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Spin and charge inhomogeneities in high-Tccuprates:

Evidence from NMR and neutron scattering experiments

Dirk K. Morr 1, Jorg Schmalian 2, and David Pines 1,3

1 University of Illinois at Urbana-Champaign, Loomis Laboratory of Physics, 1110 W. Green St., Urbana, IL 618012ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX United Kingdom

3 Institute for Complex Adaptive Matter, University of California, and LANSCE-Division, Los Alamos National Laboratory,Los Alamos, NM 87545

(April 25, 2011)

In this communication we consider the doping dependence of the strong antiferromagnetic spinfluctuations in the cuprate superconductors. We investigate the effect of an incommensurate mag-netic response, as recently observed in inelastic neutron scattering (INS) experiments on severalYBa2Cu3O6+x compounds, on the spin-lattice and spin-echo relaxation rates measured in nuclearmagnetic resonance (NMR) experiments. We conclude that a consistent theoretical description ofINS and NMR can be reached if one assumes spatially inhomogeneous but locally commensuratespin correlations and that NMR and INS experiments can be described within a single theoreticalscenario. We discuss a simple scenario of spin and charge inhomogeneities which includes the mainphysical ingredients required for consistency with experiments.

PACS numbers: 74.25.Ha,74.25.Nf

I. INTRODUCTION

Understanding the doping, frequency and tempera-ture dependence of the magnetic response in the high-Tc

cuprates is one of the most challenging problems for thehigh-Tc community. Insight into the nature of the strongantiferromagnetic fluctuations very likely holds the key tothe unusual normal state properties of the cuprates [1,2],as seen in a variety of experimental techniques. Bothnuclear magnetic resonance (NMR) and inelastic neu-tron scattering (INS) experiments are important toolsto probe spin excitations in cuprates [3–5]. While INSexperiments provide insight into the momentum and fre-quency resolved imaginary part of the spin susceptibility,χ(q, ω), NMR experiments yield information on the mo-mentum averaged real and imaginary part of χ(q, ω) inthe zero frequency limit. However, in contrast to NMRexperiments, INS measurements rely on the existence oflarge single crystals and often suffer from a rather limitedexperimental resolution. Therefore, these experimentaltechniques are complementary in the information pro-vided on spin excitations in the cuprates.

One of the central questions in the cuprate supercon-ductors is whether INS and NMR experiments can be si-multaneously understood within a single theoretical sce-nario. Thus far it is not even been clear whether one canreach agreement between INS and NMR data as far asthe order of magnitude of the spin susceptibility is con-cerned. This problem is caused, in part, by the fact thatNMR is sensitive to all magnetic fluctuations, regard-less of whether they are related to a pronounced momen-tum dependence of the spin susceptibility. In contrast,INS typically probes only those parts of the susceptibil-ity which are strongly momentum dependent; the rest aretypically attributed to the “background” of the signal, asmight be caused by the scattering of neutrons on nucleiand phonons.

The situation has been further complicated by re-cent INS experiments on YBa2Cu3O6+x which observea crossover from a substantial incommensuration in themagnetic response at low frequencies, to a commensuratestructure in χ at higher frequencies [6–8]. The questionthus arises: does this incommensurate order originatefrom spatially inhomogeneous correlations between thedoped holes [9], which would preserve a locally commen-surate magnetic response, or does it reflect homogeneousincommensuration, most likely due to Fermi surface ef-fects.

In this communication we argue that since NMRexperiments probe the local spin environment arounda nucleus, they are able to distinguish between a lo-cally commensurate and incommensurate magnetic re-sponse. Earlier theoretical studies of NMR experimentson YBa2Cu3O6+x and YBa2Cu4O8 used a phenomeno-logical form of χ [10]:

χ(q, ω) =αξ2

1 + ξ2(q − Q)2 − iω/ωsf

, (1)

where ξ is the magnetic correlation length in units of thelattice constant, a0, ωsf an energy scale characterizingthe diffusive spin excitations, α an overall temperatureindependent constant, and Q is the position of the peakin momentum space which was assumed to be commen-surate, i.e. Q = (π, π). Using this expression for χ(q, ω)a rather detailed quantitative understanding of variousNMR data has been reached [11–13]. In particular, fromthe analysis of the longitudinal spin lattice relaxationrate of the 63Cu nuclei, 1/63T1, and the spin spin re-laxation time, 1/T2G, scaling laws like ωsf ∝ ξ−z withdynamical critical exponent, z, have been deduced. Itwas found that z ≈ 1 between a lower crossover temper-ature, T∗, and a higher one, Tcr, whereas z ≈ 2 aboveTcr. In the temperature range where z = 1 scaling ap-plies, it follows that T1T/T2G is independent of temper-

1

ature; a conjecture which was experimentally verified byCurro et al. for YBa2Cu4O8 [14]. Below the pseudogaptemperature, T∗, ωsf and ξ decouple and a quasiparticleand spin pseudogap emerges. Moreover, it was recentlyshown that the temperature dependence of the incom-mensurate peak width, as determined in INS experimentson La2−xSrxCuO4 [15], showed z = 1 scaling over a widerange of temperatures, in agreement with the NMR find-ings for YBa2Cu4O8. Because of its appearance in bothNMR and INS results, we will therefore assume in thefollowing analysis that z = 1 is the proper scaling be-havior for magnetically underdoped cuprates between T∗

and Tcr.One of our central results is that the attempt to un-

derstand the NMR data for YBa2Cu4O8 with homoge-neous incommensuration is inconsistent with z = 1 scal-ing. This implies that the local magnetic response, whichis probed in NMR experiments, is commensurate. ShouldINS experiments show that this compound exhibits a(globally) incommensurate magnetic response similar tothat seen in La2−xSrxCuO4, this would strongly supporta dynamic charge and spin inhomogeneity (stripe) originof the incommensuration. In Sec. V we discuss a spin

and charge inhomogeneity scenario which reconciles theglobal incommensuration with the local commensurationin χ.

In a second important result we carry out a quanti-tative comparison of the strength of the antiferromag-netic spin fluctuations, as measured by NMR and INSexperiments. We find that agreement between the re-sults obtained from these quite different experimentaltechniques, which explore not only a different frequencyrange, but different wavevector regimes, can be obtainedwithin a factor of 2. Given the large uncertainties in de-termining the absolute value of χ′′, we believe that thisresult demonstrates that a consistent description of INSand NMR data can be achieved within the framework ofEq.(1).

Although the form of χ in Eq.(1) was originally in-vented to understand the spin response at very low fre-quencies and above the superconducting transition tem-perature [10], it has proved interesting to investigate towhat extent one can understand INS data at higher fre-quencies within the same framework. The results by Aep-pli et al. support the picture of a unique incommensuratespin response from zero energy up to 15 meV, the high-est energy used in Ref. [15]. Within the error bars ofthe experiment, only for momentum values away fromthe peak maximum do systematic deviations from Eq.(1)occur. Such deviations indicate the presence of latticecorrections to the continuum limit used in Eq.(1). Lat-tice corrections are expected to be extremely importantfor the local, momentum averaged, susceptibility:

χ′′

loc(ω) =1

4π2

∫

BZ

d2qχ′′(q, ω) , (2)

since the phase space of momenta away from the peakmaxima is considerable in two dimensions. For higher

frequencies spin excitations away from the antiferromag-netic peak and energetically large compared to ωsf comeinto play and we therefore expect a poor description ofχ′′

loc(ω) in terms of Eq.(1). This conclusion is indepen-dent of whether the peak position commensurate or in-commensurate peaks.

Deviations from the universal continuum limit of χ areexpected to be also of relevance for the relaxation ratesmeasured in NMR experiments since these are weightedmomentum averages of χ(q, ω). Thus, it is worthwhileto study whether one can find indications for lattice cor-rections from an analysis of NMR and INS data. Suchcorrections to χ are also of relevance for our understand-ing of the lifetime of so-called cold quasiparticles if oneassumes that the lifetime of these quasiparticles is alsodominated by scattering off spin fluctuations [16–19].

Our paper is organized as follows. In Sec. II we give abrief overview of the theoretical framework in which weanalyze the INS and NMR data. In Sec. III A we ana-lyze NMR data on YBa2Cu4O8 for both commensurateand incommensurate magnetic response and discuss therole of lattice corrections. In Secs. III B and III C we an-alyze NMR data on two YBa2Cu3O6+x compounds. InSec. IV we discuss the consistency between NMR andINS data, as well as the role of lattice corrections forthe local susceptibility. In Sec. V we propose a spin and

charge inhomogeneity scenario as a possible way to rec-oncile NMR and INS data. Finally, in Sec. VI we drawour conclusions.

II. THEORETICAL OVERVIEW

We briefly discuss the theoretical framework in whichwe discuss NMR and INS experiments. In order to ana-lyze the low frequency NMR data, we use the dynamicalspin susceptibility of Eq.(1), where for the commensu-rate case Q = (π, π). To allow for an incommensuratestructure of χ, we use Eq.(1) with Q = (1, 1 ± δ)π andQ = (1± δ, 1)π and sum over all four peaks [11]. The in-clusion of the correct form of lattice corrections is ratherdifficult since it requires a microscopic model which is be-yond the scope of this paper. In general we expect latticecorrections to appear in the form of an upper momentumcutoff, Λ. In the following we choose a soft cutoff proce-dure for δq ≡ q − Q in the denominator of Eq.(1):

δq2 → δq2

(

1 +δq2

Λ2

)

. (3)

We also expect a weak momentum dependence of α andωsf once the continuum description breaks down; how-ever, to keep the number of tunable parameters small weignore these effects.

In NMR experiments, one measures the spin-lattice re-laxation rate 1/T1x, with applied magnetic field in x-direction, and the spin-echo rate 1/T2G, which can beexpressed in terms of the dynamical susceptibility as:

2

1

T1xT=

kB

2h(h2γnγe)

2 1

N

∑

q

Fx(q) limω→0

χ′′(q, ω)

ω, (4)

( 1

T2G

)2

=0.69

128h2(h2γnγe)

4

{

1

N

∑

q

[

F effab (q)χ′(q, ω)

]2

−[

∑

q

F effab (q)χ′(q, ω)

]2

}

, (5)

where x = ab, c describes the direction of the externalmagnetic field. The 63Cu form factors are given by

63Fc(q) =

[

Aab + 2B(

cos(qx) + cos(qy))

]2

, (6)

63F effab (q) =

[

Ac + 2B(

cos(qx) + cos(qy))

]2

, (7)

63Fab(q) =1

2

[63

F effab (q) +63 Fc(q)

]

. (8)

where Aab, Ac and B are the on-site and transferred hy-perfine coupling constants, respectively [11].

It follows from Eq.(1) that the spin excitations in thenormal state are completely described by three parame-ters, α, ξ, and ωsf . In order to extract these parametersfrom the experimental NMR data, we also need to ob-tain the three hyperfine coupling constants, Aab, Ac, andB; hence we have six unknown parameters in the aboveequations, and require six equations to determine them.So far we have two, Eqs.(4) and (5). An additional con-straint arises from the temperature independence of the63Cu Knight shift in a magnetic field parallel to the caxis in YBa2Cu3O7 [11] which yields

Ac + 4B ≈ 0 . (9)

A fourth constraint comes from the anisotropy ofthe 63Cu spin-lattice relaxation rates [20], which forYBa2Cu3O7 was measured to be

63R =T1c

T1ab

≈ 3.7 ± 0.1 . (10)

A fifth constraint involving the hyperfine coupling con-stants can be obtained by plotting the Knight shift 63Kab

versus χ0(T ) [11], which yields

4B + Aab ≈ 200kOe

µB

. (11)

A final constraint is obtained from the earlier conjec-ture that the antiferromagnetic correlation length at thecrossover temperature Tcr is approximately two latticeconstants [13]. Recent microscopic calculations have con-firmed this conjecture, showing that indeed ξ(Tcr) is ofthe order of a few lattice constants [21].

To the extent that experimental data for T1 and T2G

are available up to Tcr, one can fit the above set of

six equations self-consistently to the data, and thus ob-tain not only the temperature independent parametersα, Aab, Ac, and B, but also the temperature dependenceof ξ and ωsf . It turns out, however, that this analysisis only possible for YBa2Cu4O8, since this is the singlecompound for which data up to sufficiently high tempera-tures have been obtained [14]. Our corresponding resultswill be presented in section III A. In order to extractthe relevant parameters from NMR data on the related,widely studied, YBa2Cu3O6+x compounds, and thus tomake a comparison between NMR and INS experimentspossible, we need to make two assumptions, which, atleast partly, will be supported by the experimental datafor YBa2Cu4O8. First, we assume a relation between ωsf

and ξ, given by [12,13]

ωsf = c ξ−z , (12)

where c is a temperature independent constant. ForYBa2Cu4O8, where we can independently extract ωsf

and ξ, we will show that there is indeed a crossover fromz = 1 behavior at low temperature to z = 2 behavior athigher temperatures. The second assumption concernsthe temperature dependence of the magnetic correlationlength, ξ(T ), which we take to be that obtained by oneof us using a renormalization group (RG) approach [22]:for z = 1 the temperature dependence of the magneticcorrelation length is given by

ξ−2 = ξ−20 + bT 2 , (13)

a result which is in good agreement with INS experimentson La2−xSrxCuO4 [15]. Having all the necessary theoret-ical tools in place, we now address the questions raisedin the introduction.

III. ANALYSIS OF NMR EXPERIMENTS

A. YBa2Cu4O8

YBa2Cu4O8 is a benchmark system for our analy-sis, since Curro et al. [14] have measured the relaxationtimes T1 and T2G over a wide temperature range from80K to 750K, and in particular between T∗approx200K and Tcrapprox500 K. We are therefore able to ex-tract the relevant parameters which we discussed abovefrom a self-consistent fit of Eqs.(4)- (11) to the T1 andT2G data. In doing so, we assume that the constraintsgiven by Eqs.(9)-(11) for YBa2Cu3O7 are also valid forYBa2Cu4O8. Since we can determine ξ and ωsf inde-pendently, we are able to study the effect of both an in-commensurate magnetic response, and of non-universallattice corrections, on the scaling law ωsf = c/ξz.

We first reconsider the case of a commensurate struc-ture of χ, and set Λ = 2

√π, i.e. we choose a momentum

cut-off which corresponds to the linear size of the BZ. Wepresent the temperature dependence of both ωsf and ξ−1,

3

0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.00.0

0.2

0.4

0.6

0.8

1.0Curro Corey et al.

et al.

Temperature [K]

(in meV

)

ωsf

ξ-1

FIG. 1. The temperature dependence of ωsf and ξ−1 inYBa2Cu4O8 for the commensurate case with Λ = 2π1/2.

which results from a self-consistent fit to the experimen-tal data, in Fig. 1. Between T∗ = 200 K and Tcr = 500K, ωsf and ξ−1 clearly scale linearly with temperature.In order to study the extent to which ωsf and ξ obey theabove scaling law, we plot in Fig. 2, ln(ωsf ) as a functionof ln(ξ) (lower curve). The dynamical scaling range isof course too limited to prove the existence of a scalingrelation. However, assuming ωsf ∝ ξ−z, we find z ≈ 1below 500 K, which we identify with Tcr and z ≈ 2 aboveTcr. These results are consistent with an earlier analysisof the scaling behavior [12,13] and the microscopic sce-nario of Ref. [21]. To study the effect of stronger latticecorrections, we decreased Λ to Λ = 2

√π/4 (upper curve

in Fig. 2 which for clarity is offset). Our conclusions con-cerning the scaling relations remain unchanged. We canthus conclude that the dynamical scaling behavior of spinexcitations is robust against even sizeable lattice correc-tions, as long as the structure of χ′′ is commensurate.

We next examine the effect of incommensuration onthe dynamical scaling. It was earlier argued that themagnetic response in YBa2Cu4O8 should be very sim-ilar to YBa2Cu3O6.8 [13]. Using the observed dopingdependence of the incommensuration in YBa2Cu3O6+x,we estimate the incommensuration in YBa2Cu4O8 to beδ = 0.23. Setting Λ = 2

√π we plot in Fig. 3 ln(ωsf ) as

a function of ln(ξ) (lower curve) and find, following thesame argumentation as above, that the dynamical scalingexponent below Tcr ≈ 500 K is now z ≈ 0.75. Increas-ing the strength of the lattice corrections by decreasingΛ we find that z is increased. However, in order to ob-tain again z ≈ 1 (the upper curve in Fig. 3), which onewould expect from the INS data on La2−xSrxCuO4, weneed a very small momentum cut-off, Λ ≈ 2

√π/15. Such

a small cut-off of order O(1/ξ) is inconsistent with thecontinuum theory which is the basis for a dynamical scal-ing approach. We therefore conclude that a spatially ho-

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8-4.4

-4.2

-4.0

-3.8

-3.6

-3.4

-3.2

-3.0

-2.8

-2.6

-2.4

-2.2

ln(ξ)

ln(ω

sf)

Curro et al.

Corey et al.

z=1.0

z=2.0

FIG. 2. ln(ωsf ) as a function of ln(ξ) for the commensu-rate case. Upper data set: Λ = 2π1/2/4; Lower data set:Λ = 2π1/2. Solid line: z=1; dashed line: z=2.

mogeneous incommensurate magnetic response is in con-tradiction with the available NMR data. The physicalorigin of this sensitivity of the magnetic response of ahomogeneously incommensurate system, compared to alocally commensurate one, results from the fact that theformer effectively decreases the spatial extent of the spin-spin correlations, thus increasing the role of large valuesof δq ∼ Λ. It should also be noted that our argumentsusing lattice corrections to discriminate between thesetwo scenarios works only for intermediate values of thecorrelation length. For very large ξ the system should beinsensitive to the cut-off procedure regardless of whetherit is commensurate or incommensurate.

In what follows we assume that the low frequency mag-netic response measured in an NMR experiment is indeedlocally commensurate which implies that z = 1 scalingprevails between T∗ and Tcr. In Sec. V we present a pos-sible theoretical scenario to resolve this apparent contra-diction between the incommensuration seen in INS exper-iments and our conclusions based on the available NMRdata. Finally, using the commensurate form of Eq.(1),and a momentum cut-off Λ = 2

√π we find the following

parameters from the solution of the above self-consistentequations: α = 10eV −1, Aab = 20.2 kOe/µB, Ac =−182.8 kOe/µB, and B = 45.7 kOe/µB for YBa2Cu4O8.

B. YBa2Cu3O7

Since it was earlier argued that for the optimally dopedYBa2Cu3O7, Tcr ≈ 125 K, only slightly above Tc, thereis no significant temperature dependence of the relax-ation rates between Tc and Tcr. We will therefore per-form our analysis of the NMR data only at Tcr. Assum-ing that ξ(Tcr) = 2, and setting Λ = 2

√π, we utilize

T2G data by Itoh et al. [23] and Stern et al. [24], as

4

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1-4.4

-4.2

-4.0

-3.8

-3.6

-3.4

-3.2

-3.0

-2.8

-2.6

-2.4

z=1.0

z=0.75

Curro

Corey et al. et al.

ln(ξ)

ln(ω

sf)

FIG. 3. ln(ωsf ) as a function of ln(ξ) for the incommen-surate case. Lower data set: Λ = 2π1/2; Upper data set:Λ = 2π1/2/15. Solid line: z=1; dashed line: z=0.75.

well as T1c data by Hammel et al. [25] to find the fol-lowing hyperfine coupling constants Aab = 27.8 kOe/µB,Ac = −175.2 kOe/µB, and B = 43.9 kOe/µB. Note thatthese values agree well with the hyperfine coupling con-stants we extracted for YBa2Cu4O8. With 1/T2G = 104

s−1 and T1cT ≈ 0.15 Ks at Tcr, we obtain α = 18.5 eV−1,ωsf ≈ 19.8 meV and c ≈ 39.6 meV. It turns out that theabove parameter set is rather robust against changes inΛ. When Λ is decreased from Λ = 2

√π to Λ = 2

√π/4, Ac

and B decrease by about 1.5%, Aab increases by about8.5%, α increases by 11%, and ωsf decreases by about15%. A change of the momentum cut-off by a factor of4, which decreases the area of integration in the BZ bya factor of 16, thus leads only to moderate changes inthe parameter set. However, as we show in Sec. IV, thecorresponding changes in the local susceptibility at highfrequencies are much more dramatic.

C. YBa2Cu3O6.63

In what follows we show that, though only a limitedset of NMR data on YBa2Cu3O6.63 is available, we never-theless reach the same conclusions regarding the scalingbehavior as in YBa2Cu4O8. In particular, since no NMRdata on YBa2Cu3O6.63 are available above T = 300 K,and since we do not know the exact value of Tcr, we can-not extract the parameter set from a self-consistent fitto the NMR data as we did in the case of YBa2Cu4O8.However, it is in this doping range that INS data areavailable and therefore a consistency check between theparameter sets extracted from NMR and INS measure-ments might be the most promising. It turns out thatthough we are not able to perform a fully self-consistentfit, we can still extract the parameter set if we make sev-

α (eV−1) c (meV) b (10−7 K2) ξ−2

0 (10−2)

Tcr = 650 K 11.0 68.1 5.1 3.5Λ = 2

√

π

Tcr = 650 K 12.2 61.2 5.0 3.9Λ = 2

√

π/4

Tcr = 550 K 13.0 67.3 6.7 4.7Λ = 2

√

π

Tcr = 550 K 14.5 59.9 6.5 5.3Λ = 2

√

π/4

TABLE I. Parameter sets for two different values of Tcr

and Λ.

eral additional assumptions which are supported by ourresults on YBa2Cu4O8. First, we assume that z = 1scaling is present between T∗ and Tcr and that we candescribe the temperature dependence of ξ and the re-lation between ωsf and ξ by Eqs.(12) and (13), re-spectively. Second, we assume that the hyperfine cou-pling constants are only weakly doping dependent, andthat to good approximation we can use the same con-stants we extracted from the analysis of YBa2Cu3O7 forYBa2Cu3O6.63. Third, we need an estimate for Tcr,which is unknown for YBa2Cu3O6.63. However, sinceTcr ≈ 500 K for YBa2Cu4O8, and since Tcr increaseswith decreasing doping, we assume Tcr ≈ 550−650 K forYBa2Cu3O6.63. To demonstrate the effect of the uncer-tainty in the latter assumption, we calculate he parame-ters α, ωsf (T ) and ξ(T ) for Tcr = 650 K and Tcr = 550 Kas well as for two different values of the momentum cut-off Λ. The resulting parameter sets based on experimen-tal data by Takigawa et al. [26] are shown in Table I. Notethat the parameters in Table I are rather robust againstlarge variations in Tcr and/or Λ and only differ at themost by about 30%. In Fig. 4 we compare our theoreti-cal results for the relaxation rates, 1/T1 and 1/T2G, withthe experimental data by Takigawa et al. [26]. We findthat the temperature dependence of the relaxation ratescalculated for each of the four different parameter sets inTable I, is practically indistinguishable; we therefore onlyplot the results for the parameter set with Tcr = 650 Kand Λ = 2

√π. This result is consistent with the ro-

bustness of the scaling behavior in the commensuratecase against non-universal lattice corrections as found forYBa2Cu4O8. We find a consistent description of the ex-perimental data for YBa2Cu3O6.63 using the assumptionof a z = 1 scaling relationship and local commensurationin the temperature regime T∗ ≈ 230 K < T < 300 K, and,as was the case for YBa2Cu4O8, assuming commensuratebehavior leads to results which are practically indepen-dent of non-universal lattice corrections.

5

Temperature T (K)

1/T

1T [

(Ks)

-1]

1/T

2G (

104 s

-1)

0.0 100.0 200.0 300.0 400.0 500.0 600.00.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.0 100.0 200.0 300.0 400.0 500.0 600.00.5

1.0

1.5

2.0

Takigawa et al.z=1.0, Tcr=650 K

Takigawa et al.z=1.0, Tcr=650 K

FIG. 4. Comparison of our theoretical results for the tem-perature dependence of 1/T1T (upper figure), and 1/T2G

(lower figure) in YBa2Cu3O6.63 with the experimental databy Takigawa et al. [26].

IV. INELASTIC NEUTRON SCATTERING

In this section we address the following two ques-tions:(i) What is the effect of lattice corrections on thelocal susceptibility, and (ii) can one describe the avail-able INS data with the parameter set extracted in theprevious section ?

As we discussed in the introduction we expect latticecorrections to lead to substantial corrections of χ′′

loc atfrequencies larger than ωsf . Above this energy scale wealso expect that the typical energy, ∆sw, of propagat-ing spin waves, which, at low frequencies are completelyoverdamped by particle hole excitations, comes into play,leading to a modified form of the spin susceptibility

χ(q, ω) =αξ2

1 + ξ2(q − Q)2 − iω/ωsf − (ω/∆sw)2. (14)

For ω < ∆2sw/ωsf no sign of a propagating peak exists

due to the diffusive character of the spin excitations. Forω > ∆2

sw/ωsf and q = Q the consequence of a propa-

χ loc(

ω)

(µB

2 /eV

)

YBa2Cu3O6.63

T=230 K

0.0 25.0 50.0 75.0 100.0 125.0 150.00.0

0.5

1.0

1.5

2.0

2.5

Λ= 2 πΛ= 2 π/4

ω (in meV)

FIG. 5. The local susceptibility χ′′

loc(ω) as a function offrequency for two different values of Λ.

gating mode is a pronounced pole in the excitation spec-trum if ∆sw < ωsf and a soft upper cut off in energy if∆sw > ωsf . The form of χ(q, ω) in Eq.(14) can be shownto describe the propagating spin mode in YBa2Cu3O6.5

above Tc [27].Using the parameter sets (1) and (2) in Table I, ex-

tracted in the previous section from NMR experiments,we plot in Fig. 5 the local susceptibility of YBa2Cu3O6.63

for two different values of Λ. We clearly see that upondecreasing the momentum cut-off, the maximum in χ′′

loc

moves towards lower frequencies. Experimentally, how-ever, the intensity at higher frequencies drops much fasterthan in Fig. 5, even for Λ = 2

√π/4 [28]. In order to

explain this discrepancy, we note that at higher frequen-cies, the dominant contribution to χ′′

loc comes from re-gions in momentum space which are far away from thepeak position at (π, π). In these regions, χ′′(q, ω) is onlyweakly momentum dependent in which case its contribu-tion might be easily attributed to the experimental back-ground. In other words, we believe that the origin of thediscrepancy lies in an underestimate of the experimentalintensity at higher frequencies due to the problems onehas in resolving it from the background [29].

In Fig. 6 we plot the calculated INS intensity, i.e.,χ′′(Q, ω) as a function of frequency at Q = (π, π). Sincethe calculated intensity is practically the same for all fourparameter sets in Table I, we only present χ′′(Q, ω) calcu-lated with the second parameter set in Table I (solid line).In the same figure, we also include the experimental databy Fong et al. for YBa2Cu3O6.7 [30]. For the tempera-ture range of interest, this material is the closest matchfor YBa2Cu3O6.63. Calculating the overall INS intensitywith the parameters extracted from NMR experiments inthe previous section, we find that we underestimate theexperimental INS intensity by about 56 %, or a factor of2.3 (the dashed line shows the calculated INS intensity,multiplied by an overall factor of 2.3). Given the uncer-

6

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.00.0

100.0

200.0

300.0

400.0

ω (in meV)

Fong et al.

T=200 K χ’

’ (Q, ω

) ( µ

B2/e

V)

YBa2Cu3O6.63

YBa2Cu3O6.7

Λ= 2 π/4

FIG. 6. χ′′(Q, ω) for YBa2Cu3O6.63. (a) Solid line:χ′′(Q, ω) calculated with parameter set (2) in Table I fromNMR experiments. (b) Dashed line: χ′′(Q, ω) from (a) mul-tiplied by an overall factor of 2.3. The experimental data arefrom Fong et al. [30].

tainties in the experimental determination of the absolutescale of χ′′, we believe that the above result representsreasonable agreement between the INS and NMR dataand thus demonstrates consistency in the description ofspin excitations based on these two experimental tech-niques.

V. SPIN AND CHARGE INHOMOGENEITIES

We saw in Sec. III A that under the assumption ofz = 1 scaling, supported by recent INS experiments [15],a locally incommensurate magnetic response is inconsis-tent with the available experimental NMR data. Wetherefore concluded that NMR data support a locallycommensurate magnetic structure. The question thusarises of how one can understand a locally commensu-rate, but globally incommensurate magnetic response asseen by INS? It has been suggested that charge stripesseparated by an average distance l0 = 2π/δ are the originof the incommensurate magnetic response seen in INS ex-periments [31]. For a large part of the sample this wouldleave the locally commensurate structure intact. Exceptfor the Nd-doped 214 compounds [32] and probably forthe particular doping value x = 1

8in La2−xSrxCuO2,

these stripes are believed to be dynamic rather thanstatic in nature.

Considering a spatial charge variation:

ρ(r) = ρ0 + δρ(r) , (15)

where ρ0 is independent of r. The question arises whetherthe system is in a regime with ρ0 ≪ δρ(r), as suggested inRef. [31], or rather characterized by rather small charge

variations, i.e. δρ(r) < ρ0. The particular appeal of theformer assumption of strong charge modulations is animmediate explanation of the doping dependence of theposition of the incommensurate peak, δ = 2x. Stripeswith one hole every second lattice site in a hole free an-tiferromagnetic environment move closer together upondoping, leading to the above x-dependence of δ. Also,the exceptional behavior of various magnetic and trans-port properties for the doping value x = 1

8can be easily

understood in terms of a stable commensurate arrange-ment of such stripes and the underlying lattice. Anotherargument in favor of this scenario are the recent resultsby Vojta and Sachdev [33] who investigated the meanfield behavior of a system with competing magnetic andlong range Coulomb interactions and found areas witha strong charge modulation separated by regions withbarely disturbed antiferromagnetic correlations.

Nevertheless, there are several conceptional problemswith such a strong charge modulation, which lead usto present some arguments favoring a more moderatecharge variation, δρ(r) < ρ0, under circumstances thatthe transverse mobility of the charge carriers with respectto the averaged position of the stripes is large. First,uncorrelated spatial stripes of width r0, which fluctuatetransversely over a distance d, will give rise to a broaden-ing d/l20 of the magnetic peaks observed by INS. The ex-perimental INS results, however, provide strong supportfor a scenario in which the width of the magnetic peaksis determined by ξ−1 [15]. Thus, in order to observe sep-arated incommensurate peaks, we need at least ξ ≫ d toestablish a well defined antiphase domain wall, implyingd ≪ l0. In other words, assuming uncorrelated stripes,the width d over which stripes fluctuate must be verysmall to account for the existing INS data. Such ”stiff”but uncorrelated stripes seem to be in contradiction tothe notion of stripes as a dynamical entity. Therefore, wearrive at the conclusion that spatial stripe fluctuations,if they exist, must be strongly correlated.

Second, at the characteristic temperatures wherestripes with ρ0 ≪ δρ(r) appear, strong modificationsof the resistivity and other transport properties are ex-pected to occur. None or only moderate changes of thiskind have been observed and no indication for a dimen-sional crossover from quasi one-dimensional to quasi two-dimensional dynamics seems to be present.

Third, for low frequencies, the spin excitations indoped cuprates are overdamped rather than propagatingspin waves. The suppression of the spin damping uponopening of a quasiparticle gap in the superconducting aswell as in the pseudogap state implies that the dominantsource of the spin damping, γ, are particle-hole excita-tions. Assuming weak charge modulations, we then findImχ−1(ω) ∝ γω, in agreement with the results of INSexperiments [15].

On the other hand, rigid stripes separated by one di-mensional charge carriers, which are effectively bosonicin character, lead to strong deviations from the abovefrequency dependence of Imχ−1(ω) [34,35], in disagree-

7

ρ0

magnetic stripes

magnetic clusters

d l0

elec

tron

cha

rge

dens

ity ρ

(r)

∆ρ

r0

FIG. 7. Schematic picture of charge and spin inhomo-geneities.

ment with INS experiments. Thus, Imχ−1(ω) does notarise from spatially varying stripes even though they cer-tainly affect the spin degrees of freedom and can causethe decay of these excitations.

Finally, NMR and other magnetic measurements onvarious cuprates which provide strong support for spatialinhomogeneities [36–41] suggest mostly spatially varyingspin degrees of freedom but not necessarily a strong mod-ulation of the charge background.

The above points suggest that the inhomogeneities ob-served in the high-Tc cuprates might be predominantlymagnetic in origin with only moderate modulations of thecharge density. This is not unrealistic since in stronglycorrelated systems small variations in the charge den-sity can bring about substantial changes in the magneticproperties of the system. This scenario of a predomi-nantly magnetic character of the inhomogeneities, lead-ing to magnetic stripes and clusters, is schematically pre-sented in Fig. 7. Note, that the magnetic stripes, inwhich the electronic charge density is lower than that inthe magnetic clusters, represent magnetic domain wallswith a phase slip of π in the ordering of the spins. Theo-retically, we believe that this more homogeneous chargearrangement is a result of quantum fluctuations beyondthe mean field level investigated in Ref. [33].

The formation of magnetic clusters and stripes requirescompeting interactions on different length scales. Be-sides the local Coulomb interaction, we need to identifya longer length scale interaction. It was argued earlierthat in the cuprate superconductors an increase of themagnetic correlation length leads to strong vertex cor-rections, which in turn give rise to a gradient couplingbetween the fermionic and spin degrees of freedom [42].For a system with an ordered ground state, this takes theform of a dipole-dipole coupling which has been shownto cause inhomogeneities and domain formation in twodimensional magnetic systems. The length scale of thisgradient coupling should roughly speaking be the mag-netic correlation length, ξ. We thus need a minimumvalue of ξ to (a) generate the gradient coupling, and to(b) observe the incommensuration. The temperature at

which incommensuration should appear is roughly set byξ(T ) ≈ l0 = 4, where the last equality arises from the mo-mentum position of the incommensurate peaks. Since, aswe argued earlier, ξ(Tcr) ≈ 2, Tcr thus presents an up-per bound for the formation of magnetic clusters. Thisanomalous coupling is of course enhanced once magneticclusters are formed; the creation of magnetic inhomo-geneities is a self-consistent process. The magnetic in-homogeneity scenario presented here is admittedly veryqualitative and further microscopic investigations will berequired to verify or disprove it.

VI. CONCLUSIONS

In this communication, we investigated whether it ispossible to understand theoretically INS and NMR datawithin a single theoretical framework. Based on the ob-servation of an incommensurate structure of the mag-netic response by INS experiments [8,15], we consideredwhether this reflects a locally or globally incommensu-rate ordering. On analyzing NMR data on YBa2Cu4O8

we found with the condition of z = 1 scaling that a ho-mogeneous incommensuration is, within the frameworkgiven by Eq.(1), inconsistent with the available experi-mental data. We thus concluded that the local magneticstructure as probed by NMR is likely commensurate.

We then investigated the effect of lattice correctionson both the parameter sets extracted from NMR dataand on the local susceptibility measured in INS exper-iments. We found that while the resulting correctionsto the NMR parameters are only weak, the most pro-nounced effect of such corrections appears in the localsusceptibility determined in INS measurements at fre-quencies above ωsf . Specifically, we found that decreas-ing the momentum cut-off leads to a suppression of χ′′

loc

at higher frequencies, thus improving the agreement withthe experimental data. We argued that the remainingdiscrepancies can be explained by an experimental un-derestimate of the INS intensity at higher frequencies.Furthermore, we quantitatively compared INS and NMRdata and found agreement within a factor of 2. Given thelarge uncertainties in resolving χ′′ from the large back-ground and in determining the absolute intensity of χ′′,we believe that this result demonstrates that a consistentdescription of INS and NMR data can be obtained usingthe expression for χ given in Eq.(1).

Finally, we discussed a spin and charge inhomogeneityscenario to reconcile the local commensurations, as seenin NMR experiments, and the incommensurate peaks,seen by INS. This scenario, even though similar in spiritto earlier charge stripe pictures, is based on a moder-ate to weak modulation of the charge density, causinga pronounced inhomogeneity in the magnetic propertiesand leading to magnetically coupled clusters separatedby weakly correlated stripes.

We would like to thank A.V. Chubukov, P. Dai, B.

8

Keimer, T. Mason, H. Mook, R. Stern, C.P. Slichter andB. Stojkovic for valuable discussions. This work has beensupported in part by the Science and Technology Centerfor Superconductivity through NSF-grant DMR91-20000(D.K.M.), and by DOE at Los Alamos (D.P.).

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9