explored by inelastic neutron scattering

Spin dynamics of the frustrated easy-axis triangular antiferromagnet 2H-AgNiO2 explored byinelastic neutron scattering

E. M. Wheeler,1,2,* R. Coldea,3,1 E. Wawrzyńska,3 T. Sörgel,4 M. Jansen,4 M. M. Koza,2 J. Taylor,5

P. Adroguer,3,6 and N. Shannon3

1Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom2Institut Laue-Langevin, BP 156, 38042 Grenoble Cedex 9, France

3H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom4Max-Planck Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

5ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom6Laboratoire de Physique, École Normale Supérieure de Lyon, 46 Allée d’Italie, 69364 Lyon Cedex 07, France

�Received 5 September 2008; revised manuscript received 29 January 2009; published 20 March 2009�

We report inelastic neutron-scattering measurements of the spin dynamics in the layered hexagonal magnet2H-AgNiO2, which has stacked triangular layers of antiferromagnetically coupled Ni2+ spins �S=1� ordered ina collinear alternating stripe pattern. We observe a broad band of magnetic excitations above a small gap of 1.8meV and extending up to 7.5 meV, indicating strongly dispersive excitations. The measured dispersions of theboundaries of the powder-averaged spectrum can be quantitatively explained by a linear spin-wave dispersionfor triangular layers with antiferromagnetic nearest- and weak next-nearest-neighbor couplings, a strong easy-axis anisotropy, and additional weak interlayer couplings. The resulting dispersion relation has global minimanot at magnetic Bragg wave vectors but at symmetry-related soft points and we attribute this anomalous featureto the strong competition between the easy-axis anisotropy and the frustrated antiferromagnetic couplings. Wehave also calculated the quantum corrections to the dispersion relation to order 1 /S in spin-wave theory byextending the work of Chubukov and Jolicoeur �Phys. Rev. B 46, 11137 �1992�� and find that the presence ofeasy-axis anisotropy significantly reduces the quantum renormalizations predicted for the isotropic model.

DOI: 10.1103/PhysRevB.79.104421 PACS number�s�: 75.25.�z, 75.10.Jm

I. INTRODUCTION

Quantum antiferromagnets on triangular lattices providemodel systems for investigating the effects of quantum fluc-tuations and geometric frustration. Zero-point fluctuationsare expected to be strongly enhanced for low spin and frus-trated lattice geometry and may stabilize nonclassical or-dered or spin liquid phases, or unconventional spin dynam-ics. The triangular Heisenberg antiferromagnet with first- andsecond-neighbor couplings J1 and J2 �see Fig. 1� showsstrong frustration effects and a macroscopically degenerateclassical ground state for 1 /8�J2 /J1�1. This was initiallyproposed theoretically as a candidate for a chiral spin liquidstate.1 However, perturbative expansions using a spin-wavebasis2–4 predicted that in this range of couplings quantumfluctuations do not stabilize a spin liquid state but instead liftthe classical degeneracy to select a collinear stripe order pat-tern �see Fig. 1�. Moreover, large quantum renormalizationsof the spin-wave dispersion relations compared to classicalhave been predicted, but not probed experimentally in theabsence of a good experimental model system. The recentobservation of a collinear stripe-ordered ground state in thelayered hexagonal antiferromagnet 2H-AgNiO2 �Ref. 5� sug-gests a potential realization of the experimentally-unexploredfrustrated J1−J2 model in the collinear phase and here wepresent first inelastic neutron-scattering measurements whichgive information about the spin gap and dispersion relations.We find that the data can be parametrized by a spin-wavedispersion relation for a J1−J2 triangular antiferromagnetwith strong easy-axis anisotropy and weak interlayer cou-plings.

Delafossite materials of the type XNiO2 have been gener-ally thought of as possible two-dimensional frustrated mag-nets. A network of edge-sharing NiO6 octahedra leads to atriangular lattice arrangement of Ni ions in planes spaced bylayers of X+ ions. There have been a variety of studies inves-tigating the properties of compounds where X=Li,Na.

J2

J1

120° collinear

~1/8 ~1

Inco

mm

ensu

rate

helix

J2/J1

FIG. 1. �Color online� Phases of the Heisenberg triangular lat-tice antiferromagnet �120° coplanar order, collinear stripes and in-commensurate spiral� as a function of the ratio between the first-and second-neighbor couplings, J1 and J2, respectively. In the col-linear phase the common spin direction is spontaneously chosenand can point anywhere, it is shown here in-plane for ease ofvisualization.

PHYSICAL REVIEW B 79, 104421 �2009�

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http://dx.doi.org/10.1103/PhysRevB.79.104421

LiNiO2 has shown no long-range magnetic order but experi-ments are hindered by the difficulty in preparing stoichio-metric samples.6 From elastic and inelastic neutron-scattering measurements, NaNiO2 was found to be a spin-1/2system with in-plane �unfrustrated� ferromagnetic interac-tions and weak antiferromagnetic interlayer couplings.7

We have recently started exploring the delafossiteAgNiO2, which shows frustrated antiferromagnetic in-planeinteractions.8,9 Detailed structural studies have been per-formed on the hexagonal polytype of AgNiO2, so-called2H-AgNiO2 with two NiO2 layers per unit cell �see Fig.2�b��, as opposed to the earlier-synthesized rhombohedral 3Rpolytype8 with a three-layer stacking sequence along c axis.High-resolution neutron-diffraction measurements of2H-AgNiO2 have observed a structural transition upon cool-ing below 365 K, which leads to a periodic arrangement ofexpanded and contracted NiO6 octahedra.10 This wasproposed5 to be a consequence of spontaneous charge orderon the Ni sites driven by the need to lift a twofold orbitaldegeneracy. This leads to a strongly magnetic Ni1 site�spin-1 Ni2+, 1/3 of sites� arranged in an ideal triangularlattice �see Fig. 2�a��, with the remaining 2/3 of Ni sitesbeing Ni3.5+ with an itinerant character.

Magnetic order occurs below TN=19.7�3� K �Ref. 10�when the magnetic moments of Ni1 sites order in a collinearpattern of alternating ferromagnetic rows �stripes� in the tri-angular layers with spins pointing along the c axis, as shownin Fig. 3�a�. A large ordered moment is found on the Ni1sites, 1.552�7��B at 4 K, and no significant ordered momentcould be detected on the remaining Ni2 and Ni3 sites, pro-posed by band-structure calculations5 to be strongly itinerantand possibly ordered but with only a very small ��0.1�B�moment. So to a first approximation the coherent spin dy-namics in the magnetically ordered phase is expected to bedominated by the S=1 Ni1 spins arranged on an ideal trian-gular lattice with antiferromagnetic couplings.

The experimentally observed collinear stripe order in Fig.3�a� is rather unusual for a triangular antiferromagnet�TAFM�, but has been proposed to occur for the classicalIsing model for finite second-neighbor couplings J2�0 �Ref.11�. For the Heisenberg TAFM the collinear stripe order hasalso been proposed to occur as a ground state for moderate J2in the range 1 /8�J2 /J1�1, Ref. 3. In this range the classi-cal ground state is macroscopically degenerate and there aremany other noncollinear states degenerate with the two-sublattice collinear stripe order shown in Fig. 3�a�, but zero-point quantum fluctuations are predicted to select the latterthrough the “order by disorder” mechanism. The large clas-sical degeneracy is manifested also in the linear spin-wavedispersion which has many soft points with zero energy, andproper inclusion of quantum corrections leads to renormal-izations of the semiclassical dispersion relation and a gap-ping of the nonphysical zero modes.3 Those effects have re-mained experimentally largely unexplored. The recentobservation5 of a collinear stripe ordered phase in the hex-agonal magnet AgNiO2 suggested a possible experimentalrealization of the S=1 TAFM in the range of moderate frus-tration and motivated us to measure its spin dynamics insome detail. We note that the physics we are exploring hereappears to be different to that in the recently explored S=5 /2 TAFM CuFeO2 which has a related but different col-linear structure of alternating “double stripes” �↑↑ ↓↓� stabi-lized by strong three-dimensional �3D� and third-neighborin-plane couplings.12 In 2H-AgNiO2 the magnetic structureis a �simpler� single-stripe pattern ↑↓ with only two in-planesublattices possibly stabilized by fewer exchanges �only firstand second in-plane neighbors� and easy-axis anisotropy.

The plan of this paper is as follows. The following Sec. IIgives details of the inelastic neutron-scattering experimentsto probe the powder-averaged spin dynamics and the resultsare presented in Sec. III. In the following Sec. IV A the dataare parametrized in terms of an empirical sinusoidal disper-sion model with minima at magnetic Bragg wave vectors anddifferent zone boundary energies along the three orthogonaldirections in the Brillouin zone �BZ�; this parametrization

FIG. 2. �Color online� �a� Triangular lattice formed by Ni1 sites�dark brown spheres� in a hexagonal NiO2 layer in 2H-AgNiO2;thick arrows indicate the paths for the in-plane NN and NNN ex-changes J1 and J2. Dashed diamond shows the crystallographic unitcell. Gray spheres are Ni2 and Ni3 sites, assumed nonmagnetic. �b�3D view of the crystal and magnetic structure. There are twosymmetry-equivalent NiO2 layers per unit cell. Layer stacking issuch that Ni1 ions sit above the center of a Ni1 triangle in the layerbelow. We consider two natural candidates for interlayer couplings:Jz between Ni1 ions in adjacent layers �three neighbors above andthree below� and Jz� between Ni1 ions at two layers apart.

FIG. 3. �Color online� �a� Magnetic structure in a triangularlayer showing the alternating stripe order of the Ni1 spins �� sym-bols indicate the projection of the ordered spin moments along the caxis�. x and y labels indicate a natural orthogonal coordinate systemwith x along and y transverse to stripes. Solid diamond indicates themagnetic unit cell doubled along the crystallographic a axis com-pared to the chemical unit cell �light gray shaded area�. The de-picted magnetic structure has stripes running along the b axis andordering wave vector k= �1 /2,0 ,0�; equivalent domains are ob-tained by rotating this pattern by �60° around the c axis. Furtherneighbor couplings are required to stabilize this structure with re-spect to the ferrimagnetic honeycomb up-up-down pattern shownin �b�.

WHEELER et al. PHYSICAL REVIEW B 79, 104421 �2009�

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gives a gapped and predominantly two-dimensional disper-sion relation. In the following Sec. IV B the data are com-pared with linear spin-wave theory for a microscopic spinHamiltonian for the localized Ni spins that includes bothfirst- and second-neighbor antiferromagnetic couplings in thetriangular layers, easy-axis anisotropy modeled by a single-ion term, and different models for interlayer couplings con-sistent with the crystal structure; a minimal spin Hamiltonianis proposed. In the following Sec. IV C we consider howmagnon interactions included at order 1 /S in spin-wavetheory renormalize the dispersion relation and find slightlyrenormalized values for the proposed spin interactions. Fi-nally, the results for the spin dynamics are summarized anddiscussed in the concluding Sec. V. Technical details of thespin wave calculations for coupled easy-axis triangular lay-ers are given in Appendix A. Details of the derivation of the1 /S quantum corrections to the dispersion relation for aneasy-axis triangular antiferromagnet are given in AppendixB.

II. EXPERIMENTAL DETAILS

Powder samples of the hexagonal polytype of AgNiO2were prepared from Ag2O and Ni�OH�2 using high oxygenpressures �130 MPa� as described in Ref. 9. The samplesused for the inelastic neutron experiments were part of thesame batch used in previous neutron-diffractionmeasurements10 performed on about half of the total samplequantity, which revealed a high-purity hexagonal phase��1% admixture of the rhombohedral polytype�. The crystalstructure of 2H-AgNiO2 is shown in Fig. 2�b� and is hexago-nal with space group P6322 �no. 182� and lattice parametersa=5.0908�1� Å and c=12.2498�1� Å �Ref. 5�. Ni ions arelocated inside hexagonal �ab� layers and there are three dis-tinct crystallographic sites: Ni1 which sit inside slightly ex-panded NiO6 octahedra arranged in a periodic triangular lat-tice of spacing a, surrounded by a honeycomb of contractedNiO6 octahedra which contain the Ni2 and Ni3 sites, see Fig.2�a�.

The powder-averaged magnetic excitation spectrum wasprobed using two direct-geometry time-of-flight neutronspectrometers: MARI, at the ISIS Facility in the U.K., andIN6, at the Institute Laue-Langevin in France. Measurementson MARI showed that the full dynamic range of the spinexcitations extended only up to 7.5 meV so could be ac-cessed with incident neutrons of energy Ei=18 meV, whichgave an energy resolution of 0.61�1� meV �FWHM� on theelastic line. Higher-resolution measurements to probe thelow-energy part of the spectrum were made using IN6 oper-ated with incident neutrons of energy Ei=3.86 meV, whichgave a measured energy resolution of 0.142�1� meV�FWHM� on the elastic line. The sample was cooled usingeither a closed-cycle refrigerator �MARI, base temperature4.7 K� or orange cryostat �IN6, base temperature 1.8 K�.Measurements were made at the lowest temperatures in themagnetically ordered phase, near the ordering transition andat high temperatures in the paramagnetic phase.

For the MARI experiment 23 g of powder were placed inan annular-shaped sachet to minimize absorption. For the

low-energy IN6 measurements a similar powder quantitywas placed in a plate-shaped container angled at 45° to theincident beam and the data were corrected for neutron ab-sorption effects using a numerical calculation for plate-shaped samples. The scattering intensities from both instru-ments have been converted into absolute units of S�Q ,�� ofmbarns/meV/sr/Ni1 by normalizing the raw counts to thesample mass and to the measured scattering intensities froma vanadium standard.

III. MEASUREMENTS AND RESULTS

An overview of the measured inelastic neutron-scatteringdata is shown in Fig. 4. Below the Néel ordering temperatureTN=19.7�3� K a strong band of scattering is clearly ob-served at low energies, E, and low wave vectors, Q= �Q�,

FIG. 4. �Color online� Powder-averaged magnetic excitationspectrum in 2H-AgNiO2 in the magnetically ordered phase �toppanels �a� and �c�� showing a band of spin excitations above a smallgap, �b� in the paramagnetic phase at high temperature showing anoverdamped signal and �d� slightly above the magnetic orderingtemperature showing a filling-in of the spin gap �compare with �c��.Data are raw counts collected after 35 h �a� and 4.5 h �c� andnormalized to absolute units as described in the text. The streak ofintensity in panel �c� near Q=1.1 Å−1 is nonmagnetic and attrib-uted to an acoustic phonon dispersing out of the �002� structuralBragg peak. Middle panels show the intensity distribution for theempirical dispersion model in Eqs. �1� and �3�. The calculationsinclude convolution with the instrumental resolution, the magneticform factor, polarization factor, and an estimate of the nonmagneticbackground to be directly compared with the data in the panelsimmediately above. Thick bold arrows labeled A–E indicate thelocation of scans plotted in Figs. 5�A�–5�E� and the gray dottedlines in all plots indicate the low-Q edge of the measured region.

SPIN DYNAMICS OF THE FRUSTRATED EASY-AXIS… PHYSICAL REVIEW B 79, 104421 �2009�

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��Figs. 4�a� and 4�c��, attributed to magnetic excitations.The intensity of the signal decreases with increasing wavevector Q, confirming its magnetic character. The intensitystarts above a small gap of 1.8 meV and extends up to 7.5meV, above which the signal decreases to background level.This energy scale for the magnetic excitations spectrum isconsistent with susceptibility measurements which observeda Curie-Weiss temperature of kB=−9.2 meV �Ref. 10�. En-ergy scans through the data at constant wave vectors show-ing the low energy gap and the full extent of the scatteringare shown in Figs. 5�A� and 5�E� �solid points�. The dashedline in those plots indicates the estimated nonmagnetic back-ground, obtained by a smooth interpolation of the signal ob-served at energies below and above the magnetic signal. Theenergy gap to magnetic excitations fills up upon heatingthrough the Néel temperature �compare Fig. 4�c� with Fig.4�d� and scans shown in Fig. 6�. The structured band ofmagnetic scattering is replaced as expected by an over-damped signal in the paramagnetic phase at high tempera-tures; compare Fig. 4�a� with Fig. 4�b� and various scans inFig. 5�E�.

At the lowest temperatures the boundaries of the magneticscattering show considerable structure which can be used toimpose constraints on the underlying dispersion relation. Inparticular information is contained in the gap, extent of

bandwidth, slope of low-Q dispersion up to the first mini-mum gap �see Fig. 4�a�� and the low-energy dispersion��10% of bandwidth� between subsequent minimum gapwave vectors �see Fig. 4�c��. The powder data are a sphericalaverage of the dispersions along all directions in reciprocalspace weighted by the neutron structure factor, so the low-energy boundary of the powder data corresponds to the dis-persion along some direction in reciprocal space which hasthe minimum energy at a given Q. In general the spin-wavedispersion has global minima at magnetic Bragg wave vec-tors and here strong low-energy scattering is expected. Thisis consistent with the data in Fig. 4�c� showing clear lobes ofmagnetic scattering intensity coming down in energy nearwave vectors 0.7 and 0.9 Å−1 where the first two magneticBragg peaks �1/2,0,0� and �1/2,0,1� occur; a constant energyscan near the gap minimum is shown in Fig. 5�D�. Inelasticintensity occurs near the same wave vectors as magneticBragg peaks in the diffraction pattern shown in Fig. 5�D�bottom panel �data from the GEM diffractometer at ISIS�.The low-energy boundary of the scattering is clearly modu-lated as a function of wave vector and the onset energy var-ies as a function of wave vector Q as illustrated by compar-ing energy scans in Figs. 5�B� and 5�C�.

IV. ANALYSIS

The modulations observed in the lower boundary of themagnetic scattering indicate strongly dispersive excitations,however, in principle it is not possible to extract precise dis-persion relations from the powder data, which represents anaverage along all directions in reciprocal space. Neverthe-less, if the dispersion is highly asymmetric along differentdirections and if in-plane and out-of-plane lattice parametersare sufficiently different then the lower boundary of the scat-tering can be identified over certain Q ranges with the dis-persion along specific directions in reciprocal space, so onemay impose certain constraints on the dispersion model. Fol-lowing this approach we have found two models for the dis-persion relation that could both explain the observed wave-vector dependence of the boundaries of the powder-averagedspectrum and present them in the following.

Inte

nsit

y

0

10

20

30

DataEmpirical model

Energy (meV)2 4 6 8

Inte

nsit

y

0

10

20

30

40

50

604.70 K215K

Energy (meV)2 4 6 8

0

10

20

30

40

50

60DataEmpirical model

0

10

20

30

40

DataEmpirical model

Energy (meV)1.0 1.5 2.0 2.5

0

10

20

30DataEmpirical model

Q (Å-1)0.6 0.8 1.0 1.2 1.4

Inte

nsit

y(a

rb.u

nits

)

30

35

40

Inte

nsity

Inte

nsity

Inte

nsity

Energy=2±0.05meV

A

E

B

D

C

(1/2

,0,0

)

(1/2

,0,1

)

(1/2

,0,2

)

(1/2

,-1,

1)

(0,0

,2)

Q = 0.8±0.2 Å-1

2Θ = 5.5±2.5o

Q = 0.7±0.025 Å-1

Q = 1.050±0.025 Å-1

Diffractionprofile GEM

FIG. 5. �Color online� Scans through the powder data alongspecial directions labeled A–E in Fig. 4. Scan D shows that there isstructure in the low-energy inelastic signal for wave vectors whereBragg peaks occur �lower panel in D shows diffraction data indi-cating magnetic Bragg peaks�. The energy scan E integrates over abroad Q range and shows the full bandwidth of the magnetic exci-tations: low temperature data �solid points� are replaced in the para-magnetic phase at high temperatures by an overdamped signal �tri-angles, solid line is a guide for the eyes�. In panels A–D the thicksolid lines show calculations for the sinusoidal dispersion model inEqs. �1� and �3� with dispersion relations plotted in Fig. 7. Thegreen dashed lines indicate the estimated nonmagnetic background.

Energy (meV)1.0 1.5 2.0 2.5 3.00

20

40

601.79 K21.5 K31.1 K

Q=0.7 0.1 Α−1

Intensity

+�

FIG. 6. �Color online� Energy scan at the minimum gap wavevector showing a filling of the gap upon heating above the magneticordering temperature. Solid lines are guides for the eyes and thedashed line indicates the estimated nonmagnetic background.


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A. Parametrization by an empirical sinusoidal dispersionmodel

We first consider a parametrization of the data by an em-pirical sinusoidal dispersion model with a finite gap abovethe magnetic Bragg peaks and different zone boundary ener-gies along the three orthogonal directions in the Brillouinzone: E1 along the magnetic stripe direction, E2 transverse tostripes in the plane, and Ez in the interlayer direction. Indetail we consider the following empirical form for the dis-persion relation �Q:

��Q�2 = 2 + �E12 − 2�sin2�Qxa/2� + �E2

2 − 2�sin2�Qya�3/2�

+ �Ez2 − 2�sin2�Qzc/2� , �1�

where Qx, Qy, and Qz are components �in Å−1� of the wavevector transfer Q in an orthogonal coordinate system with xalong the magnetic stripes, y transverse to stripes in plane,and z normal to the planes. The transformation to wave vec-tor components �h ,k , l� in units of the reciprocal lattice of thehexagonal unit cell is given by

Qx = −2�

ak ,

Qy =2�

a�3�2h + k� ,

Qz =2�

cl . �2�

Figure 7 shows a plot of this dispersion relation along vari-ous symmetry directions in the Brillouin zone and Fig. 8�b�shows a 2D contour map of the dispersion. The minimumgap is reached at the origin and at all magnetic Bragg peakpositions �H ,K ,L��k with k= �1 /2,0 ,0� the ordering wavevector and H, K and L integers. In order to compare thisdispersion relation with the data one needs also a model forthe neutron-scattering cross section. We assume the follow-

ing simple form for the one-magnon cross section:

Sxx�Q,�� = Syy�Q,�� = CS

2��k�max

1 − �Q

�G�� − �Q� . �3�

Here C is an overall scale factor and �1−�Q� is a geometricfactor to concentrate the intensity near the antiferromagneticBragg peak wave vectors �H�1 /2,K ,L� with H, K, L inte-gers, �Q=cos�Qxa /2�cos�Qya�3 /2�. The 1 /� factor is acharacteristic energy dependence of the spin-wave structure

0

1

2

3

4

5

6

7

Ene

rgy

(meV

)

(12,0,1

) (12,0,0

)

M

(0,0,0)

Γ

(12,−1,0

) (12,−1

2,0

)

M′

∆

E1

E2

Ez

FIG. 7. Sinusoidal dispersion relation in Eq. �1� plotted alongsymmetry directions in the Brillouin zone in Fig. 8�b� �dashedlines�. Dispersion parameters are =1.78 meV, E1=5.73 meV,E2=4.15 meV, and Ez=2.33 meV.

FIG. 8. �Color online� �a� Reciprocal basal plane of 2H-AgNiO2

showing the Brillouin zone edges �thick line hexagons�, zone cen-ters �solid points�, and locations �stars� of magnetic Bragg peaksfrom the ordered domain shown in Fig. 3�a� with stripes along the baxis. Labels , M, and K indicate special symmetry points in thehexagonal Brillouin zone �M� and M� are Bragg peak positions forequivalent ordered domains with stripes along a+b and a, respec-tively�. Dashed gray lines show directions along which the disper-sion is plotted in Fig. 10. Arrows labeled x and y indicate the natu-ral orthogonal coordinate system used for the empirical dispersionmodel in Eq. �1�. ��b� and �c�� Color contour maps of the dispersionin the �ab� plane in the two models consistent with the data: �b� theempirical parametrization with a sinusoidal dispersion plotted inFig. 7 and �c� the linear spin-wave model for J1−J2 triangular lay-ers with easy axis and interlayer couplings in Fig. 10�d� �dashedline�. Note that in model �b� the minimum gap occurs at the mag-netic Bragg wave vector M, whereas in model �c� the minimum gapoccurs at the soft point M�.


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factor for antiferromagnets and ��k�max is the maximummagnon energy for all wave vectors k, and is introduced herefor dimensionality purposes. S=1 is the spin quantum num-ber. Equation �3� includes the fact that spin fluctuations oc-cur in the plane transverse to the ordered spin direction in acollinear antiferromagnet, so x and y refer to two orthogonaldirections in the hexagonal ab plane and z is along c. Thisform is a generic structure factor for many simple spin-wavemodels of collinear antiferromagnets. Here G��−�Q� is aGaussian function which models the instrumental resolution.

A spherical average of the intensity distribution of Eq. �3�including the neutron polarization factor, magnetic form fac-tor for Ni2+ ions �as in Eq. �A5��, and an estimated additivenonmagnetic background �including the incoherent elasticline� was compared to the data and the overall comparison isshown in the middle panels of Fig. 4. The overall bandwidthand the observed structure in the data at low energies is wellreproduced by this model for gap =1.78�5� meV, E1=5.73�4� meV, E2=4.15�6� meV and Ez=2.33�5� meV inEq. �1�. The maximum magnon energy is ��k�max=7.0 meV and an overall scale factor C�0.35 in Eq. �3�gives intensities comparable to data.

To analyze the agreement quantitatively we first show inFig. 9�a� how the lower boundary of the powder-averagedmagnetic signal can be identified over certain wave-vectorranges with the dispersion along particular directions in re-ciprocal space. In particular the observed sharp low-Q dis-persion up to the first gap minimum near QM =0.7 Å−1 in

Fig. 4�a� is attributed to the dispersion between the origin �000� and the closest magnetic Bragg peak at M�1 /2,0 ,0�;scan A along the direction indicated by the bold arrow in Fig.4�a� is therefore very sensitive �at low energies� to the Mdispersion and a fit to this scan shown in Fig. 5�A� givesE2=4.15�0.06 meV, the zone boundary along the directiontransverse to stripes in plane. To explain the observed fullextent of the bandwidth up to 7.5 meV another dispersionlarger in magnitude is required and this is assigned to thedirection along the stripes, E1, since the interlayer dispersionbandwidth Ez− is assumed to be small. Quantitative fitsgive E1=5.73�0.04 meV. The energy scan in Fig. 5�B� issensitive to the gap above the magnetic Bragg wave vector,obtained as =1.78�5� meV. The energy scan in Fig. 5�C� isat a wave vector corresponding to the midpoint between twosubsequent Bragg wave vectors along the c axis, �1/2,0,1�and �1/2,0,2� �see Fig. 9�a��, so this scan is sensitive to theinterlayer zone boundary and fits give Ez=2.33�0.05 meV,thus obtaining the last parameter of the empirical dispersionrelation.

Figure 9 shows in more detail the agreement between theexperimentally determined onset of the magnetic scattering�stars� and the empirical model �various solid lines� in panel�a� and the calculated powder-averaged spectrum in panel�b�. The onset points were determined from scans throughthe measured powder data at fixed wave vector Q and locat-ing the energy where the magnetic intensity was above aminimum threshold value. The model reproduces well theshape of the lower edge of the scattering. Fits to specificscans shown in Figs. 5�A�–5�D� give good account of theonset and upper boundary of the scattering. The form of theintensity distribution within the excitation band is not repli-cated in detail, as seen for example in Fig. 5�A� where thesimulation gives a more pronounced trough in the intensityin the middle of the band. This is almost certainly due to theoversimplification of the model.

B. Parametrization in terms of an easy-axis Heisenberg modelon a triangular lattice

The empirical sinusoidal dispersion model considered inSec. IV A is the simplest dispersion model that can be usedto parametrize and explain quantitatively the observed dis-persion of the boundaries of the powder-averaged magneticinelastic scattering. This analysis suggested that the disper-sion relation is gapped and predominantly two dimensional.Here we compare the data quantitatively with predictions oflinear spin-wave theory for a microscopic model of coupledtriangular layers where we include antiferromagnetic in-plane couplings for the first- and second-neighbor exchangesJ1 and J2, an easy-axis anisotropy and allow for both first-and second-layer couplings Jz and Jz�, as shown in Figs. 2�a�and 2�b�. Specifically, we consider the spin Hamiltonian

H0 = NN

J1Si · S j + NNN

J2Si · Sl − Di

�Siz�2

+ InterlayerNN

JzSi · Sk + InterlayerNNN

Jz�Si · Sn, �4�

where the spin is S=1, NN and NNN stand for nearest-

FIG. 9. �Color online� �a� Dispersion relations for the empiricalsinusoidal model in Eq. �1� along various directions involved indefining the low-energy edge of the powder-averaged spectrum. �b�Corresponding powder-averaged spectrum. Stars in both panels arethe experimentally determined low-energy edge of the magneticscattering. Thick arrows indicate location of scans B, C, and D inFig. 5.


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neighbor and next-nearest-neighbor and each spin pair iscounted once in the summation. The third term is an easy-axis anisotropy, needed to select the ordering spin directionalong c, and to generate a spin gap in the spectrum. In theabsence of a clear physical picture of the origin of the aniso-tropy, we have chosen a single-ion form for this term. Weanticipate that an exchange anisotropy ��J�zSi

zSjz would have

qualitatively similar effects. Both in-plane exchanges are an-tiferromagnetic J1 ,J2�0 whereas the interlayer couplingsare taken to be ferromagnetic Jz ,Jz��0, to stabilize the ob-served magnetic structure shown in Fig. 2�b�.

We have calculated the spin-wave dispersion and dynami-cal correlations for the spin Hamiltonian �4� in the large Slimit using the standard Holstein-Primakoff formalism anddetails are given in Appendix A with the dispersion relationsin Eq. �A2�. To illustrate the effect of the various terms in theHamiltonian we plot the dispersion for a triangular latticewith no anisotropy in Fig. 10�a� �dashed line�. The linearspin-wave dispersion has zero modes not only at the mag-netic Bragg peak position M for the specific ordered mag-netic domain, but also at soft points such as M� in Fig. 8which are Bragg peak positions for the symmetry-equivalentdomains �there are three possible magnetic domains relatedby a threefold rotation around the c axis�.

The presence of such unphysical gapless modes is a con-sequence of the macroscopic ground-state degeneracy at theclassical level and including 1 /S corrections �Fig. 10�a�,solid line� generates a gap at the soft points but preserves agapless Goldstone mode at the Bragg wave vectors.3 Thesituation is completely different in the presence of an easy-axis anisotropy which generates a gap everywhere �see Fig.10�b��; the gap at the Bragg wave vector increases very rap-idly as a power law, whereas at the soft point increases onlylinearly with anisotropy, so above some threshold anisotropyvalue the minimum gap is no longer at the magnetic Braggwave vector M, but moves to the soft point M�.

Coupling the layers by a NN exchange Jz�0 �ferromag-netic� has the effect of splitting the dispersion into twomodes �see Fig. 10�c��, corresponding to acoustic and opticmagnon modes between the two layers in the unit cell. Thegap minimum is no longer at the commensurate soft pointM� but at incommensurate positions symmetrically displacedfrom M� along the magnetic stripe direction; see Fig. 10�c�.The incommensurate position of the gap minimum varieslinearly with �Jz� for small Jz and it originates physicallyfrom the fact that hopping of magnons is somewhat frus-trated as an up spin in the bottom layer interacts through Jzwith two up and one down spin in the layers above and

0

2

4

6

Ene

rgy

(meV

)M Γ M�

SWT+1/SLSWT

Triangular J1

− J2

antiferromagnet D = 0

(a)0

2

4

6

M�L = 1M�

(b)

0

2

4

6

Ene

rgy

(meV

)

�12,0,0

�(0,0,0)

�12,−1

2,0

� �34,−1,0

��12,−1,0

�

D = 1.78

(d)

0

2

4

6

En

erg

y(m

eV)

�12,0,1

� �12,0,0

�(0,0,0)

�12,−1

2,0

� �34,−1,0

� �12,−1,0

�

D = 1.78, Jz′ = −0.14

SWT+1/Scorrected parameters

LSWT

0

2

4

6

�12,−1

2,0

� �12,−1

2,1

�

0

2

4

6

En

erg

y(m

eV)

ML = 1 M Γ M�

D = 1.34, Jz= −0.3

ω−

ω+

(c)

FIG. 10. �Color online� Dispersion relation for the stacked triangular antiferromagnetic layers within LSWT �dashed lines� and SWT+1 /S �solid lines� for various values of the parameters. The dispersion is plotted along high-symmetry directions in reciprocal space �dashedline path in Fig. 8�a��, top labels are high symmetry points in the Brillouin zone. �a� Triangular lattice with antiferromagnetic couplingsJ1=1.32 meV, J2=0.15J1 and no anisotropy. 1 /S quantum corrections have a large effect at the soft point M�. �b� Same as �a� but witheasy-axis anisotropy D=1.78 meV; 1 /S corrections are much smaller. �c� Triangular layers with easy-axis anisotropy and coupled by NNinterlayer exchange Jz=−0.3 meV. This leads to two incommensurate spin-wave modes with very little interlayer dispersion near the gapminimum �last panel in the row�, not consistent with data. �d� Triangular layers with anisotropy coupled by second-layer exchangeJz�=−0.14 meV; this model gives the best fit to the data. Dashed �solid� line shows best fit using LSWT �SWT+1 /S� with parameter valuesgiven in Table I.


104421-7

below; see Fig. 15�b�. Moreover, there is very little interlayerdispersion of the gap minimum, i.e., between soft points M�with different L values �see Fig. 10�c� last panel�, also due tofrustration in the magnon interlayer hopping at those wavevectors.

Both of these features, incommensurate minima in thedispersion displaced �in-plane� away from M� and very weakinterlayer dispersion of the gap minimum, are inconsistentwith the data, which shows minima in the scattering at wavevectors very close to those corresponding to the commensu-rate M� point and a significant ��10% of the total band-width� interlayer dispersion of the gap minimum. This sug-gests that the observed interlayer dispersion is not due to thenearest-neighbor interlayer exchange Jz but is due to anotherinterlayer coupling.

To find a possible explanation for the observed interlayerdispersion we now assume Jz=0 and consider the effects of aferromagnetic second-layer coupling Jz� straight up along thec axis; see Fig. 2�b�. The resulting dispersion is plotted inFig. 10�d� �dashed line�; there is a single mode with essen-tially the same features as for decoupled triangular layers�see Fig. 10�b�� with global minima at the commensurate softpoint M� but now there is a modulation in the energy as afunction of L value to first order in Jz�. This model can ac-

count for all observed dispersions in the boundaries of thepowder-averaged spectrum and the calculation for the best fitparameter set is plotted in Figs. 11�a� and 11�b� to be com-pared with data in Figs. 4�a�–4�c�.

In the calculation we used the dynamical correlations inEq. �A3�, included polarization factor and magnetic formfactor as in Eq. �A5�, and performed a spherical average. Anoverall intensity scale factor C�0.57 gives intensities com-parable to data; we do not attach special significance to thisvalue except that it is comparable to the LSWT prediction ofC=1. The intensity distribution is also reasonably well cap-tured by the model as shown by the typical scans in Fig. 12;any differences compared to the data may be due to interac-tions beyond the minimal spin Hamiltonian considered.

To obtain quantitative values for the model parameters wehave used the fact that various parts of the spectrum aremostly sensitive to only one parameter; in particular the gapdepends only on anisotropy via M�=2SD�1− 1

2S �, the upperboundary of the spectrum is mostly sensitive to the maincoupling J1, the magnitude of the low-energy dispersion be-tween the gap minima with L=1 and 2 is given by 4S�Jz��.The remaining parameter J2 is extracted from the slope of thelow-Q dispersion up to the first gap minimum. Specifically, afit to scan B at the minimum gap �Fig. 12�B�� gives the

(b)(a)

(c) (d)

FIG. 11. �Color online� Powder averaged spin-wave spectrum for easy-axis triangular layers with/without �top/bottom� interlayer cou-plings. Left panels to be compared with Fig. 4�a� and right panels to Fig. 4�c�. The parameters are J1=1.32 meV, J2=0.15J1, D=1.78 meV, Jz=0, and Jz�=−0.14 meV �top panels� or Jz�=0 meV �bottom panels�. The calculations contain an additive flat background.The various lines overplotted onto the intensity color maps show the dispersion along the directions from the origin point to the closest softpoints at M��1 /2,−1 /2,0� �and the L=1 and L=2 versions of this� and M��1,−1 /2,0�, with minima at wave vectors indicated by thevertical arrows in the right panels. The dispersion at the lower boundary of the scattering in �b� is very similar to the data in Fig. 4�c� andare attributed to the weak interlayer couplings �no interlayer dispersion is present in �d� calculated for decoupled layers�. The dotted lines atlow-Q mark the left edge of the region probed by the experiments.


104421-8

anisotropy D, scan C probes the interlayer dispersion sogives Jz�, fits to scans A and E upper boundary �Fig. 12�E��give the main coupling J1 and fits to scan A lower boundarygive the secondary coupling J2.

The best fit values for the parameters obtained this wayare shown in Table I. The calculated powder-averaged spec-trum is shown in Figs. 11�a� and 11�b� and compares wellwith the data in Figs. 4�a�–4�c�. To emphasize the sensitivityof the data to the 3D couplings we also plot in Figs. 11�c�and 11�d� the powder-averaged spectrum assuming decou-pled layers, which reproduces the bandwidth, gap, and slopeof the low-Q dispersion up to the first gap minimum but notthe low-energy dispersion ��10% of bandwidth� betweensubsequent minimum gap wave vectors.

In the above analysis we neglected the NN interlayer cou-pling Jz since the powder data are not very sensitive to itspresence; in particular a finite Jz produces only a negligibleinterlayer dispersion of the gap minimum �dispersion appearsonly at order Jz

2 /D for small Jz�. Using the fact that a finite Jzproduces a small incommensurate shift of the minimum gapwave vector an upper bound for the maximum Jz that wouldstill be consistent with the data can be estimated as �Jz��0.25 meV.

Within the minimal model we considered in Eq. �4� theestimated values of the interaction parameters would give aCurie-Weiss constant expected in a high-temperature suscep-tibility measurement on a powder sample as13

kB = −1

3S�S + 1��6J1 + 6J2 + 2Jz�� = − 5.9 meV.

This is smaller than, but comparable to, the experimentalvalue of kB=−9.2 meV extracted from high-temperaturesusceptibility data10 and the difference may be due to other

interactions beyond the minimal model we have consideredto explain the dispersive boundaries of the powder data.

C. Spin-wave theory with 1 ÕS corrections for easy-axistriangular layers

In the absence of anisotropy the 2D triangular J1−J2 an-tiferromagnet in the collinear phase has been shown to ex-hibit a strong renormalization of the dispersion relation dueto quantum fluctuations. Including the effects of fluctuationsto order 1 /S in spin-wave theory3 generates a gap at the softpoints where the LSWT calculation predicts unphysical gap-less modes, as illustrated in Fig. 10�a� solid line is SWT+1 /S and dashed is LSWT.

In Appendix B we have calculated the 1 /S corrections tothe dispersion relation for the triangular antiferromagnetwhen an easy-axis anisotropy is also present and the resultsare plotted in Fig. 10�b�. The energy at the soft point is nowonly marginally renormalized up in energy because the easy-axis anisotropy generates a gap in the spectrum and this re-duces significantly the zero-point quantum fluctuation ef-fects. Using the 2D dispersion relation at order 1 /S andincluding also the interlayer coupling the best parameter val-ues obtained from comparison to the data are very similar tothose found using LSWT and are shown in Table I. Thedispersion relation at order 1 /S for these parameters is plot-ted in Fig. 10�d� �solid line� and is very similar to the LSWTdispersion for the uncorrected parameters �dashed line�. Thepowder-averaged spectrum is also similar to Figs. 11�a� and11�b� for the uncorrected parameters and is therefore notreproduced here.

In converging to the above set of parameter values for themicroscopic Hamiltonian we have used two constraints im-posed by the data �i� the interlayer dispersion is relativelysmall, �10% of the bandwidth, as inferred from the ob-served small dispersion in the low-energy boundary of themagnetic inelastic scattering, and �ii� the spin gap is rela-tively large, �30% of the bandwidth, which in turn implies arelatively large easy-axis anisotropy. Specifically, subtractingthe estimated interlayer dispersion magnitude from the ob-served maximum energy leads us to search for a 2D disper-sion shape where the gap energy divided by the maximum,i.e., the reduced gap /�max, is �25%.

The dispersion of the microscopic Hamiltonian in Eq. �4�has the minimum either at the Bragg peak position M, or atthe soft point M�. This places a strong constraint on modelparameters, since both reduced gaps must be at or above thethreshold value of 25%, with at least one gap precisely at the

TABLE I. Table of best fit values for easy-axis Heisenberg model applied to AgNiO2. In both cases fitsare obtained by comparing the spin-wave dispersion calculated within linear spin-wave theory, and includingleading interaction corrections, with the envelope of the powder-averaged spin structure factor S�Q ,E�measured by inelastic neutron scattering. We set S=1 and Jz0 in all fits. All exchange parameters are givenin meV.

S J1 J2 D Jz Jz�

LSWT 1 1.32�5� 0.15�3� 1.78�5� 0 −0.14�2�SWT+1 /S 1 1.35 0.15 1.59 0 −0.14

Energy (meV)2 4 6 8

Inte

nsity

0

10

20

30

40

50DataLSWT

Energy (meV)1.0 1.5 2.0 2.5

0

10

20

30

DataLSWT

Inte

nsity

EBQ = 0.8±0.2 ÅQ = 0.7±0.025 Å-1 -1

FIG. 12. �Color online� Comparison between data �filled points�and the linear spin-wave model for easy-axis triangular layers �solidlines�. Dashed lines show the estimated nonmagnetic backgroundand parameters are as in Fig. 11�a�.


104421-9

threshold. In Fig. 13 we map out these constraints for�J2 /J1 ,D /J1� parameter space, with gaps calculated usingspin-wave theory with 1 /S corrections, as described in Ap-pendix B. Possible solutions lie on the solid line dividingsolutions where the reduced gap M� /�max�25% from thosewhere M� /�max�25%.

On this line other features in the data such as the observedslope of the low-Q dispersion further narrow down possibleparameter values, leading to the best fit values. So within theminimal microscopic Hamiltonian considered the require-ment for quantitative agreement for the absolute gap relativeto the maximum energy requires a relatively large anisotropy,and in this case the dispersion minimum is at the soft pointM� as illustrated in Fig. 10�d� �solid line�.

V. DISCUSSION AND CONCLUSIONS

We have reported inelastic neutron-scattering measure-ments of the powder-averaged spin dynamics in the layeredhexagonal antiferromagnet 2H-AgNiO2, which has a collin-ear alternating stripe-ordered ground state. A broad band ofmagnetic inelastic scattering is observed above a finite gapindicating strongly dispersive magnetic excitations. The ob-served modulations in the boundaries of the magnetic scat-tering could be well explained by a gapped and predomi-nantly two-dimensional dispersion relation, suggestingmagnetically weakly coupled triangular layers. For a quanti-tative analysis we have considered a minimal spin Hamil-tonian in Eq. �4� for the localized Ni1 ions �S=1� located onan ideal triangular lattice and have neglected the coupling tothe itinerant electrons on the surrounding Ni sites. Withinthis localized model we considered first- and second-neighbor antiferromagnetic couplings J1 and J2=0.15J1 in

the triangular layers, a strong easy axis anisotropy modeledby a single ion term D�1.3J1, and weak interlayer cou-plings Jz�=−0.1J1 and have found that this minimal modelcan account quantitatively for all the observed wave-vectordependence of the boundaries of the powder-averaged spec-trum.

The deduced values of the magnetic interactions provide anatural explanation for the stability of the observed collinearalternating stripe magnetic order. The large value for theeasy-axis anisotropy strongly modifies the physics of an an-tiferromagnet on a triangular lattice. First, it stabilizes collin-ear order at the expense of coplanar states like the 120°spiral ground state of a simplest nearest-neighbor Heisenbergmodel. Second, it opens a gap to spin excitations whichstrongly suppresses quantum zero-point fluctuations. Theseare essential for the “order from disorder” selection of col-linear order in the isotropic J1−J2 model considered by Chu-bukov and Jolicoeur.3 But in the easy-axis J1−J2 model rel-evant to AgNiO2, fluctuations lead only to small quantitativecorrections to the spin-wave dispersion.

We have calculated these corrections explicitly to O�1 /S�and find that, for most purposes, they can safely be neglectedas they only lead to a very modest renormalization of thespin-wave dispersion—and the parameters inferred fromit—as illustrated in Table I. Competing interactions such asJ2 also play a secondary role, selecting the observed form ofcollinear order from the vast manifold of Ising ground stateson a triangular lattice. The extent of the collinear AF phaseas a function of �J2 /J1 ,D /J1� can easily be estimated withinmean-field theory. The results of this analysis are shown inFig. 14. The parameter set deduced above, with strong aniso-tropy D but relatively small second-neighbor interaction J2,places AgNiO2 within the collinear AF phase—as required—but not far from the boundary with a coplanar three-sublattice state.

At present, the origin of this strong easy-axisanisotropy—which may be of single-ion or exchangecharacter—is unclear. AgNiO2 is a highly anisotropic mate-

FIG. 13. �Color online� Constraints on the parameters J2 /J1 andD /J1 determined from the spin gap measured in inelastic neutron-scattering experiments. Gaps at M and M� are calculated withinSWT+1 /S discussed in Appendix B, and normalized to the maxi-mum energy for spin excitations. The upper solid line divides solu-tions where the calculated gap at M� is larger than that seen inexperiment, from those where it is smaller. The point J2 /J1�0.15,D /J1�1.2 which gives the best fit overall fit to experiment ismarked with an open circle. The region of small D for which nei-ther the gap at M nor M� are large enough to fit experiment isbounded by a dashed line. Linear spin-wave calculations give simi-lar lines, slightly shifted. �The small region of parameter spaceD /J1�0, J2 /J1�1 /8 for which spin-wave theory is ill-conditionedis shown by the red hatched area.�

FIG. 14. �Color online� Ground-state phase diagram of the easy-axis Heisenberg model on a triangular lattice as a function of J2 /J1

and D /J1, calculated from mean-field theory. The parametersJ2 /J1�0.15, D /J1�1.3 deduced from linear spin-wave theory fitsto inelastic neutron-scattering data place AgNiO2 within the collin-ear AF phase, not far from the boundary with a coplanar three-sublattice state.


104421-10

rial, and the crystal fields at the magnetic sites will reflectthis. However, relativistic �i.e., spin-orbit coupling� effects inNi are not usually strong, so the absolute size of anisotropyis surprising. The more massive Ag ions, which have stron-ger spin-orbit coupling, are nonmagnetic, and thought to playa negligible role in interactions between the Ni2+ sites. How-ever, given the partially itinerant nature of Ni electrons, asimple local-moment model such as Eq. �4� is probably un-likely ever to offer a complete description of the magnetismin AgNiO2.

Further band-structure calculations based on the modeldeveloped in Ref. 5 could potentially test the proposed mini-mal Hamiltonian and give further insight into the micro-scopic origin of the exchange interactions and easy-axis an-isotropy. Of particular interest would be the contribution ofthe itinerant Ni2 and Ni3 sites which surround the stronglymagnetic Ni1 sites to magnetic excitations, and their role inmediating interactions between localized Ni1 sites. Eventhough the dispersions of the boundaries of the powder-averaged data are well explained by Eq. �4�, there are dis-crepancies in the intensity distribution as a function of en-ergy and wave vector �see Fig. 12�E��, and these may containinformation about the itinerant electrons. It is also interestingto ask whether the weak but long-range interlayer interactionJz� could be of RKKY origin. This is a promising avenue forfuture research, and recent transport experiments on AgNiO2in high magnetic field reinforce the idea that charge carrierscouple strongly to the magnetic order.16

Even without these complications, the minimal micro-scopic model we have proposed has a number of very inter-esting features. In particular it predicts a spin-wave disper-sion with global minima not at magnetic Bragg wave vectorsbut at symmetry-related soft points. This prediction could betested by future inelastic neutron-scattering experiments onsingle crystals or by ESR experiments, which probe the ex-citations at the point, which by periodicity have the sameenergy as those at the magnetic Bragg wave vector. We notethat the unusual dispersion relation we propose with globalminima at multiple soft points may lead to nontrivial mag-netically ordered phases in applied magnetic fields that over-come the spin gap and future studies of the phase diagram inhigh applied magnetic field along the easy axis could show arich behavior that would be fruitful to exploreexperimentally16 and theoretically.17

ACKNOWLEDGMENTS

We acknowledge useful discussions with A. V. Chubukov,A. I. Coldea, K. Damle, M. Enderle, L. Seabra, and A. Vish-wanath. The research was supported in part by EPSRC-GBU.K. Grants No. GR/R76714/02 �R.C.�, No. EP/C51078X/2�E.W.�, and No. EP/C539974/1 �N.S.�, a CASE award fromthe EPSRC-GB and ILL �E.M.W.�, and the EU program atthe ILL. N.S. and P.A. gratefully acknowledge the support ofthe guest program of MPI-PKS Dresden, where part of thiswork was carried out.

APPENDIX A: LINEAR SPIN-WAVE DISPERSIONS FORCOUPLED TRIANGULAR LAYERS

Here we outline the derivation of the linear spin-wavedispersion relations for a model of stacked antiferromagnetic

triangular layers with collinear stripe order. The experimen-tally determined magnetic structure in 2H-AgNiO2 is shownin Fig. 15. In each triangular Ni layer the ordered spins formalternating ferromagnetic rows described by two magneticsublattices �up/down�, leading to a total of four magneticsublattices for the two layers in the magnetic unit cell �box inFig. 15�b��. After a Holstein-Primakoff transformation fromspin to magnon operators and Fourier transformation the spinHamiltonian in Eq. �4� becomes

H = q

X†HX + E0,

where E0 is a constant and terms higher than quadratic areneglected. The sum extends over all wave vectors q in thefirst magnetic Brillouin zone. The q dependence of the op-erator matrix X and of the Hamiltonian matrix H is implicit.The operator matrix X† is given by

X† = ��q†,�q

†,�−q,�−q� ,

where �, �, �, and � refer to the magnetic sublattices 1, 2, 3,4 �in this order� and �q

† creates a plane-wave magnon modeon sublattice 1, and so on. The Hamiltonian matrix in thisoperator basis is

H = �A B C D�

B� A D C

C D� A B

D C B� A , �A1�

where

A = 2S�J1 cos�2�k� + J1 + J2 cos�2��2h + k�� + J2

+ Jz� cos�2�l� − Jz� − Jz� + 2SD�1 −1

2S� ,

FIG. 15. �Color online� Magnetic structure of 2H-AgNiO2. �a�In each layer spins form alternating ferromagnetic stripes �dashedlines�, � symbols indicate the projection of the spin moments alongthe c axis. Stripes are parallel between adjacent layers �thick blacksymbols for layer 1 and faint gray for layer 2� but have an offsetbecause of the structural arrangement of Ni1 ions. Labels 1–4 referto the four magnetic sublattices �two for each layer� in the magneticunit cell �solid contour�, which is doubled along a axis compared tothe crystallographic unit cell. �b� 3D view of the magnetic structureshowing the orientation of the spin moments �thick arrows� of theNi1 ions �dark brown spheres� in the two layers in the unit cell.Light gray spheres are Ni2 and Ni3 ions, assumed nonmagnetic.The box shows the 3D magnetic unit cell.


104421-11

B = 4SJz cos��k�cos��l�� ,

C = 4SJ1 cos��k�cos��2h + k��

+ 4SJ2 cos�3�k�cos��2h + k�� ,

D = 2SJz cos��l��2,

� = e−�2h+k��i/3,

where �h ,k , l� are components of the wave vector q in unitsof the reciprocal lattice of the hexagonal unit cell shown inFig. 2�b�. Note that the very last term in function A above,2SD�

−12S , formally comes in spin-wave theory as a higher

order �1 /S� correction term, but we have included it herefollowing common convention,14 since it originates from aquadratic term in boson operators.

Diagonalizing the Hamiltonian in Eq. �A1� using standardmethods15 gives two doubly degenerate modes with disper-sions given by

��q��2 = A2 + BB� − C2

− DD� � sgn�B/��4�AB − CD��2 − �B�D� − BD�2,

�A2�

where � stands for optic and acoustic modes between thetwo layers and sgn�x� is the sign function. The above expres-sions for the dispersion relations also hold by periodicity fora general wave vector Q= �h ,k , l� outside the first magneticBrillouin zone �the two modes are degenerate when B=0 andthe prefactor sgn�B /�� in front of the square root is requiredto ensure continuity of the two distinct modes on both sidesof a degenerate point�.

For decoupled triangular layers with no anisotropy �Jz=Jz�=D=0 the dispersion is plotted in Fig. 10�a�� dashedline� and for the general case is plotted in Fig. 10�c� �dashedand dash-dotted lines�. The finite easy-axis anisotropy D�0 leads to a gap in the spectrum. The gap above the mag-netic Bragg peaks increases as a power law with increasing

anisotropy �M− =2S�D�4J1+4J2+2Jz+D�, where D=D�1

− 12S �. However, the gap is near linear in anisotropy at the soft

point M�, �M�� =2S�D�D−2Jz�, so in the limit of weak cou-

plings between the layers the minimum gap is always at thesoft point M� and not at the Bragg wave vector M �see Fig.10�b��.

The dynamical correlations �per Ni1 spin� are obtained as

Sxx�Q,�� = CS

8� �W�− �+� + X�− �+� + Y�− �+� + Z�− �+��2

N�− �+�

+�W��+� + X��+� + Y��+� + Z��+��2

N��+� �G�� − �+�

+ CS

8� �W�− �−� + X�− �−� + Y�− �−� + Z�− �−��2

N�− �−�

+�W��−� + X��−� + Y��−� + Z��−��2

N��−� �G�� − �−�

�A3�

where G��−�� is a Gaussian with a finite width to modelthe instrumental resolution, we use the shorthand notation��=�Q

� and the functions W, X, Y, Z, and N are given by

W�� = − �A + ��A2 + BB� − C2 − DD� − �2� + 2ABB�

− C�B�D� + BD� ,

X�� = C�A2 + BB� − C2 + DD� − �2� − A�B�D� + BD�

− ��B�D� − BD� ,

Y�� = B��A + ��2 − BB� + C2� − 2C�A + ��D + BD2,

Z�� = D�A2 + C2 − DD� − �2� + B�2D� − 2AB�C ,

N�� = �− WW� + XX� − YY� + ZZ�� .

The overall intensity prefactor C in Eq. �A3� is 1 in LSWTand is considered here as a variable parameter in the com-parison with the experimental data in order to account �in afirst approximation� for a possible overall intensity renormal-ization compared to LSWT. We note that the above expres-sions simplify considerably in the absence of the interlayercoupling Jz; in that case the two magnon modes in Eq. �A2�become degenerate with dispersion �Q=�A2−C2 and the dy-namical correlations in Eq. �A3� become

Sxx�Q,�� = CS

2

A − C

�G�� − �Q� . �A4�

For completeness we also quote the obtained transforma-tion matrix S to the normal operator basis X�=S−1X in whichthe Hamiltonian is diagonal

S = �W��−� W��+� W�− �−� W�− �+�

Y��−� Y��+� Y�− �−� Y�− �+�

X��−� X��+� X�− �−� X�− �+�

Z��−� Z��+� Z�− �−� Z�− �+� ,

where we use the shorthand notation W��=W�� /�N��and so on. This matrix satisfies the eigenvalue equationgHS=SgH� and the normalization condition SgS†=g, whereg is the commutator matrix for the operator basis defined by

g = X�X��T − �X�XT�T = �1 0 0 0

0 1 0 0

0 0 − 1 0

0 0 0 − 1

and

H� = ��− 0 0 0

0 �+ 0 0

0 0 �− 0

0 0 0 �+

is the Hamiltonian representation in the normal operator ba-sis X�.


104421-12

The one-magnon neutron-scattering intensity includingthe polarization factor and the magnetic form factor is

S�Q,�� = ��ro�2�1 +Qz

2

Q2��g

2f�Q��2

Sxx�Q,�� , �A5�

where ��ro�2=290.6 mbarns /sr is a conversion factor tobring the intensity into absolute units of mbarns/meV/sr,f�Q� is the magnetic form factor for Ni2+ ions and we takethe g factor g=2. Qz is the component of the wave-vectortransfer Q along the c axis. A numerical average of Eq. �A5�over a spherical distribution of orientations of the wave-vector transfer Q was performed in order to obtain theorientational-averaged intensity S�Q ,�� to be compared withthe measured powder data.

APPENDIX B: SPIN-WAVE THEORY WITH 1 ÕSQUANTUM CORRECTIONS FOR THE EASY-AXIS J1−J2

TRIANGULAR ANTIFERROMAGNET

Here we outline calculations of corrections to spin-wavedispersion arising from interactions between magnons atO�1 /S�. Calculations to this order have previously been car-ried out by Chubukov and Jolicoeur3 for the isotropic J1−J2 Heisenberg model on a triangular lattice. Here we extendthis work to the case of a finite easy-axis anisotropy D. Tomake contact with this earlier work we use here an orthogo-nal coordinate system with x along the magnetic stripes andy transverse to stripes in plane as in Fig. 8�b� and label the2D wave vectors as �kx ,ky�= �Qxa ,Qya�, where Qx,y are or-thogonal components �in Å−1� and a is the hexagonal latticeparameter.

In the usual linear semiclassical approximation �i.e., ne-glecting terms which contain more than two Bose operators�we obtain a dispersion

�k = �Ak2 − Bk

2,

with coefficients

Ak = 2SJ1�1 + cos kx + � + � cos �3ky� + 2SD ,

Bk = 4SJ1 cos�3ky

2�cos

kx

2+ � cos

3kx

2� , �B1�

where �=J2 /J1 and D=D�1−1 / �2S��. We note that our unitsdiffer by an overall factor of two from those in Ref. 3, andthat single-ion easy-axis anisotropy contributes terms to thelinearized Hamiltonian which are formally of order O�1 /S�relative to usual LSWT.

The leading effect of interactions between spin waves,treated at a one-loop �i.e., Hartree-Fock� level, is to renor-malize the coefficients of this dispersion to give

�k = ��Ak + �Ak�2 − �Bk + �Bk�2,

where

�Ak =2J1

N

p� Ap − �p

2�pF�k,p� +

Bp2

2�p� ,

�Bk = −2J1

N

p� Ap − �p

2�pBk −

Bp

2�pG�k,p�� . �B2�

Here, N is the total number of spins in the triangular plane,and the sum on internal momenta p extends over the entirenuclear Brillouin zone �large thick line hexagon in Fig. 8�a��.Following Ref. 3, we write Z=Z / �2SJ1��Z=A ,B ,�� and in-troduce the functions

F�k,p� = �1 − cos kx��1 − cos px� − 2�1 + ��

+ ��1 − cos �3ky��1 − cos �3py� −2DJ1

,

G�k,p� = cos�3ky + kx

2cos

�3py + px

2

+ cos�3ky − kx

2cos

�3py − px

2

+ ��cos�3ky + 3kx

2cos

�3py + 3px

2

+ cos�3ky − 3kx

2cos

�3py − 3px

2� . �B3�

Anisotropy D enters �Ak explicitly through F�k ,p�, but itsmain effect is to introduce a gap into the noninteracting dis-persion �k, which acts as an infrared cutoff in all denomina-

tors. In order not to mix different orders in 1 /S, we set D→D in calculations of �Ak and �Bk.

To obtain quantitative results for comparison with experi-ment, the sums in Eq. �B2� were evaluated numerically. Theresulting corrections to the linear spin-wave dispersion areillustrated first for the case of no anisotropy in Fig. 10�a��solid line�. In this case we exactly reproduce the results ofRef. 3: LSWT �dashed line� predicts a gapless mode at thesoft point M� �as an indirect consequence of the macroscopicclassical ground-state degeneracy�; however, quantum cor-rections lift the classical degeneracy by stabilizing the col-linear stripe order and generating a spontaneous gap at thesoft points while maintaining a gapless Goldstone mode atthe magnetic Bragg wave vectors.

In the presence of anisotropy the situation is somewhatdifferent. First the collinear order observed in AgNiO2 isnow stable at a classical level for a wide range of D and J2�cf. Fig. 14�. And second, all spin excitations are nowgapped, even in the absence of fluctuation effects �see Fig.10�b��. Expanding about collinear order, at a semiclassicallevel, the gap at the Bragg wave vectors M = �0,2� /�3�grows as

M � 2S�D�4J1 + 4J2 + D� ,

whereas at the soft points M�= �� ,� /�3� it increases only

linearly as 2SD. As a result, for finite D, the minimum in thespin-wave dispersion occurs on the zone boundary at the softpoint M�, and not at the magnetic ordering vector M. Once1 /S corrections are included, a gap is generated dynamically


104421-13

at M� for D=0, and there is a threshold anisotropy value Dcabove which the global minimum gap moves from M to thesoft point M�.

For parameters relevant to 2H-AgNiO2 �see Table I� thegap to spin excitations is large throughout the BZ, with aminimum at M� �Dc�0.21 meV for this parameter set�.Quantum fluctuations lead only to small quantitative correc-tions to the dispersion—compare the dashed to the solid linein Fig. 10�b�—and fits to LSWT are therefore justified aposteriori. A further indication that quantum fluctuations aresmall is the fact that the zero-point reduction in the sublatticemagnetization

�mS =2

N

p

Ap − �p

2�p�B4�

is calculated to be small: �mS=0.038 for the parametersgiven in Table I.

We note that this is not true in the limit of small aniso-tropy D→0 and small NNN couplings J2 /J1→1 /8, inwhich case the system is close to a quantum critical point. Inthis case zero-point fluctuations are very strong and the cor-rection to the sublattice magnetization diverges withinLSWT. A consequence of this is that 1 /S corrections drivethe gap

� 4��J1S + J2S�D�S − 1/2 − �mS� �B5�

at the point k= �0,0� imaginary for a small but finite re-

gion of parameters D /J1�0, J2 /J1�1 /8—cf. Fig. 13. Ac-curate treatment of this small D regime would require a moresophisticated self-consistent spin-wave theory, which is notwarranted for AgNiO2.

For the purposes of comparing the spin-wave calculationsat O�1 /S� with the envelope of scattering in ��Q� ,�� it issufficient to replace Ak, Bk, and �k by their corrected valuesin the expression for the scattering intensity

Sxx�Q,�� = CS

2

Ak − Bk

�k�� − �k� . �B6�

Finally, the interlayer coupling Jz� can easily be includedin these calculations by modifying Eq. �B1� as follows:

Ak → Ak + 2SJz��cos kz − 1� .

Here kz=Qzc where Qz is the c-axis wave vector componentin Å−1. At order O�1 /S� the interlayer couplings modify Eq.�B3� to read

F�k,p� → F�k,p� +Jz�

J1�1 − cos 2kz��1 − cos 2pz� .

However since this change is relatively very small for sim-plicity we neglect it in the analysis and only consider theexplicit effects of Jz� through changes in the dispersion rela-tion at linear order in SWT. The resulting three-dimensionalspin-wave dispersion and corrections are shown in Fig.10�d�.

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explored by inelastic neutron scattering

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