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PHYSICAL REVIEW B 72, 064437 共2005兲

Mapping spin-wave dispersions in stripe-ordered La2−xSrxNiO4 (x = 0.275, 0.333) Hyungje Woo,1,2,* A. T. Boothroyd,3 K. Nakajima,4 T. G. Perring,2 C. D. Frost,2 P. G. Freeman,3 D. Prabhakaran,3 K. Yamada,5 and J. M. Tranquada1 1Physics

Department, Brookhaven National Laboratory, Upton, New York 11973, USA ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom 3Department of Physics, Oxford University, Oxford, OX1 3PU, United Kingdom 4Neutron Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan 5Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 共Received 29 April 2005; published 18 August 2005兲 2

Using the MAPS spectrometer at the ISIS spallation source, we have measured the magnetic excitations of single-crystal samples of stripe-ordered La2−xSrxNiO4 with x = 0.333 and 0.275. The full two-dimensional spin-wave dispersions were obtained using incident energies of 60 and 160 meV. To analyze the excitations, we have evaluated a spin-only Hamiltonian describing diagonal, site-centered stripes in the linear spin-wave approximation. Besides the superexchange energy J within antiferromagnetic domains, we have considered effective exchange couplings J1 and J2 across a charge stripe coupling second-neighbor Ni sites along Ni-O bond directions and along the plaquette diagonal, respectively. From least-squares fits of the model to the measurements on the x = 1 / 3 sample at T = 10 K, we find that the dispersions are well described by a model using just J and J1, but not J and J2. Consistent with an analysis of previous measurements, we find that J is about 90% of the superexchange energy of undoped La2NiO4 and J1 / J ⬇ 0.5. The excitations observed for x = 0.275 are surprisingly similar to those for x = 1 / 3, despite the differing magnetic-ordering wave vectors; the main difference is a broadening of the excitations for x = 0.275. For both samples, we find that one spin-wave branch has a gap of ⬃20 meV, confirming a previous observation for x = 1 / 3. We discuss the possible origin of this gap. DOI: 10.1103/PhysRevB.72.064437

PACS number共s兲: 75.30.Ds, 61.12.Ex, 75.30.Fv, 71.45.Lr

I. INTRODUCTION

The nature of the magnetic excitations in cuprate superconductors remains a topic of considerable interest and controversy.1–5 One school of thought associates the excitations with stripe correlations.6–9 While there have been a number of theoretical analyses of the spin fluctuations associated with a stripe-ordered state,9–19 there have been relatively few experimental tests of such models. One useful test system that exhibits well characterized diagonal stripe order is La2−xSrxNiO4+␦.20–24 Experimental measurements of spin waves, initially covering just low-energy excitations25 in La2NiO4.125, have more recently been extended to cover the full spin-wave bandwidth26–28 in La2−xSrxNiO4 with x ⬇ 0.33 and 0.5. The studies26,27 of x ⬇ 0.33 have shown that the spin waves observed along high-symmetry directions seem to be consistent with the predictions of linear spinwave theory. Here we present a more thorough mapping of the spinwave dispersion for samples of La2−xSrxNiO4 with x = 1 / 3 and x = 0.275. In analyzing the measured dispersions for x = 1 / 3, we consider a spin-wave model for site-centered stripes that allows for two possible effective couplings across a charge stripe 关see Fig. 1共a兲兴. Overall, we find that a good description of the data is obtained with a cross-stripe coupling along the Ni-O bond directions. The value of the superexchange between nearest-neighbor Ni sites within an antiferromagnetic domain obtained from a fit to the data, 27.5± 0.4 meV, is only slightly reduced relative to the value of 31± 1 reported for undoped La2NiO4.29 Comparison with calculations by Carlson et al.16 suggests that our results are 1098-0121/2005/72共6兲/064437共13兲/$23.00

incompatible with bond-centered stripes. One feature that is not properly captured by the spin-wave model is a gap near 20 meV that was previously identified by Boothroyd et al.26 It is tempting to look for a possible connection between this feature and the recently discovered quasi-one-dimensional magnetic excitations associated with the charge stripes;30 however, a proper understanding of the gap will require further research. For x = 1 / 3, there are only two magnetically ordered Ni sites per unit cell, and hence ignoring spin anisotropy, there should be a single, doubly degenerate branch of spin waves. In contrast, for x = 1 / 4 there are six magnetically ordered Ni sites per unit cell, and one expects, in general, to have a gap between acoustic and optic spin-wave branches.16 Our sample of x = 0.275 is effectively incommensurate; nevertheless, one might expect the excitations to look a bit more like the x = 1 / 4 case. Surprisingly, we find that the dispersions look very similar to those measured for x = 1 / 3. The main difference is a broadening of the spectra, which is large compared to the elastic magnetic peak widths.23,24,31 This effect has been attributed28 to the lack of simple commensurability for x ⫽ 1 / n.32 The rest of this paper is organized as follows. Experimental details are given in the following section. The spin-wave model is described in Sec. III. The experimental results are presented and analyzed in Sec. IV. Section V contains some discussion, while conclusions are presented in Sec. VI. We note that a very preliminary version of our experimental results was published previously in Ref. 33.

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FIG. 1. 共Color online兲共a兲 Schematic diagram of spin and charge stripe order within a NiO2 plane for the case of x = 1 / 3. Arrows indicate ordered S = 1 spins on Ni2+ sites, open circles are nominally Ni3+ sites within charge stripes. J, J1 , J2 label the effective exchange interactions discussed in the text. 共b兲 Similar to 共a兲, but showing a twin domain with stripes rotated by 90ⴰ. 共c兲, 共d兲 Reciprocal lattices corresponding to the order shown in 共a兲, 共b兲, respectively. Circles 共squares兲 indicate superlattice Bragg peak positions. A subset of these 共filled symbols兲 coincide with fundamental Bragg peaks from the average crystal structure. 共e兲 Diagram indicating experimental configuration, with superimposed reciprocal lattices of orthogonal stripe domains. Axes are rotated by 45° relative to 共c兲 and 共d兲. II. EXPERIMENTAL DETAILS

The x = 1 / 3 sample is a single crystal with a mass of 15 g and a mosaic of ⬃1° 共full width half maximum兲 grown at Oxford.34 Similar crystals have been characterized in recent triple-axis neutron-scattering studies.26,30 The x = 0.275 sample consisted of four coaligned crystals with a total mass of 45 g and a net mosaic of ⬃1°. The latter crystals were grown at Kyoto University; a similar crystal was studied in Ref. 31. The inelastic neutron-scattering experiments were performed on the direct-geometry chopper spectrometer MAPS 共Ref. 35兲 at the ISIS Facility, a pulsed spallation source operating at a frequency of 50 Hz. The neutrons that reach MAPS come from an ambient water moderator. The incident energy Ei of neutrons reaching the sample is controlled by a

Fermi chopper. The phasing of the chopper’s “open” state with respect to the time t0 at which the neutron pulse is created determines Ei, while rotation frequency of the chopper impacts the energy resolution. The sample temperature is controlled by a displex closed-cycle refrigerator. Neutrons scattered by the sample are counted by a position-sensitive detector covering an area of approximately 16 m2 at a radial distance of 6 m from the sample. The detector consists of 576 linear position-sensitive detectors, each 1 m long, 2.5 cm in diameter, and filled with 3He gas at 10 bars. Under typical conditions, the data are collected in a manner that effectively divides the detector area into approximately 40 000 pixels. The neutron arrival time is histogrammed into ⬃2500 time channels. Using the two position coordinates on the detector plus the neutron flight time, software converts the distribution to the four-dimensional 共4D兲 phase space of momentum transfer, qQ, and energy transfer, E = q␻. For a fixed orientation of the sample, kinematic constraints limit the measurement to a 3D surface within the 4D phase space. In the present case, the layered nature of the nickelate structure allows a simplified visualization of the data. We oriented each sample so that its c axis 共perpendicular to the NiO2 planes兲 was parallel to the incident beam. As a consequence, positions on the detector map onto Q储 within the NiO2 layers; however, for a fixed energy transfer, there is in general a finite momentum component Qz, perpendicular to the planes, which varies with position across the detector. In a study using a triple-axis spectrometer,36 a check was made for dispersive effects dependent on Qz, and none were found for energies above ⬃5 meV. Hence it is reasonable to plot the data in terms of Q储 and q␻, ignoring Qz for visualization purposes. On the other hand, to make a quantitative comparison of a model cross section with the data, it is necessary to take account of the full 共Q , ␻兲 dependence. The range of Q储 and ␻ obtained in a single measurement are controlled by the choice of Ei, with the useful data range extending up to q␻ ⬃ 32 Ei. For a fixed chopper frequency, the energy resolution becomes more coarse as Ei increases. Of course, improved energy resolution competes with neutron flux, so that one must generally find a reasonable compromise in selecting parameters. Our choices are listed in Table I. To convert the measured intensities to absolute units and to correct for variations in detector sensitivity, the raw data were normalized to measurements of white beam 共Fermi chopper removed兲 from a standard vanadium sample using the facility-provided program HOMER. This program also performs the conversion from detector-position-time binning to 共Q , ␻兲 space. The normalized data set corresponds to 2 ˜S共Q, ␻兲 = ki d ␴ , k f d⍀ f dE f

共1兲

with units of mbarn/共steradian meV formula unit兲. Here the subscripts i and f label the initial and final neutron wave vector k, solid angle of scattering ⍀, and energy E. The multidimensional data sets were then visualized with the program MSLICE,37 which was also used to prepare many of the

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MAPPING SPIN-WAVE DISPERSIONS IN STRIPE-…

TABLE I. List of the data sets measured for samples of La2−xSrxNiO4 on MAPS, together with associated parameter values. In all cases, the c axis of the sample was parallel to the incident beam. The proton beam energy was 800 MeV, and the spallation target was tungsten. The beam monitor is located slightly upstream of the Fermi chopper, and it is calibrated to give a measure of the time-integrated proton beam current.

Data set

Sample x

Tsample 共K兲

Ei 共meV兲

f chopper 共Hz兲

Counting time 共h兲

Beam monitor 共mA h兲

1 2 3 4 5

0.333 0.333 0.275 0.275 0.275

10 10 10 10 300

60 160 60 160 162

300 350 300 350 350

47.1 72.4 39.1 61.4 35.5

7.85 12.07 6.51 10.23 5.92

figures presented in the following. MSLICE can provide 2D slices and 1D cuts through the data, with integration of data over a specified range in directions transverse to the slice or cut. For quantitative fitting of the model scattering function 共see next section兲 to the data, the program TOBYFIT 共Ref. 38兲 was used. This program takes into account the resolution function of the spectrometer. Figure 2 shows an example of a constant-energy “slice” of data 共upper panel兲 measured with Ei = 160 meV, integrated over the energy-transfer range 50-55 meV, and plotted as a function of in-plane momentum transfer, where Q = 共h , k , l兲 in units of 共2␲ / a , 2␲ / a , 2␲ / c兲 with a = 3.8 Å and c = 12.7 Å. The white regions indicate areas not covered by detectors. The lower panel shows a simulation of the magnetic scattering using TOBYFIT and a model that will be discussed shortly. The latter shows a slice through cones of spin waves centered on the magnetic Bragg points 关see Fig. 1共d兲兴. Note that several magnetic Brillouin zones fit within the detector. The data in the upper panel look like a noisy version of the simulation; there is also enhanced signal at larqer Q due to overlapping phonon scattering 共not included in the simulation兲. Given the fourfold symmetry of the pattern, we can improve the statistics by reflecting all of the data into one quadrant and summing it. That is how we will present data in the rest of the paper. One final note: our labeling of the axes in momentum space leads to some awkwardness. As indicated in Fig. 1共d兲 and in Fig. 2, the crystal is oriented such that the horizontal component of Q is 共h , h , 0兲 while the vertical component is 共k , −k , 0兲. We will label the distance along each axis in terms of h, so that the corresponding magnitude of Q is 冑2ha* = h ⫻ 2.34 Å−1. 共The only deviation from this rule occurs in Fig. 9, where the scan is along a diagonal direction, and Q = ha*.兲 To describe some of the slices and cuts, it is necessary to specify the range of Q over which data are integrated. To make this unambiguous, we will use Qh = h along the 关h , h , 0兴 direction and Qk = k along 关k , −k , 0兴.

sites within an antiferromagnetic domain. There are two possible couplings between Ni2+ ions across a stripe as illustrated in Fig. 1共a兲: J1 connects third-nearest neighbors through a linear series of Ni-O bonds, while J2 connects next-nearest neighbors along a diagonal path that is shorter but with 90° bonding paths. Boothroyd et al.26 evaluated a model including J and J1, plus an out-of-plane spin anistropy energy Kc. Alternatively, Krüger and Scheidl15 considered a model with J and J2, while Carlson et al. used J , J1, and J2. Here we evaluate a Hamiltonian containing all of these terms: H=

1 2

兺 Jr,r⬘Sr · Sr⬘ + Kc兺r 共Szr兲2 ,

共2兲

r,r⬘

where the indices r and r⬘ run over all spin sites, and the couplings Jr,r⬘ are as defined above. Here each Jr,r⬘ represents the exchange energy per bond. 共In contrast, the J’s in Ref. 26 are defined as exchange energy per spin.兲 We note that in writing down this model we are ignoring spin degrees of freedom that exist on the charge stripes themselves, and we will come back to this issue later. The energy dispersion and scattering cross section are calculated using the linear spin-wave approximation. There are two spin-wave modes, whose squared frequencies can be expressed as 2 ␻q± = ␻21 − 共兩␻2兩 ± ␻3兲2 ,

共3兲

␻1 = ␻0共1 + ␭1 + 21 ␭2 + ␭c兲 ,

共4兲

where

␻2 = ␻0关ei␲共h+k兲cos ␲共h − k兲 + ␭1e−i2␲共h+k兲cos 2␲共h − k兲 + 21 ␭2e−i2␲共h+k兲兴 ,

共5兲

III. MODEL: LINEAR SPIN-WAVE THEORY

Several analytic treatments have been reported for period =3, diagonal, site-centered stripes;15,16,26 they differ in the choice of effective exchange couplings across the stripes. All assume a superexchange J between nearest-neighbor Ni2+

␻ 3 = ␻ 0␭ c .

共6兲

¯ 0兴 diHere we have assumed that the stripes run in the 关11 rection, as in Fig. 1共a兲, and we have set

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FIG. 2. 共Color兲共upper panel兲 Representative constant-energy slice for the x = 1 / 3 sample, measured with Ei = 160 meV and integration of data over the energy range 50⬍ E ⬍ 55meV. 共lower panel兲 Simulation, using the spin-wave model and parameter values of fit 1 in Table II, with instrumental resolution taken into account. The coordinate system corresponds to Fig. 1共e兲. The circles correspond to a slice through cones of spin waves centered on the Bragg vectors of the magnetic reciprocal lattice.

␻0 = 2JS/q,

␣,␤

with ␭1 = J1 / J , ␭2 = J2 / J, and ␭c = Kc / 共2J兲. The scattering function of Eq. 共1兲 can be written as39 ˜S共Q, ␻兲 = 共 where

兲

1 2 2 2 −2W共Q兲 S共Q, ␻兲, 2 ␥r0 g f 共Q兲e

␣␤ ˆ Q ˆ S共Q, ␻兲 = 兺共␦␣␤ − Q ␣ ␤兲S 共Q, ␻兲.

共7兲

共8兲

共9兲

Here f共Q兲 is the 共approximately兲 isotropic magnetic form factor for Ni2+ , W共Q兲 is a Debye-Waller factor, ␥r0 / 2 = 0.2695⫻ 10−12 cm, ␣ and ␤ label directions in real space, ˆ = Q / 兩Q兩. 关Note that in applying this formula to fit the and Q ␣ ␣

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data, we will set exp共−2W共Q兲 = 1.兴 We know that the spins lie within the NiO2 planes.40 In La2NiO4.133, the spins are collinear and point parallel to the stripes,41 whereas the spin direction seems to be rotated in Sr-doped samples,21,31 although it has not been proved whether the spins are collinear in the latter cases. For simplicity, we will assume a distribution of spin directions in the plane, and average over scattering factors Sxx and Syy. Also, the gap for out-of-plane fluctuations is only ⬃7 meV,26 so for most of the spin-wave bandwidth it is reasonable to take Syy = Szz. 共This applies for a local region with ordered spins that point along the x direction. For a region with spins along y, we take Sxx = Szz.兲 With these approximations we have

共 23 − 21 Qˆz2兲Szz , ˆ 2兲共S+− + S−+兲. = 41 共 23 − 21 Q z

S共Q, ␻兲 =

共10兲

Evaluating the correlation functions, we obtain S共Q, ␻兲 = 21 S

共2␲兲3 V0

共 23 − 21 Qˆz2兲兺 ␦共Q + q − ␶兲

FIG. 3. 共Color online兲 Diagram indicating paths of cuts through the data used in the analysis, superimposed on the simulation of Fig. 2 after folding the “data” into a single quadrant.

q,␶

⫻ 兺共1 − e

−q␻/kBT −1

兲兩uqj − vqje

IV. RESULTS

i␶·a 2

兩

A. x = 1 / 3

j=1,2

⫻关␦共␻ − ␻qj兲 − ␦共␻ + ␻qj兲兴, where uqj =

vqj =

冑

冑

共11兲

␻1 + ␻qj , 2␻qj

共12兲

␻1 − ␻qj −i␾ e , 2␻qj

共13兲

and tan ␾ = Im共␻2兲/Re共␻2兲.

共14兲

Equation 共11兲 gives the dynamic structure factor for a single orientation of stripes. To compare with experiment, we have to average over both possible stripe domains. Also, to allow for damping, we make the replacement

␦共␻ − ␻qj兲 →

1 A⌫ , ␲ 共␻ − ␻qj兲2 + ⌫2

共15兲

where A is a scale factor and ⌫ is the energy width. In applying Eq. 共11兲 to model the data, we note that the factor 共2␲兲3 兺 ␦共Q + q − ␶兲 V0 q,␶

共16兲

is a formal way of expressing the fact that the spin waves appear in every Brillouin zone. For calculations, it can be replaced by a factor N, equal to the number of Ni ions in the sample. We divide this factor out to compare with the data, which have already been normalized to scattering per Ni ion. We must also average Eq. 共11兲 over the two orientations of stripe domains.

We will begin by considering the results for the sample of La2−xSrxNiO4 with x = 1 / 3. Figure 3 provides a road map to guide much of our discussion of the data and analysis. It corresponds to the simulation of Fig. 2 after reflecting the data back to a single quadrant of reciprocal space. The arrows indicate various directions along which we will analyze the spin-wave dispersion. It is common to plot dispersion curves in the form of energy vs Q along symmetry directions. Thus it is natural to try to plot the data in a similar form. In order to compensate for the decrease in the scattering cross section with increasing energy, we plot the data in the form of intensity ⫻ energy in Fig. 4. Panels 共a兲 and 共c兲 correspond to the direction labeled Cut A in Fig. 3, while 共b兲 and 共d兲 correspond to Cut B. The vertical “V”s of the spin waves are fairly obvious in 共a兲 and 共b兲, measured with Ei = 60 meV. In 共c兲 and 共d兲, measured with Ei = 160 meV, the relatively stronger signal of the phonon bands 共running roughly horizontally兲 makes it more challenging to pick out the spin waves. 关The analysis leading to the superimposed spin-wave curves in 共c兲, and the simulations in 共e兲 and 共f兲, will be explained shortly.兴 The one feature we wish to point out here is the enhancement in the spin-wave intensity above 20 meV, as can be seen in 共a兲 and 共b兲. That this feature is definitely due to spin waves and not to superimposed phonons was demonstrated by previous measurements using polarization analysis.26 Given that many of the phonon branches show relatively little dispersion, we can get clearer images of the spin-wave scattering by plotting constant-energy slices, as in Fig. 5. Here we have restricted the Q range to the first antiferromagnetic Brillouin zone. Up to about 60 meV 关panels 共a兲–共c兲兴, the slices appear to cut through nearly isotropic cones of spin waves. In the 60–65-meV interval, panel 共d兲, the cones start to overlap, and the intensity is enhanced at the points of intersection. Continuing to higher energies, the dispersions

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FIG. 4. 共Color online兲 Data slices in E vs Q planes, with Q along Cut A of Fig. 3 in 共a兲 and 共c兲, and along Cut B in 共b兲 and 共d兲, illustrating the superposition of spin-wave and phonon dispersions. The intensity is multiplied by the energy to compensate for the energy dependence of the spin-wave and phonon cross sections. The incident energy was 60 meV for 共a兲 and 共b兲; 160 meV for 共c兲 and 共d兲. 共e兲 and 共f兲 are simulations of the magnetic scattering corresponding to 共c兲 and 共d兲, respectively, calculated using the spin-wave model discussed in the text and the parameter values of fit 1 in Table II.

merge at the antiferromagnetic wave vector, QAF = 共 21 , 21 , 0兲, in the 75–80-meV window 共e兲, while the dispersions reach the corners of the antiferromagnetic Brillouin zone in the 85–90-meV interval 共f兲. 共We will delay the discussion of the x = 0.275 results to later.兲 We can determine the values for the parameters of the spin-wave model that best describe the data by performing a least-squares fit. It would be ideal if we could perform fits to 2D constant-energy slices; however, it would be challenging to handle the “background” due to the phonons, as well as a challenge to present the results. Instead, we have chosen a number of constant-energy cuts along the directions indicated in Fig. 3. The selected data sets are plotted in Figs. 6–9. For incident energies below 50 meV, we made use of the data obtained with Ei = 60 meV, which have better resolution. For higher energies, we used the Ei = 160-meV data. For the Ni2+ magnetic form factor we have used the 具j0典 form factor from Ref. 42. 共We note that previous studies of magnetic Bragg intensities in pure La2NiO4, Ref. 43, and La2NiO4.13, Ref. 41, have found that the experimentally determined magnetic form factor shows considerably less variation with Q at small Q than does 具j0典; this is presumably due to effects of hybridization with oxygen ligands.43 Here, for simplicity, we have ignored this difference. The main impact of this choice is likely to be in the fitted amplitude for the spin waves.兲 To get a good fit to the data at lower energies, we found that it was essential to take account of the gap of ⬃20 meV. Now, in our spin-wave model there is only one gap, that for out-of-plane spin fluctuations, and we know26 that the magnitude of the gap is about ⬃7 meV ; the in-plane fluctuations

have a negligible gap. As we have only two spin-wave branches in our model, we have no degrees of freedom left to describe the 20-meV gap; nevertheless, we have found that by applying the formulas in an ad hoc manner we can obtain a reasonable fit to the data for energies greater than 7 meV. As we are interested here in characterizing the dispersions over larger energy scales, we chose to ignore data at the lowest energies 共below 7 meV兲 where the proper gap for out-of-plane fluctuations is relevant. We then adjusted the anisotropy parameter Kc to model the larger gap. The energy of the gap at a magnetic Bragg wave vector is 关8Kc共J + J1 1/2 + 21 J2兲兴 . In early stages of the fitting process we adjusted Kc to yield a gap of about 23 meV, and then fixed the value of Kc. The “background” signal is largely due to scattering by phonons, as is apparent in Fig. 4. We attempted to pick energy windows where the phonon contribution was relatively constant with Q, and then constrained the fitted background to be linear in Q. 共For cuts B and C, where the cross section is expected to be symmetric about Qk = 0, the slope was fixed at zero.兲 A parameter that can become correlated with the background is the energy width ⌫. To minimize this effect, we performed fits with ⌫ fixed to several small values. We found that we could obtain quite reasonable fits with ⌫ fixed at 3 meV, and that the fits were not very sensitive to changes on the order of ±1 meV. In principle, ⌫ may vary with Q within the Brillouin zone corresponding to the magnetic stripe order; however, correlations with other parameters would make it difficult to obtain meaningful information about such Q dependence from the present data. In the spin-wave model described in the last section, we have included two different couplings across a stripe, J1 and

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FIG. 5. 共Color兲 Comparison of constant energy slices for the x = 1 / 3, 共a兲-共f兲, and x = 0.275, 共g兲共l兲, samples. Each panel is labeled by the energy range over which the data have been integrated, with the excitation energy increasing from bottom to top. 共a兲, 共b兲, 共g兲, and 共h兲 correspond to Ei = 60 meV, and the rest to Ei = 160 meV. The Q range is restricted to the area of an antiferromagnetic Brillouin zone.

J2. To test whether both couplings are necessary to describe the data, we have performed several different fits; in each, all of the cuts shown in Figs. 6–9 共and only those cuts兲 were fit simultaneously. 共Note that a single amplitude factor A applies to all cuts, but the background parameters are different for each cut.兲 The parameter values obtained in the various fits are shown in Table II. In fit 1, we set J2 = 0, and just varied J and J1. The calculated results are indicated by the

solid lines in Figs. 6–9 and were used for the simulations shown in Figs. 4共e兲 and 4共f兲. While the fit is not perfect, it does successfully describe most of the major features seen in the data. In fit 2, we allowed J2 to vary along with J and J1. The extra degree of freedom allowed a very small decrease in ␹2; however, the improvement in the quality of fit is not significant. For fit 3, we set J1 = 0, but here there was a very large

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HYUNGJE WOO et al.

FIG. 6. Constant-energy cuts for the x = 1 / 3 sample along the direction labeled Cut A in Fig. 3. Data have been averaged over the indicated energy ranges; all were measured with Ei = 60 meV. Solid lines indicate fitted spin-wave model 共including convolution with instrumental resolution function兲 using the parameter values of fit 1 in Table II. Dashed lines indicate fitted background.

FIG. 7. Data and fits along Cut B 共see caption of Fig. 6 for description兲. Incident energies of 60 meV 共160 meV兲 were used for energy transfers below 共above兲 50 meV.

FIG. 8. Data and fits along Cut C 共see caption of Fig. 6 for description兲. The incident energy was 160 meV for all cuts.

increase in ␹2. We conclude that an adequate description of the data can be obtained using exchange parameters J and J1 共and J1 ⬇ 0.5J兲, but not with J and J2 alone. 共Differences in the dispersions resulting from these two parameter sets are illustrated in Fig. 9 of Ref. 16.兲 As a further test of the different models, we compare in Fig. 10 several constant-energy slices of data with simulations using the parameters of fits 1 and 3. The selected slices, panels 共a兲–共c兲, are at energies close to the dispersion maximum, which is the range most sensitive to the model differ-

FIG. 9. Data and fits along Cut D 共see caption of Fig. 6 for description兲. The incident energy was 160 meV for all cuts.

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TABLE II. Parameter values obtained for several different fits to the data for the x = 1 / 3 sample, as discussed in the text.

Fit

J 共meV兲

J1 共meV兲

J2 共meV兲

Kc 共meV兲

⌫ 共meV兲

A

␹2

1 2 3

27.5± 0.4 27.9± 0.6 45.9± 1.0

13.6± 0.3 11.0± 0.5 0.0共fixed兲

0.0 共fixed兲 5.4± 0.6 6.4± 0.4

1.58 1.58 1.58

3 3 3

0.318± 0.008 0.311± 0.009 0.138± 0.007

2.16 2.12 5.27

ences. As one can see, the simulations for fit 1, panels 共d兲– 共f兲, give a remarkably good match to the experimental results, while for fit 3, panels 共g兲–共i兲, the agreement is quite poor for E ⬎ 70 meV. B. x = 0.275

A series of constant energy slices for the x = 0.275 sample are presented in panels 共g兲–共l兲 of Fig. 5. While the splitting of the magnetic peaks about QAF is somewhat smaller than for x = 1 / 3, the excitations look remarkably similar. The main difference is that the features observed for x = 0.275 appear broadened compared to x = 1 / 3.

If the spin-wave velocity is roughly independent of doping and the incommensurate splitting is reduced, then one would expect the excitation energy at QAF to decrease.12,15,16 Such an effect is not apparent in Fig. 5; however, to get a better measurement of this, we plot in Fig. 11 the intensity, multiplied by energy, as a function of the excitation energy for Q integrated over a small area about QAF. In the lowtemperature spectra, the peaks at ⬃22 and 47 meV are due to phonons. At 10 K, the feature at about 80 meV corresponds to spin waves; note that it is at the same energy for both samples. The high-energy feature disappears at 300 K, and weight appears to move down to ⬃45 meV. For comparison, we plot in Fig. 12 similar spectra centered at 共2,2,0兲, where

FIG. 10. 共Color兲 Comparison of measured and simulated constant energy slices for the x = 1 / 3 sample. 共a兲–共c兲 Data, 共d兲–共f兲 simulations using the parameter values of fit 1, 共g兲–共i兲 simulations using fit 3. 064437-9


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FIG. 11. 共Color online兲 Constant-Q cuts at QAF; data averaged over 0.495⬍ Qh ⬍ 0.505 and −0.05⬍ Qk ⬍ 0.05兲. The intensity is multiplied by the energy. Filled squares: x = 1 / 3 sample at 10 K; filled circles: x = 0.275 sample at 10 K; open circles: x = 0.275 sample at 300 K. The low-temperature features at 5, 20, and 45 meV are due to phonons, while the peak at about 80 meV is due to spin waves.

we expect to see only phonons, due to the facts that the magnetic cross section decreases with Q because of the magnetic form factor, while the phonon cross section, on average, increases as Q2. Note that the particular phonons present in each figure are different because the scans are at different points in the first Brillouin zone. In particular, the 47-meV mode in Fig. 11 has dispersed to lower energy in Fig. 12, and the 85-meV zone-center mode in Fig. 12 is known44 to split and weaken as one moves towards Qh = 0.5, so that it would not be detectable in Fig. 11. The changes with temperature in Fig. 12 are minimal,45 supporting the interpretation of softened spin fluctuations in Fig. 11. This result is consistent with the temperature dependence reported by Bourges et al.27 It is also interesting to check whether the gap at ⬃20 meV is also present in the x = 0.275 sample. To overcome the broadening of the spectra, we integrated the inten-

FIG. 13. 共Color online兲 Intensity vs energy at a magnetic zone center. For x = 1 / 3 共squares兲 the data have been integrated over the window 0.25⬍ Qh ⬍ 0.41, −0.15⬍ Qk ⬍ 0.15; for x = 0.275 共circles兲 the window is 0.27⬍ Qh ⬍ 0.43, −0.15⬍ Qk ⬍ 0.15. The step at ⬃20 meV is evidence of a gapped spin-wave mode in both samples.

sity over an area in reciprocal space centered on a magnetic superlattice point; the integrated intensities are plotted as a function of energy in Fig. 13. The figure shows that the gap is present in both samples. Another issue concerns the question of whether the spin waves might split into acoustic and optic branches, with an energy gap in between. For example, in the case of diagonal site-centered stripes separated by four atomic rows 共the idealized x = 1 / 4 situation兲, and taking J1 ⬇ 0.5J as we have found for x = 1 / 3, one would expect to find a gap at about half of the maximum spin-wave energy.15,16 共The gap is a consequence of the larger magnetic unit cell required for x = 1 / 4.兲 Of course, our x = 0.275 is “incommensurate”—it should locally have a long-period commensurability similar to that observed32 in La2NiO4.133. A well-ordered long-period structure would likely have multiple spin-wave gaps. In any case, we compare cuts at a number of energies for the x = 0.275 and 1 / 3 samples in Fig. 14. From these data it is difficult to identify any feature resembling a gap. Aside from the differences in incommensurability and peak broadening, the results for x = 0.275 look remarkably similar to those for x = 1 / 3. V. DISCUSSION

FIG. 12. 共Color online兲 Similar to Fig. 11, but at a fundamental Bragg point, Q = 共2 , 0 , 0兲; data averaged over 1.95⬍ Qh ⬍ 2.05 and −0.05⬍ Qk ⬍ 0.05兲.

We have seen that the spin-wave spectrum of the x = 1 / 3 sample is described rather well by the J − J1 model 共but not by the J − J2 model兲, confirming the previous analysis of Boothroyd et al.26 The value of J obtained in the fit 共27.5± 0.4 meV兲 is about 90% of the superexchange energy found in the undoped antiferromagnet.29 This is a remarkably small change considering the large density of holes doped into the NiO2 planes. It provides strong support for the interpretation of this system as a doped antiferromagnet.46,47

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FIG. 14. 共Color online兲 Constant energy cuts 共Cut A兲 comparing data for x = 0.275 共open circles兲 with x = 1 / 3 共filled squares兲; the lines are guides to the eye. Data are averaged over −0.05⬍ Qk ⬍ 0.05. The incident energy was 60 共160兲 meV for excitation energies less than 共greater than兲 50 meV.

The value of J1 is fairly large, considering that it describes the effective coupling between next-nearest-neighbor Ni sites along the bonding direction across a charge stripe. Note that nothing more than nearest-neighbor coupling is necessary to describe the spin waves in pure La2NiO4,29 and we have not included any couplings beyond nearest neighbor within the antiferromagnetic domains of the present model. Carlson et al.16 have provided formulas for the anisotropic spin-wave velocity appropriate for our J − J1 model. For the directions perpendicular and parallel to the stripes the respective spin-wave velocities, relative to the undoped antiferromagnet, are v⬜ / vAF = 23 冑␭1 and v储 / vAF = 21 冑1 + 5␭1 + 4␭21. For our case of ␭1 ⬅ J1 / J ⬇ 0.5, we obtain v⬜ = v储 = 1.06vAF. Thus the spin-wave velocity is effectively isotropic, and close to that of the antiferromagnet,27 despite the significant difference between J and J1. It is also of interest to consider the bandwidth of the spin waves. For our stripe-ordered x = 1 / 3 sample, the spin-wave energy at QAF is approximately equal to 4J冑␭1 = 77 meV. This compares to q␻max = 4J = 128 meV for the antiferromagnet. Thus the difference between J and J1 is reflected in the bandwidth but not the velocity. The fitted amplitude indicates that the measured signal appears to be only ⬃1 / 3 of that predicted by the spin-wave model. Given how well defined the measured excitations are, and the substantial charge excitation gap,48 a discrepancy of this magnitude comes as a bit of a surprise. As a check on the data normalization, we considered the nuclear incoherent elastic scattering. The calculated value for La2−xSrxNiO4

with x = 1 / 3 is 570共40兲 mbarn sr−1f.u.−1, due mostly to the La and Ni, while the measured value is 948共20兲关968共18兲兴 mbarn sr−1f.u.−1 for Ei = 60 meV关Ei = 160 meV兴. The incoherent signal from the sample should be much larger than that from the aluminum sample holder, can, and shields; however, we did not measure the scattering from the cryostat without the sample, so we are uncertain of the contribution from the sample environment. In any case, the measured signal is as large as the expected value and then some, so there is no evidence that the normalization process has underestimated the scattering from the sample. A small part of the reduced amplitude could come from our use of an inexact magnetic form factor, as discussed earlier. It is also possible that a significant part of the missing signal has gone into two-magnon excitations.49 Such excitations would form a diffuse continuum extending to twice the highest magnon energy 共⬃170 meV兲. It would be interesting to test for such a continuum, but such efforts are beyond the scope of the present study. The spin gap of about 20 meV is not properly described by the spin-wave model considered here. Of course, we have ignored the spin degrees of freedom within each charge stripe, and simply considered an effective cross-stripe exchange. Recently, spin excitations associated with the charge stripes have been discovered and characterized.30 They have been observed to disperse up to 10 meV, and their intensity disappears into the background as the temperature approaches 100 K. There must be a coupling between the spins on the charge stripes and those in the intervening antiferromagnetic domains. One might hope that taking proper account of the coupling would help to account for the 20-meV gap; however, there is difficulty with this idea. For the collinear arrangement of spins shown in Fig. 1, the coupling between Ni2+ and Ni3+ sites is frustrated by the antiparallel alignment of Ni2+ spins across a charge stripe. Due to this frustration, any Heisenberg coupling to the Ni3+ spins will have no impact on the spin-wave spectrum at energies comparable to the 20-meV gap. One possibility is that the spin configuration shown is not quite correct. A change in spin orientation is known to occur in La5/3Sr1/3NiO4 below a temperature of ⬃50 K.31 If, instead of the uniform rotation that has been considered, the change had a spiral-like component,50 then the coupling to the Ni3+ moments would no longer be frustrated. Intriguingly, a large drop in the dielectric constant has recently been reported to occur near the spin-canting temperature,51 and this is also the region where magnetization measurements indicate the onset of spin-glass-like behavior.52 As these changes occur well below the charge and spin ordering temperatures, it appears that there might be an adjustment in the charge order coupled to the magnetic changes. If these phenomena have a connection to the spin gap, then one might expect the gap to show a change near 50 K. This will be studied in a future experiment. Another possible explanation for the spin gap is associated with the zigzag chainlike nature of the S = 1 spins within an antiferromagnetic domain. An isolated S = 1 chain should have a Haldane spin gap. For example, Y2BaNiO5 contains spin chains with no static magnetic order and a gap of about 9 meV.53,54 The size of the gap is theoretically predicted to be

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0.4J, so in our stripe-ordered sample with J ⬃ 28 meV, we might expect a gap of 11 meV. Coupling Haldane spin chains to an independent magnetic sublattice can actually increase the gap energy.55 Furthermore, Nd2BaNiO5 is an example56 where, well into the ordered state, the intensity of the longitudinal Haldane excitation goes to zero, leaving only gapped transverse excitations. Such behavior would be consistent with the polarized-neutron studies36 of La2−xSrxNiO4. If such ideas were to turn out to be relevant, it would make a significant parallel with the quantum excitations discovered recently in a stripe-ordered cuprate.2 It seems somewhat surprising that the spin-wave dispersions in the x = 0.275 sample are so similar to those found for x = 1 / 3. In particular, one might expect to see a slightly different bandwidth; however, this experimental result is consistent with Raman scattering studies57–59 which show that the two-magnon excitation energy does not change between x = 0.225 and x = 1 / 3 共an even larger doping range than we have studied here兲. There is a second peak observed in Raman scattering, at about 735 cm−1 共92 meV兲 in La2−xSrxNiO4 共685 cm−1 in La2NiO4+␦,60 that has also been interpreted as a two-magnon feature.57,58 The temperature dependence of this feature is quite similar to that of the higher-energy two-magnon peak; however, a peak of about the same energy, though narrower, is also found in undoped La2NiO4.60 As noted previously,27 there are no obvious features in the spin-wave spectrum that would suggest the likelihood of a second two-magnon feature. On the other hand, the energy of ⬃90 meV is about twice that of a strong phonon feature that can be seen at QAF and 45 meV in Fig. 4共c兲. Thus it seems possible that the ⬃90-meV feature might be a two-phonon excitation that becomes Raman-active due to some interaction involving the magnetic order. 共We note that a complementary process involving phonon-assisted multimagnon optical absorption61 has been observed62 in La2NiO4.兲 Finally, we note that the spin-wave dispersion found here in La2−xSrxNiO4 with diagonal stripe order is quite different from that observed in 1 / 8-doped La2−xBaxCuO4 with vertical stripe order.2 The differences are particularly clear if one compares our Fig. 5 with Fig. 2 of Ref. 2. The difference is not explained by the difference in orientation of the stripes,

*Present address: Department of Physics & Astronomy, University of Tennessee, Knoxville, TN 37996-1200, USA. 1 S. M. Hayden, H. A. Mook, P. Dai, T. G. Perring, and F. Doğan, Nature 共London兲 429, 531 共2004兲. 2 J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, Nature 共London兲 429, 534 共2004兲. 3 V. Hinkov, S. Pailhès, P. Bourges, Y. Sidis, A. Ivanov, A. Kulakov, C. T. Lin, D. P. Chen, C. Bernhard, and B. Keimer, Nature 共London兲 430, 650 共2004兲. 4 C. Stock, W. J. L. Buyers, R. A. Cowley, P. S. Clegg, R. Coldea, C. D. Frost, R. Liang, D. Peets, D. Bonn, W. N. Hardy, and R. J. Birgeneau, Phys. Rev. B 71, 024522 共2005兲.

as spin-wave calculations using appropriate models12,13,15,16 also fail to describe the cuprate results. Models that take account of singlet correlations, which are enhanced by the stripe modulation, provide a better description of the cuprate excitations.2,9,17 One feature that appears to be common to the cuprates and nickelates is the robustness with respect to doping of the superexchange within antiferromagnetic domains. VI. CONCLUSIONS

We have mapped out the two-dimensional dispersions of magnetic excitations of single-crystal samples of stripeordered La2−xSrxNiO4 with x = 1 / 3 and 0.275. We have shown that the excitations measured for x = 1 / 3 are well described by a linear spin-wave model characterized by a superexchange energy J within antiferromagnetic domains and an effective coupling J1 across a charge stripe acting along the Ni-O bond direction. If the effective coupling is changed to act between Ni sites along a plaquette diagonal, then the fit to the data is quite poor. The fitted value of J is about 90% of that in the undoped parent antiferromagnet, and J1 / J ⬇ 0.5. The excitations measured for x = 0.275 are surprisingly similar to those of the x = 1 / 3 sample, despite the differing magnetic-ordering wave vectors. Even the bandwidth of the excitations is the same; the main difference is a peak broadening in the case of x = 0.275. In both samples, one spin-wave branch has a gap of ⬃20 meV. We have discussed possible origins for this gap, but it remains a mystery at the moment. Measuring the temperature dependence of the gap may provide a useful clue. ACKNOWLEDGMENTS

It is a great pleaure to acknowledge many helpful discussions with D. T. Adroja and S.-H. Lee. Work at Brookhaven was supported by the U.S. Department of Energy’s Office of Science under Contract No. DE-AC02-98CH10886. This work was also supported by grants from the MEXT, Japan, and it has benefited from the U.S.-Japan Cooperative Program on Neutron Scattering.

5 N.

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