Table Of ContentSpin excitations used to probe the nature of the exchange coupling in the
magnetically ordered ground state of Pr Ca MnO
0.5 0.5 3
R. A. Ewings,1,∗ T. G. Perring,1,2 O. Sikora,3 D. L. Abernathy,4 Y. Tomioka,5 and Y. Tokura6,7
1ISIS Facility, STFC Rutherford Appleton Laboratory,
Harwell Oxford, Didcot OX11 0QX, United Kingdom
2London Centre for Nanotechnology, 17-19 Gordon Street, London WC1H 0AH, United Kingdom
3H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom†
4Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
5Electronics and Photonics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST),
6 Tsukuba Central 4, 1-1-1 Higashi Tsukuba 305-8562, Japan
1 6Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
0 7RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
2 (Dated: July 25, 2016)
l We have used time-of-flight inelastic neutron scattering to measure the spin wave spectrum of
u
J the canonical half-doped manganite Pr0.5Ca0.5MnO3 in its magnetic and orbitally ordered phase.
The data, which cover multiple Brillouin zones and the entire energy range of the excitations, are
2
compared with several different models that are all consistent with the CE-type magnetic order,
2
but arise through different exchange coupling schemes. The Goodenough model, i.e. an ordered
state comprising strong nearest neighbor ferromagnetic interactions along zig-zag chains with anti-
]
l ferromagnetic inter-chain coupling, providesthebest description of thedata, providedthat further
e
neighbor interactions along the chains are included. We are able to rule out a coupling scheme in-
-
r volvingformationofstronglyboundferromagneticdimers,i.e. Zenerpolarons,onthebasisofgross
t
s features of the observed spin wave spectrum. A model with weaker dimerization reproduces the
. observed dispersion but can be ruled out on the basis of discrepancies between the calculated and
t
a observedstructurefactorsatcertainpositionsinreciprocalspace. Addingfurtherneighborinterac-
m tions results in almost no dimerization, i.e. recovery of the Goodenough model. These results are
- consistent with theoretical analysis of the degenerate double exchange model for half-doping, and
d providearecipeforhowtointerpretfuturemeasurementsawayfromhalf-doping,wheredegenerate
n doubleexchange models predict more complex ground states.
o
c
[
I. INTRODUCTION strongintrasiteexchange. Theessentialphysicsisacom-
2 petitionbetweendelocalizationoftheeg electrons,favor-
v ing a homogeneous, fully spin polarized, ferromagnetic
8 A characteristic of many materials exhibiting strong metalviadoubleexchange,andlocalizationduetolattice
4 electroniccorrelationsis adelicatebalancebetweenmul- distortions trapping the e electrons to form polarons8.
g
1 tiple competing phases which can often be tuned with
For x 0.3 the materials are typically ferromagnetic
1 relatively modest changes in external parameters such ∼
metals,withaseaofpolaronsintheinsulatingparamag-
0 as temperature, magnetic field, pressure or chemical
netic phase. The most marked CMR effects often occur
1. doping1. Preeminent amongst strongly correlated elec-
forthosematerialsinwhichthepolaronsorderinthehalf
0 tron systems are transition metal oxides, which dis-
doped state, x 0.5, to give a charge and orbitally or-
5 playphenomenaasdiverseassuperconductivity2,orbital ∼
dered insulating state for T < T , the charge-ordering
1 order3, and complex magnetic order4. Most families of CO
: temperature. In the case of Pr0.5Ca0.5MnO3(PCMO)
v transition metal oxides have been studied extensively,
T = 260K and the transition from a paramagnetic
CO
Xi and manganites are no exception. to antiferromagnetic phase occurs at TN =180K9.
r Interest in the manganites in recenttimes has broadly Despite its relevance to the defining properties of the
a centeredontwophenomena–colossalmagnetoresistance CMRmanganites,thenatureofthecharge-orderedstate
(CMR)5 and multiferroicity6. CMR manganites are in is not yet well-understood. At half-doping, the pic-
many ways typical of strongly correlated electron sys- ture widely assumed is of CE-type charge and mag-
tems, in that they exist at the cusp of several compet- netic ordering, illustrated in Fig. 1(a), first proposed
ing phases7. For the particular case of (nearly) cubic by Goodenough10. In this original picture, half the Mn
R1−xAxMnO3(whereRisarare-earthorLa,andAisCa, ions are Mn3+ (hence spin 2) and half are Mn4+ (spin
Sr, Ba, or Pb) and its layered analogs (R,A) MnO and 3/2) in a plane of the pseudocubic lattice; the coupling
2 4
(R,A) Mn O such phases include charge-ordered, an- is ferromagnetic (FM) between ions along the zig-zag
3 2 7
tiferromagnetic, ferromagnetic metal, and ferromagnetic chains, with antiferromagnetic (AFM) inter-chain cou-
insulator. In these perovskite manganites each Mn ion pling. Along the zig-zag chains the nearest neighbor Mn
has three electrons in the t d-orbitals, and a fraction sites are occupied by alternating Mn3+ and Mn4+ ions,
2g
(1 x) have one in the higher energy eg orbitals, whose with the lobes of the d3z2−r2-type orbitals on the Mn3+
spi−ns are aligned parallel to the local t moment by sites pointing at the Mn4+ ions on the zig-zags. These
2g
2
planes are then stacked antiferromagnetically. There is Mn sites in Pr Ca MnO itself25, which in the full
0.6 0.4 3
considerableexperimentalevidenceinfavoroftheGood- analysis is shown not to be consistent with the form of
enough model. Numerous diffraction and x-ray resonant ZPdefinedinref. 21. Indeed,ofthestudiescitedinfavor
scattering results point to two inequivalent Mn sites in of the Goodenough model11–14, many explicitly rule out
various half-doped manganites4,11–14, and the CE-type the existence of Zener polarons. On the other hand, re-
magnetic structure is universally accepted4,15,16. The portsofhighresolutiontransmissionelectronmicroscopy
chargemodulationismarkedlylessthanexpectedinthis and electron diffraction experiments claim to confirm
simple picture in various half-doped manganites, how- the existence of ZP-type ordering in Pr1−xCaxMnO3,
ever, including La Sr MnO , Nd Sr MnO and x = 0.3 0.5, with phase coexistence of the ZP and
0.5 1.5 4 0.5 0.5 3
Pr Ca MnO . Bond valence sums applied to the Mn the conve−ntionalcharge-orderedstructure26,27.
0.5 0.5 3
ions typically reveal Mn valences of approximately 3.4+ Theexistenceoftwoinequivalentsitesdiscussedabove
and 3.6+ rather than the expected 3+ and 4+ at the as evidence against ZP formation in itself does not pre-
two sites4,14, and X-ray resonant scattering at the Mn clude the existence of dimers made from adjacent Mn
K-edgealsosuggeststhatthe valencesdiffer bythe same ions. Thereareacontinuumofpossibilitiesforhole dop-
amount12,13. ingbetweentheZP(bondcentered)andtheGoodenough
Theformationofthecharge-orderedandCE-typemag- (site centered) limits, in which the two Mn sites in the
neticstructureathalf-dopingcanbeunderstoodtheoreti- ferromagneticdimers thatthe DDEXmodelpredicts are
callyintermsofthedegeneratedoubleexchange(DDEX) inequivalent, with different occupancy of the two e or-
g
model17,18. The DDEXmodelis ageneralizationofdou- bitals at the two positions. The dimers break inversion
ble exchange that allows for different occupancies of the symmetryandthisleadstothepossibilityofferroelectric-
two-fold degenerate e orbitals on a lattice of Mn sites, ity in manganites; indeed there have been reports28,29 of
g
finite Hund’s coupling between the eg and t2g orbitals, spontaneous electric polarization in Pr1−xCaxMnO3 for
andsuperexchangebetweenthe coremomentsofthe Mn a range of x, including x=0.5. Motivated by the possi-
sites. Frustration between double exchange, which fa- bility of multiferroicity, more sophisticated theories that
vors delocalization and ferromagnetic alignment of the extendtheDDEXmodeltoincorporateon-siteCoulomb
Mn core moments, and the antiferromagnetic superex- repulsion together with MnO tilting, and / or Jahn-
6
change results in a cooperative orbitaland chargeorder- Teller distortions have been developed, which have been
ing that favors ferromagnetism along zig-zag chains and analysedintermsofmodelcalculationsoracombination
antiferromagnetic stacking of the chains. ofthesewithDFTforhalf-dopedmanganites30,31. These
At half-doping and for suitable choice of (renormal- approachesrevealinstabilitytowardssomedegreeofMn-
ized) superexchange and electron hopping energies, the Mn dimerizationfor a rangeof parametersin the model,
DDEXmodelpredictssite-centeredchargeordering,with as do full ab initio calculations32,33 for Pr0.5Ca0.5MnO3.
charge disproportionation of 0.1e which is similar to Because the origin of the ferromagnetic interaction is
∼
the experimental values. Below half-doping there is a double exchange, the characteristic strength of the mag-
range of hole doping, x, over which ferromagnetic dimer netic exchange can reasonably be expected to be that
formation takes place in the ground state. The limiting typical of ferromagnetic manganites34–37, which in turn
case of bond-centered charge ordering, in which the Mn is consistent with estimates made from realistic values
ions in a dimer are equivalent, occurs for x 0.4 . This for the band widths36,38. The values of the magnetic ex-
limiting caseof a dimer resemblesthe Zener∼polaron19,20 changeconstants within and betweendimers will in gen-
(ZP), in which adjacent Mn ions pair up via delocaliza- eral be different because of the inequivalent e orbital
g
tion of the lone e electron to produce a single strongly occupancies on the two Mn sites. In the extensions to
g
bound unit with spin 7/2. The Zener polarons form a the DDEX model that incorporate interaction with the
herringbone pattern, as shown in Fig. 1(b), with AFM lattice,Jahn-Tellerdistortions(whichcouplewiththe e
g
coupling between parallel units, and FM coupling be- orbital occupancies) and alternating modulation of Mn-
tween perpendicular units. O-Mn bond angles also lead to different intra and inter
The idea of the ZP came to prominence following a dimer exchange.
key single crystal neutron diffraction experiment21 on The motivation for the current work is two-fold. First
Pr Ca MnO performed at a temperature below T is the question of whether or not the Mn ions are paired
0.6 0.4 3 CO
in which the orientation of the elongated MnO octahe- in a dimer-like fashion in Pr Ca MnO . On the
6 0.5 0.5 3
draandthetwosimilar(althoughnotidentical)Mnsites basis of previous inelastic neutron scattering measure-
inthestructuralrefinementsuggesttheformationofMn– ments of other half-doped manganites39,41,42 we might
Mn pairs. The concept of Zener polarons also received expect the Goodenough model to provide a better de-
theoreticalsupportfromab initiocalculationsperformed scription than models involving dimerization. However,
at the time in half-doped manganites22–24. However, since Pr1−xCaxMnO3 is the materialin which the possi-
the existence ofZener polaronshas provedcontroversial. ble existence of Zener polarons came to prominence, we
Notwithstanding the evidence in support of the Goode- believe that it is important to confirm whether or not
nough model for half-doped manganites cited above, X- this is the case. This then leads to our second motiva-
ray resonant scattering results point to two inequivalent tion, which is to present an exhaustive account of how
3
FIG. 1. (Color online) (a) Illustration of the Goodenough model (see text). The spins in different one-dimensional chains
(zig-zags) are indicated by either red (dark) or purple (light) arrows. The orbitals on the Mn3+ sites are d3x2−r2 or d3y2−r2
in character, where x and y point along the two perpendicular Mn-Mn directions along the zig-zags. In the model as orig-
inally proposed by Goodenough10 there is a ferromagnetic interaction between nearest neighbors along a zig-zag, JF1, and
antiferromagnetic coupling between nearest neighbors in adjacent zig-zags, J . To explain our data it is necessary to allow
A
for second neighbor interactions, JF2 and JF3, within zig-zags (see main text). Nearest neighbor antiferromagnetic coupling
betweenplanes,J ,isnotshown. (b)TheZPmodel,indicatingexchangeinteractionsbetweenrigidlycoupledS =7/2unitsin
c
theab-plane. TheZPunitsareindicatedbydashedovals. ParallelunitshaveAFMcoupling,J ,andperpendicularunitshave
A
ferromagnetic couplings, either J or J . (c) The dimer model, as proposed by Johnstone et al39, in which spins are not
FU FD
rigidly coupled as in the ZP case, but rather havea stronger ferromagnetic intra-dimercoupling, J , and weaker inter-dimer
FS
couplings, FM JFW, and AFM JA1 and JA2. The orbitals are shown to have a mixed character of a superposition of dy2−z2
(dx2−z2) and d3x2−r2 (d3y2−r2), following the conventions used in model calculations for extensions to the DDEX model31,40
discussedinthetext. Asoriginally proposedthemodelwaspurelyphenomenological withoutreferencetoanelectronicmodel,
and the orbitals should be considered purely schematic in this panel. (d) An alternative more general dimer model (see text)
in which we distinguish between the Mn sites with different proportions of dy2−z2 (dx2−z2) and d3x2−r2 (d3y2−r2) character,
indicatedherebypurpleandlight bluecoloring respectively. TheorbitalcharactertogetherwithGoodenough-Kanamori rules
lead to FM JFW2, and AFMJA, different to themodel shown in (c).
to distinguish between the various scenarios that arise definingthe energyoffeaturesinthe dispersionrelations
within DDEX models from an investigation of the spin suchasoverallbandwidthandsizesofgaps,thedirections
waves. The measurement of the spin wave excitation anddistancesofthe interactionsdefinetheperiodicityin
spectra using neutron scattering directly yields informa- wavevector of those features. These periodicities can be
tion about the strength of the magnetic interactions in used to distinguish between the Goodenough model, the
a material. In addition to the strength of interactions case of strongly bound dimers (which we refer to as the
4
’ZP model’) and weakly dimerized models (’dimer mod-
els’), all of which exhibit the same magnetic structure
but arise due to different sets of exchange interactions.
As discussed briefly above, in two earlier studies of
the spin waves in CE-type half-doped manganites41,42,
single-layered La Sr MnO and pseudo-cubic
1/2 3/2 4
Nd Sr MnO , the lower part of the spin wave disper-
0.5 0.5 3
sionwasmeasuredandfoundtobewell-explainedbythe
Goodenough model including second neighbor exchange
J along the FM zig-zags (see Fig. 1(a)). The limiting
F2
ZP model of strongly bound dimers, that is, when the
intra-dimer exchange is overwhelmingly stronger than
anyotherinteractionsinthesystem,wasdismissed. The
authors argued that of the nearest-neighbor inter-dimer
exchange interactions in the ZP model, only that along
theAFstackingdirectionresultsinsignificantdispersion FIG. 2. Calculated dispersion relations for theGoodenough
because the other two (labeled by J within a chain model (dashed red line) and ZP model (solid blue line). The
FU
parameters for the Goodenough model are those obtained as
and J between chains in Fig. 1(b)) are frustrated,
FD
theglobalbestfitinouranalysis(seemaintext). Theparam-
resulting in weak or no dispersion perpendicular to the
etersfortheZPmodelareJ =1.6meV,J = 0.11meV,
AFstackingdirection. Thismaybethecaseforthenon- FU FD −
J = 0.11meV and the inter-planar coupling is 1.05meV.
A
collinear structure proposed in ref. 18 for bond-centered
Theseparameterswerechosentoshowthatthereexistsaset
charge ordering on the basis of symmetry, in which the of parameters for the ZP model such that at low energies it
pairs of spins in adjacent ferromagnetic dimers along is practically indistinguishable from the Goodenough model,
the zig-zag structure are orthogonal. However, for the butthatthedispersionathigherenergiesservesasaneffective
case of the known CE-type magnetic structure4,15,16, discriminant. The x-axis labels denote high-symmetry posi-
there is no such a-priori reason to be confident that tions in the first Brillouin zone - G=(0,0,0), X = (1,0,0),
4
J and J are similar. The importance of measuring Y =(0,1,0), Z =(0,0,1), M =(1,1,0), and L=(1,1,1).
FU FD 4 2 4 4 4 4 2
the full set of dispersion relations is illustrated in Fig. Notethatwehavechosentoindexreciprocalspaceaccording
2. Considering still the limiting case of strongly bound to space group Pbnm,described in themain text.
dimers, Fig. 2 shows theoretical spin wave dispersion
relations for the Goodenough model (dashed line) and
ZP model (solid line)43 At low energies the dispersion model and the Goodenough model could be resolved in
relations near the zone center are qualitatively very favor of the latter by inspecting maps of the intensity
similar. The main difference, however, is the absence S(Q,ω)ataparticularkeyenergy,namelythetopofthe
of the higher branches for the ZP model, because the lower band of excitations.
binding of spins in pairs halves the number of spin wave
In this paper we present an extensive investigation of
modes (see Section IV. Discussion).
the spin excitation spectrum of Pr Ca MnO . This
0.5 0.5 3
In light of the discussion above of the origin of both material was chosen since it is perhaps the canonical ex-
the intra and inter dimer magnetic exchange along the ampleofaCE-typehalf-dopedmanganite. Furthermore,
zig-zags originating from double exchange in the DDEX the whole debate surrounding whether the Goodenough
description, a more realistic dimer model needs to allow orZPmodelis anappropriatedescriptionofthe interac-
for weak dimerization, that is, the case when the two tions in the groundstate arosebecause ofthe pioneering
exchange parameters are similar. In a recent time-of- work of Daoud-Aladine et al21 on PCMO close to half-
flight (ToF) neutron spectroscopy study39 of the bilayer doping. We compare our time-of-flight inelastic neutron
manganite Pr(Ca Sr ) Mn O (PCSMO) the higher scatteringmeasurementsqualitativelyandquantitatively
0.9 0.1 2 2 7
branchesofthedispersionwhicharepresentintheGood- to each of the Goodenough, ZP and generalised weak
enoughmodelbut not inthe ZP modelwere successfully dimer model shown in Fig. 1(d). We consider in detail
measured. This enabled the authors to show that the the dispersion relations and dynamical structure factor,
Goodenough model clearly provides the better descrip- S(Q,ω),elucidatingthespecificfeaturesinthedatathat
tion of the ground state in this material. Furthermore, allow us to distinguish between the models. Despite the
the observed dispersion required both second neighbor complication of six-fold twinning we are able to show
interactions J and J along the FM zig-zags, which that the Goodenough model provides the best descrip-
F2 F3
was ascribed to indicating some itinerant electron char- tion of our data. It is our intention that this detailed
acter. The authorsalsoconsideredanalternativeweakly description of how inelastic neutron scattering data can
dimerized model, shown in Fig. 1(c). The model allows beanalyzedtodistinguishbetweenthedifferentscenarios
foralternatingstrongerandweakernearestneighborFM that can arise in DDEX models will serve as a reference
exchange along the zig-zags on a purely phenomenologi- for future work on manganites away from half-doping,
cal basis. In PCSMO the differences between this dimer for which the calculations indicate a greater instability
5
towards the formation of dimer-like ground states.
II. EXPERIMENTAL METHODS
The ToF neutron scattering measurements were per-
formedontheARCSchopperspectrometerattheSpalla-
tion Neutron Source, Oak Ridge National Laboratory44.
Datawerecollectedwithincidentneutronenergies(E )of
i
35, 70, and 140meV, with the instrument’s correspond-
ingFermichopperfrequencieschoseninordertogiveen-
ergy resolution 0.05Ei (full-width half-maximum) at FIG. 3. (Color online) (a) The reciprocal space coverage of
≈
theelasticline. Thesampleusedforthesemeasurements thedetectorsinthe(equatorial)(H,0,0)/(0,0,L)-plane,with
comprised a single crystal of mass 1.57g, grown as de- E = 70meV, zero energy transfer, and for a multi-angle
i
scribed previously45. It was mounted in an aluminium dataset with thesample c-axisrotated stepwise from parallel
can containing helium exchange gas. The can was then to perpendicular to the incident beam. (b) Reciprocal space
mounted on the cold finger of a closed-cycle refrigera- coverage for a single-shot dataset with the orientation fixed
to bethec-axis parallel to theincident neutron beam.
tor (CCR) andcooledto 5.5K,atwhich temperature all
measurementswereperformed. Thesamplewasmounted
with the a- and c-axes horizontal. For the data collected
a = 3.85˚A, by a b √2a , c 2a , thus the mag-
with E = 35meV and E = 140meV the sample was P P P
i i netic unit cellhas s≃ize 2≃√2a 2√≃2a 2a . We there-
heldinasingleorientation,withthe c-axisofthe sample P P P
× ×
fore expect magnetic Bragg peaks, in the Pbnm conven-
parallelto the incidentneutronbeam. The data inthese
tion that we use, with wavevectors k = (0,1/2,L) and
configurations were collected for 4 hours and 29 hours 1
k = (1/2,1/2,L), and structural peaks with wavevec-
respectively. We hereafter refer to measurements taken 2
tor q = (1/2,0,L′), where L is an odd integer and L′ is
in this configuration as ‘single-shot’ datasets.
anyinteger,arisingasaresultofaJahn-Tellerdistortion
ForthedatacollectedwithE =70meV,weperformed
i of the Mn-O octahedra and orbital ordering4. In PCMO
a single-shot measurement with the c-axis of the sample
weexpectsixfoldtwinningofthecrystalthatisdescribed
parallel to the incident neutron beam, in which config-
by a systematic permutation of the principal axes of the
uration we collected data for 14 hours. In addition we
pseudocubic lattice. In order to demonstrate the effect
performedameasurementinwhichthesamplec-axiswas
of the twinning we show in Fig. 4 the various peaks we
rotated about the vertical axis from parallel to perpen-
◦ expect to observe for a series of planes with L = 0, 0.5
dicular to the incident neutron beam, in 0.5 steps, with
and 1 respectively. In the analysis of our INS data we
spectra recorded for 1hour at each orientation. The
∼ willaccountforthetwinninginthemodelingexplainedin
spectra at each orientation were then combined into a
singlelargedatasetusingthe Horacesoftware46, allow- SectionIV.Inaddition,weexpecttoobservecrystalelec-
tric fieldtransitionsoriginatingfromthe Pr ions. Inelas-
ing us to make a broad survey of the cross-section in
tic neutron scattering experiments48 on Pr Sr MnO
thefull4-dimensionalwavevector-energyspace. Wehere- 0.5 0.5 3
show that there are transitions with energies 12.8meV
after refer to measurements taken using this method as
and 15.35meV. We did not observe strong transitions
‘multi-angle’datasets. Such a broadsurveyis crucialfor
(compared to the other sources of magnetic scattering)
a complete understanding of this material since, unlike
at these energies, however.
bilayer PCSMO for example, the magnetic interactions
have appreciable strength in all three dimensions. In a
ToF measurement of this nature energy transfer is cou-
pled to the component of the wavevector Q parallel to III. RESULTS
the incident neutron wavevectorki. Only by performing
measurements in multiple orientations (or alternatively Figure5showsanoverviewofthedispersionfrommea-
with multiple incident energies)can one decouple Q and surements with all three incident energies used. The
energy46. Toillustratethis,we showinFig. 3the ARCS essential features shown here are that the dispersion is
spectrometer’sdetectorcoverageoftheac-planeofrecip- highly structured over the full energy range, but most
rocal space for a single-shot run (c-axis parallel to the importantlytherearetwobandsofdispersiveexcitations
◦
incident neutron beam) and for a 90 angle scan with with an energy gap between them. Qualitatively, we
E =70meV. estimate the two bands to cover energies in the range
i
We have chosen to index reciprocal space according 0.E .37meVand 42.E .75meV. The data shown
to space group Pbnm47. In this convention, the zig-zag in panels (a) - (c) are taken from single-shot datasets,
chains propagate along the (1,0,0)-direction, as shown so the component of Q parallel to the incident neutron
in Fig. 1(a). This crystal structure is related to the beam,(0,0,L),variesasafunctionofenergy. Thismakes
cubic perovskite system, which has lattice parameter qualitative statements difficult to make, other than the
6
Intensity [mb / sr / f.u.]
40 60 80 100 50 100 150
30 60
(a) (b)
25 50
V] V]
me20 me40
ergy [15 ergy [30
En10 En20
5
10
−1 0 1 −2 −1 0 1 2
(H,0) [r.l.u.] (H,0) [r.l.u.]
FIG. 4. (Color online) The magnetic Bragg peaks k1 =
50 100 150 200 60 80 100 120
(0,1/2,L)andk2 =(1/2,1/2,L),andstructuralsuperlattice
peak q = (1/2,0,L′), indicated by red circles, blue squares 60
(c) (d)
and green triangles respectively. Herewehavetakenaccount 80 50
V] V]
of thesix-fold structural twinningof thePCMO crystal. No- e e
m60 m40
tice that the k2 peaks do not overlap with the structural y [ y [
superlattice peaks, whereas thek1 peaks do. erg erg30
n40 n
E E20
20 10
obvious presence of two bands of spin waves, but the −2 −1 0 1 2 0 1 2
(H,0) [r.l.u.] (H,0,2.5) [r.l.u.]
value of L is known for each energy so these data can
beusedcollectivelytoconstraintheparametersofdiffer-
FIG. 5. (Color online). Neutron scattering intensity maps of
ent spin wave models. In panel (d) we show data from the dispersion along the (H,0)-direction, which is parallel to
the multi-angle dataset,so the value of L is the same for the zig-zag chains. The maps are taken from the single-shot
all of the data presented. This serves to illustrate that datasets with (a) E = 35meV, (b) E = 70meV and (c)
i i
the gap observed between the lower and upper bands of Ei=140meV.Thedatahavebeen smoothed byconvolution
the spin wave dispersion is not an artefact of the way with a Gaussian with FWHM equal to two bin sizes along
in which a gapless dispersion might project on to the each dimension for clarity. The map in panel (d) is taken
from the multi-angle dataset with E = 70meV, hence why
curved(Q,E)hypersurfaceofmeasuredreciprocalspace i
the value of L can be explicitly given. The white dashed
coordinates from a single-shot run.
rectangles in panels (b)-(d) indicate the position of the gap
To follow the evolution of the spin waves with energy,
betweenupperandlowerbranchesoftheexcitations,whichis
we show in Fig. 6 a series of intensity maps at fixed consistent with the Goodenough model. Note that the data
energy from single-shot runs, together with simulations in all four panels have been multiplied by f(E)= E/E0 ,
usingthe Goodenoughmodelwiththe globalbestfitpa- 1−e−E/E0
whereE0=kB 10K.f(E) istherefore constant whenE
rametersobtainedinthenextSection. Becausethevalue × ≪
E0 and f(E) E when E E0. This rescaling is todisplay
ofLchangeswithincidentenergyandenergytransferfor thefullspectr∝ummoreclea≫rlyonasingleintensityscale. For
a given Q coordinate in these plots, we are mapping the panel (a) the signal was integrated over 0.05 r.l.u. in the
±
L dispersion as well as the in-plane dispersion. The first (0,K) direction, whereas for panels (b)-(d) the integration
fourpanels((a)to(d))thereforeshowaspinwavewhich was 0.1 r.l.u.
±
disperses from (1,1), with the fourth panel, (d), show-
ing the top of the lower band of the spin waves. Panels
(e) and (f) show the upper band of spin waves, with the not seen in any single twin.
dispersion convergingon (1,0) as the band maximum at
75meV is approached. This part of the spectrum can
∼be understood, with reference in addition to the period- IV. DISCUSSION
icity evident in Fig. 5(c), as arising from dominant FM
exchange between nearest neighbor Mn ions along the The DDEX model17,18, which is a more general ver-
zig-zags. sion of double exchange, would be an appropriate start-
We incidentally note that the gross features described ing point for analyzing our data. However, calculation
aboveare notdependent on a considerationofthe struc- of the excitation spectra within such a framework is,
tural twinning that is present in this material. That is though possible, far from straightforward49,50. In the
to say, the presence of a gap must be a feature of all the absence of such calculations, we therefore use an effec-
twins present in our sample, and is independent of their tive Heisenberg Hamiltonian to model our data, just as
relativepopulations. Similarly, the periodicity ofthe ob- was done for the half-doped manganites mentioned in
served signal in momentum space can only arise from the introduction39,41,42.The use of an effective Heisen-
the periodicity of the spin chains. This is as opposed berg Hamiltonian is a well-established procedure to an-
to a situation where contributions from different twins alyze the spin wave spectra in magnetically ordered sys-
that might each have different periodicity that taken to- tems, and not just those with purely integer or half-
gether combine to give an overallsignal with periodicity integer spins, nor even solely insulators. The exchange
7
Intensity [mb / sr / meV / f.u.] amples include the ferromagnetic metallic-like materials
1.5
(a) (g) 10 with x 0.2 0.48(e.g. refs. 34–37), where softening of
1 8 E=35meV thedisp∼ersion−towardsthezoneboundaryandthedoping
i
0.5 6 9<E<12meV dependence of intra-bilayer exchange could be qualita-
0 4 L=1.2 tivelyexplainedbymappingtheexcitationsofthedouble
5 exchangemodelontoanequivalentferromagneticHeisen-
(b) (h)
1 4 E=70meV bergHamiltonian (validatthe quasi-classicallevel)with
i corrections beyond 1/S and incorporating the e orbital
0.5 3 15<E<20meV g
u.] L=1.2 occupancies49,53,54.
(0,K) [r.l. 01 (c) (i) 234.5 Ei=70meV dthiseTpoeHraesiinosaenlnybrzeeelragtthioHenadsmaatinlatdoqnduiyaannntaimtaitc=iavlelsytrPwucetucarelcJufalScattoerdSstfhoer
0.5 3 20<E<25meV H − <ij> ij i · j −
2.5 L=1.6 D(Sz)2 using linear spin wave theory for each of the
i
0 Goodenough, ZP and dimer models illustrated in Fig.
(d) (j) 2 1(a-d). In the present analysis appropriate choice of ex-
1 1.8 E=70meV
i change interactions yields the known magnetic ground
0.5 1.6 35<E<40meV state for all of the models. The procedure for calculat-
1.4 L=2.6
0 1.2 ing the dispersion relations for the first two cases is ex-
0 0.5 1 1.50 0.5 1 1.5 plained in refs. 55 and 56, in which textbook linear spin
wavetheory57 is appliedto this particularlycomplex ex-
2 (e) (k) 2.2 ample. An extended explanation, including calculation
2 E=140meV
1.5 i
1.8 of the structure factors, and will be the subject of a fu-
1 1.6 50<E<60meV
u.] 0.5 1.4 L=2.9 ture publication. For the generalized dimer model the
K) [r.l. 02 (f) (l) 11..24 liicnaelasrtrsupcitnuwreafvaecttohresorwyerdeiscpaelrcsuiolanterdeluatsiionngstahnedMdcyPnhaamse-
0, 1.2 E=140meV
( 1.5 1 i package58. We confirmed that in the special case when
1 70<E<80meV
0.8 this model reduces to the Goodenough model that it re-
0.5 0.6 L=3.8 produces the results of ref. 55. In the first part of the
0 0.4
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 analysis procedure for each model, we obtained robust
initialestimatesforexchangeconstantsfromexamination
(H,0) [r.l.u.]
of the periodicities and energies of key spectral features
detailed below. We then fitted the calculated S(Q,ω),
FIG. 6. (Color online). Panels (a) – (f) show intensity
maps taken with energies in the range 9 E 12, 15 E 20, averaged with equal weight from each twin, to the data
20 E 25, 35 E 40, 50 E 60, and 70 ≤E ≤80meV,≤an≤d in- byleast-squaresrefiningtheexchangeconstantsJij. The
≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤
cident energies of 35, 70, 70, 70, 140, and 140meV respec- term D is a single-ion anisotropy term to constrain the
tively. The data have been smoothed by convolution with spins to the ab-plane. This term was included with a
a Gaussian with FWHM equal to two bin sizes along each fixed value of 0.07meV in all of the models discussed in
dimension for clarity. Panels (g) – (l) show corresponding our analysis, to be consistent with the literature39,41,42.
simulations of those data using the Goodenough model and
Its small value means that it has negligible effect on the
theglobalbestfitparameters(seemaintext). Allofthedata
cross-section at the energies that we probed in our ex-
shown hereare from single-shot datasetswith thec-axis par-
periment and hence does not affect any of the results of
alleltotheincidentbeam. Wehavemadeuseofthesymmetry
our fits or our conclusions. The data that we fitted were
of the crystal, and the symmetry and extent of the detector
array byfolding thedataabout the(1,0,0) and(0,1,0) axes all either one-dimensional cuts or two-dimensional slices
toeffectivelyincreasetheoverallcountratebyafactoroffour. taken at constant energy, so that the energy-dependent
Thecalculated S(Q,ω)wereconvolutedwithGaussianswith resolutionofthe spectrometer could be accountedfor by
FWHM matched to the spectrometer’s energy resolution at using a fixed Gaussian broadening of the calculated in-
each energy and Ei . tensity,thewidthofwhichwaschosentomatchthereso-
lutionateachenergy. Twentyeightone-dimensionalcuts
andtwelvetwo-dimensionalsliceswereusedinthefitting
interactions in the model Hamiltonian that are required process, with the fit being performed simultaneously on
to reproduce the observed dispersion relations and in- all of them to determine a single set of best-fit parame-
tensities establish the strength, range and symmetries ters. The robustness of the fit parameters obtained was
of the magnetic interactions. Examples where such checked by using numerous different sets of starting val-
an analysis is done include the parents of iron based ues,toconfirmthatthefitalwaysconvergedonthesame
superconductors51,52 in which the signs and magnitudes result.
of the nearest neighbor and further neighbor exchange At this point we reiterate the distinction between the
constants elucidated the dominant second neighbor an- different models we will consider. In all cases they have
tiferromagnetic exchange. In the case of manganites ex- the same magnetic ground state, as shown in Fig. 1,
8
whichiscommonlyreferredtoastheCEmagneticstruc- odicity which arises from the antiferromagnetic coupling
ture. What distinguishes what we refer to as the ‘Good- betweenzig-zagchains,J . Variations in this parameter
A
enough model’ and the ‘ZP model’ and dimer models is havenegligibly smalleffects onfeaturesofthe dispersion
thus the nature of the electronic environments around above 10meV,soinpracticeitcanbedeterminedfrom
∼
the Mn ions, which give rise to different sets of ex- fits to the E = 35meV dataset alone, where the back-
i
change interactions between them. Our INS measure- ground arising from the tails of the elastic line at these
ments, andanalysiswith the effective Heisenbergmodel, low energies is minimized. The global best fit yielded a
therefore provide insight into the nature of this aspect value of J = 0.45 0.01meV.
A
− ±
of the ground state, rather than the magnetic structure
itself, which is well established16. Intensity [mb / sr / meV / f.u.]
13
12 12
A. Goodenough model 11
10 10
V]
We start with the Goodenough model. We use this me 9
term to apply to a model which has the CE magnetic gy [ 8 8
structure and a set of exchange parameters as shown in er
n 7
Fig. 1(a). Although we consider during our analysis the E
6
possibility ofchargedisproportionation,we donota pri- 6
oriassumethatitispresent. The orbitalorderisimplic- 5
ityassumedtobethatshowninFig. 1(a),sincethisthen 4
4
provides a natural explanation of how the signs of the
−0.5 0 0.5
nearestneighborexchangeinteractionsweconsidercome (0.5,K,1) [r.l.u.]
about. We also allow for the possibility of next-nearest-
neighborinteractionsalongthe FMzig-zags,ashasbeen FIG. 7. (Color online). Low energy dispersion, the extent
requiredinpreviousstudiesofsimilarsystems39,41,42 and in energy of which allows a reasonable estimate of J to be
A
istakentoindicateadegreeofelectronicitinerancyalong obtained before theglobal fittingprocedure.
the zig-zag chains.
To simplify the analysis we initially assumed that the The existence of a gap in the dispersion at (1/2,0)
charge disproportionation between notional Mn3+ and (Fig. 5(d)) immediately implies a halving of the period-
Mn4+ sites is zero. This meant that all spins were set to icity in real space along the zig-zags. In the framework
an average value of S = 7/4. It is well known that the of an effective Heisenberg Hamiltonian this implies one
charge disproportionation is substantially smaller than or both of the second neighbor interactions J and J
F2 F3
the ideal value4,14,16 and close to Mn3.4+ and Mn3.6+. is non-zero. The position in energy and size of the gap
Bychoosingtouseequalspinsonallsites,weareableto between the lower and upper spin wave bands, shown
drawconclusionsmorestraightforwardlyaboutthe mag- schematically in Fig. 2, define two energies which deter-
nitudeandsignofexchangeinteractionsfromqualitative mine the values of both the next-nearest-neighborterms
analysis of the periodicity and extent of the dispersion. J and J . In order to reproduce the measured values
F2 F3
Later in our analysis we will revisit this assumption and J and J must both be non-zero, and we find that
F2 F3
will show that the effect of introducing a magnetic mo- in fact they must have opposite signs, as was similarly
ment disproportionationequal to the experimentally de- concluded for PCSMO39. The size and position of the
terminedvalue16 doesnotaltertheprincipalconclusions. gap alone can be reproduced identically by exchanging
As has already been noted, the dominant exchange is the signs of J and J , that is, with J < 0 <J
F2 F3 F2 F3
thenearest-neighborinteractionJF1 linkingadjacentMn or JF2 > 0 >JF3. However the details of the structure
ions along the zig-zag chains. Examination of Fig. 5(c) factor aroundthe gapcan distinguish between these two
showsthatthemaximumofthedispersion,whichhasthe scenarios,specificallyspectralweightisswappedbetween
periodicity of a ferromagnet, is 75meV, and thus the two relatively flat branches of the dispersion that lie at
∼
strongest constraint on the size of JF1 comes from this the top of the lower set of spin wave bands and the bot-
part of the dispersion. We determined from our global tom of the upper set of spin wave branches. Also, the
bestfitthatJF1 =9.01 0.02meV.Thisvalueiscompa- periodicity of peaks in the structure factor at the top of
rable to thatmeasuredi±nPCSMO (11.39 0.05meV)39, the lower set of spin wave branches is doubled along a
La0.5Sr1.5MnO4 (9.98meV)41, the AFM±CO phases of key symmetry direction for the case of JF3 >0>JF2, at
Nd0.5Sr0.5MnO3 (6.76meV)42, and a little larger than variance with the data. This is illustrated in Fig. 8. For
the typical values for the FM double exchange in metal- the global best fit we find that J = 0.93 0.02meV
F2
lic FM manganites37,59. In other words, it is consistent and J = 0.97 0.02meV. Note that J ±is therefore
F3 F3
− ±
with its origin being due to double exchange. antiferromagnetic.
At much lower energies ( 10meV) there is a branch A particular difference between PCMO and PCSMO
∼
of the dispersion, shownin Fig. 7, with double this peri- is that in the former the coupling between planes is ex-
9
of 3.18µ /2.75µ = 1.16. Alternatively, if the differ-
B B
2 ence in the spins is considered to arise from charge dis-
J < 0, J > 0
F2 F3 proportionation in a purely ionic model (in which Mn3+
u.] 28<E<34meV JF2 > 0, JF3 < 0 has S = 2 and Mn4+ has S = 3/2) this corresponds
V / f.1.5 Data to Mn2.7+ and Mn4.3+. In contrast, bond valence sums
e
m for PCMO from joint refinement of neutron and x-ray
b / sr / 1 dteiffntrawctiitohntydpaitcaa1l4dyisieplrdopMornt3io.4n6+atioanndofM.n3.05.11+e,incohnasilsf--
m doped manganites4,12–14. The gap therefore cannot be
y [
nsit0.5 accounted for by the experimental moment or charge
e disproportionation. The contribution that it can make
nt
I can be determined as follows. If we fix the ratio of the
spins to that of the measured magnetic moments, then
0
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 this changes the values of J and J to 0.70meV and
F2 F3
(−1/4,K,3) [r.l.u.]
0.64meV c.f. 0.93meV and 0.97meV respectively, a
− −
reduction in the values of about 30%. Alternatively, fix-
FIG.8. (Coloronline). Cutacrosspartofthetopofthelower
ing the ratio of the spins by the values corresponding to
bandofexcitations. Thered(solid)lineshowstheglobalbest
typical charge disproportionation of 0.1e changes J
fit to the data in this region, whereas the blue (dashed) line ± F2
and J to 0.84meV and 0.84meV, a reduction com-
shows the structure factor if the signs of both JF2 and JF3 F3 −
are reversed. Such a reversal preserves the magnitude of the pared to the values assuming no disproportionation of
gap between the lower and upper bands of excitations, and lessthan14%. Tosummarize,thedifferencebetweenthe
also the dispersion relation, but changes the structure factor Mn moments or charge disproportionation on the two
as shown. sites can therefore only account for a small fraction of
themeasuredgap. Thepositioninenergyandthesizeof
thegapthereforerobustlyrequiresnon-zeroJ andJ
F2 F3
pected to be significant,whereasin the latter itis rather intheGoodenoughmodel. Weadditionallynotethatac-
small. Indeed, previous low energy INS measurements counting for the small charge / spin disproportionation
of Nd Sr MnO (ref. 42), which one would expect to as discussed above only results in a change of the values
0.5 0.5 3
besimilartoPCMO,foundthattheinter-planecoupling of JF2 and JF3 ; the other fitted exchange parameters,
was about a factor of four larger than the coupling be- JF1, JA and Jc, remain unaltered within the size of the
tween zig-zag chains. The complex twinning, together errorbars.
with an intermediate strength coupling between planes, Our analysis has up to this point has also assumed
means there is not a single feature which can be used an equal population of the six structural twins that are
to determine Jc easily. The strength and position of the possible in this system. It is simplest to describe the
signal around ( 0.5,0) at 22meV in Fig. 5(a), with twins with respect to the pseudo-cubic perovskite unit
± ∼
Ei =35meV, exhibits qualitative dependence on Jc, but cell. Twin1istheunrotatedcoordinatesystem,andtwin
overall the best method for refining Jc is to use all of 2 corresponds to a 90◦ rotation of this system about its
the data when performing the final global fit. We found pseudo-cubic c-axis. Twin 3 is a 90◦ rotation of twin 1
Jc = 2.09 0.03meV, which like in Nd0.5Sr0.5MnO3 is about its a-axis, and then twin 4 is equivalent to twin 3
− ±
about a factor four larger than the in-plane antiferrog- rotated by 90◦ about the c-axis of twin 1. Twin 5 is a
magnetic coupling between zig-zag chains. 90◦ rotation of twin 1 about its b-axis, and finally twin
◦
Let us now return to the question of charge / spin 6 is a further 90 rotation of twin 5 about the c-axis of
disproportionation between the different Mn sites. The twin 1.
effect of this being non-zero would be to have different We now illustrate the validity of the approach of as-
values of the spin on alternate Mn sites along the zig- suming an equal population of twins in two ways. First,
zags, which would halve the real-space periodicity and in Fig. 9 we show simulations of some of the constant-
thereby open up a gap to create lower and upper disper- energy slices that have already been shown in Fig. 6.
sion branches, even with JF2 =JF3 =0. Specifically, panels (a) and (b), and (c) and (d), should
To account for different moments on the Mn sites we be respectively compared to panels (g) and (h) of Fig.
can set S = 1.75+δS on the notional Mn3+ sites and 6, and panels (e) and (f) should be compared to panel
S = 1.75 δS on the notional Mn4+ sites, instead of (l) of Fig. 6. We have re-calculated the scattering in-
−
the uniform values of S = 1.75 used up to now. In tensity for the global best fit to the Goodenough model,
fact, the dispersion relations throughout the Brillouin decomposed into the contribution from each of the six
zone for our global best fit parameters for the Goode- twins, and then formed combinations of them to show
nough model can be almost identically reproduced with that there is no permutation of fewer than all six twins
J =J = 0, but this requires δS = 0.42. This corre- that will allow us to reproduce the fits to our dataset
F2 F3
sponds to a ratio of moments on the Mn3+ / Mn4+ sites acrossthe entireenergyrange. Forexample,inFig. 9(a)
of 1.63, significantly at odds with the measured ratio16 we can qualitatively reproduce the global best fit with
10
energygapdeterminedbythe muchstrongerintra-dimer
Intensity [mb / sr / meV / f.u.]
1.5 exchange. That is, we expect a gap that could be sev-
10
(a) (b) eral multiples of the bandwidth of the lower energy set
1
8 Ei=35meV of spin wave branches, which in the limiting case of the
0.5 Twin 1 + 2 + 3 + 4 Twin 5 + 6 6 9<E<12meV ratio of intra-dimer to inter-dimer exchange tending to
L=1.2
4 infinity eliminates half the spin wave branches. In con-
0
1.5 5 trast, the Goodenough model permits a small (or even
K) [r.l.u.] 0.15 (Tcw)in 1 + 2 + 5 + 6 (Tdw)in 3 + 4 34 E15i=<7E0<m2e0VmeV zeaexlorisnot)gegntachpee(bfoeerrtrwooemtehanegrtwnweiosteics)eztoisfgo-szfeacbgornacdnhcanhineeissg.dhAbeposerinlildnuitsnetgrraaotcnetdiothnines
(0, L=1.2 Fig. 2, judicious choice of J , J and J in this lim-
A FU FD
0 2
0 0.5 1 1.5 0.5 1 1.5 iting ZP model (the model shown in Fig. 1(b)) results
2.5 1.4 in spin wave branches that closely follow the lower en-
2 (e) (f) 1.2 E=140meV ergy half of the spin wave branches of the Goodenough
1.5 Twin 3 + 4 + 5 + 6 Twin 1 + 2 1 i model, but in general the reverse is not possible. This is
1 0.8 70<E<80meV
0.5 0.6 L=3.8 because the limiting ZP model has only half the number
ofspinwavemodesbyvirtueoftherigidcouplingofspins
0 0.4
0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 withinthe dimers. Moreover,the periodicity ofthe most
(H,0) [r.l.u.]
energetic branch in our data corresponds to the nearest-
neighbor Mn–Mn distance projected along the direction
FIG.9. (Coloronline). Decompositionofsomeofthefitspre-
of the zig-zags. The nearest-neighbor separation of ZP
sentedinFig. 6intocontributionsfromspecificsetsoftwins.
dimers is twice that, which would result in a doubling
(a) Contribution from twins 1-4, which should be compared
of the periodicity of the spin wave branch compared to
to Fig. 6(a) and (g); (b) contribution from twins 5 and 6,
i.e. those which are not included in panel (a); (c) contribu- whatweobserve. Insummary,becausewehavebeenable
tion from twins 1, 2, 5, and 6, which should be compared to to measure the higher energy branches, which are sepa-
Fig. 6(b) and (h); (d) contribution from twins 3 and 4, i.e. rated from the lower branches by a relatively small gap,
those not included in panel (c); (e) contribution from twins we are able to discount a model of weakly interacting
3-6, which should be compared to Fig. 6(f) and (l); (f) con- dimers without making assumptions about the relative
tributionfrom twins1and2,i.e. thosenot includedinpanel magnitudesofJ ,J andJ . Infact, this conclusion
A FU FD
(e). Ourconvention for labeling thetwins is described in the
shouldapplytoanycaseofinteractingdimers,solongas
main text.
theintra-dimerexchangeisdominantoverallotherinter-
actions (nearest neighbor and further neighbor)between
a spin in one dimer and those in other dimers.
contributions from the twins we label 1 4, whereas in
−
Fig. 9(c) we require contributions from twins 1, 2, 5
and 6, and in Fig. 9(e) all of the features in the data
can be accounted for with just twins 3 6. Second, we C. Dimer model
−
repeated the entire fitting procedure with additional fit
parameters of the relative contributions from each twin. As discussed earlier, it is more realistic to consider
We found that, irrespectiveof the initial values of the fit a model of weak dimerization as an alternative to the
parameterschosen,withintheerroronthefitparameters weakly interacting strongly bound dimers of the ZP
the populations of all six twin were essentially equal. model. AmodelalongtheselineswasdiscussedbyJohn-
stone et al in their work on PCSMO39, and is shown in
Fig. 1(c). The empirical rationale for this model was
B. Zener polaron model that the dominant exchange is the FM nearest-neighbor
interaction along the zig-zags, but allowing for a small
Let us now turn to the ZP model (that is, of strongly modulation between weaker and stronger exchange will
bound dimers) for the sake of completeness, before con- create a gap at (1/2,0) by virtue of doubling the repeat
sidering the more realistic dimer model in Sec. IVC. As distance along the zig-zags. In PCSMO, with a suitable
shown in Fig. 2, a key feature that distinguishes the choice of parameters, this model produces an essentially
Goodenough and ZP models is the presence or absence indistinguishable dispersion relation to the Goodenough
ofa secondbandofexcitations. By farthe dominantex- model, in addition to the accepted CE magnetic struc-
change constant in the ZP model is expected to be that ture. However, the two models can be distinguished by
withinthedimers. Accordingly,weexpectonesetofspin differences in the structure factor at the top of the lower
wave branches associated with coupling of the rigid ZP band of the dispersion.
dimers that will have a band width determined by the The weak dimer model of ref. 39 was purely phe-
characteristic inter-dimer exchange energies. These will nomenological without reference to an electronic model.
be separated from a second set of branches associated However, it is in fact a special case of a more general
with excitation of the pair of spins within the ZP by an weakdimermodel,showninFig. 1(d), whichwewilluse
