Spin fluctuations in insulating, weakly metallic and superconducting La2−xSrxCuO4

ELSEVIER

Physica B 197 (1994) 158-174

Spin fluctuations in insulating, weakly metallic and superconducting La2_xSrxCuO 4 G. Shirane a'*, R.J. Birgeneau b, Y. Endoh c, M.A. Kastner b "Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA bDepartment of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA CDepartment of Physics, Tohoku University, Sendai, 980 Japan

Abstract

We review the results of a comprehensive set of neutron scattering experiments on the spin fluctuations in La2_xSrxCuO 4 with x varying between 0 and 0.15. For 0 < x ~<0.0175 the materials exhibit 3D Nrel order; for x = 0 the spin correlations above TN are accurately described by recent theory for the 2D S = 1/2 Heisenberg antiferromagnet. For 0.02~0.06 the spin fluctuations are incommensurate; we find in a sample with x = 0.15 and Tc = 33 K that the spin fluctuations at low energies for T ~< 100 K are quite sharp in momentum space. Further, for energies as low as 1.5 meV the spin fluctuations persist into the superconducting state with no significant change in the incommensurate geometry down to 2 K.

1. Introduction

As is by now well known, the lamellar copper oxides, both pure and doped, exhibit unusually interesting and rich microscopic magnetic properties [1]. The antiferromagnetic spin fluctuations in the copper oxides have been studied with a variety of probes. These may be local, such as N M R , N Q R [2] or p~SR [3], or extended such as neutron scattering [1,4]. Each of these techniques has contributed in important ways to our understanding of the magnetism in the copper oxides. In some respects the most informative of

* Corresponding author.

these probes has been neutron scattering since it is capable of providing information about the spin fluctuations for wave vectors throughout the Brillouin zone and for energies ranging from 10 -5 to 1 eV. Our early work in this field involved a general characterization of the lattice dynamical and spin fluctuation properties of varied lamellar copper oxides including especially the model one-layer material La2_xSrxCuO 4. These results are reviewed in Ref. [1]. In this initial work a variety of interesting results were obtained. This included the discovery and characterization of the two-dimensional (2D) spin fluctuations in pure and lightly doped La2CuO 4. This was followed by our observation of low energy antiferromag-

0921-4526/94/$07.00 ~ 1994 Elsevier Science B.V. All rights reserved SSDI: 0921-4526(93)E0479-Z

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G. Shirane et al. / Physica B 197 (1994) 158-174

netic spin fluctuations in the superconducting materials La2_xSrxCuO 4 with x ~>0.06. This resuit, albeit initially controversial, has since been confirmed by others using both neutron scattering [4,5] and N M R techniques [2] in the YBa2Cu306+y system. For the past year our efforts have focussed on obtaining extremely detailed information about the static and dynamic antiferromagnetic spin fluctuations in the La2_xSrxCuO 4 system for insulating, weakly metallic and superconducting samples. In this paper we shall review the results of these recent measurements and their implications for models of the CuO 2 layers, both pure and doped. As we shall discuss below, we have also recently carried out a comprehensive study of the instantaneous spin-spin correlations in the material Sr2CuO2C12 which is the best realization to-date of the S = 1/2 2D square lattice Heisenberg antiferromagnet ( 2 D S L H A ) [6]. These measurements make possible a rigorous comparison with both Monte Carlo simulations [7] and theory [8,9]. The format of this paper is as follows. In Section 2 we discuss the basic phase diagram of La2_xSr~CuO 4. Section 3 reviews the results on the instantaneous spin-spin correlations in Sr2CuO2CI 2 as well as data from Monte-Carlo simulations and theoretical predictions. In Section 4 we discuss the evolution of the magnetism in La2_xSr~CuO 4 as x is varied from 0 to 0.04 with emphasis on the 'spin-glass' samples 0.02 ~< x~<0.04. Section 5 reviews the results of a comprehensive study of the dynamic spin fluctuations in a sample Lal.85Sr0.15CuO 4 which has a superconducting transition temperature of 33 K. A discussion of the results and suggestions for future experiments and theory are given in Section 6.

2. Phase diagram T h e r e are four basic regimes in the phase diagram of all lamellar copper oxides [1]. For the sake of definiteness we discuss explicitly La:_xSr~CuO 4. A schematic phase diagram is shown in Fig. 1. For 0 ~
TIK] 530

'

~

a2_xSrxCUO4

325 i

~TeU'agonal

0.02

0.05

0.21 0.25 x

Fig. 1. Schematicphase diagram of La2 xSrxCuO4.Note that the temperature axis is not to scale. exhibits three-dimensional (3D) N t e l order at T = 0 . The N t e l temperature which at zero doping is --325 K decreases linearly with increasing x with the N t e l order vanishing by x = 0.02 [10,11,12]. As we shall discuss later in this review, there is also some evidence for reentrancy in the phase boundary at the extremum of the N t e l state [12]. Simultaneous with this evolution of the N t e l order, the transport evolves from semiconducting for very small x to weakly metallic for x ~>0.02 [10,13]. Indeed for x ~> 0.02 and T ~> 100 K the resistivity varies linearly, with T with a conductance per carrier the same as that found in the normal state of the highest T c superconductors [10,11]. Below 100K with decreasing temperature the conductance evolves from logarithmic in T to exponential in T -1/z indicating successive crossovers from metallic to weakly localized to strongly localized regimes [11]. For 0 . 0 2 ~ x ~<0.05, La2_xSrxCuO 4 exhibits commensurate spin fluctuations with a finite 2D correlation length at T = 0 [11]. T h e r e is, in addition, evidence for a spin glass-like freezing transition at low temperatures [11,14]. The resistivity behavior in this regime is as described above. This concentration regime is normally labelled the 'spin glass' regime [15]. We will discuss the magnetic fluctuations in this concentration range in Section 4 of this paper.

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G. Shirane et al. / Physica B 197 (1994) 158-174

For 0.05 ~0.05 to 0.07 the spin fluctuations in La2_xSrxCuO 4 become incommensurate [16,17]. The relationship between the low energy antiferromagnetic spin fluctuations and the superconductivity is, of course, a central focus of research by ourselves and many other groups. Finally, for x ~>0.25 the material no longer exhibits superconductivity [18]. There may be some interplay between the superconducting phase boundary and the tetragonal-orthorhombic structural phase transition boundary although this region is currently the subject of continuing investigation [18]. Rather little is understood about the overdoped regime although transport measurements suggest that the material is an unusual metal. There are currently no magnetic neutron scattering data for samples in this 'overdoped' concentration regime.

3. Instantaneous spin-spin correlations in Sr2Cu02CI 2

The discovery of high temperature superconductivity in the lamellar copper oxides in 1986 [19] led to a renaissance not only in the field of superconductivity but also in the field of lowerdimensional quantum magnetism. The reason for the latter is that the parent compounds such as LaeCuO 4 correspond to rather good approximations to the S = 1/2 two-dimensional square lattice Heisenberg antiferromagnet [1]. Prior to 1986 this 2D quantum system represented one of the major unsolved problems in quantum statistical physics. As a result of symbiotic interactions between neutron scattering experiments [1], Monte Carlo simulations [7] and theory [8,9], a coherent picture has emerged for the low temperature properties of the S = 1/2 2DSLHA. It is now generally agreed that the spin-spin correlation length diverges exponentially with decreasing temperature leading to true long-range order at T = 0. Furthermore, as we shall review in the next section, the correlation lengths in LazCuO 4 are predicted reasonably well in absolute units by theory [11]. However, because of the ortho-

rhombic distortion the spin Hamiltonian of L a 2 C u O 4 is rather complicated [20] and this compromises any critical comparison between experiment and theory. Thus in order to test the theories properly one requires precise correlation length data covering as wide a temperature range as possible in a system which is described accurately by the ideal 2D S = 1/2 Heisenberg Hamiltonian. Recently, we have carried out energy-integrating neutron scattering studies [21] of the 2D Cu 2÷ S = 1/2 instantaneous spin-spin correlations in the material Sr2CuO2CI 2 [6]. This system is the best experimental realization found to-date of the S = 1/22DSLHA. Further, the exchange coupling J is known reasonably well from two-magnon Raman scattering measurements [22]. We review these Sr2CuOzC12 results in this section. Sr2CuO2C12 has the K2NiF4 crystal structure, space group I4/mmm, with square sheets of CuO 2 separated by two intervening sheets of SrCI. The material is tetragonal down to at least 10K; this high symmetry makes the magnetic properties much simpler than those of L a 2 f u O 4 [23]. The room temperature lattice constants are a = 3.967/~ and c = 15.59~. The Cu 2+ S = 1/2 2D spin Hamiltonian is given simply by H = ~ (i.j)

Ji/S, • Si + ~

(i,j)

l , v Y--t-1 sfsf " J (xi .rj ) .-u

(1)

From overlap considerations one expects that the isotropic and anisotropic exchange interactions are overwhelmingly between nearest-neighbor Cu e+ spins alone. From the two-magnon Raman scattering measurements of Tokura et al. [22], one deduces for the nearest-neighbor exchange J = 125 --- 6 meV. From measurements of the outof-plane spin wave gap one has j x r / j = 1.4 × 10 -4. Because of the perfect tetragonal symmetry with its resultant cancellation of the nearestneighbor interplanar isotropic exchange, the net 3D coupling is reduced by several orders of magnitude below the X Y anisotropy. Thus Sr2CuO2C12 is indeed an extremely good approximation to the ideal S = 1/22DSLHA. Finally, we note that Sr:CuOeCI 2 is very difficult to dope chemically with either electrons or holes. Thus compared with, for example, L a 2 f u O 4 the mag-

G. Shirane et al. / Physica B 197 (1994) 158-174

netism in the CuO 2 sheets in SrzCuO2C12 is relatively immune to the effects of intrinsic or extrinsic carriers. Greven et al. [6] report energy integrating two-axis measurements of the instantaneous spin correlations in SrECUOECl 2 for temperatures between T N = 2 5 6 K and 600K. In order to ensure proper energy integration over the dynamic fluctuations the incoming neutron energy was increased progressively with increasing temperature. The measured profiles generally were accurately described by 2D Lorentzians with half-width-at-half-maximum (HWHM) of K where K is the inverse correlation length. The results of Ref. [6] for K are shown in Fig. 2. We will discuss the comparison with theory below. A comprehensive theory for the phase diagram and spin correlations of the 2DSLHA has been given by Chakravarty, Halperin and Nelson (CHN) [8]. Their theory involves a mapping of the 2DSLHA into the 2D quantum nonlinear sigma model (2DQNLtrM). We will not discuss this model in any detail here. We note only that, by analogy with Fig. 1, the 2DQNLo-M has three Inverse Magnetic Correlation Length i i i ,

0.05

--HN 0.04

/i

(2/rPs = 144 msV)

a E~ = 5 meV •

Ei = 14.7 meV

v[i = 30.5 meV • El = 41 meV

0.03

_ t j/~

161

basic regimes: (a) the renormalized classical regime where the correlation length diverges exponentially in 1/T achieving long-range order (LRO) at T = 0; (b) the quantum critical regime with ~ - c / T where c is the spin wave velocity; (c) the quantum disordered regime where remains finite as T---~0. In the quantum disordered phase theory predicts a gap in the excitation spectrum. The expected behavior of the inverse correlation length in these three regimes, calculated by CHN [8] in the one-loop approximation, is shown in Fig. 3. C H N predict that the nearest-neighbor S = 1/22DSLHA should lie in the renormalized classical regime of the 2DQNLo'M phase diagram. Thus the correlation length should evolve with temperature like the lower curve in Fig. 3. The CHN model has been refined by Hasenfratz and Niedermayer (HN) [9]. They show that in the renormalized classical, region the correlation length is given rigorously by e c / a exp(2~ro~/r) [ 1 - ~ -1 (2~p~) 8 2"fro S

(~/a

+

(2wps) I

(2)

/]

where Ps is the spin stiffness. In order to cornInverse Correlation

0.10

-

Length in the QNLoM

0.08

~,o.o6 0.02 T "*'0.04 0.01

S

0.02

N

¢ 0 250

350

I

i

450

550

Temperature

650

(K)

Fig. 2, Inverse correlation length versus temperature in Sr2CuO2C12. The data were obtained with varying incoming neutron energies as indicated in the figure. The solid line is Eq. (3) with J = 125meV and the lattice constant a is measured at each temperature (a = 3.974 A at 420 K).

0

.

,

i

~

i

300

600 Temperature

i 900

[K]

Fig. 3. Qualitative behavior of the inverse magnetic correlation length in the quantum disordered (upper curve), quantum critical (middle curve) and renormalized classical (lower curve) regimes of the 2DQNLtTM as calculated by CHN [8] from a one-loop renormalization group analysis.

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G. Shirane et al. / Physica B 197 (1994) 158-174

pare the predictions of Eq. (2) for the 2DQNLtrM with the data shown in Fig. 2 it is necessary to know the relationships between c, Ps and J. Fortunately these have been determined rather well by recent theory and Monte Carlo simulations. The spin wave velocity of the S = 1/2 2 D S L H A is given simply by c = ZcV~Ja where in the spin wave approximation Z c = 1.18 + ~(1/2S) 3 [24] and J is the nearest-neighbor isotropic exchange. An identical result for Z c is found in Monte Carlo simulations [25]. Related calculations give 2":rps = 1.15J with an uncertainty in the proportionality constant of about 2% [24,26]. Substitution of these values into Eq. (2) then yields ~/a = 0.493 el'lSJ/r[1 - 0 . 4 3 ( T )

Magnetic Correlation Length 1500 900 I

The solid line in Fig. 2 corresponds to Eq. (3) with J fixed at the measured value of 125 meV. Otherwise there are no adjustable parameters. Clearly Eq. (3) describes the inverse correlation length K = ~-1 in SrzCuO2C12 extremely well. To extend the effective range of the measurements it is of value to combine results in Sr2CuOzCI 2 with data from the Monte-Carlo simulations of Makivi6 and Ding [7] on the S = 1 / 2 2 D S L H A with pure nearest-neighbor interactions. The data may be compared quantitatively in reduced form without any adjustable parameters by plotting ~/a vs J/T. The results soobtained are shown in Fig. 4. There is a substantial region of overlap covering the length range from about 6 to 28 lattice constants. As is evident in Fig. 4 the Sr2CuO2CI 2 and Monte Carlo results agree remarkably well. We should also note that recently Sokol et al. [27] have calculated the correlation length ~: for the t - J model using high-temperature series expansion techniques. For the half-filled band case which corresponds.to the S = 1 / 2 2 D S L H A they find quantitative agreement with the Monte Carlo results and, accordingly, with our Sr2CuO2CI 2 data; their calculations extend to correlation lengths of 17 lattice constants. The solid line in Fig. 4 is Eq. (3). It is evident

T [K] 450

I

500

i

i

--HN o o

100

SrzCuOlCl2

* MD

1

(3)

600

I

2

3

4

5

J/T Fig. 4. Semilog plot of the reduced magnetic correlation length (~/a versus J/T. The open circles are the data for Sr2CuO2CI 2 from Fig. 2 plotted with J = 125 meV and the lattice constant as measured; the filled circles are the results of the Monte Carlo simulations by Makivi6 and Ding [7]. The solid line is the theoretical prediction of the S - ~ 2DQNL~rM, Eq. (3). The dashed line is the simple exponential form ~/a = 0.276 e l25Jjr suggested by Makivi6 and Ding [7]. --

1

that the C H N - H N theory describes the combined Sr2CuO2C12-Monte Carlo results very well for length scales from 200 lattice constants down to nearly 1 lattice constant or, in temperature units using as a scale the measured exchange for S r z f u O z C 1 2 , from --275K to --1450K. We emphasize that technically Eq. (3) is the result for the 2DQNL(rM rather than the S = 1 / 2 2 D S L H A . Figure 4 thus demonstrates that the isomorphism between the two models is valid down to very short length scales. We should also emphasize that Eq. (3) is only valid to terms of order (T/2"rrps). We have found empirically that by including small higher-order corrections in (T/2"nPs)2 and (T/2~pJ the agreement between the data shown in Fig. 4 and Eq. (3) can be made essentially perfect. Clearly, it would be invaluable if these terms were calculated analytically. Recently there have been suggestions that the

G. Shirane et al. / Physica B 197 (1994) 158-174

S = 1/22DSLHA may exhibit a crossover from renormalized classical to quantum critical behavior at temperatures as low as T---2ps = 0.36J [27,28,29]. This would imply, for example, a change in the temperature dependence of K from exponential e -2~ps/r to linear in T, K = Coc(T Tps) where Coc should be universal [30]. Unfortunately the explicit crossover function is not yet known beyond the one-loop approximation. We include in Fig. 4 as the dashed line the simple exponential form suggested heuristically by Makivi6 and Ding [7], ~ / a = 0.27 e 125s/~. It is evident that this form describes the correlation length data over the entire temperature range very well. Most importantly, there is no evidence for a change in the slope of In ~ vs 1/T over the complete range of 1 < J / T < 5.2 and accordingly there is no evidence of a crossover from renormalized classical to quantum critical behavior. It should be noted that if one instead plots ~-1 vs T the exponential form becomes sigmoidal in shape and between - 6 0 0 K and 1000 K, K indeed appears to vary linearly with T - Tpx. However, the slope Coc thus obtained differs from the universal quantum critical value by a factor of - 2 [28,30]. We conclude, therefore, that this apparent 'quantum critical behavior' in ~ is an artifact arising from fits of the data over too limited a temperature range. We suspect, although we have not proven, that the same criticism also applies to analyses of l/T1, and 1 / T 2 obtained from NMR and NQR measurements in La2CuO 4 [31]. At low temperatures both of these quantities should be proportional to some power of T times the correlation length ~ [32] and hence must reflect the basic exponential behavior of ~ shown in Fig. 4 over the complete temperature range from --350 K to 1500. We should note however, that, as argued by Sachdev and co-workers [28], other quantities such as the uniform susceptibility which do not involve ~ directly, may exhibit the influence of quantum critical effects at lower temperatures. Finally, it is appropriate to ask whether or not there is any feature in the Sr2CuO2C12 or La2CuO 4 data which is purely quantum rather than renormalized classical in nature since most quantities such as ~ or the T = 0 sublattice

163

magnetization are simply renormalized from their classical values by quantum fluctuations. One dramatic example is the mid-infrared exciton-multimagnon absorption recently discovered by Perkins et al. [33]. Classically, only exciton-one magnon absorption processes are allowed. However, in both Sr2CuO2CI 2 and La2CuO 4 Perkins et al. [33] observe a welldefined peak at the exciton-2-magnon position as well as more diffuse absorption at higher energies due to 3- and 4-magnon processes. Within the current theoretical framework these processes are only made allowed by the quantum nature of the Cu 2÷ spins. Thus much of the optical absorption in the mid-infrared region in the pure materials is exclusively quantum in nature.

4. Spin fluctuations in La2_xSrxCu04+8 for O~
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G. Shirane et al. / Physica B 197 (1994) 158-174

[11] on La2_xSrxCuO 4 we first review earlier results in oxygenated La2CuO4+ 8. As is by now well-known, as-grown samples of La2CuO 4 typically contain excess oxygen atoms. These excess oxygen atoms which are incorporated as interstitials donate holes to the CuO 2 sheets. Chen et al. [13] have determined that T N ( 0 ) -- TN(t~ ) varies linearly with the hole concentration. In the discussion that follows we will assume that the properties of doped La2CuO 4 depend uniquely on the hole concentration, p, independent of the actual origin of the holes. Results for the sublattice magnetization and the inverse correlation length for three samples of La2CuO4+ ~ with T N -- 325 K ( p = 0 ) [11], 1 9 0 K ( p - - 0 . 0 1 ) and 9 0 K (p~<0.02) [12] are

La2Cu04+ 6

1.25

(°)

1.00

~%0.75 o

~0.50

TN=325K

"~

0.25 0

I'= 0.25

0

I 0.50

I 0.75

~'bO-o1.00

01.25

T/TN

0.04

i

i

(b) 0.03

oo2 0.01

TN

TN

1oo

200

I

, 300

I ,oo

500

600

Temperoture [K]

Fig. 5. (a) Square of the normalized sublattice magnetization of undoped (T N = 325 K) and oxygen-doped (T N = 190 K and 90 K) samples. The solid lines are guides to the eye. (b) Inverse magnetic correlation lengths of the same samples. The solid lines are calculated from K(x, T)I=K(x, 0 ) + K(0, T) as discussed in the text. 1 rlu = 1.175 A - .

shown in Fig. 5. For the stoichiometric T N :325 K crystal the inverse correlation length K follows the H N form Eq. (2) reasonably well with c = 850 meV ~ and 2Wps fitted as 1.11J; the latter is satisfactorily close to the current theoretical expression 2rrps = (1.15-+0.02)J. It is evident that Eq. (2) describes the measured K reasonably well although there are some systematic discrepancies. These probably arise from the fact that because of the orthorhombic symmetry there are significant antisymmetric exchange and interplanar exchange terms in La2CuO 4 [20]. The data for K in the lower T N samples are similar to those in the pure system except that at a given T, K seems to be shifted up uniformly. We shall discuss this in more detail after reviewing the data for the Sr-doped sampies. The sublattice magnetization curves shown in the top panel of Fig. 5 are themselves very interesting. First, although not explicitly illustrated, we have shown that the sublattice magnetization in La2CuO 4 is accurately predicted by self-consistent spin wave theory up to --260 K with no adjustable parameters [35]. This, of course, is also consistent with the renormalized classical behavior. This good agreement at the level of a few percent necessitates that spin wave theory predicts the zero-point spin deviation to that same accuracy. An interesting feature of Fig. 5(a) is that as 8 is increased the sublattice magnetization curve flattens out at low temperatures and indeed for the 90 K sample it becomes re-entrant. This reentrancy is probably connected to the spin glass freezing in the more heavily doped samples; specifically, it reflects the effects of 10w energy 'central peak' excitations arising from the doped holes which are in addition to the spin waves [11,12,14]. Results for the inverse correlation length K from 10 K to 550 K in La2_xSrxCuO 4 with x = 0, 0.02, 0.03 and 0.04 are shown in Fig. 6. As discussed in Ref. [11], the x = 0.02 sample is just on the borderline of ordering three-dimensionally. One important result, not stated explicitly in Ref. [11], is that to within the uncertainty connected with the energy integration, the peak intensity S(xr, ~r)T = A~2(T) where A is a constant; this holds for all three doped samples.

G. Shirane et al. / Physica B 197 (1994) 158-174

Magneti cCorrela, tionLength , , o La2Cu04. 0 / 0.05 n Lo1.98Sro.02Cu04 / v La,.gTSro.03CuO 4 T~ / 0.04 ~ ~! Lali~.96Sr,!o.04Cu04 ~ ~ ~'T~=0.03 0.02 ~/ 0.06

Inverse

0.01~ 0

e<~ 1O0

I

200

300

I

i

400

500

Temperafure[K]

600

Fig. 6. Inverse magnetic correlation lengths of four Laz_xSrxCuO 4 samples. The solid lines are calculated from K(x, T) = K(x, 0) + K(0, T) as discussed in the text.

Thus, there is no prefactor scaling like T to some power as is predicted to occur in the pure material according to CHN [8]. The data shown in Fig. 6 are quite striking, in all four samples the inverse correlation length is given simply by the heuristic formula

K(x, T) = K(x, 0) + K(0, T ) .

(4)

The temperature independent inverse length K(X, 0) corresponds to correlation lengths of 150, 65 and 42/~ for the x = 0 . 0 2 , 0.03 and 0.04 samples respectively. K(0, T) is Eq. (2) with 2"rrps = 150 meV. The solid lines in Fig. 5(b) also correspond to Eq. (4) with K-I(x, 0) = 275 and 140 ~ for the T N = 190 K and 90 K samples respectively. Clearly Eq. (4) describes the instantaneous correlation length data extremely well for hole concentrations of 0, 0.01, 0.02, 0.03 and 0.04. Evidently the borderline between the 3D N6el region and the spin glass region in Fig. 1 occurs when the 2D correlation length determined by the doped holes is ~150A. For longer lengths the residual anisotropic and interplanar interactions precipitate a transition to 3D LRO while for shorter lengths only 2D shortrange order occurs down to 10 K. As emphasized in Ref. [11] these results seem

165

incompatible with the CHN scenario as illustrated in Fig. 3. The onset of the temperature dependence of K(x, T) at -250 K seems to be independent of x for x ~>0.02 and there is no concentration where quantum critical behavior, K - CocT, is observed. Thus we believe that a different approach which goes beyond the translationally invariant 2DQNLo-M is required to understand the generic phase diagram of the doped copper oxides. A promising approach, not yet fully developed, is that of Aharony et al. [15]. Specifically, those authors assume that the holes are localized on the oxygen atoms in the CuO 2 planes; the resultant frustration of the Cu2+-Cu 2+ antiferromagnetic bond leads to a long-range dipolar distortion of the antiferromagnetic order. Glazman and Ioselevich [36], following the earlier work of Villain [37], argue that in 2D such a distortion invariably leads to the destruction of LRO at T = 0 for any concentration of holes. Further, very recently, Gooding and Mailhot [38] have carried out a classical simulation of the frustrated bond model of Ref. [15] for average hole densities of x = 0.02, 0.03 and 0.04. These calculations indeed reproduce the empirical behavior, Eq. (4), quite well; further they find £ ( T = 0 ) = 103A, 53A and 4 2 ~ compared with the measured values of 150 A, 65 ~ and 42 A for x = 0.02, 0.03 and 0.04 respectively. This agreement is surprisingly good given the classical nature of the simulations and the uncertainty in the actual hole concentrations for the measured samples. An alternative approach which also seems capable of explaining the observed behavior is the microscopic phase separation model of Emery and Kivelson [39]. The underlying physics is closely related to that of Ref. [15] as developed in Birgeneau et al. [40] and indeed both long-range dipolar distortion and microscopic phase separation effects could be present in the real materials. It seems, therefore, that the magnetic frustration model of Ref. [15] is indeed capable of explaining quantitatively the phase diagram and the measured correlation lengths as a function of temperature in La2_~SrxCuO 4 for x up to 0.04. To summarize , in this picture the 2D multicritical point occurs at T = 0, x = 0. However, for x <~0.02 the 2D correlations become sufficiently

166

G. Shirane et al. / Physica B 197 (1994) 158-174

well developed such that the X Y , antisymmetric and interplanar exchange terms are able to initiate a phase transition to 3D LRO. However, for x ~>0.02 these residual interactions are ineffective and the system remains purely 2D down to 0 K. As we shall comment briefly later there is, however, evidence for a 2D Heisenberg to 2D XY crossover below 20 K as well as a spin-glass freezing at similar temperatures. It is our view that it would be extremely difficult to reproduce the rich phenomenology in the La2_xSrxCuO 4 system with any translationally invariant model such as the t - J model. Application to the electron-doped materials produces similar difficulties [11]. This completes our discussion of the instantaneous spin correlation in the N6el and spin-glass regions of the phase diagram, Fig. 1. In Ref. [11] a detailed study of the spin dynamics in the x = 0.04 sample is also reported. Related experiments which give equivalent results have since been reported for Lal.98Sr0.02CuO 4 [41], YBa2Cu306. 5 [42], YBa2Cu306. 6 [43] and YBa2Cuz.9Zn0.106. 6 [44]. The experiment involved measuring at fixed energy the intensity integrated around the (1, 0, 0) position (or equivalently (rr,~r) in square lattice, unit lattice constant notation) as a function of temperature. Experiments were carried out for energies between 0 and 45 meV and temperatures between 1.5 K and 500 K. Figure 7 shows the energy dependence of the q-integrated intensity at T = 10 K for energies between 2 and 45 meV. For the energies shown in Fig. 7 the intensity is independent of temperature below 10 K and the plot therefore represents the zero-temperature response I(Itol,0). The inset in Fig. 7 shows the intrinsic width of the scattering profiles as a function of energy. The width at low energies, as expected, equals that measured in the energy-integrating experiments. The increasing linewidth as a function of to reflects a crossover to propagating spin waves as expected for to ~>cK -----20 meV. From the main figure it is evident that the integrated intensity at low temperatures is enhanced at low energies and becomes nearly independent of energy above --10 meV. It is important to note that for simple classical spin waves the intensity inte-

La1.96Sro.o4CuO4

5

'

i

i

i

i

~4

~ 0.02 o o ,'o ~2

[meV]

--1 0

0

20

30 [meV]

40

50

Fig. 7. Energy dependence of the Q-integrated intensity at T = 10 K in Lal.96Sro.04CIIO 4. The inset shows the intrinsic HWHM of the scattering profiles.

grated around the ('rr, "rr) position is independent of energy, the 1/to = 1/cq structure factor being cancelled out by the factor of q in the 2D Jacobian (d2q = qdq). Thus the flattening out of I(1'ol, 0) above - 1 0 m e V simply reflects a crossover to normal spin wave behavior at higher energies. The q-integrated intensity as a function of temperature is shown in Fig. 8 for energies between 2 and 45 meV and temperatures between 10 K and 500 K. As may be seen from Fig. 8 the intensity peaks at T = 2 t o and then decreases with further increase in temperature. This immediately suggests consideration of models involving scaling in to/T. The simplest of these is the marginal Fermi liquid picture of Varma and co-workers [45] which assumes X"(q, to) ~- (o/T for to/T ~ 1 and =1 for to/T > 1 with little or no q-dependence. However, it is immediately evident from Fig. 7 that this model is too simplified to explain the La1.96Sr0.04CuO 4 data. In Ref. [11] the more general form g"(to) =

f d2q X"(q, to)

2 to = l(]to], 0 ) - ~ - a r c t a n [ a l ( T ) q- a3(-~) 3]

(5) is instead proposed. Thus

x"(,o)/zgltol, 0)

should

G. Shirane et al. / Physica B 197 (1994) 158-174

167

I_a1.96Sro.o4CU04

Lo1.96Sr0.04Cu04 ~.T ~?

1.25

o 2meV ~, 9meV • 3rMV * IZmW

• ~*o

o

o 4~,meV v 20meV

, eneV • ~.~neV

V o

0 45ewV

,

7

1.00

AlX

-I

•

O

o

i:

"o

3

3 2

1 0

1 0

3 2 •

r-

•

|

l

0

0

•

o

0

-m.V

i

0.25 1

•~.

o 2meV

0

c

,

0 V

6

4 3 2

"r-

,

. o .¢

1

2

o

4.5meV

A. • v • O

9meV 6meV 12meV 20meV 55meV 45rneV

3

4

Fig. 9. Normalized integrated spin susceptibility as a function of the scaling variable to/T ( K e i m e r scaling). T h e solid line is the function (2/~r) tan-l[al(m/T) + a3(to/T) 3] with a 1 = 0.43 and

a 3 = 10.5.

1 1 i

o

,oo

20o

Temperdure (K) Fig. 8. Scattering intensity integrated around the 0r, It) position in reciprocal space for 2meV~
be a universal function. The data of Fig. 8 with the Bose factor removed are plotted in this reduced form in Fig. 9. The solid line is Eq. (5) with a 1 = 0.43 and a 3 = 10.5. Clearly the scaling function, Eq. (5), works extremely well. The solid curves in Fig. 8 are calculated from Eq. (5) multiplied by the Bose factor ( 1 - e-'°/r) -1. As may be seen in Fig. 8, Eq. (5) also gives a very good description of the integrated intensity except for some systematic deviations for the lowest energies to = 2 and 3 meV. These discrepancies become more dramatic at energies of 1 and 0.75 meV. It is argued in Ref. [11] that the breakdown of the scaling at low energies reflects the influence of the XY anisotropy. Recall that in pure La2CuO 4 the XY spin wave gap is 5 meV [46]. As noted above, the scaling, Eq. (5), has also been found in a variety of other materials. In La1.98Sr0.02CuO 4 [41] and YBa2Cu2.9Zn0.104

[44] I(JtoJ,0) behaves as in Fig. 7 whereas ih YBa2Cu306. 5 (T c = 50 K) [42] and YBa2fu306. 6 (T c = 53 K) [43] I(Itol, 0) ---Itol for to 15 meV. Nevertheless the to/T temperature scaling holds quite well. It should also be noted that in YBa2fu306. 6 the scaling breaks down at low energies at temperatures somewhat above T c [43]. Presumably, this reflects the influence of the superconducting gap, by analogy with the effects of the XY anisotropy gap in Lal.96Sr0.04CuO 4 [11]. Keimer et al. [11] have also carried out studies of the quasielastic response in La1.96Sr0.04CuO 4 with varied values of the energy resolution. They find a behavior similar to that observed in canonical spin glasses, that is, an to = 0 central peak appears at a temperature which decreases with decreasing energy resolution. The onset temperatures are 4 0 K and 2 0 K for energy windows of 0.5 meV and 0.1 meV H W H M respectively. Spin freezing is also observed in I~SR experiments [3,12]. These aspects of the problem, however, still require further research. Finally, we should note that in the energy region where I(Ito I, 0) is approximately constant, Eq. (5) is homogeneous in to/T. This then leads to a natural explanation of various properties of the normal state including the DC and A C conductivities and the N M R [11].

168

G. Shirane et al. / Physica B 197 (1994) 158-174

5. Spin fluctuations in superconducting Lal.ssSro.lsCU04 One of the interesting features of the data discussed in the previous section is that for hole dopings up to p = 0.06 the spin fluctuation scattering remains commensurate at the (at, 70 position. This commensurability is not yet properly understood. However, as p is increased above 0.06 there is a commensurate-incommensurate transition in the spin fluctuation geometry. This was originally discovered independently by Yoshizawa et al. and Birgeneau et al. [16] and the detailed geometry of the incommensurability has recently been mapped out by Cheong et al. [47]. The geometry of the scattering in the incommensurate phase, p ~>0.06, is illustrated in Fig. 10. Neutron scattering data on the spin fluctuations above and below T c exist for a variety of superconducting La2_xSrxCuO 4 samples. How(a) (0,~,0) (0,0,1)

o,o,0)

(b) (h,0,0)

(0,n)

~ 0 , 0 ) 1

(0,0)

(n,~)

(Tt,0)

Fig. 10. The scattering configuration in the (h, 0, l) geometry. (a) The four rods and the ellipse represent the incommensurate magnetic rods and the resolution function of the spectrometer respectively. (b) The solid line illustrates a scan trajectory and the black circles represent incommensurate scattering peaks.

ever, recently Matsuda et al. [48] have reported a rather complete study of the energy and temperature dependences of the spin fluctuation scattering in two high-quality single crystals of Lal.85Sr0.~sCuO4, both with T c = 33 K. (Here T c is taken as the temperature at which the sample is fully superconducting; the onset temperature is closer to 38K.) We therefore will limit our discussion primarily to a review of these recent results since they represent the most comprehensive set of data available to datel In this section, we use the crystallographic notation pertaining to the low temperature orthorhombic space group Bmab; reflections in the CuO 2 planes then have Miller indices (h, k, 0) while [00l] is the direction perpendicular to the CuO 2 planes. We will treat the material as being effectively tetragonal. For clarity we note that the rods (1,0, l) and (1, 7,1 l) in Bmab notation correspond at I = 0 to the points ('rr, ~r) and (0, ~) respectively in the two-dimensional square lattice, unit lattice constant notation favored by theorists. Data to be presented in Figs. 11, 12, 13 and 14 were all taken with the sample oriented for scattering in the (h, 0, l) zone and tilted about the (0, 0, 1) axis by an angle ~b, so that scans with qx = h cos ~b, qy = h sin ~b, and qz = l arbitrary were possible; such tilt scans are parameterized by the variable h. When ~b is set at 6° , the scattering near the incommensurate peak positions, (0.89, 0.11, l) and (1.11, 0.11, l) or, equivalently, (0.78"rr, rr) and ('rr, 1.22"rr), could then be probed. This scattering configuration is illustrated in Fig. 10. Shown in Fig. 11 are a series of scans at T = 1.9 K and 35 K at energies of 1.5 and 3 meV together with scans at 20 K at energies of 15 and 20 meV. The solid lines were all calculated using 2D Lorentzian profiles convolved with the instrumental resolution function. These data contain a number of important features. First, the excitations are remarkably sharp in momentum space at low temperatures and energies. Quantitatively similar results were obtained by Mason et al. [49] on a sample Lal.86Sr0.14CuO 4. Second, the excitation momentum width increased with increasing energy. Most importantly, from the

169

G. Shirane et al. / Physica B 197 (1994) 158-174 La1.85Sr0AsCuO 4 ( K O S 1+2) •"~ 160 f

......

ET/510r~V

t

8

T c = 33 K H7 40'-80'-80'-80' 200 . . . . . . . . . .

400

0.8 .

, , 14.7E~

300

1.0

1.2

1.4

0.8

8C

2

4C

0 "r-, 6

0 0.6

0.8

1.0

1,2

1.4

o 6.0meV • 6.0meV o 3.5meV • 3.0meV

, ....... E ~ 3meV T = L9K ( h ~s~*, h sin6*,-0.6)

72

1.0

1.2

1.4

0.6

t 0.8

1.0

1.2

~

~- ~ o

o

t

6

~ ,

• ~

I

J

~

t

I

~

J

,

• I

• ~

~

t

lo• . meV

1.4

OOOQ

'~-

~

n , , ,

6

401)

~

n , , ,

4

!:t 0 0.6

,

160 ~ 120

0.6

,

14.7E i

300

T =

I,gK

( h cos6~,h sin6°,d),6)

0

.o

h (r.l.u.)

.4

0 0.6

0.8

,

1.0 h (r.l.u.)

80

Fig. 12. Temperature dependence of X"(q,~o) at q = (0.78~r, ~r) as measured by Matsuda et al. [48] (closed circles) in Lal 8sSr0.15CuO~ and by Mason et al. [49] (open circles) in

o o.,

20 40 60 Temperature (K)

1.2

1.4

Inelastic neutron scattering spectra of T = 1.9 K and 35 K at energies of 1.5 and 3 meV together with scans at 10 K at 15 and 20 meV. The scans all follow the trajectory (hcos6 °, h sin6 °, -0.6) illustrated in Fig. 10(b). The lines are the results of fits to four Lorentzians centered at positions 0r(1-+6),~r) and (~r, ~r(1 +-6)) with 6 = 0.22 c0nvolved with the instrumental resolution function.

Lal.s6Sro.laCUO4.

Fig. 11.

LaL85Sr0AsCuO4 at

scans at 1.9 K in Fig. 11 it is e v i d e n t t h a t t h e s e s h a r p , l o w - e n e r g y e x c i t a t i o n s persist in t h e s u p e r c o n d u c t i n g state. T h e g e n e r a l i z e d s u s c e p t i b i l i t y X"(q, to) at the ( ~ ( 1 - 6 ) , w) p o s i t i o n as a f u n c t i o n o f t e m p e r a t u r e for t e m p e r a t u r e s 1.9 K ~< T ~< 80 K a n d e n e r gies 1.5 m e V ~< to ~< 6 m e V is s h o w n in Fig. 12. W e i n c l u d e , in a d d i t i o n , r e c e n t results b y M a s o n et al. [49] w h i c h c o r r o b o r a t e t h e B r o o k h a v e n experiments. We remind the reader that because of the anisotropic resolution function these data in fact r e p r e s e n t o n e - d i m e n s i o n a l i n t e g r a t i o n s o v e r t h e i n c o m m e n s u r a t e p e a k s . X" r e a c h e s its m a x i m u m v a l u e for e n e r g i e s o f 1.5 a n d 3 m e V at a t e m p e r a t u r e w h i c h to within t h e e r r o r s coinc i d e s with T~ = 33 K a n d t h e n is e i t h e r a c o n s t a n t

o r d e c r e a s e s s o m e w h a t b e l o w T c. H o w e v e r , at 6 m e V t h e r e a p p e a r s to b e n o significant effect o f the superconducting transition on the amplitude o f t h e spin fluctuations. W e s h o w in Fig. 13 t h e H W H M as a f u n c t i o n o f e n e r g y for t h e s c a t t e r i n g in Lal.ssSr0.15CuO 4 for T = 10 a n d 35 K a n d e n e r g i e s 1.5 m e V ~< to ~< 20 meV. T h e H W H M is 0.034 -+ 0.006 ,~-1 at to = 1.5 m e V a n d T = 10 K a n d i n c r e a s e s b y a f a c t o r o f f o u r b e t w e e n to = 1.5 m e V a n d 20 meV. T h e b e h a v i o r is a l m o s t t h e s a m e at T = 35 K , w h i c h is j u s t a b o v e T c. D a t a for a s a m p l e with x = 0.02 at T = 35 K a r e also s h o w n in t h e l o w e r p a n e l o f Fig. 13. T h e e n e r g y d e p e n d e n c e o f X" i n t e g r a t e d a r o u n d t h e (-rr, 70 p o s i t i o n is s h o w n in Fig. 14. A t b o t h 10 K a n d 35 K t h e r e is e v i d e n c e f o r p s e u d o g a p b e h a v i o r b e l o w 10 m e V w h i c h , p e r h a p s coinc i d e n t a l l y , for T c = 33 K is t h e w e a k c o u p l i n g BCS gap energy. W e n o w c o m p a r e t h e results o f R e f . [48] in Lal.85Sr0.15CuO 4 with results in s u p e r c o n d u c t i n g YBaECU306. 6 [43] (To = 53 K ) as well as with t h o s e in Lal.98Sr0.02CuO4, a crystal with just e n o u g h Sr to d e s t r o y t h r e e - d i m e n s i o n a l long-

170

G.

Shirane

et al.

/

Physica

B

197

(1994)

LaL85Sro.15CuO4 (KOS 1+2) T c = 33 K 0.004 . . . . . . . . . . . . . . . . . . . . . . . . T-- 10K

Lal.85Sro.]sCuO4 (KOS 1+2) T c = 33 K 0.15 . . . . , .... , .... , .... , ....

"7

158-174

T=10K =

0.003

*< 0.10

=

0.002

,z

0.05

~t,,,,,,,,,,,,,,,,ion~

0.001 0.000

0.00

5

10 15 co (meV)

20

25

Lal.85Sr0.15CuO4 (KOS 1+2) Tc = 33 K 0.15 . . . . , . . . . , . . . . , . . . . . ' ' ' '

0

5

10 15 co (meV)

20

25

Lal.85Sr035CuO 4 (KOS 1+2) Tc = 33 K 0.004 . . . . , . . . . . . . . . . ...., .... T=35K

:Z °.°°' ~< 0.10

z~

0.002

,' ooo

0

x.00

.................. 5 10 15 o) (meV)

20

0.001 _t 25

Fig. 13. Energy dependences of the H W H M of S ( q , o J ) in m o m e n t u m space for temperatures of 10 and 3 5 K for La~.85SroAsCuO 4 (open circles) and for Lal.98Sr0.02CuO 4 (closed circles). T h e solid lines are guides to the eye.

range magnetic order as discussed in Section 4. T h e hole concentration in the Y B a 2 f u 4 0 6 . 6 sample should be comparable to that in Lax.85Sr0.15CuO 4 whereas, of course, the x = 0.02 sample has of order 2% holes per Cu. First, the t e m p e r a t u r e dependence of X"(q,~o) in L a 1 . 8 5 S r 0 . 1 5 C u O 4 shown in Fig. 12 is qualitatively similar to that observed by Sternlieb et al. [43] in YBa3fu306. 6 albeit with the energy scale smaller by factor of order 1.5 to 2 which, in fact, is also the ratio of the superconducting To's; there are also some minor differences in the behavior below T c. F u r t h e r m o r e , above T c the data are qualitatively consistent with the ~o/T scaling model, Eq. (5), discussed in the previous section; that is, X" increases with decreasing temp e r a t u r e at a rate which increases with decreasing energy. It should also be noted that in samples of YBa2Cu306+ x with T c/> 60 K a gap

0.000

0

........................

5

10 15 0) (meV)

20

25

Fig. 14. Energy dependences of the 2D integrated susceptibility X"(co)= S(~,~)dq2D X " ( q , w ) for t e m p e r a t u r e s of 10 and 3 5 K for Lal.85Sr0.15CuO 4 (open circles) and for Lal.98Sr0.ozCuO 4 (closed circles). T h e data for the two samples have been scaled using p h o n o n m e a s u r e m e n t s so that this figure gives the correct relative values. T h e solid lines are guides to the eye.

appears below T c which increases with increasing x and hence increasing hole concentration [4]. No such gap is found in Lal.85Sro.15CuO4. In YBa2Cu306. 6 the scattering is nearly commensurate at the (~r, "rr) position and the scattering m o m e n t u m width is almost t e m p e r a t u r e independent for temperatures 10 K ~< T ~< 100 K. The difference in the geometry of the spin fluctuations in La1.85Sro.15CuO 4 and YBaaCu3fuO6. 6 is believed to arise f r o m differences in the Fermi surfaces [50]. The widths of the scans in q-space in Y B a a f u 3 0 6 . 6 a t T = 10 K are 0.10/~ - I for ~ o = 5 m e V and 0.13/~k -1 for w = 18 meV [43]. The width at 18 meV is similar to the values reported in Ref. [48] for Laa.85Sr0.sCuO4, but at low energies the width measured in Y B a 2 f u 3 0 6 . 6 is a factor of three

G. Shirane et al. / Physica B 197 (1994) 158-174 larger than that of Lal.85Sr0.~sCuO 4. Generally various calculations including especially those by Levin and coworkers based on a nearly localized Fermi liquid model [50,51] seem to account very well overall for the measured spin fluctuation spectra both in La1.85Sr0.15CuO 4 and in YBa2fu306. 6 in the normal state including the temperature and energy dependences. These theories have not, however, yet addressed the marked sharpening the H W H M at low energies which is observed for temperatures below - 7 5 K in La1.85Sr0.15CuO4. Also, none of these theories appear to predict the incommensurate-commensurate transition in La2_xSrxCuO 4 as x is decreased below 0.06. The energy dependence of the integrated susceptibility X"(to) = f(-~,-~)dq2D X"(q, to), for La1.98Sro.o2CuO4 is compared with that for La~.85Sr0.15CuO 4 in Fig. 14. The data for the two samples have been scaled by the relative volumes, which were determined by phonon measurements. The integrated susceptibility for the 2% sample decreases with increasing energy as discussed in Section 3. This behavior is quite different from that of Lal.85Sro.15CuO 4, in which a pseudo-gap exists. Specifically, as noted above, in the superconducting sample the integrated susceptibility is almost constant above t o = 10 meV and then begins to decrease with decreasing to at to = 10 meV. In YBa2fu306. 6 the pseudo-gap behavior is more pronounced and the intensities below 5 meV are near zero, although strong magnetic scattering is observed well below the weak coupling BCS 2A of ~18 meV [43]. Perhaps the most surprising feature of Fig. 14 is that for energies above --10 meV the q-integrated susceptibility X"(to) in Lal.85Sr0.15CuO 4 is about a factor of 3 larger than that in La1.98Sr0.02CuO 4 at the same energies. The latter in turn should be comparable to the spin wave value for X"(to) in the pure material La2CuO 4. This enhancement of the integrated susceptibility above that characterizing the lightly doped system is to us a very surprising result. We should note that exactly the same enhancement effect has been observed in YBa2Cu306. 6 (T c = 53 K) compared with YBazfu306.15 (T N = 415 K) [52]. Once more, the similarities in the energy and

171

temperature dependence of X"(q, to) in La1.85Sr0.15CuO 4 (T c = 33 K) and YBa2fu306.6 (T c = 53 K) are striking. Theoretical calculations of x"(q, to) with conventional s-wave BCS models predict that the spectral weight for to <2/1, where A is the superconducting gap, observed near (,rr(18), -rr) should be rapidly suppressed to zero as T drops below T c [50,51]. However, the spectral weight for a dx:_y2 wave gap can remain significant for T < T c. As shown in Fig. 11 strong magnetic scattering is observed well below the weak-coupling gap energy. It is also predicted by some theoretical calculations that for a dx2_y2 superconductivity wave function the geometry of X"(q, to) should change below To, and specifically new peaks at the (~r(1---6), "rr(1---6)) and (-rr(1 * 6 ), ~r(1 --- 6 )) positions should become dominant at low temperatures. However, as reported first in Ref. [53] for Lal.85Sr0.15Cu0.988Zn0.0120 4 (T c = 16 K) this component is unobservable within statistical error even at 1.9 K for 1.5 meV ~< T ~<4 meV. A similar result was obtained down to 10 K by Thurston et al. [54] in Lal.ssSr0.1sCuO 4. Consistent with these results, Matsuda et al. [48] find that in Lal.85Sr0.~sCuO 4 at low energies the line shapes and positions of the incommensurate peaks are nearly independent of temperature below T~ down to 1.9K (cf. Fig. 11). Thus it appears that the simplest dx2 y2 model for the superconductivity cannot be correct. This conundrum may, however, be resolved by inclusion of the effects of disorder.

6. Conclusions

In reviewing the phase diagram and magnetic properties of La2_xSrxCuO 4 it is clear that as x is progressively increased our understanding of the basic physics progressively decreases. The measurements in Sr2CuO2C12 show that we now have both excellent experimental and Monte Carlo data on the spin correlations in the S = ½ 2DSLHA together with a sophisticated and highly successful theory. The lightly doped region with its concomitant 3D N6el order is now well

172

G. Shirane et al. / Physica B 197 (1994) 158-174

characterized and a model based on quenched frustrated random bonds due to the doped holes seems capable of explaining the rich phenomenology. The magnetic properties in the 'spin glass' region of the phase diagram 0.02 ~
based on antiferromagnetic spin fluctuations. As we have noted in the previous section our results rule out both simple s-wave and simple d x 2 y ~ models for the superconductivity. Closely similar data and conclusions have recently been reported by Aeppli and co-workers [49]. There clearly is a need for much more experimental information on the spin fluctuations in the lamellar copper oxides. However, two regions deserve special a t t e n t i o n - t h e region around x - 0.06 to 0.07 where the spin fluctuations become incommensurate and the overdoped region. Currently no information at all exists about the ant~ferromagnetic spin fluctuations, or lack thereof, in this large-x region of the phase diagram.

Acknowledgements We would like to thank our collaborators A. Aharony, N. Belk, A. Cassanho, C.Y. Chen, R.W. Erwin, M. Greven, B. Keimer, H. Kojima, M. Mastuda, I. Tanaka, T.R. Thurston and K. Yamada for their role in this research. We would also like to thank G. Aeppli, S. Chakravarty, V.J. Emery, P.A. Lee, K. Levin, S. Sachdev, D.J. Scalapino, S.K. Sinha, A. Sokol, B.J. Sternlieb and J.M. Tranquada for helpful discussions. This work was suppaorted by the USJapan Cooperative Neutron Scattering Program. The work at Tohoku University was supported by a Grant-In-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture. The work at BNL was supported by the Division of Materials Science, the Office of Basic Energy Science of the US Department of Energy, under contract No. DE-AC0276CH00016. The work at MIT was supported by the US National Science Foundation under Contracts No. DMR 90-22933 and DMP 90-07825.

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G. Shirane et al. / Physica B 197 (1994) 158-174

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