PHYSICAL REVIEW B 72, 014405 共2005兲

Spin waves and magnetic interactions in LiCu2O2
T. Masuda,1 A. Zheludev,1 B. Roessli,2 A. Bush,3 M. Markina,4 and A. Vasiliev4
1Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6393, USA
2Laboratory for Neutron Scattering, ETH Zurich and Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
3

Moscow Institute of Radiotechnics, Electronics and Automation, Moscow 117464, Russia

4Low Temperature Physics Department, Moscow State University, Moscow 119992, Russia

共Received 21 December 2004; published 1 July 2005兲
The quasi-one-dimensional helimagnet LiCu2O2 was studied by single crystal inelastic neutron scattering.
The dispersion relation of spin wave excitations was measured in the vicinity of the principal magnetic Braggreflection. A spin wave theoretical analysis of the data yields an estimate of the relevant exchange constantsand explains the mechanism of geometric frustration that leads to helimagnetism. It is found that the simple
antiferromagnetic J1-J2 model that was previously proposed is inadequate for LiCu2O2. The experimental
findings are generally in a qualitative agreement with first principles calculations of 关A. A. Gippius, E. N.
Morozova, A. S. Moskvin, A. V. Zalessky, A. A. Bush, M. Baenitz, H. Rosner, and S.-L. Drechsler, Phys. Rev.
B 70, 020406 共2004兲兴, though certain important discrepancies remain to be explained.
DOI: 10.1103/PhysRevB.72.014405

PACS number共s兲: 75.50.⫺y, 75.30.Ds, 75.30.Et, 75.25.⫹z

I. INTRODUCTION

LiCu2O2 is a quasi-one-dimensional magnetic material
that has recently attracted a great deal of attention.1–4 Muon
spin resonance,5 neutron scattering experiments,6 and NMR
studies7 clearly demonstrated that this S = 1 / 2 system has anincommensurate helimagnetic ground state. In our previous
work6 we postulated a simple model of magnetic interactions
that seemed to account well for the observed properties. It
was proposed that LiCu2O2 features weakly coupled zigzag
S = 1 / 2 chain with competing nearest-neighbor J1 and nextnearest-neighbor J2 antiferromagnetic interactions. Due togeometric frustration this model exhibits an unusual competition between a quantum-disordered gapped9 ground statewith commensurate spin correlations,10 and semiclassical helimagnetic incommensurate state. However, independent
first-principles numerical studies7,11 suggested a totally different model for the geometric frustration in LiCu2O2. According to that work, the nearest-neighbor J1 interactions are
actually ferromagnetic, and it is the unusually large fourthnearest-neighbor antiferromagnetic coupling constant J4 thatcauses geometric frustration and favors helimagnetism. A
similar scenario was recently proposed for a very similar
isostructural helimagnet NaCu2O2.12
Bulk magnetic susceptibility data for LiCu2O2 seem to becompatible with both models,6,11 and are thus inconclusive.
The only direct way to determine 共effective兲 exchange constants is by means of single crystal inelastic neutron spectroscopy. This technique can probe the dispersion relation ofspin wave excitations in the system. In the present paper we
describe the results of such a study. Preliminary results werereported in Ref. 8. We find that the actual topology of magnetic coupling is indeed more complex than originally envisioned.
A. Structural considerations

tal

LiCu2O2 has a layered charge-ordered orthorhombic crysstructure 关space group Pnma, a = 5.730共1兲 Å, b

1098-0121/2005/72共1兲/014405共7兲/$23.00= 2.8606共4兲 Å, and c = 12.417共2兲 Å兴, as described in detail in
Refs. 3 and 5. The magnetism is due to double chains of
Cu2+ ions that run along the crystallographic b axis. The
period of each “leg” of the double spin chains is equal to b.
The two legs are offset by the b / 2 relative to each other andthus form a “triangular ladder,” as illustrated in Fig. 1共a兲.
These double chains are arranged in layers parallel to the
共a , b兲 plane and are effectively separated along the c direction by planes of non-magnetic Cu+ ions.
The approximate magnetic structure was determined inRef. 6 and is illustrated in Fig. 1 of that paper. The ordering

FIG. 1. 共a兲 A schematic view of exchange interactions between
magnetic Cu2+ ions in LiCu2O2. 共b兲 An equivalent Bravais lattice ofspins obtained by displacing every other Cu2+ ion in the original
non-Bravais lattice by the vector d. It is assumed that the spin
Hamiltonian remains intact upon this transformation.

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temperature is Tc = 24 K. Each double spin chain carries aplanar helix of spins. The magnetic propagation vector is
q0 = 共0.5, ␨ , 0兲 with ␨ ⬇ 0.827. Consecutive spins on eachrung are almost parallel, being rotated relative to each otherby an angle ␾ = 2␲␨. Within each double chain any nearestneighbor spins from opposite legs are almost antiparallel andform an angle ␾ / 2 = ␲␨. Preliminary neutron diffraction experiments were consistent with 共a , b兲 being the spin rotation
plane, but independent NMR studies clearly show an out-ofplane component.7 This suggests that the spin rotation planes
are, in fact, tilted relative to a high-symmetry orientation.
The issue deserves further investigation, perhaps involving
more accurate diffraction measurements. However, if magnetic anisotropy effects are negligible compared to Heisenberg exchange interactions 共a reasonable assumption for
Cu2+ ions兲, the actual arrangement of spin rotation planes
should not affect spin wave dispersion relations that are thefocus of the present study.
The main goal of the inelastic neutron scattering experiments described below is to establish the hierarchy of exchange interactions. The corresponding coupling constants
are schematically shown in Fig. 1共a兲, and define a model
Heisenberg Hamiltonian for LiCu2O2
H̃ = 兺 关J1Si,jSi+1,j + J2Si,jSi+2,j + J4Si,jSi+4,j + J⬜Si,jSi,j+1兴,i,j共1兲where the index i labels consecutive spins in each double
chain, as shown in Fig. 1共a兲, and j labels the double chains.8
II. EXPERIMENTIn the present study we employed a 5 g single-crystal
sample of LiCu2O2. Crystal mosaic was not particularly
good: irregular, with as much as a 4° full width at half maximum 共FWHM兲 spread. This circumstance imposed certain
constraints as will be discussed below. The measurements
were carried out in two separate series of experiments. Neutrons with a fixed final energy of E f = 13.5 meV were used at
the HB-1 thermal three-axis spectrometer at the High Flux
Isotope Reactor at Oak Ridge National Laboratory, Oak
Ridge, TN 共ORNL兲 共Setup 1兲. Pyrolitic graphite PG 共002兲
reflections were used for a monochromator and an analyzer.
A PG filter was installed after the sample to eliminate higherorder beam contamination. The setup employed 48⬘ − 80⬘− 80⬘ − 240 collimators. Alternatively, we utilized cold neutrons with a final energy fixed at E f = 5 meV at the TASP
three-axis instrument installed at the SINQ spallation source,
Villigen PSI, Switzerland 共Setup 2兲. A PG monochromatorand analyzer were used with no additional collimation and acold Be filter positioned after the sample. In all cases the
sample was mounted with the 共a , b兲 plane coinciding withthe horizontal scattering plane of the instrument. The sample
environment was a standard “Orange”-type flow cryostat.
Most of the data were taken at T = 1.5 K.
Data collection procedures were aimed at minimizing theadverse effects of the broad mosaic spread of the sample. A
large mosaic translated to a poor wave vector resolution per-

FIG. 2. A constant-E scan along the 共−0.5, k , 0兲 reciprocal-space
rod measured in LiCu2O2 at T = 1.5 K at an energy transfer of
5 meV 共symbols兲 using Setup 1. The lines are a simulation based on
the measured spin wave dispersion relation and the known resolution function of the instrument.

pendicular to the momentum transfer q. Since the spin wave
dispersion is steepest along b*, the best focusing conditions
are obtained for q pointing close to that direction. Considering the intensity reduction due to the effect of magnetic form
factors at large 兩q兩, an optimal “window” for low-energy spinwave measurements was identified in the vicinity of the 共0.5,
0.827, 0兲 and 共0.5, 1.173, 0兲 magnetic Bragg peaks. Anotheradvantage of this reciprocal-space region is that it is relatively clear of spurious scattering and phonons, that were
carefully checked for. Most scans were repeated at T = 75 K,
i.e., well above Tc, to verify that the signal observed at low
temperature is indeed of magnetic origin. In some casessmoothed high-temperature scans were used as backgroundfor point-by-point subtraction from the corresponding lowtemperature data sets.
Another important technical problem that had to be dealt
with is crystal twinning. In LiCu2O2 twinning occurs at a
microscopic level.3 The a axis of one type of domain coincides with the b axis of the other domain type, since a ⬇ 2b.
The immediate consequence is that neutron diffraction andinelastic scattering necessarily detect a joint signal originating from both domain types. As will be discussed in detailbelow, the contributions from different domains and different
spin wave branches within each domain could be reliably
separated only in the direct proximity of the 共0.5, 0.827, 0兲
and 共0.5, 1.173, 0兲 peaks. Even so, twinning reduces theeffective sample volume by a factor of two. To avoid any
confusion we will use the indexes h, k, and l to label wave
vector components in the coordinate system defined by thereciprocal lattice of one particular domain type 共type A兲. Indexes hB = 2k, kB = h / 2, and lB = l will refer to the other crystallographic domain 共type B兲.
III. RESULTS

A typical constant-energy scan collected using Setup 1 isshown in Fig. 2. The non-magnetic background for this scan
共featureless and typically about 80 counts per 5 min兲 was
measured at T = 75 K. A linear fit to the measured background was subtracted from the data shown. The four prominent peaks seen in this scan can be attributed to an acoustic

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SPIN WAVES AND MAGNETIC INTERACTIONS IN LiCu2O2

FIG. 3. Typical constant-q scans measured in LiCu2O2 共symbols兲 at T = 1.5 K. The lines are as in Fig. 2. The shaded area in thelower scan is an “accidental Bragg” spurious peak originating from2ki scattering in the monochromator, 共0, 2, 0兲 Bragg scattering inthe sample, and inelastic thermal-diffuse scattering in the analyzer.
The spurious peak appears much narrower than the experimental
resolution 共about 2 meV FWHM兲.

spin wave emanating from two magnetic Bragg peaks at 共0.5,
0.827, 0兲 and 共0.5, 1.173, 0兲. Representative constant-q scanscollected using the same setup are shown in Fig. 3, where aflat background has been subtracted from the data. All inelastic peaks observed for 0.75ⱗ k ⱗ 1.25 and ប␻ ⱗ 8 meV canbe associated with a single branch of excitations. The corresponding dispersion relation along the crystallographic b axis
was extracted from the measured scans using Gaussian fits.
The result is plotted in circles in Fig. 4共a兲, where solid andopen symbols indicate constant-q and constant-E measurements, respectively. Any attempts to follow the observed spinwave branch to higher energies were not successful. The
mode’s intensity drops progressively, while the backgroundincreases and becomes structured. As will be explained below, part of this problem may be due to a multitude of additional spin wave branches from both types of domains. The
situation was further aggravated by limited resolution, phonon scattering, and instrument-related spurious peaks. For
this reason, throughout this paper we shall limit the discussion to experimental data collected at low energies in direct
proximity of the two above-mentioned magnetic Bragg
peaks.
Additional measurements were performed using Setup 2
to determine the dispersion along the a* axis. Typical scans
and the measured dispersion curve are shown in Figs. 5 and
4共b兲, respectively. The important zone-boundary scan at 共0,0.827, 0兲 was also measured using Setup 1 in the range

FIG. 4. Spin wave dispersion measured in LiCu2O2 at T
= 1.5 K 共symbols兲. Shaded and open circles are data points obtained
from constant-E and constant-q scans, respectively. The lines are a
fit to the data, as described in the text. Heavy solid, heavy dashed,
and thin dotted lines correspond to Models 1, 2, and 3, respectively.
In both panels the ellipses represent the FWHM of the instrument
resolution function in the appropriate projection 共open ellipse兲 andsection 共shaded ellipse兲.

2 – 15 meV and found to be fully consistent with that shownin Fig. 5共c兲.
A. Data analysis

In order to extract the relevant exchange constants fromthe measured dispersion curves the data were analyzed in theframework of semiclassical spin wave theory 共SWT兲.13
1. Equivalent Bravais lattice

In LiCu2O2 there are four magnetic ions per unit cell.
However, within the approximation Hamiltonian 共1兲, any interactions between the double-chain layers are ignored, andthe spin lattice becomes topologically a Bravais one. The
transformation to a spatial Bravais lattice is made by displacing the atoms as shown in Fig. 1共b兲. The dynamic structure
factor S0共q , ␻兲 of this “straightened out” spin network is directly related to that of the original model throughS共q, ␻兲 = S0共q, ␻兲cos2

014405-3

qd
qd+ S0„q + 共0,1,0兲, ␻…sin2,22

qd = 2␲h␦x + 2␲l␦z ,

共2a兲
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The requirement that q0 minimizes J共q兲 fixes the relation
between J1, J2, J4, and ␨
J1 + 4J2 cos共␲␨兲 + 16J4 cos共␲␨兲cos共2␲␨兲 = 0.

共5兲Conveniently, J⬜ does not enter this expression. The singlemagnon cross section has three contributions
S Aq + Bq
␦共␻ − ␻q兲,
2 gaq

共6a兲

S−0 共q, ␻兲 =S Aq−q0 − Bq−q0
␦共␻ − ␻q−q0兲,
8
␻q−q0

共6b兲

S+0 共q, ␻兲 =S Aq+q0 − Bq+q0
␦共␻ − ␻q+q0兲.
8
␻q+q0

共6c兲

S⬜
0 共q, ␻兲 =FIG. 5. Typical scans measured in LiCu2O2 using Setup 2 共symbols兲. The lines are as in Fig. 2.

where q = 共h , k , l兲 is the wave vector transfer and d is therelative displacement of the two legs in each double chain:d = a␦x + c␦z. For the crystal structure of LiCu2O2 we have
␦x = 0.24 and ␦z = 0.19. It is important to emphasize that Eq.
共2兲 is exact: It does not imply any particular properties of
S共0兲共q , ␻兲 and S共q , ␻兲, and relies only on the definitions ofthese correlation functions. While it is possible to derive theexcitation energies in the the general 共non-Bravais兲 case, theuse of Eq. 共2a兲 is fully valid for our model and greatly facilitates the identification of individual spin wave branches
in LiCu2O2.
2. SWT dynamic structure factor

A very useful recipe for calculating the SWT dynamic
structure factor can be found in Ref. 14. The magnon dispersion relation is given by:

␻q2 = Aq2 − Bq2 ,Aq = 2S

再

共3a兲

冎

Jq 1
+ 关Jq−q0 + Jq+q0兴 − Jq0 ,2 4

Bq = 2S再冎Jq 1− 关Jq−q0 + Jq+q0兴 ,
2 4

共3b兲

共3c兲

where, for our model HamiltonianJh,k,l = 2J1 cos共␲k兲 + 2J2 cos共2␲k兲 + 2J4 cos共4␲k兲+ 2J⬜ cos共2␲h兲.

共4兲Here S⬜共q , ␻兲 represents fluctuations of the spin component
perpendicular to the plane of the helix, while S+共q , ␻兲 and
S−共q , ␻兲 are in-plane excitations. The total SWT cross section for LiCu2O2 is related to S0共q , ␻兲 through Eq. 共2兲.
It now becomes clear why measuring spin waves in
LiCu2O2 is so technically challenging: There are six spinwave branches in each crystallographic domain. As a result,in any experiment one has to deal with a total of 12 spinwave branches, all of which are acoustic and therefore contribute to scattering at low energies. Also, compared to asimple antiferromagnet on a Bravais lattice, the scattering by
each individual branch is typically much weaker, since intensity is effectively redistributed between twelve modes. The
existence of a suitable, albeit narrow, measurement windowthat was exploited in our experiments is a fortunate coincidence.
B. Fits to experimental data

The energy of the spin wave branch observed in our experiments clearly goes to zero at the principle magnetic
Bragg peaks of the A-type domain. It must therefore be associated with the S⬜共q , ␻兲 mode and the first term in Eq.
共2a兲. An analysis of the polarization of spin wave excitations
is beyond the scope of this work. Nevertheless, we can pointout that 共a , b兲 being the spin rotation plane6 is consistent
with a strong S⬜共q , ␻兲. Indeed, the corresponding polarization factor for unpolarized neutrons is a maximum for inplane momentum transfers.14 As a first step in our analysis
we ignored all other spin wave branches and used Eq. 共3a兲 to
fit the experimental data. It is easy to show that there areexactly two sets of exchange parameters that exactly reproduce 共i兲 the experimentally determined incommensurability
parameter ␨, 共ii兲 the excitation energies at the zone boundaries 共0.5, 1 , 0兲 and 共0,0.827,0兲, and 共iii兲 the spin wave velocity along the b* direction near the magnetic Bragg peaks.
Using these two sets of parameters as initial points, we employed a least-squares algorithm to best-fit the data under the
rigid constraint set by Eq. 共5兲. The two resulting optimized
sets of parameters are listed in Table I and will be referred toas Models 1 and 2, respectively. For a purpose that will bemade clear in the next section, we also attempted a fit to the014405-4

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SPIN WAVES AND MAGNETIC INTERACTIONS IN LiCu2O2
TABLE I. Sets of exchange constants obtained by fitting the
calculated spin wave dispersion relation to the experimental curvesin comparison with first-principle calculations.a

J1
J 2c
J4
J⬜

Model 1
共meV兲Model 2
共meV兲Model 3

LDAb
共meV兲

3.2 共0.5兲
−5.95
3.7 共0.3兲
0.9 共0.1兲

52.8 共4.0兲
16.9−0.8 共0.1兲
0.12 共0.01兲

0 共fixed兲
−7.0 meV3.75 共0.05兲 meV
3.4 共0.2兲 meV

0.4
−8.1
14.4
5.7

aNote the difference in the definition of J compared to Ref. 8.
bFrom Ref. 7.

Not refined: fixed by Eq. 共5兲.

cdata taken only along the b axis while fixing J1 ⬅ 0. The
measured a-axis dispersion was not included in this fit. The
resulting set of exchange constants is also listed in Table I
and will be referred to as Model 3. Dispersion relations forthe out-of-plane branch and the first term in Eq. 共2a兲 calculated for type-A domains using Models 1 through 3 are plotted in lines in Fig. 4.
The next step is to understand why the other 11 spin wave
branches were not observed in the studied wave vector andenergy ranges. All 12 dispersion curves calculated using
Model 1 are plotted in Fig. 6共a兲. The intensities of eachbranch were scaled by the energy transfer 关to get rid of the
1 / ␻ factors in Eq. 共6兲兴 and are plotted in Fig. 6共b兲. The
neutron polarization factor was not included in this calculation, and the maximum value 共unity兲 was assumed for eachbranch. One readily sees that all branches originating fromtype-B domains and the first term in Eq. 共2a兲 are extremely
weak in our area of interest. The same is true for all spinwaves associated with type-A domains and the second termin Eq. 共2a兲. Contributions from type-B domains and the second term in Eq. 共2a兲 are either weak or at very high energy.
In the end, only the “principal” modes originating fromtype-A domains and the first term in Eq. 共2a兲 are relevant toour experiments. Of these three branches the two in-planemodes are almost entirely outside the studied energy range
and do not affect any of the scans collected. As a result, only
one out-of-plane mode is seen. Similar arguments can bemade regarding the dispersion measured along the a axis.
The simulation also explains why we were unable to follow
the “principal” out-of-plane mode to higher energies. At
around 8 meV energy transfer any inelastic scans become
crowded with a host of peaks associated with other spinwave branches that can not be reliably resolved due to limited experimental resolution.
From Fig. 4 it is apparent that the studied energy andwave vector transfer range the dispersion relations produced
by Model 2 are virtually identical to those resulting from
Model 1. Both models fit the available experimental datarather well. The only small discrepancy is seen along the adirection where the observed spin wave velocity near q = 共−0.5, 0.827, 0兲 appears slightly larger than calculated, while
the zone-boundary energy at q = 共0 , 0.827, 0兲 is slightly
smaller. This discrepancy is easily explained if one assumesa small anisotropy-induced gap ⌬ in the spin wave spectrum

FIG. 6. 共Color online兲 共a兲 Dispersion of all 12 spin wave
branches along the 共0.5, k , 0兲 reciprocal-space rod in twinned
LiCu2O2 crystals calculated using Model 1. 共b兲 The corresponding
structure factors scaled by energy transfer. In both panels the outof-plane modes are plotted in red. Modes shown in blue and green
lines are polarized in the plane of spin rotation. Thick and thin linesof all types correspond to the first and second terms in Eq. 共2a兲,
respectively. Solid and dashed lines refer to type-A and type-B crystallographic domains. Symbols are as in the top panel of Fig. 4.

at the magnetic Bragg peak position. From the existing scans
it is possible to tentatively estimate ⌬ ⬇ 1.5 meV, thoughmore high-resolution data will be needed for a reliable determination. As far as only the b-axis dispersion is concerned,
Model 3 seems to work just as well as the other two sets ofparameters, and for the 共0.5, k , 0兲 reciprocal-space rod produces almost identical curves. However, this good agreement
under the constraint J1 = 0 necessarily results in a very poor
agreement with the a-axis dispersion relation. The corresponding bandwidth calculated using Model 3 关thin dottedline in Fig. 4共b兲兴 is considerably larger than the observed
value. In other words, assuming J1 = 0 in Hamiltonian 共1兲 can
not be made consistent with the bulk of the experimental
data.
The resulting parameter values for Model 1 were used tosimulate the measured inelastic neutron scans. For this purpose we utilized the magnetic dynamic structure factor as
given by Eq. 共6a兲. This model cross section was scaled by
the magnetic form factor for Cu2+ and numerically convoluted with the four-dimensional spectrometer resolutionfunction. The latter was calculated in Gaussian form using
the Cooper-Nathans approximation. The evolution of the cor-

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responding FWHM resolution ellipsoid in the surveyed areasof 共E , q兲 space is shown in Fig. 4. An overall scaling factor
was chosen to best fit the measured scans. The results of such
simulations are shown in solid lines in Figs. 2–5. Overall, theexperimental peak shapes are well explained by resolution
effects alone.
IV. DISCUSSIONAs mentioned in the Introduction, the knowledge of spinwave dispersion in an ordered magnetic material can, in principle, provide comprehensive information on the exchangeinteractions. This requires measurements of dispersion
curves across the entire Brillouin zone. In the particular caseof LiCu2O2, twinning, less than perfect sample mosaic, and acomplex spectrum limit us to only observing spin waves inthe direct proximity of the principal magnetic Bragg reflections. For this reason the resulting sets of parameters can be
regarded as unique only under the assumption of the validityof Hamiltonian 共1兲 and the SWT approximation.
The Hamiltonian has a solid justification in the crystal
structure, and is well supported by first-principles calculations. However, a vital question is to what extent an analysis
based on essentially classical spin wave theory can be valid
in the case of a quasi-low-dimensional frustrated magnet
such as LiCu2O2? At the very least, the exchange constants
obtained from such an analysis will be renormalized compared to their actual values due to quantum corrections. Thisis particularly true for interchain interactions, for which
SWT exchange parameters are typically much smaller than
the actual ones. Nevertheless, even for highly anisotropic or
frustrated systems, acoustic modes can be associated withwaves in the order parameter field. In this case a renormalized SWT with some effective set of exchange parameters
typically works rather well. Just two examples are theweakly coupled S = 1 / 2 chains compound KCuF3 共Ref. 15兲
and the quasi-2D frustrated helimagnet Cs2CuCl4.14 Deviations from classical behavior in LiCu2O2 do not appear particularly severe. The typical energy scale of exchange interactions computed by local density approximation 共LDA兲
exceeds the temperature of three-dimensional 共3D兲 longrange ordering by less than an order of magnitude. While theabsolute value of the ordered moment has not been determined to date, that in the very similar helimagnet NaCu2O2
is quite large 共0.64 ␮B兲.12 We therefore have good reason toassume that renormalized SWT will work well for our system.
Turning now to the two solutions that were obtained, we
note that Model 2 almost exactly corresponds to the simple
J1-J2 model that we originally proposed.6 However, unlessthe renormalization of exchange constants is severe, theirlarge values appear unrealistic. Indeed, these parameters are
an order of magnitude larger than those estimated by fitting
the measured magnetic susceptibility to the J1-J2 model.6 In
addition, a straightforward estimate of the Curie-Weiss temperature TC-W = 2S共S + 1兲J共0兲 / 共3kB兲 = 共J1 + J2 + J4 + J⬜兲 / kB for
Model 2 yields TCuW ⬃ 800 K. This value is vastly greaterthan the actual TCuW = 40 K,16 as estimated from the susceptibility data of Ref. 6.

We thus conclude that Model 1 is much more likely to be
an adequate description of LiCu2O2. Considering that theSWT exchange constants are renormalized compared to theiractual values, the predicted TC-W ⬃ 21 K is consistent withexperimental observation. Additional support for Model 1
could be potentially found in measurements of the dispersion
relation along the c axis, that were not performed in thepresent study. However, since there are no obvious dominantinterlayer exchange pathways in the LiCu2O2 structure, ananalysis of c-axis dispersion would necessarily introduce
several small interlayer exchange parameters. This, in turn,would result in an over-parametrized problem and would notnecessarily help to unambiguously establish J1, J2, and J4.
The key features of Model 1, namely, a ferromagnetic J2bond and a substantial J4 coupling constant, are similar to
those of the LDA calculations of Ref. 7. The mechanism ofgeometric frustration in LiCu2O2 is thus similar to that in
NaCu2O2.12 The main difference between the exchange parameters in Model 1 and those emerging from LDA calculations of Ref. 7 is in the magnitude of J1. As illustrated byModel 3, assuming that J1 is negligibly small makes theSWT calculation incompatible with the measurements. We
conclude that either LDA severely underestimate J1, or thisexchange constant is renormalized by an order of magnitudein the SWT. Of course, one can never entirely rule out thepossibility that both the LDA calculations and our SWT
analysis are missing some crucial terms in the Hamiltonian.
Further experimental and theoretical insight will be required
to fully resolve this remaining inconsistency.

V. CONCLUSION

In summary, based on inelastic neutron scattering measurements we can conclude that the simple J1-J2 model that
we originally proposed does not apply to LiCu2O2. The frustration mechanism is more complex and resembles that proposed in Refs. 7 and 11. It involves a competition between a
combination of antiferromagnetic J1 and ferromagnetic J2
interactions against an additional antiferromagnetic longrange J4 coupling. We find the three corresponding exchange
constants to be of comparable absolute strength. This discrepancy between a SWT-based interpretation of the experimental data and first-principles calculations remains to be
explained.

ACKNOWLEDGMENTS

One of the authors 共A.Z.兲 would like to thank A. Drechsler for numerous insightful discussions on the subject andfor his constructive comments 共Ref. 11兲 on Ref. 6. This work
was partially supported by the Civilian Research and Development Foundataion Project No. RU-P1-2599-04. Work at
ORNL was carried out under Contract No. DE-AC0500OR22725, U.S. Department of Energy. Part of this work
was done at the Swiss Neutron Spallation Source SINQ,
Switzerland.

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SPIN WAVES AND MAGNETIC INTERACTIONS IN LiCu2O2
1 A. M. Vorotynov, A. I. Pankrats, G. A. Petrakovkii, K. A. Sa-

blina, W. Paszkowicz, and H. Szymczak, JETP 86, 064424
共1998兲; J. Magn. Magn. Mater. 188, 233 共1998兲.
2 F. Fritschij, H. Brom, and R. Berger, Solid State Commun. 107,
719 共1998兲.
3
S. Zvyagin, G. cao, Y. Xin, S. McCall, T. Caldwell, W. Moulton,
L.-C. Brunel, A. Angerhofer, and J. E. Crow, Phys. Rev. B 66,
064424 共2002兲.
4
D. A. Zatsepin, V. R. Galakhov, M. A. Korotin, V. V. Fedorenko,
E. Z. Kurmaev, S. Rartkowski, M. Neumann, and R. Berger,
Phys. Rev. B 57, 4377 共1998兲.
5 B. Roessli, U. Staub, A. Amato, D. Herlach, P. Pattison, K. Sablina, and G. A. Petrakovskii, Physica B 296, 306 共2001兲.
6 T. Masuda, A. Zheludev, A. Bush, M. Markina, and A. Vasiliev,
Phys. Rev. Lett. 92, 177201 共2004兲.
7
A. A. Gippius, E. N. Morozova, A. S. Moskvin, A. V. Zalessky,
A. A. Bush, M. Baenitz, H. Rosner, and S.-L. Drechsler, Phys.
Rev. B 70, 020406 共2004兲.
8 T. Masuda, A. Zheludev, A. Bush, M. Markina, and A. Vasiliev,
Phys. Rev. Lett. 94, 039706 共2005兲. Note: There is a factor of
two difference in the definition of exchange constants relative tothe present work and to Ref. 7. The discrepnacy is due to thedefinition of J in the Hamiltonian, where each bond is counted
twice, as opposed to the convention used in this work, where
each bond enters the sum once only.

9 C.

K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, 1388
共1969兲; B. S. Shastry and B. Sutherland, Phys. Rev. Lett. 47,
964 共1981兲; F. D. M. Haldane, Phys. Rev. B 25, 4925 共1982兲; S.
R. White and I. Affleck, ibid. 54, 9862 共1996兲; A. A. Aligia, C.
D. Batista, and F. H. L. Essler, ibid. 62, 3259 共2000兲 and references therein.
10 T. Tonegawa and I. Harada, J. Phys. Soc. Jpn. 56, 2153 共1987兲;R. Chitra, Swapan Pati, H. R. Krishnamurthy, Diptiman Sen,and S. Ramasesha, Phys. Rev. B 52, 6581 共1995兲.
11 S.-L. Drechsler, J. Mlek, J. Richter, A. S. Moskvin, A. A. Gippius, and H. Rosner, Phys. Rev. Lett. 94, 039705 共2005兲.
12 L. Capogna, M. Mayr, P. Horsch, M. Raichle, R. K. Kremer, M.
Sofin, A. Maljuk, M. Jansen, and B. Keimer, cond-mat/0411753
共unpublished兲.
13
See, for example, T. Nagamiya, Solid State Physics 共Academic
Press, New York, 1967兲, Vol. 20, p. 305.
14 R. Coldea, D. A. Tennant, and Z. Tylczynski, Phys. Rev. B 68,
134424 共2003兲.
15 D. A. Tennant, S. Nagler, D. Welz, G. Shirane, and K. Yamada,
Phys. Rev. B 52, 13381 共1995兲.
16 In Ref. 3 T
C-W is cited as 80 K. As discussed in Ref. 6, thisdiscrepancy is due to the contamination of the samples used inRef. 3 with the Li2CuO2 phase. This impurity phase also produces a sharp peak in susceptibility at T = 9 K that is absent inour samples.

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