard or tJ model may correctly describe the initial phase separation, when second phase islands contain only two or three holes. In the particular case of the cuprates, this second phase should fall almost exactly at the vHs (Markiewicz, 1990a). This arises from a strong electron-phonon coupling, which is also driven by the large dos associated with the vHs (incipient CDW formation, discussed further below). The sharp dos peak produces a giant Kohn anomaly, with a free energy minimum at the vHs concentration, which leads to a miscibility gap between the Mott insulating phase and the vHs phase, and a second phase separation when the material is overdoped beyond the concentration corresponding to the vHs. This electronically-driven instability is closely related to the phase separations in Hume-Rothery alloys (Blandin, 1967, Friedel, 1977, Hafner, 1987). Since the phase transition is electronically driven, it is possible that the holes could phase separate by themselves, without accompanying ionic motion. Since the resulting domains would be electrically charged, Coulomb effects preclude an ordinary macroscopic phase separation, but can still allow a state of ‘mesoscopic’ (more accurately, nanoscopic) disorder. In this state, a single domain may contain only a few particles - indeed, for light doping, the stable phase may correspond to real-space pairing of holes. In many cases, this pair-phase is found to be met&able with respect to a phase in which the dense holes form ‘grain-boundaries’ between antiferromagnetic domains (Zaanen and Gunnarson, 1989, Schulz, 1990, Markiewicz, 199Oa). In three-dimensions, there is strong pinning of these phases to impurities, thereby obscuring their direct observation. In the layered cuprates, the CuOa-planes are modulation doped, so free
1366
R. S. MARKIEWICZ
hole-domains may be expected to form. In this case, the phase diagram (Markiewicz, 1990a, 1991c) should be characterized by three transitions as a function of doping (Fig. 3): first, from an antiferromagnetic insulating phase to a spin-glass-like phase, with antiferromagnetic domains surrounded by domain walls of the higher-hole-density phase; secondly, a metal-insulator transition from this phase to a metallic phase, with intrinsic weak links (corresponding to domain walls of the antiferromagnetic phase); and finally to a uniform phase at or near to the vHs. Whereas high-T, superconductivity is associated with this latter phase, it persists well into the two-phase regime.
AFM d
Spin
Glass
Metal x----+
t vHs
Fig. 3 Schematic free energy (AF) d ia g ram for LSCO, showing two-phase coexistence regime for intermediate doping between the Mott insulator (x=0) and the vHs (x-0.17). Solid line = uniform phase; short dashed line = islands of antiferromagnetic (AFM) insulator; longdashed lines = islands of metallic phase at vHs. Lowest energy phases are labelled on horizontal axis [from Markiewicz, 1991c]. There is considerable experimental evidence for this scenario (Markiewicz, 1991a). Perhaps the clearest example arises in LSCO, which can be hole-doped either by substituting Sr for La, or by incorporating interstitial 0. Since the Sr is essentially immobile at low temperatures, phase separation is on the nanoscopic scale discussed above. On the other hand, the interstitial oxygen is highly mobile at low temperatures, so the electronic instability leads to macroscopic phase separation (Jorgensen, et al., 1988). There is considerable evidence for a nanoscopic phase separation in LSCO and YBCO, but most of it is indirect (Phillips, 1990, Markiewicz, 1991a). For instance, the Meissner fraction is maximized at the composition of highest Tc, and falls smoothly to zero with variation of hole content (van Dover, et al, 1987). Recently, Song, et al. (1991) interpreted l/f-noise measurements in YBCO as evidence for the presence of phase separation on a N 10 nm scale. Bi-2212 provides a quite different example of electronically driven transitions. In this compound, substitution of Pb for Bi would be expected to change the average valence of the Bi(Pb)Olayer, and hence the average number of holes in the CuOz planes. However, this is not what is observed to happen (Zhang, et al., 1990). For Pb substitution of up to 35% of the Bi, T, is found to be essentially unchanged. Instead, there is a large increase of the b-axis superlattice period associated with the BiO layers. This is interpreted as follows: the superlattice is accompanied by the incorporation of excess oxygen, which provides a hole doping of the CuOs planes. When Pb is substituted for Bi, the superlattice period increases (less 0 is incorporated) in such a way that the net doping of the CuO2 planes remains unchanged. This is a remarkable result: the material ‘knows’, at the temperature it is prepared, the correct hole-doping required to produce the optimum T,. This result is readily understood in terms of the vHs model - doping at the
The van Hove singularity model
1367
vHs minimizes the free energy. A similar phenomenon has been reported in the Tl-based superconductors (Hibble, et al., 1988). In stoichiometric single-T1 layer superconductors (e.g., TlBazCazCusOs - Tl-1223(g)), the average Cu valence is 2.33, whereas in the double-T1 layer compounds (e.g., Tl-2223(10)) the Cu valence is only 2. It was found that the double-T1 layer compounds contain cation vacancies (Tl, Ca deficits), which are absent in the single-T1 layer materials, in such a way that the average CU valence is in the range 2.3-2.6 for both families of compounds. Thus, the disorder can take on a large variety of forms, but is always associated with a preferred doping level in the CuOz planes. Something similar occurs in the YBCO superconductors, and has been used to make a yttrium-free Pr-based superconductor, Pro.sCao.sBazCusOr-a (Norton, et al., 1991). Substituting Pr for Y depresses T,, ultimately destroying superconductivity when more than 50% of the Y is replaced. The T, reduction is believed to have two causes: Pr acts both as a pair-breaker and as a hole-filler. Norton, et al. showed that the hole-filling action could be reversed by substituting Ca for Pr, and that the optimum T, corresponded to the same average (Pr-Ca) valence as in YBCO - i.e., the same hole-doping of the CuOz planes. (The fact that T, is not as large as in YBCO is presumably due to Pr pairbreaking.) These phase separations are similar to those predicted by Phillips’ quantum percolation theory (Phillips, 1987, 1990), but the microscopic origins are totally different. The present phase separation is electronically driven, and hence (a) associated with a preferred hole density in the Cu02 planes and (b) related to the phase separation associated with the Mott transition. In Phillips case, the phase separation is driven by the impurities, and hence should be more sensitive to states off of the CuOz planes. Furthermore, Phillips is applying a theory of strong localization (metal-insulator transition) to a situation in which the intrinsic localization effects would appear to be quite weak. The equivalent sheet resistance of a single layer of YBCO is about 800R at T_90K, and most of this is inelastic scattering. Extrapolating the normal state resistance to T=O suggests that the elastic sheet resistance is 5 lOOs1. For such small sheet resistance, localization effects would show up as a weak logarithmic correction to the resistance, measurable only below N 5K. Even a weak interlayer coupling, as discussed above, would be sufficient to wash out this localization correction: in graphite intercalation compounds, no such corrections are observed unless strong disorder is intentionally introduced, even though the interlayer couplings can be 100x weaker than in Bi-2212. Hence, localization effects due to twodimensionality in these materials should be negligible; in contrast, the strong scattering near the vHs produces a different form of quasi-localization, discussed below. CDW AND SDW FLUCTUATIONS The strong electron-phonon coupling which leads to phase separation away from the optimum doping concentration also has profound effects on material doped at the correct vHs composition - namely, strong fluctuations of short-range CDW or SDW (spin density wave) order. These arise from a special form of nesting associated with the vHs (Rice and Scott, 1975). A CDW or SDW is stabilized because the transition opens a gap at the Fermi level, lowering the average energy of the occupied electronic states. Since the vHs is associated with a large dos, this stabilization can be achieved by any transition that drives the vHs below the Fermi level. Thus, in LSCO, there are two vHs, associated with the X and Y points of the Brillouin zone, so any density wave oscillation which couples them (i.e., has a component of symmetry (z/a, z/o)) can open up the appropriate gap. Indeed, such a phase transition to long-range CDW order has been observed in the closely related compound Laz_,Ba,CuOd (LBCO) (Axe, et al., 1989). The transition splits the two vHs (Markiewicz, 1990d, Cohen, et al., 1991), and severely degrades T,, essentially to T=O. Significantly, the transition is most complete at the hole concentration which would otherwise have had the highest T, (i.e., the Fermi level coincides with the vHs). In LSCO, strong fluctuations to this phase exist, but long-range CDW order is not observed above T,. There is a small T, depression, and the onset of long-range order below T, (Fukase, et al., 1990), both effects occuring at nearly the same hole-doping as in LBCO. Similar competition between CDW order and superconductivity is found in the Al5 compounds. Analogous phenomena should arise in the spin susceptibility, which also has logarithmically diverging peaks at the same q-vector, (z/u, z/u). Indeed, this form of spin susceptibility has
I368
R. S. MARKIEWICZ
been introduced phenomenologically to account for the anomalous nuclear relaxation rates in YBCO (Millis, et al., 1990, Lu, et al., 1990). The cutoff parameter T, 11 115K introduced by Millis, et al., can be interpreted as the interlayer coupling t, which cuts off the susceptibility divergence. The strong fluctuations toward short-range density wave order can explain a number of transport anomalies associated with the cuprates, having to do with the discrepancy between the small carrier density revealed by Hall effect measurements (proportional to the doping, x) versus the much larger density expected from the large Fermi surface sections found theoretically and experimentally (proportional to 1 &z). In the presence of long-range density wave order, the large sections of Fermi surface would be nested (h la Rice and Scott), leaving only small pockets of holes of the requisite area oc x. In the presence of short-range density-wave order, carriers near the vHs would be quasi-localized, and hence not contribute to transport measurements. Transport would be dominated by carriers from other regions of the Fermi surface, and can be thought of as ‘shadow Fermi surfaces’ of the hole pockets (Markiewicz, 1988, Lee, 1990, Kampf and Schrieffer, 1990). These shadow Fermi surfaces should be observable by, e.g., angle resolved photoemission (Kampf and Schrieffer, 1990) or the de Haas-van Alphen effect (Markiewicz, 1991d). The near instability toward charge or spin density wave formation can be looked on as a form of excitonic instability (Halperin and Rice, 1968, Markiewicz, 1990b, 1991e), wherein there is a net Coulombic attraction between holes at one vHs and electrons at a second. The polarizability of these excitons may be identified with the ‘phenomenological polarizability’ of Varma, et al. (1989). The fact that a vHs can explain the anomalous electron-electron scattering rate, re;l cx T,w, had been pointed out earlier (Lee and Read, 1987, Markiewicz, 1989a). CONCLUSIONS The paradox of the vHs model is that, whereas it provides a transparently simple means of enhancing T, (Hirsch and Scalapino, 1986, Labbe and Bok, 1987, Dzyaloshinskii, 1987) and explaining the small isotope effect (Tsuei et al. 1990), a closer examination suggests that the underlying physics is immensely richer and more complicated. The actual superconducting transition may involve a two stage process, wherein strong electron-phonon coupling produces short-range CDW order (?bipolaron formation), and these electronic fluctuations drive a superconducting transition of the shadow Fermi pockets (Markiewicz, 199Oc, Toby, et al., 1991). In conclusion, the vHs model seems flexible enough to explain many of the varied phenomena observed in the new layered cuprates, while relying on the same underlying mechanism as in more conventional high-T, materials, such as the Al5 compounds, the Bi-oxides, and perhaps The model requires a strong electron-phonon the heavy-fermion and organic superconductors. coupling, evidence for which was amply demonstrated at the recent March A.P.S. Meeting, and it predicts a strong intrinsic disorder involved with the vHs. The importance of such microscopic disorder has recently been stressed by a number of experimental groups (Sleight, 1990, Jorgensen, et al., 1991). This work was supported by the Department of Energy under subcontract from Intermagnetics General Corporation. Publication 480 from the Barnett Institute. REFERENCES J.D. Axe, A.H. Moudden, D. Hohlwein, D.E. Cox, K.M. Mohanty, A.R. Moodenbaugh, Xu, Phys. Rev. Lett. 62, 2751 (1989).
and Y.
C.A. Balseiro and L.M. Falicov, Phys. Rev. B20, 4457 (1979). A.P. Blandin, in “Phase Stability of Metals and Alloys”, ed. by P.S. Rudman, J. Stringer, and R.I. Jaffee, (McGraw-Hill, NY, 1967), p. 115. R.E. Cohen, H. Krakauer, and W.E. Pickett, to be published, (1991).
Miami Workshop
I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 93, 1487 (1987) [Sov. Phys.-JETP V.J. Emery, S.A. Kivelson, and II.&. Lin, Phys. Rev. Lett. 64,475
(1990).
on High-T,,
66, 848 (1987)J.
1369
The vanHovesingularity model J. Friedel, J. Phys. Condens. Matt. 1, 7757 (1989), and unpublished. J. Friedel, in “Electron-Phonon
Interactions and Phase Transitions”,
ed. by T. Riste (Plenum,
NY, 1977) p. 1. T. Fukase, T. Nomoto, T. Hanagari, T. Goto, and Y. Koike, Physica B165-66,
1289 (1990).
M. Grilli, B.G. Kotliar, and A.J. Millis, Phys. Rev. B42, 329 (1990). J. Hafner, “From Hamiltonians to Phase Diagrams”,
(Springer, Berlin, 1987), Ch. 7.2.
B.I. Halperin and T.M. Rice, in “Solid State Physics, Vol. 21”, ed. by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic, NY, 1968), p. 115. S.J. Hibble, A.K. Cheetham, A.M. Chippindale, (1988).
P. Day, and J.A. Hriljak, Physica C156,
604
J.E. Hirsch and D.J. Scalapino, Phys. Rev. Lett. 56, 2732 (1986). M.S. Hybertsen, M. Schluter, and N.E. Christensen, Phys. Rev. B39, 9028 (1989). J.D. Jorgensen, B. Dabrowski, S. Pei, D.G. Hinks, L. Sonderholm, E.L. Venturini, and D.S. Ginley, Phys. Rev. B38, 11337 (1988).
B. Morosin, J.E. Schirber,
J.D. Jorgensen, S. Pei, P. Lightfoot, B. Dabrowski, D.G. Hinks, and D.R. Richards, Third Int. Symp. on Supercond., Sendai, Japan, Nov., 1990, to be published (Springer, Tokyo, 1991). A.P. Kampf and J.R. Schrieffer, Phys. Rev. B42, 7967 (1990). J. Labbe and J. Bok, Europhys. Lett. 3, 1225 (1987). P.A. Lee, in “High Temperature Superconductivity”, ed. by K.S. Bedell, D. Coffey, D.E. Meltzer, D. Pines, and J.R. Schrieffer (Addison-Wesley, Redwood City, 1990), p. 96. P.A. Lee and N. Read, Phys. Rev. Lett. 58, 2691 (1987). K. Levin, J.H. Kim, J.P. Lu, and Q. Si, unpublished (1991). J.P. Lu, Q. Si, J.H. Kim, and K. Levin, Phys. Rev. Lett. 65, 2466 (1990). R.S. Markiewicz, in “High-Temperature Superconductors”, ed. by M.B. Brodsky, R.C. Dynes, K. Kitazawa, and H.L. Tuller (MRS, Pittsburgh, 1988), p. 411. R.S. Markiewicz, Phys. Rev. Lett. 62, 603 (1989a). R.S. Markiewicz, J. Phys. Condens. Matt. 1, 8911,8931 (1989b). R.S. Markiewicz, J. Phys. Condens. Matt. 2, 665 (1990a). R.S. Markiewicz, Physica Cl68,
195 (1990b).
R.S. Markiewicz, Physica C169, 63 (199Oc). R.S. Markiewicz, J. Phys. Condens. Matt. 2, 6223 (199Od). R.S. Markiewicz, to be published, Int. J. Mod. Phys. B (199Ia). R.S. Markiewicz, to be published, Physica C (1991b). R.S. Markiewicz, to be published, Miami Workshop on High-T,, (1991c). R.S. Markiewicz, to be published, Physica C (1991d). R.S. Markiewicz, to be published, J. Phys. Condens. Matt., Letter (1991e). R.S. Markiewicz and B.C. Giessen, Physica ClSO, 497 (1989). R.M. Martin, S. Bacci, and E. Gagliano, to be published, Miami Workshop on High-T’, (199 1). L.F. Mattheis, Phys. Rev. Lett. 58, 1028 (1987). A.J. Millis, H. Monien, and D. Pines, Phys. Rev. B42, 167 (1990). E. L. Nagaev, “Physics of Magnetic Semiconductors”,
(Mir, Moscow, 1983).
D.M. Newns, P.C. Pattnaik, and C.C. Tsuei, Phys. Rev. B43, 3075 (1991). D.P. Norton, D.H. Lowndes, B.C. Sales, J.D. Budai, B.C. Chakoumakos, Phys. Rev. Lett. 66, 1537 (1991). J.C. Phillips, Phys. Rev. Lett. 59, 1856 (1987).
and H.R. Kerchner,
R. S. MARKIEWICZ
1370
J.C. Phillips, Phys. Rev. Lett. 64, 1605 (1990). W.E. Pickett, R.E. Cohen, and H. Krakauer, Phys. Rev. B42, 8764 (1990). T.M. Rice and G.K. Scott, Phys. Rev. Lett. 35, 120 (1975). C.A.R. Sa de Melo and S. Doniach, Phys. Rev. B41, 6633 (1990). H.J. Schulz, Phys. Rev. Lett. 64, 1445 (1990). Q, Si and K. Levin, Bull. A.P.S. 36, 516 (1991) and unpublished. A.W. Sleight, quoted by D.P. Hamilton, Science 250, 375 (1990). Y. Song, A. Misra, P.P. Crooker, and J.R. Gaines, Phys. Rev. Lett. 66, 825 (1991). B.H. Toby, T. Egami, C. Janot, M. de Boissieu, and M.A. Subramanian, Bull. A.P.S. 36, 990 (1991). S.A. Trugman, Phys. Scripta T27, 113 (1989). C.C. Tsuei, Physica A168, 238 (1990). C.C. Tsuei, D.M. Newns, C.C. Chi, and P.C. Pattnaik, Phys. Rev. Lett. 65, 2724 (1990). R.B. van Dover, R.J. Cava, B. Batlogg, and E.A. Rietman, Phys. Rev. B35, 5337 (1987). C.M. Varma, P.B. Littlewood, Lett. 63, 1996 (1989).
S. Schmitt-Rink,
E. Abrahams, and A.E. Ruckenstein, Phys. Rev.
P.B. Visscher, Phys. Rev. BlO, 943 (1974). J.H. Xu, T.J. Watson-Yang,
J. Yu, and A.J. Freeman, Phys. Lett. 120A,
489 (1987).
J. Zaanen and 0. Gunnarsson, Phys. Rev. B40, 7391 (1989). Z. Zhang, Y.L. Wang, X.L. Wu, J.-L. Huang, and C.M. Lieber, Phys. Rev. B42, 1082 (1990).
Chrm.
Solids
Vol.
52.
No.
11112.
pp.
1363-1370.
1991
0022.3697/91
Printed in Great Britain.
$3.00 + 0.00 Pergamon Press plc
THEVANHOVESINGULARITY AND HIGH-T, SUPERCONDUCTIVITY: THEROLEOFNANOSCALEDISORDER ANDINTERLAYERCOUPLING R.S. Markiewicz Physics Department and Barn&t
Abstract
Institute, Northeastern University, Boston, MA 02115 USA
- The van Hove singularity (vHs) model provides a unifying picture for understanding many
anomalous features of high-temperature
superconductors.
calculated Fermi levels in most high-Tc
cuprates.
The vHs is found to occur very close to the
The large doa peak (the logarithmic
singularity is
only modestly broadened by interlayer coupling) provides a simple explanation of enhanced superconductivity,
but also leads to striking intrinsic disorder, both as the material is doped away from the
vHs (potentially
Keywords:
nanoscale phase separation),
superconductivity,
and at the vHs (short-range
charge density wave, phase separation,
density wave order).
interlayer. Fermi surface
INTRODUCTION All two-dimensional band structures have a saddle-point van-Hove singularity (vHs), at which the density-of-states (dos) diverges logarithmically. Prior to the discovery of high-temperature superconductivity, a vHs model was introduced to study the competition between superconductivity and charge-density wave (CDW) formation (Balseiro and Falicov, 1979). If CDW formation could be suppressed, Hirsch and Scalapino (1986) showed that the dos peak provided a simple means of attaining high Tc’s , and a vHs model was very soon applied to the high-T, cuprates (Labbe and Bok, 1987, Dzyaloshinskii, 1987). The earlier theories (Labbe and Bok, 1987, Dzyaloshinskii, 1987, Friedel, 1989) used a simple tight-binding model, in which the vHs falls exactly at half-filling of the band; this leads to complications associated with the Mott-like insulating state found at half-filling. More recent theories (Markiewicz, 1988-1991e, Tsuei, 1990, Tsuei, et al. 1990, Newns, et al. 1991, Levin, et al. 1991, Si and Levin, 1991, Toby, et al., 1991) assume that the vHs occurs away from half-filling, as found in band-structure calculations, and corresponds to the hole doping of optimum T,. INTERLAYER COUPLING Because the BCS equation averages the dos over an interval of 4kr~T,, the vHs singularity can be smeared out over the same interval without significantly degrading T,. Thus, pairbreaking due to real phonon scattering is unlikely to produce significant T, reduction, since the electronphonon scattering rate deduced from dc conductivity is TV&’ N 2k~T. Thus, the factor most harmful to high Te is interlayer coupling, which broadens the vHs dos peak over an energy interval 4t,, where t, is the interlayer hopping parameter. It can be shown (Markiewicz, 1991b) that as long as t, < T,, there is no T, reduction, although there is a substantial reduction for larger t, values. Moreover, if the hopping involves interband interlayer hopping, the effective value of t, involved in broadening the dos is considerably reduced from the full t, value. For example, in BizSrzCaCuzOs (Bi-2212), interlayer hopping may involve hopping from a CuOz band on one layer to a BiO band on the next. In this case, the effective value t: involved in broadening the vHs peak is 2: N t i/AE, where AE is the separation of the bare CuOz and BiO bands at the vHs. This is illustrated in Fig. la (Markiewicz, 1991b): The single vHs peak associated with a CuOz plane is split by coupling with a BiO plane into three components. Since a value t, = 1OOmeV was chosen, the splitting of the dos peaks is of order SOOmeV, but the 1363
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broadening of the individual peaks is only of order t: N 10meV. Fig. lb illustrates another interesting aspect of interlayer coupling: while the dos peak C is derived from the CuOz plane vHs, the interlayer coupling has shifted the peak from the a-point of the Brillouin zone to a point along the T - it? line.
Fig. 1 (a) Normalized density of states for a simplified model of Bi-2212, showing the splitting and broadening of the vHs peak produced by interlayer coupling. (b) Fermi surface of model Bi-2212, near the energy of feature C in the dos. Solid lines show Fermi surface at k, = 0, dashed lines at k, = 7r/2c, with c the c-axis lattice constant. Letter C denotes the (incipient) topological change which gives rise to feature C in Fig. la. [After Markiewicz, 1991b] LOCATION OF THE vHs It has long been known that the vHs falls at the hole doping of optimum T, in local density approximation calculations for La2 _,Sr,CuOs (LSCO) (Mattheis, 1987, Xu, et al. 1987). For most other high-T, cuprates and bismuthates, the vHs also falls close to the calculated Fermi level (Markiewicz, 1991a). A conspicuous exception had been YBazCusOr_6 (YBCO), for which early band structure calculations did not place the vHs at the Fermi level of the stoichiometric material (5 = 0). However, the CuOz plane hole concentration can be systematically varied by doping or by deviations from oxygen stoichiometry (6 > 0). If this is accounted for by rigidly shifting the CuOz band with respect to the Fermi level, it is found that the vHs falls at the Fermi level at the hole doping which experimentally corresponds to optimum T, (Markiewicz, 1991a). Moreover, the most recent band structure calculations (Pickett, et al., 1990) do place the vHs at the Fermi level of the 07 compound, as illustrated in Fig. 2. The solid lines are CuOzplane like bands, the dashed line is a CuO chain band, which crosses a CuO:! plane band and strongly hybridizes with it near the X-point. The arrow points to the X-point vHs, which falls precisely at the Fermi level, near the center of the X-U-line, just as in LSCO. The feature along the r - Y line, labelled with an asterisk is also generated by plane-chain interaction, and is strikingly reminiscent of the vHs feature of Fig. lb. It should be noted that the band structure calculations predict a c-axis dispersion, due to interlayer coupling, of m 150meV, which would wash out the sharp dos peaks. However, interlayer coupling should be very sensitive to electron correlation effects and to strong electron-phonon coupling, both of which can act to sharpen the dos peak.
I365
The van Hove singularity model
Y
r Z
S
R
xv
r,t
Fig. 2 Fermi surface of YBCO, from Pickett, et al., 1990. Solid lines = predominantly CuOa planelike sections; dashed lines = predominantly chain-like sections. Arrows indicate position of the vHs, along the X-U-line. Asterisk indicates possible second vHs feature; compare point C of Fig. lb. INTRINSIC DISORDER At half filling of the CuOa plane band, LSCO is an antiferromagnetic insulator. This can be explained as a Mott transition (more appropriately, a transition to a charge-transfer insulator), and the transition can be described within the slave boson formulation (Sa de Melo and Doniach, 1990, Grilli, et al., 1990, Markiewicz, 1990a), using realistic band parameters (Hybertsen, et al., 1989, Martin, et al., 1991). Doping away from half filling leads to an instability against phase separation (Nagaev, 1974, Visscher, 1974, Zaanen and Gunnarson, 1989, Trugman, 1989, Emery, et al., 1990, Schulz, 1990, Markiewicz, 1990a). This transition can be understood as an intrinsic feature of the Mott transition. Mott’s picture is that the transition occurs at exactly half filling, to avoid the strong on-site Coulomb repulsion of two holes, independently of the band structure. In this case, many different normal states would converge to the same antiferromagnetic insulator at half filling, which is only possible if the transition is first order. In the presence of such a transition, it is impossible to infer the nature of the doped phase from the properties of the antiferromagnetic phase at half filling. At best, a Hubbard or tJ model may correctly describe the initial phase separation, when second phase islands contain only two or three holes. In the particular case of the cuprates, this second phase should fall almost exactly at the vHs (Markiewicz, 1990a). This arises from a strong electron-phonon coupling, which is also driven by the large dos associated with the vHs (incipient CDW formation, discussed further below). The sharp dos peak produces a giant Kohn anomaly, with a free energy minimum at the vHs concentration, which leads to a miscibility gap between the Mott insulating phase and the vHs phase, and a second phase separation when the material is overdoped beyond the concentration corresponding to the vHs. This electronically-driven instability is closely related to the phase separations in Hume-Rothery alloys (Blandin, 1967, Friedel, 1977, Hafner, 1987). Since the phase transition is electronically driven, it is possible that the holes could phase separate by themselves, without accompanying ionic motion. Since the resulting domains would be electrically charged, Coulomb effects preclude an ordinary macroscopic phase separation, but can still allow a state of ‘mesoscopic’ (more accurately, nanoscopic) disorder. In this state, a single domain may contain only a few particles - indeed, for light doping, the stable phase may correspond to real-space pairing of holes. In many cases, this pair-phase is found to be met&able with respect to a phase in which the dense holes form ‘grain-boundaries’ between antiferromagnetic domains (Zaanen and Gunnarson, 1989, Schulz, 1990, Markiewicz, 199Oa). In three-dimensions, there is strong pinning of these phases to impurities, thereby obscuring their direct observation. In the layered cuprates, the CuOa-planes are modulation doped, so free
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hole-domains may be expected to form. In this case, the phase diagram (Markiewicz, 1990a, 1991c) should be characterized by three transitions as a function of doping (Fig. 3): first, from an antiferromagnetic insulating phase to a spin-glass-like phase, with antiferromagnetic domains surrounded by domain walls of the higher-hole-density phase; secondly, a metal-insulator transition from this phase to a metallic phase, with intrinsic weak links (corresponding to domain walls of the antiferromagnetic phase); and finally to a uniform phase at or near to the vHs. Whereas high-T, superconductivity is associated with this latter phase, it persists well into the two-phase regime.
AFM d
Spin
Glass
Metal x----+
t vHs
Fig. 3 Schematic free energy (AF) d ia g ram for LSCO, showing two-phase coexistence regime for intermediate doping between the Mott insulator (x=0) and the vHs (x-0.17). Solid line = uniform phase; short dashed line = islands of antiferromagnetic (AFM) insulator; longdashed lines = islands of metallic phase at vHs. Lowest energy phases are labelled on horizontal axis [from Markiewicz, 1991c]. There is considerable experimental evidence for this scenario (Markiewicz, 1991a). Perhaps the clearest example arises in LSCO, which can be hole-doped either by substituting Sr for La, or by incorporating interstitial 0. Since the Sr is essentially immobile at low temperatures, phase separation is on the nanoscopic scale discussed above. On the other hand, the interstitial oxygen is highly mobile at low temperatures, so the electronic instability leads to macroscopic phase separation (Jorgensen, et al., 1988). There is considerable evidence for a nanoscopic phase separation in LSCO and YBCO, but most of it is indirect (Phillips, 1990, Markiewicz, 1991a). For instance, the Meissner fraction is maximized at the composition of highest Tc, and falls smoothly to zero with variation of hole content (van Dover, et al, 1987). Recently, Song, et al. (1991) interpreted l/f-noise measurements in YBCO as evidence for the presence of phase separation on a N 10 nm scale. Bi-2212 provides a quite different example of electronically driven transitions. In this compound, substitution of Pb for Bi would be expected to change the average valence of the Bi(Pb)Olayer, and hence the average number of holes in the CuOz planes. However, this is not what is observed to happen (Zhang, et al., 1990). For Pb substitution of up to 35% of the Bi, T, is found to be essentially unchanged. Instead, there is a large increase of the b-axis superlattice period associated with the BiO layers. This is interpreted as follows: the superlattice is accompanied by the incorporation of excess oxygen, which provides a hole doping of the CuOs planes. When Pb is substituted for Bi, the superlattice period increases (less 0 is incorporated) in such a way that the net doping of the CuO2 planes remains unchanged. This is a remarkable result: the material ‘knows’, at the temperature it is prepared, the correct hole-doping required to produce the optimum T,. This result is readily understood in terms of the vHs model - doping at the
The van Hove singularity model
1367
vHs minimizes the free energy. A similar phenomenon has been reported in the Tl-based superconductors (Hibble, et al., 1988). In stoichiometric single-T1 layer superconductors (e.g., TlBazCazCusOs - Tl-1223(g)), the average Cu valence is 2.33, whereas in the double-T1 layer compounds (e.g., Tl-2223(10)) the Cu valence is only 2. It was found that the double-T1 layer compounds contain cation vacancies (Tl, Ca deficits), which are absent in the single-T1 layer materials, in such a way that the average CU valence is in the range 2.3-2.6 for both families of compounds. Thus, the disorder can take on a large variety of forms, but is always associated with a preferred doping level in the CuOz planes. Something similar occurs in the YBCO superconductors, and has been used to make a yttrium-free Pr-based superconductor, Pro.sCao.sBazCusOr-a (Norton, et al., 1991). Substituting Pr for Y depresses T,, ultimately destroying superconductivity when more than 50% of the Y is replaced. The T, reduction is believed to have two causes: Pr acts both as a pair-breaker and as a hole-filler. Norton, et al. showed that the hole-filling action could be reversed by substituting Ca for Pr, and that the optimum T, corresponded to the same average (Pr-Ca) valence as in YBCO - i.e., the same hole-doping of the CuOz planes. (The fact that T, is not as large as in YBCO is presumably due to Pr pairbreaking.) These phase separations are similar to those predicted by Phillips’ quantum percolation theory (Phillips, 1987, 1990), but the microscopic origins are totally different. The present phase separation is electronically driven, and hence (a) associated with a preferred hole density in the Cu02 planes and (b) related to the phase separation associated with the Mott transition. In Phillips case, the phase separation is driven by the impurities, and hence should be more sensitive to states off of the CuOz planes. Furthermore, Phillips is applying a theory of strong localization (metal-insulator transition) to a situation in which the intrinsic localization effects would appear to be quite weak. The equivalent sheet resistance of a single layer of YBCO is about 800R at T_90K, and most of this is inelastic scattering. Extrapolating the normal state resistance to T=O suggests that the elastic sheet resistance is 5 lOOs1. For such small sheet resistance, localization effects would show up as a weak logarithmic correction to the resistance, measurable only below N 5K. Even a weak interlayer coupling, as discussed above, would be sufficient to wash out this localization correction: in graphite intercalation compounds, no such corrections are observed unless strong disorder is intentionally introduced, even though the interlayer couplings can be 100x weaker than in Bi-2212. Hence, localization effects due to twodimensionality in these materials should be negligible; in contrast, the strong scattering near the vHs produces a different form of quasi-localization, discussed below. CDW AND SDW FLUCTUATIONS The strong electron-phonon coupling which leads to phase separation away from the optimum doping concentration also has profound effects on material doped at the correct vHs composition - namely, strong fluctuations of short-range CDW or SDW (spin density wave) order. These arise from a special form of nesting associated with the vHs (Rice and Scott, 1975). A CDW or SDW is stabilized because the transition opens a gap at the Fermi level, lowering the average energy of the occupied electronic states. Since the vHs is associated with a large dos, this stabilization can be achieved by any transition that drives the vHs below the Fermi level. Thus, in LSCO, there are two vHs, associated with the X and Y points of the Brillouin zone, so any density wave oscillation which couples them (i.e., has a component of symmetry (z/a, z/o)) can open up the appropriate gap. Indeed, such a phase transition to long-range CDW order has been observed in the closely related compound Laz_,Ba,CuOd (LBCO) (Axe, et al., 1989). The transition splits the two vHs (Markiewicz, 1990d, Cohen, et al., 1991), and severely degrades T,, essentially to T=O. Significantly, the transition is most complete at the hole concentration which would otherwise have had the highest T, (i.e., the Fermi level coincides with the vHs). In LSCO, strong fluctuations to this phase exist, but long-range CDW order is not observed above T,. There is a small T, depression, and the onset of long-range order below T, (Fukase, et al., 1990), both effects occuring at nearly the same hole-doping as in LBCO. Similar competition between CDW order and superconductivity is found in the Al5 compounds. Analogous phenomena should arise in the spin susceptibility, which also has logarithmically diverging peaks at the same q-vector, (z/u, z/u). Indeed, this form of spin susceptibility has
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been introduced phenomenologically to account for the anomalous nuclear relaxation rates in YBCO (Millis, et al., 1990, Lu, et al., 1990). The cutoff parameter T, 11 115K introduced by Millis, et al., can be interpreted as the interlayer coupling t, which cuts off the susceptibility divergence. The strong fluctuations toward short-range density wave order can explain a number of transport anomalies associated with the cuprates, having to do with the discrepancy between the small carrier density revealed by Hall effect measurements (proportional to the doping, x) versus the much larger density expected from the large Fermi surface sections found theoretically and experimentally (proportional to 1 &z). In the presence of long-range density wave order, the large sections of Fermi surface would be nested (h la Rice and Scott), leaving only small pockets of holes of the requisite area oc x. In the presence of short-range density-wave order, carriers near the vHs would be quasi-localized, and hence not contribute to transport measurements. Transport would be dominated by carriers from other regions of the Fermi surface, and can be thought of as ‘shadow Fermi surfaces’ of the hole pockets (Markiewicz, 1988, Lee, 1990, Kampf and Schrieffer, 1990). These shadow Fermi surfaces should be observable by, e.g., angle resolved photoemission (Kampf and Schrieffer, 1990) or the de Haas-van Alphen effect (Markiewicz, 1991d). The near instability toward charge or spin density wave formation can be looked on as a form of excitonic instability (Halperin and Rice, 1968, Markiewicz, 1990b, 1991e), wherein there is a net Coulombic attraction between holes at one vHs and electrons at a second. The polarizability of these excitons may be identified with the ‘phenomenological polarizability’ of Varma, et al. (1989). The fact that a vHs can explain the anomalous electron-electron scattering rate, re;l cx T,w, had been pointed out earlier (Lee and Read, 1987, Markiewicz, 1989a). CONCLUSIONS The paradox of the vHs model is that, whereas it provides a transparently simple means of enhancing T, (Hirsch and Scalapino, 1986, Labbe and Bok, 1987, Dzyaloshinskii, 1987) and explaining the small isotope effect (Tsuei et al. 1990), a closer examination suggests that the underlying physics is immensely richer and more complicated. The actual superconducting transition may involve a two stage process, wherein strong electron-phonon coupling produces short-range CDW order (?bipolaron formation), and these electronic fluctuations drive a superconducting transition of the shadow Fermi pockets (Markiewicz, 199Oc, Toby, et al., 1991). In conclusion, the vHs model seems flexible enough to explain many of the varied phenomena observed in the new layered cuprates, while relying on the same underlying mechanism as in more conventional high-T, materials, such as the Al5 compounds, the Bi-oxides, and perhaps The model requires a strong electron-phonon the heavy-fermion and organic superconductors. coupling, evidence for which was amply demonstrated at the recent March A.P.S. Meeting, and it predicts a strong intrinsic disorder involved with the vHs. The importance of such microscopic disorder has recently been stressed by a number of experimental groups (Sleight, 1990, Jorgensen, et al., 1991). This work was supported by the Department of Energy under subcontract from Intermagnetics General Corporation. Publication 480 from the Barnett Institute. REFERENCES J.D. Axe, A.H. Moudden, D. Hohlwein, D.E. Cox, K.M. Mohanty, A.R. Moodenbaugh, Xu, Phys. Rev. Lett. 62, 2751 (1989).
and Y.
C.A. Balseiro and L.M. Falicov, Phys. Rev. B20, 4457 (1979). A.P. Blandin, in “Phase Stability of Metals and Alloys”, ed. by P.S. Rudman, J. Stringer, and R.I. Jaffee, (McGraw-Hill, NY, 1967), p. 115. R.E. Cohen, H. Krakauer, and W.E. Pickett, to be published, (1991).
Miami Workshop
I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 93, 1487 (1987) [Sov. Phys.-JETP V.J. Emery, S.A. Kivelson, and II.&. Lin, Phys. Rev. Lett. 64,475
(1990).
on High-T,,
66, 848 (1987)J.
1369
The vanHovesingularity model J. Friedel, J. Phys. Condens. Matt. 1, 7757 (1989), and unpublished. J. Friedel, in “Electron-Phonon
Interactions and Phase Transitions”,
ed. by T. Riste (Plenum,
NY, 1977) p. 1. T. Fukase, T. Nomoto, T. Hanagari, T. Goto, and Y. Koike, Physica B165-66,
1289 (1990).
M. Grilli, B.G. Kotliar, and A.J. Millis, Phys. Rev. B42, 329 (1990). J. Hafner, “From Hamiltonians to Phase Diagrams”,
(Springer, Berlin, 1987), Ch. 7.2.
B.I. Halperin and T.M. Rice, in “Solid State Physics, Vol. 21”, ed. by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic, NY, 1968), p. 115. S.J. Hibble, A.K. Cheetham, A.M. Chippindale, (1988).
P. Day, and J.A. Hriljak, Physica C156,
604
J.E. Hirsch and D.J. Scalapino, Phys. Rev. Lett. 56, 2732 (1986). M.S. Hybertsen, M. Schluter, and N.E. Christensen, Phys. Rev. B39, 9028 (1989). J.D. Jorgensen, B. Dabrowski, S. Pei, D.G. Hinks, L. Sonderholm, E.L. Venturini, and D.S. Ginley, Phys. Rev. B38, 11337 (1988).
B. Morosin, J.E. Schirber,
J.D. Jorgensen, S. Pei, P. Lightfoot, B. Dabrowski, D.G. Hinks, and D.R. Richards, Third Int. Symp. on Supercond., Sendai, Japan, Nov., 1990, to be published (Springer, Tokyo, 1991). A.P. Kampf and J.R. Schrieffer, Phys. Rev. B42, 7967 (1990). J. Labbe and J. Bok, Europhys. Lett. 3, 1225 (1987). P.A. Lee, in “High Temperature Superconductivity”, ed. by K.S. Bedell, D. Coffey, D.E. Meltzer, D. Pines, and J.R. Schrieffer (Addison-Wesley, Redwood City, 1990), p. 96. P.A. Lee and N. Read, Phys. Rev. Lett. 58, 2691 (1987). K. Levin, J.H. Kim, J.P. Lu, and Q. Si, unpublished (1991). J.P. Lu, Q. Si, J.H. Kim, and K. Levin, Phys. Rev. Lett. 65, 2466 (1990). R.S. Markiewicz, in “High-Temperature Superconductors”, ed. by M.B. Brodsky, R.C. Dynes, K. Kitazawa, and H.L. Tuller (MRS, Pittsburgh, 1988), p. 411. R.S. Markiewicz, Phys. Rev. Lett. 62, 603 (1989a). R.S. Markiewicz, J. Phys. Condens. Matt. 1, 8911,8931 (1989b). R.S. Markiewicz, J. Phys. Condens. Matt. 2, 665 (1990a). R.S. Markiewicz, Physica Cl68,
195 (1990b).
R.S. Markiewicz, Physica C169, 63 (199Oc). R.S. Markiewicz, J. Phys. Condens. Matt. 2, 6223 (199Od). R.S. Markiewicz, to be published, Int. J. Mod. Phys. B (199Ia). R.S. Markiewicz, to be published, Physica C (1991b). R.S. Markiewicz, to be published, Miami Workshop on High-T,, (1991c). R.S. Markiewicz, to be published, Physica C (1991d). R.S. Markiewicz, to be published, J. Phys. Condens. Matt., Letter (1991e). R.S. Markiewicz and B.C. Giessen, Physica ClSO, 497 (1989). R.M. Martin, S. Bacci, and E. Gagliano, to be published, Miami Workshop on High-T’, (199 1). L.F. Mattheis, Phys. Rev. Lett. 58, 1028 (1987). A.J. Millis, H. Monien, and D. Pines, Phys. Rev. B42, 167 (1990). E. L. Nagaev, “Physics of Magnetic Semiconductors”,
(Mir, Moscow, 1983).
D.M. Newns, P.C. Pattnaik, and C.C. Tsuei, Phys. Rev. B43, 3075 (1991). D.P. Norton, D.H. Lowndes, B.C. Sales, J.D. Budai, B.C. Chakoumakos, Phys. Rev. Lett. 66, 1537 (1991). J.C. Phillips, Phys. Rev. Lett. 59, 1856 (1987).
and H.R. Kerchner,
R. S. MARKIEWICZ
1370
J.C. Phillips, Phys. Rev. Lett. 64, 1605 (1990). W.E. Pickett, R.E. Cohen, and H. Krakauer, Phys. Rev. B42, 8764 (1990). T.M. Rice and G.K. Scott, Phys. Rev. Lett. 35, 120 (1975). C.A.R. Sa de Melo and S. Doniach, Phys. Rev. B41, 6633 (1990). H.J. Schulz, Phys. Rev. Lett. 64, 1445 (1990). Q, Si and K. Levin, Bull. A.P.S. 36, 516 (1991) and unpublished. A.W. Sleight, quoted by D.P. Hamilton, Science 250, 375 (1990). Y. Song, A. Misra, P.P. Crooker, and J.R. Gaines, Phys. Rev. Lett. 66, 825 (1991). B.H. Toby, T. Egami, C. Janot, M. de Boissieu, and M.A. Subramanian, Bull. A.P.S. 36, 990 (1991). S.A. Trugman, Phys. Scripta T27, 113 (1989). C.C. Tsuei, Physica A168, 238 (1990). C.C. Tsuei, D.M. Newns, C.C. Chi, and P.C. Pattnaik, Phys. Rev. Lett. 65, 2724 (1990). R.B. van Dover, R.J. Cava, B. Batlogg, and E.A. Rietman, Phys. Rev. B35, 5337 (1987). C.M. Varma, P.B. Littlewood, Lett. 63, 1996 (1989).
S. Schmitt-Rink,
E. Abrahams, and A.E. Ruckenstein, Phys. Rev.
P.B. Visscher, Phys. Rev. BlO, 943 (1974). J.H. Xu, T.J. Watson-Yang,
J. Yu, and A.J. Freeman, Phys. Lett. 120A,
489 (1987).
J. Zaanen and 0. Gunnarsson, Phys. Rev. B40, 7391 (1989). Z. Zhang, Y.L. Wang, X.L. Wu, J.-L. Huang, and C.M. Lieber, Phys. Rev. B42, 1082 (1990).