Spin fluctuations and high temperature superconductivity in the antiferromagnetically correlated oxides: YBa2Cu3O7; YBa2Cu3O6.63; La1.85Sr0.15CuO4

Spin fluctuations and high temperature superconductivity in the antiferromagnetically correlated oxides: YBa2Cu3O7; YBa2Cu3O6.63; La1.85Sr0.15CuO4

HIYSICA Physiea C 185-189 (1991) 120-129 North-Holland Spin Fluctuations and High Temperature Superconductivity in the Antiferromagnetically Correla...

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Physiea C 185-189 (1991) 120-129 North-Holland

Spin Fluctuations and High Temperature Superconductivity in the Antiferromagnetically Correlated Oxides: YBa2Cu307; YBa2Cu306.63; Lal.85Sr0.15CuO4 David Pines Physics Department, University of Illinois/Urbana-Champaign, 1110 W. Green Street, Urbana, Illinois 61801 The evidence from NMR experiments for the existence of strong temperature dependent antiferromagnetie correlations among nearly localized Cu2+ spins in the cuprate oxide planes is reviewed. It leads naturally to a description of the normal state as a nearly antiferromagnetie Fermi liquid in which the magnetic excitations are commensurate, or very nearly so, antiferromagnetic paramagnons whose spectr~,m at low frequencies can be obtained from fits to NMR experiments, and whose coupling to quasiparticles on a 2D square lattice is responsible for the measured anomalous transport and optical properties. Recent work by Monthoux, Balatsky and Pines demonstrates that the retarded quasiparticle interaction induced by these paramagnons can lead to a superconducting state with dx2_ y2 symmetry, a high transition temperature, and energy gap behavior comparable to those measured. NMR experiments on the superconducting state are shown to support d-wave pairing, while existing penetration depth experiments do not appear to provide conclusive evidence concerning the pairing state. 1. INTRODUCTION In this talk I describe a ~bottom-up~

first principles. ''(2), there are a large number of

experimentally based approach to understanding

novel properties from which to choose. Of these,

the cuprate oxide superconductors, in which a

the low frequency magnetic properties measured

model Hamiltonian for microscopic calculations

in NMR experiments are perhaps the most

is constructed and constrained from a

unusual, so I begin with these. Experiment(3)(4)

phenomenological description of the normal state.

shows that a single, nearly localized, Cu 2+ spin

This was the approach which John Bardeen(1)

component is responsible for both the measured

used in the course of developing the microscopic

Knight shifts and spin-lattice relaxation times. A

theory of superconductivity with Leon Cooper and

phenomenological theory makes evident the

Bob Schrieffer, and in opting for this approach, I

presence of strong temperature-dependent

have been s, ~ ea~,y ^*' - im~uenced " ~--u~ n-lylong and close

¢:LILt, LL~.,AZVLJLI¢.Z~AA~.,~,.LI..,

association with John.

spins which lead to a substantial peaking of the

Almost every aspect of the normal state


a~*..,L, TVV,.,'.~.AI



low frequency spin-spin correlation function, ~"

behavior of the cuprate oxides is anomalous, so

(q,¢o), for wavevectors near (~Ja, rJa) and a very low

that if one is to follow Richard Feynman's dictum,

characteristic energy, scale.(5) These results led

"It does not make any difference what we explain,

me to suggest that the normal state is a nearly

as long as we explain some property correctly from

antiferromagnetic FelTni liquid of coupled

Elsevier Science Publishers B.V.

D. Pines / Spin fluctuations and high temperature superconductivity quasiparticles and antiferromagnetic paramagnons, and that the spin fluctuation induced quasiparticle interaction might be the physical origin of the transition to the superconducting state.(6) In Sec. 2 I discuss NMR experiments and in


temperature dependence, one finds instead 17W(T) ~ C2a Xo (T) T F (1) where F is a magnetic Fermi energy (which may be weakly T dependent) and Ca, the transferred hyperfine coupling between the oxygen nucleus

Sec. 3 give a brief progress report on the properties

and a Cu 2+ spin localized at the nearest neighbor

of the two dimensional nearly antiferromagnetic

Cu sites, can be determined from Knight shift

Fermi liquid. Sec. 4 is devoted to a report on recent work with Philippe Monthoux and Alexander Balatsky (MBP)(7), which demonstrates that the retarded quasiparticle interaction induced by paramagnons leads uniquely to a superconducting state with dx2_y2 symmetry. With a quasiparticle paramagnon

m e a s u r e m e n t s . (9) This unusual behavior, was anticipated by Millis, Monien, and Pines.(5) •

The 63Cua spin-lattice relaxation rate, on the

other hand, is greatly enhanced over what might be expected from the hyperfine coupling constant determined by the measured Knight shifl~, and displays markedly non-Korringa behavior. It may

state transport and optical properties, high

be expressed in the form 63Wa (T) (A~ff)2Xo (T) T ~ ~'SF (T)

transition temperatures and energy gap behavior

where ACeff,a material constant independent of

comparable to those measured are obtained (7). In

temperature and hole content, takes into account

Section 5 I discuss the extent to which present

both the anisotropic direct hyperfine coupling, Aa,

NMR and penetration depth experiment support

of an on-site Cu 2+ spin, and the nearest neighbor

the calculated d-wave pairing.

transferred hyperfine coupling described by the


Mila-Rice Hamiltonian.(lo) In this notation, the

coupling consistent with the measured normal

NORMAL STATE The key results of NMR measurements on the normal state3,4, 8 may be summarized in the


low energy spin fluctuation energy, ~SF (T), is F ~SF (T) = RAF (T) (3) where RAF (T) is the temperature dependent

following way: antiferromagnetic enhancement of the 63Cu • For YBa2Cu307, where the uniform spin relaxation rate. From fits to experiment, one susceptibility Xo is temperature independent, th~ finds at 100I~ (RAF = 8, e~SF = 14meV) for 170 relaxation rate, 17W(T), displays Korringa-like YBa2Cu307; (RAF = 23, ~SF = 5.6meV) for temperature behavior (17W(T) ~ T), but for other 1YBa2Cu306.63; and RAF = 40, ~SF = 3.2meV) for 2-3 oxygen concentrations, and for the 2-1-4 system, where Xo displays considerable

La 1.85Sr0.15Cu04.

D. P/nesI Sp/n J~ctuat/o~ and h/gh temperature supe~com/ucari0,


* The temperature dependence of ~0SF(T) may be

second represents the contribution from the

obtained directly from 63Cu relaxation rate and

antiferromagnetic paramagnons of energy

Knight shift measurements, since from Eq. (2), 63T1 (T) 63KS(T) T - ~'SF (T).


Quite generally,although neither 63T1 CI0 or 63Ks (T) [the spin part of the Knight shift,for which one could equally well substitute 17Ks (T)] display a simple variation with temperature, their product with temperature does; above -125 K it takes the Curie-Weiss form, ~OSF(T) = a + bT.

~SF(T) = ~

~2 ;


is the a f correlation length which measures the range of the ay build-up of the static susceptibility at Q=(rJa, ~/a), ZQ (T) = Xo (T) (~2/~2o). In the MMP fits to experiments on YBa2Cu307, (5) YBa2Cu306.63,(11) and Lal.85Sro.15Cu04, (12) ~o is taken to be temperature independent and .- 0.56a, while


The presence of a m a x i m u m in a plot of [63W(T)fP]

~2(T) const a s = c+dT


for, say, YBa2Cu306.63 thus reflects a temperature

is responsible for the measured temperature

region in which Xo (T) varies more rapidly with

dependence of the antiferromagnetic correlations

temperature than does the comparatively

(and the low characteristic spin-fluctuation

smoothly varying ~SF (T).

energies, O)SF [T] or COSF [T]). If,on the other

A quantitative fit to the NMR experiments m a y

hand, one takes (~Ja)- I and independent of

be obtained by combining the Mila-Rice

temperature, as Rossat-Mignot has proposed from

Hamiltonian with the MMP ansatz for the

his fitsto neutron scattering measurements, 13

imaginary part of the spin-spin correlation

then ~o must be both temperature dependent and

function. One obtains a W(T) lira Y ah2(q) X" (q,co) T = co-~o q co

quite small compared to the latticespacing, a, in (6)

order to fit the N M R experiments on 63W(T). One can fit,63Wc(T), equally well with either choice of

where c~ refers to the nuclear site in question, parameters and, as shown by Millis and ah2(q) is the corresponding form factor (which includes the appropriate hypeffine coupling

Monien,(14) it is likelythat for YBa2Cu307 one can also arrive at reasonable fitsfor 17W(T), 89W(T),

constants), and X" (q,co)= x Xo (T) co )Co(T) (~2~o2) (c4COSF) [1+ (Q_q)2 ~212+ co2/co2SF F (7) The first term on the rhs of Eq. (7) is a "local moment" version of the usual Fermi liquid term; F, the magnetic Fermi energy, is - 0.4 eV and is very weakly dependent on temperature. The

and the 63Cu N M R relaxation rate for a field applied in the a-b plane, 63Wa(T). Matters appear otherwise for attempts to fitthe latterquantities for YBa2Cu306.63, or La1.85Sro.15CuO4, where the measured very much larger antiferromagnetic enhancement, RAF (T), makes it difficultto avoid "leakage" of the non-Korringa behavior of 63Wc(T)

D. Pines / Spin fluctuations and high temperature ~q,ercondu~vitv


on to 17W(T), which, in fact, maintains the "quasi-

explore the influence of these excitationson the

Korringa" local m o m e n t behavior represented by

transport and opticalproperties of the normal

Eq. (1) for both materials.(9).(ls)

state we m a y followAnderson(Is)and use a one-

A similar problem exists with the spin-spin

band description of the planar excitations of a

correlation function obtained from the analysis by

2-D square lattice which are assumed to satisfy

Cheong eta/. (16) of their neutron scattering

Luttinger's theorem; however, instead of

experiments on Lal.85Sro.15Cu04; Thelen and I(17)

introducing spinons and holons we assume that

find t h a t the degree of incommensurability which

in the absence of coupling to the spin excitations

they propose is so large t h a t the resulting

the quasiparticles obey the usual tight-binding

"leakage" of their peaks to the oxygen sites would

dispersion relation, ek = -2t (coskxa-coskya) and

produce a behavior substantially different from t h a t measured for 17W(T) by Reven et a/.(15) The

describe their coupling to the spin excitations by Z ~ (~) -~ (~). ~ (_-~) (10) Hint = ~1 ~,

message from the NMR community to the neutron

1 where ~ (q) = ~ Ya.l;.k tP+k+q,

scattering community is clear; the acid test of a

is the spin fluctuation operator whose correlation

suggested svin-svin correlation function is getting

function X (Q-q,co) is taken to be strongly peaked

the right Quantitativebehavior ofl7W(T) not getting

around i~ = (g/a, g/a) with an energy spectrum

the right temperature dependence for 63Wc(T)!

which yields quantitative argument with NMR (or

Finally, it should be noted that one is only

neutron) experiments, and -~ (q,t) is a coupling

obtaining information about the very low frequency part of X" (q,o~)in N M R

-O~c~5tlJk~ and "~ (q)

constant which may in general be both q and T

experiments; as dependent. Because ~ is commensurate while the

discussed by Millis and Monien, (14)one could, for example, replace ¢~2/c02SFin Eq. (7) by a quantity which has the same low frequency behavior, but a

Fermi surface is incommensurate, the resulting quasiparticle self energy, Y~(P,co, T) is sensitive to

larger cut-off. In this case the mean

the position of the quasiparticle on or near the

paramagnon energy would be increased.

Fermi surface (quasiparticles capable of coupling





and neutron scattering experiments

tell us that the dominant magnetic excitations for the planar Cu 2+ spins lie near Q = (rJa,g/a), and

to ~ will clearly have their properties modified co_n_s~_deroblymore than those which do not). and to co and T through the dependellce of X (Q-q,co,t) on the latter quantities. Thus far a complete self consistent calculation

that their characteristic energy is small compared

of the resulting quasiparticle spectrum has not

to the magnetic Fermi energy, r ~ 0.4 eV. To

been carried out. However in weak coupling

D. Pines ] Spin fluctua'.kms and high temperature superconductivity


calculations using the MMP expression for x(Q-

mechanism has, however, been rejected by many

q,o))(19), or an RPA expression (20) or an SCR

on the grounds that the likely transition

expression of the kind Moriya has described at this meeting(21), and taking the Fermi surface to be nearly circular, all authors find that the Fermisurface averaged imaginary part of the self energy

pairing state would be a "d-wave" state, (see, however, Schrieffer et a/.(23)), in apparent contradiction with measurements of the temperature dependence of the penetration

takes the form < I m Y.(p,co,T)> ~ M a x (co,T)

temperature would be low, and that the likely


in agreement with resistivity and optical experiments and in striking contrast to the Fermi liquid value, Im Y. - (co2+ ~- T2). Philippe Monthoux in Urb:ma is in the process of implementing a self-consistent calculation of Y. (p,e) which reflects the expected asymmetry in matching the Fermi surface to ~ in certain directions.(22) O n averaging over the Fermi surface, he obtains results of the form, Eq. (11). It remains to be seen whether the temperature dependence of Xo reflects self-energy effects or the presence of a spin pseudo gap. 4. S U P E R C O N D U C T I V I T Y In view of the similarities between heavy electron superconductors and the cuprate oxides (both are strongly correlated electron systems which exhibit antiferromagnetic behavior), and

depth.(24) Since, however, the NMR experiments establish the existence of antiferromagnetic paramagnons, and these provide, as we have seen above, a plausible candidate for the anomalous normal state behavior, it seemed worthwhile to look again at the possibility that these might give rise to high temperature superconductivity. Monthoux, Balatsky, and I have begun such a reexamination(7), and I give here a brief report on the results of the calculations. MBP consider the first order spin fluctuation exchange between quasiparticles described by the interaction, Fig. (1); they use a spin fluctuation spectrum determined by fits to NMR experiments, and find, in common with the authors ci~d

above, (23) that this leads uniquely to a superconducting state with dx2_y2 symmetry, with an energy gap, A(k) = Ao(k) [coskxa - coskya] .


the m a n y indications that the heavy electron systems are unconventional superconductors for which spin fluctuations provide the superconducting mechanism, the spin fluctuation induced interaction quasiparticle has for some time been considered an appealing candidate for superconductivity in the cuprate oxides.(23) The

From plow of its dependence on the respective energies of the pairing quasiparticles, as shown in Fig. 1, they find that the characteristic energy scale which the spin fluctuation induced interaction is effective is - Fin 2, where F, the magnetic Fermi energy, is - 0.4 eV; for realistic values of g2eff (Tc) they find transition

D. Pines / Spin fluctuations and high temperature superconductivity

{2 5

t e m p e r a t u r e s comparable to those measured.




"YBo-z C~ Oe.¢,,~|

s,.o,o 1


,j 3






, ~ - 4t



-"'-.,,,,,, .....

~ .....".. ' "


""' '"-'.



..,,. ID 2.O 5JO I/~;f (Tc)Xo (TcJ (eV"t) •









T/To • "1"


• 0.1




| ....


, ....

i ....


, ....

i ....

. ....




FIGURE 1 S t r u c t u r e of the retarded interaction between quasiparticles calculated by Monthoux, Balatsky, and Pines. (7) For a quasiparticle on the Fermi surface (E' = 0.25 ev), the maximum interaction occurs for quasiparticles of energy E = -0.25 ev, which are coupled to E' via Q.

FIGURE 2 The t e m p e r a t u r e dependence of the maximum value of the energy gap for YBa2Cu307 (Te = 95K), YBa2Cu306.63 (To = 60K), and Lal.85Sr0.15Cu04 (To = 40K); the inset gives a plot of the relationship, F(Tc) 1 T¢ = ~ exp ( ): ~geff (To) Xo(Te)N(O) (a = 1.66, T1= 1.07) for YBa2Cu307; (a = 1.49, TI = 1.07) for YBa2Cu306.63; (a = 1.51, TI = 1.21) for Lal.85Sro.15CuO4 (from Monthoux, Balatsky and Pines(7)).

As shown in Fig. 2, MBP were able to fit their computer-generated transition temperature to an In the superconducting state, because the energyexpression of the BCS form r 1 Tc a ~-~exp - k(Tc) =

dependent effective interaction depends on the (13)

energy gap, there is a feedback effect which MBP

where the dimensionless effective coupling

find leads to novel properties for A(T); as may be

constant, ~.(Tc) = TI g2eff (Tc) Xo(Tc) N(0), varies

seen in Fig. 2, even for weak to intermediate

from 0.48 to 0.33 depending on the compound, and

coupling strengths, the enera~v _~-apovens uv very

a and T1 are material constants of order unity.

ratfidlv below T~, and reaches a maximum value which is large comvared to kT¢, in good . . . . ~¢;xxx~aal, .. * with cxper]ment. The q t l a l.... , ~ l v ": . .¢:. .

D. Pines

/ Spin fluctuations and high temperaturesuperconductivity

anomalous normal state transport and optical

properties. Moreover, since superconductivity in


high Tc compounds is three dimensional, interplane coupling will establish the true 3D i



- ,

I, I,

It II ." iIIS




.- , . I . . 1: II II ::ill ii ." It ~:l~ : i

.~ll ,ll '. l l l.l , ~ , q '


'.... Jj, ll


i I "!




." ,7


~o,4L "~ I

/ /

.-* .-'.J 0~']~, (, , . , . o a z ~ 4 ( 1 6 o~l.,

~ =.~. ~ . | . ." I t

~i . ~ ~ ' " . ~ "


i x 'i,

-=o,e 3



......... .... T,T,:O.,O



Z 0.4




: # " "" : /' '//--I /#' #' i ~ /' i ' " Z #\-:j~?" ~.,r , 0,0(~ ' ,






.20 2 c,i/B

__,,,, o.= ........ T/T© = 0.50 T/T,- = 0.70 . . . . . . TIT¢ • 0.80 ----IT/T©'0.90 ---T/T¢ • 0 . 9 4 ........ T/T¢ • 0.98 T/T,, • 0,99 I I i


4o x iO"3

coherence of the order parameter. However, since the interplanar coupling is weak, it is to be expected that the in-plane symmetry of the gap will remain dx2_y 2, so I turn now to the experimental evidence concerning the pairing state.

5. IS T H E P A I R I N G S T A T E D - W A V E ? In trying to understand the nature of the superconducting state, it is again useful to begin



The density of states for the d-wave gap A(~) as a function of the variable (2o~B), where B = 2eV is the bandwidth at various values of (TPrc) for YBa2Cu3OT; the inset shows the calculated Knight shift in the superconducting state. The dotted line is the base value, the full line with the Fermi liquid correction, Foa = -0.6. The dots are the

experimental points of Barrett eta/.(28) (from Monthoux, Balatsky and Pines(7)).

with N M R

experiments which possess the great

advantage of probing the local surroundings of a given nucleus, so that these can be used to isolate the properties of the "planar" portion of the Fermi surface. Experiment(3)(4)(25) shows: * No Hebel-Slichter peak in either 63W(T) or 17W(T) in the vicinity of Tc

As M B P point out, although the qualitative

• A very rapid fall-offof Xo (T) below Tc

agreement with experiment is encouraging, these

• A very rapid fall-offof63Wc (T) below Tc

calculations represent only a firststep toward the

• A n approximate, but not identical, fall-offof

development of a consistent theory of the

17We(T) and 63We(T) below Tc

properties of the normal and superconducting

• [63Wc(T)Pr] ~ Xo(T), not IN(O)]2. (26)

state. For example, before a quantitative

These results are consistent with d-wave pairing

comparison with experiment on the transition

and antiferromagnetic correlations which do not

temperature and gap properties can be made, it is

very appreciably with temperature below Tc.(27)

important to incorporate lifetime effects and to

For example, Monien and I found it possible to fit

carry out a self-consistent calculation of the

the temperature dependence of both 63W(T) and

chemical potential and its variation with doping,

Zo(T) with a large energy gap and d-wave

which is needed as well as for the normal state

pairing(27), while M B P find a good fit to the N M R

D. Pines / Spin fluctuations and high temperature $uperconductivio.


experiments of Barrett e t a / . (28) can be obtained by

low temperature increase cannot occur, and

applying a Fermi liquid correction to their

would only be explicable as a new low temperature

calculated value of Zo for YBa2Cu307 (Fig. 3).

orbital relaxation process which somehow mimics

Added support for d-wave pairing comes from experiments involving the substitution of nonmagnetic atoms in planar Cu sites. Thus Xiao et a/. (29) find that substituting 2% Zn in Lal.85Sro.15CuO4 reduces Tc from 40K to OK, while as discussed here by Kitaoka, (4) a comparable substitution of Zn in Yba2Cu307 leads to gapless behavior.

the effect found naturally in the d-wave BulutScalapino calculation.









While all of the above results are easily explained by =d-wave" pairing, an ardent advocate of =s'-wave pairing could construct an explanation

I 02


I 0.4




T 0.6



for most of them. What appears, however, to be a "smoking-gun" for d-wave pairing comes from the experimental results on the anisotropy of 63W(T); found originally by Barrett eta/. (30), confirmed by Takigawa et al. (31), and discussed by Slichter(25) at this meeting, [63Wa(T)/63Wc(T)] displays an initial rapid drop below Tc of some 20% (from its

FIGURE 4 The anisotropic relaxation rate ratio for YBa2Cu307 as a function of (T/Te). The dots are the experimental values o~ Barrett eta/.(30), Takigawa et a/.(31) and Martindale et a/.(30); the line is the d-wave pairing calculation of Bulut and Scalapino (32) (from Bulut and Scalapino(32)). These results led Thelen, Lu, and me to

temperature independent value of-3.8 above Tc),

reexamine the information on the pairing state

followed by a gradual rise, until at low

provided by experiments on the temperature

temperatures it is some 30% above its normal state

dependence of the penetration depth, from which

value (Fig. 4). As the calculations of Bulut and

it is straightforward to extract pn(T)/p.(26) Annett,

Scalapino (32), which are compared with

Goldenfeld, and Renn (33) had earlier shown that a

experiment in Fig. 4, show, this result is uniquely

careful analysis of the experiments of Fiory et

explicable by the interplay of antiferromagnetic

ai.(24) on YBa2Cu307 leads to a T2 dependence for

correlations and d-wave pairing.

[pn(T)/p] at low temperature (inconsistent with s-

The rapid drop near Tc is produced by d-wave coherence effects (remember that the gap opens up rapidly), while the low temperature increase results from the scattering of excitations between nodes of the gap function. With s-state pairs, this

wave pairing, possibly consistent with d-wave pairing when Fermi liquid effects and impurity d-wave scattering are taken into account). Thelen et al. (26) compare the results of FioD" et al. ~ith the results of }~sr experiments and with a d-wave


D. Pines / Spin fluctuations and high temperature ~e.rconductivity

pairing calculation in which Amax (0) = 3.5 RTc and various Fermi liquid corrections are applied. As m a y be seen in Fig. 5, the different experiments do not agree, and the d-wave pairing calculation is "representative"of the range of the different experimental results which are not at present consistent with one another. From this I conclude that penetration depth of experiments do not at present provide convincing evidence for or against either s or d-state pairing. 1












FIGURE 5 Experimental measurements of [pn(T)/p]by Fiery et a/.,(24) Pumpin et al.(24) and Harshman eta/.(24) are compared with d-wave pairing calculations for Amax (0) = 3.5 kTc using various Fermi liquid corrections (from Thelen, Lu, and Pines(26)). 6. C O N C L U S I O N As M B P note, Schrieffer and his colleagues(23) share our view that af correlationsplay a key role in determining the superconducting transition, but a.,~.!ethat the coupling between quasipa~ic!es and spin fluctuationsis so strong that the normal state excitations are spin bags; they focus on the secsnd order spin fluctuation exchange to obtain s-wave superconductivity, rather than using the first-orderterm to get d-wave pairing. In the M B P approach, the results of experiments on the normal state may be used to

fix to fundamental quantities which enter the gap equation. W e are encouraged that our numerical results agree qualitativelywith m a n y experiments, and have very recently been confirmed by Millis(34) in an approximate analytic calculation which also includes self-energy effects. ACKNOWLEDGEMENTS I take this occasion to remember John Bardeen, not only for many stimulating conversations on these and related topics,but for being a constant source of inspiration and encouragement during the thirty-one years we had officesnext to one another. I should also like to thank m y collaborators and colleagues, E. Abrahams, A. Balatsky, IC Bedell, J.-P. Lu, A. MBlis, H. Monien, P. Monthoux, J. R. Schrieffer, C. P. Slichter,and D. Thelen for stimulating discussions on these and related topics, the National Science Foundation for its support, through grants DMR 88-17613, DMR 89-20538, and DMR 88-09854 through the Science and Technology Center for Superconductivity, Andy Millis for a critical reading of this manuscript, and the Aspen Center for Physics for its hospitality during its preparation. REFERENCES 1. J. Bardeen, Encyclopedia of Physics, Vol. 15, p. 274 (Springer-Verlag, 1956). 2. R.P. Feynman, Rev. Mod. Phys. 29 (1957) 205. 3. H. Yasuoka, these proceedings. 4. Y. Kitaoka, these proceedings. 5. A. Millis, H. Monien, and D. Pines, Phys. Rev. B ~42,(1990) 167. For closely related RPA-based calculations see N. Bulut et aL (20), and SCR calculations, see T. Moriya, Y. Takahashi, and K. Ueda, J. Phys. Soc. Jpn. 52 (1990) 2905. 6. D. Pines in High T~mgerature Superconductivity, eds. K. S. Bedell, D. Coffey,

D. Meltzer, D. Pines, and J. R. Schrieffer (Addison-Wesley, 1990) 392-396. 7. P. Monthoux, A. Balatsky, and D. Pines,

D. Pines / Spin fluctuations and I~igh temperature superc~nduct.ri~

submitted to Phys. Rev. Lett. (July, 1991). 8. For a review of experimental work up to 1990, see C. H. Pennington and C. P. Slichter. Physical Prouerties of High Temperature Supercondt~tors II, D. M. Ginsberg, ed. (World Scientific, NJ, 1990) 269. 9. M. Takigawa, A. P. Reyes, P. C. Hammel, J. D. Thompson, R. H. Heffner, Z. Fisk, and K. C. Ott, Phys. Rev. B 43 (1991) 247. 10. F. Mila and M. Rice, Physica C 157, (1989) 561. 11. H. Monien, D. Pines, and M. Takigawa, Phys. Rev. 13, 43 (1991) 258. 12. H. Monien, P. Monthoux, and D. Pines, Phys. Rev. B 43 (1991) 275. 13. J. Rossat-Mignot, these proceedings. 14. A. Millis and H. Monien, preprint. 15. L. Reven, J. Shore, S.-T. Yang, T. Duncan. D. Schwartz, J. Chung, and E. Oldfield, Phys. Rev. B 43 (1991) 10466. 16. S. W. Cheong, G. Aeppli, et al (preprint). 17. D. Thelen and D. Pines, in preparation. 18. P. W. Anderson, Science, 235 (1987) 1196. 19. H. Monien, P. Monthoux, and D. Pines(12); B. Arfi, preprint. 20. N. Bulut, D. Hone, D. J. Scalapino, and N. E. Bickers, Phys. Rev. B 41 (1990) 1797. 21. T. Moriya, these proceedings; T. Moriya and Y. Takahashi, preprint. 22. P. Monthoux, private communication. 23. Pioneering calculations of spin fluctuation induced superconductivity in the heavy electron superconductors include those of V. J. Emery, J. Phys. (Paris) Colloq., 43, C (1983) 3977, and Synth. Mat., 13 (1986), 21; J. E. Hirsch, Phys. Rev. Lett., 54 (1985) 1317; D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B, 34 (1986) 8190; K. Miyako, S. Schmitt-Rink and C. M. Varma, Phys. Rev. B, 34 (1986) 6654; M. Norman, Phys. Rev. B 37 (1988) 4987; in the

: :~

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