Spin-Wave Instabilities and Non-Collinear Magnetic Phases of a Geometrically-Frustrated Triangular-Lattice Antiferromagnet

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by  J. T. Haraldsen

Spin-Wave Instabilities and Non-Collinear Magnetic Phases of a
Geometrically-Frustrated Triangular-Lattice Antiferromagnet
J.T. Haraldsen,1 M. Swanson,1,2 G. Alvarez,3 and R. S. Fishman1
1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831
2Department of Physics, North Dakota State University, Fargo, ND 58105 and
3Computer Science & Mathematics Division and Center for Nanophase Materials Sciences,
Oak Ridge National Laboratory, Oak Ridge, TN 37831
Thispaperexaminestherelation betweenthespin-waveinstabilities ofcollinearmagneticphases
andtheresultingnon-collinearphasesforageometrically-frustratedtriangular-latticeantiferromag-
9
net in the high spin limit. Using a combination of phenomenological and Monte-Carlo techniques,
0
we demonstrate that the instability wave-vector with the strongest intensity in the collinear phase
0
determinesthewave-vectorofacycloidorthedominantelasticpeakofamorecomplexnon-collinear
2
phase. Ourresults are related to theobserved multi-ferroic phase of Al-doped CuFeO2.

n
a PACSnumbers: 75.30.Ds,75.50.Ee,61.05.fg
J

5
1 It is well-known that the transition between different
magnetic ground states may be signaled by the soften-

]
ing of a spin-wave (SW) mode. In the simplest case of
l
e
a conventional square-lattice antiferromagnet, the soft-
-
r ening of a SW mode at wave-vectors (π,0) and (0,π)
t
s signals the spin flop and canting of the magnetic mo-
.
t ments at a critical field. In the manganites [1], the SW
a
instabilities of the ferromagnetic state have been used J
m 3
toconstructthephasediagramfortheantiferromagnetic
J
d- (AF) phases that appear with Sr doping. The soften- J 2
n ing of a SW excitation at (π,0) signals the instability 1
o of the N´eel state and the canting of the spins in a spin-
c
1/2 union-jack lattice [2]. But the relation between the
[
SW instabilities ofa collinearphase andaresulting non-

1 collinearphaseislessclearwhenmultipleSWinstabilities
v
occursimultaneouslyorwhenthenon-collinearphasehas
6
a complex magnetic structure with severalelastic peaks. FIG. 1: (Color online) A portion of the TLA phase diagram
3
3 ThispaperexplorestherelationbetweentheSWinstabil- for large D and J1 < 0. The dashed (red) curve divides the
2 ities of the collinear 4 and 8-sublattice (SL) phases of a 4-SL phase into 4-SL I and 4-SL II regions. For the 4 and 8-
. geometrically-frustrated triangular-lattice antiferromag- SLphases,redcirclesdenoteupspinsandbluecirclesdenote
1
down spins; the solid (gray) line defines the unit cell. The
0 net(TLA)andthenon-collinearphasesthatappearwith
dashed (blue) line obeys the relation J3 = 1.3J2, the black
9 decreasinganisotropyD. WhenmultipleSWinstabilities
circleistheestimatedlocationoftheexchangeparametersfor
0 ofthecollinearphaseoccuratonce,theinstabilitywave- CuFeO2, and the white circle lies on the boundary between
:
v vectorwiththelargestintensitydeterminesthedominant the4-SL I and 4-SL II regions.
i orderingwave-vectoroftheresultingnon-collinearphase.
X
Oneofthepredictednon-collinearphasesmayberelated
r
a to the multi-ferroic phase that appears in CuFeO2 with J3/J1 0.57 [8, 9].
| |≈−
Al doping [3]. As demonstrated by the small SW gap of about 0.9
Frustrated TLAs with AF nearest-neighbor exchange meV [3, 8] on either side of the ordering wave-vector
J1 <0exhibitaremarkablenumberofcompetingground q=πx, the spin fluctuations of CuFeO2 are muchsofter
states[4]. WithinteractionsJ uptothirdnearestneigh- than would be expected for Ising spins. With Heisen-
i
bors (denoted in Fig. 1) and assuming Ising spins along berg spins, the collinear magnetic phases of a TLA be-
the z direction, Takagi and Mekata [5] obtained a phase comelocallyunstablebelowacriticalanisotropyD that
c
diagramwithferromagnetic(FM),2-SL,3-SL,4-SL,and depends on the exchange parameters J . The observed
i
8-SLphases. Aportionofthatphasediagramissketched softening of the SW modes in CuFe1−xAlxO2 with Al
inFig. 1. Forthe geometrically-frustratedTLACuFeO2 doping [3] can be reproduced by lowering D towards Dc
in fields below 7 T, the ground state is the 4-SL phase [10] in the 4-SL I region of Fig. 1. For Al concen-
[6, 7] and the black dot in Fig. 1 denotes the esti- trations above about x 0.016, the magnetic ground
c
≈
matedratioofexchangeparametersJ2/J1 0.44and state of CuFe1−xAlxO2 becomes non-collinear and dis-
| |≈−
2
parameters [14]. With the exchange parameters corre-
sponding to the black circle in region 4-SL I, we deter-
mined the stable magnetic phases as a function of D.
As shown in Fig. 2, the 4-SL phase is stable down to
y
D/J1 0.27, below which MC simulations obtain the
| | ≈
x complex non-collinear (CNC) phase shown on the top of
Fig. 2. Sincethesameupordownspinfrequentlyoccurs
at sites R=mx+n√3y and R′ =R+x/2+√3y/2 or
R′′ =R x/2+√3y/2, the CNC phase retains some of
z y the FM c−orrelations present in the 4-SL phase. Transla-
x tionally invariant in the y direction, the CNC phase has
a period in the x direction between 2 and 3 lattice con-
stants. Becauasethe MC simulations wereperformedon
a finite lattice, the energy of the CNC phase is overesti-
matedandwecannotsaywhetherthisphaseiscommen-
surate or incommensurate.
FIG. 2: (Color online) Energy as a function of D/|J1| for Below a second threshold value of D/J1 0.08, a
| | ≈
the4-SL(blackdashed),CNC(bluediamonds),andcycloidI cycloidliketheonesketchedinthebottompanelofFig. 2
(red circles) phases with J3/J2 =1.3 and J1/J2 =2.28. The [15]hasalowerenergythantheCNCphase. Asdiscussed
bottom diagram shows the the cycloid for arbitrary q. The
below, the wave-vector of the cycloid is independent of
top diagram shows the CNC phase with up spins in red and
D. If the CNC phase were neglected, then the cycloid
down spinsin blue.
would achieve a lower energy than the 4-SL phase below
D/J1 0.2,stillabovethecriticalvalueDc/J1 0.15
| |≈ | |≈
plays multi-ferroic properties [11, 12, 13]. for the local stability of the 4-SL phase.
Wehavedeterminedtheground-statemagneticphases Togainabetterunderstandingofthephasesstabilized
of the TLA using a combination of Monte-Carlo (MC) within the TLA, we have evaluated the magnetic phases
simulations andphenomenologicaltechniques. The TLA alongthe line with J3/J2 =1.3drawnthroughthe black
Hamiltonian is dot in Fig. 1. Five stable phases are presented as a
functionof J1 /D and J2 /D inFig. 3. The 4-SLphase
1 | | | |
H =−2XJijSi·Sj −DXSi2z, (1) iAslsthtaobulgehalnoontgiandsitcraiptedthrboyugthhisthfiegduiraeg,otnhael o4f-StLhisrepgliootn.
i6=j i
disappears above J1 /D 40. Close to the origin or
| | ≈
whereJ includesfirst,second,andthird-neighborinter- for large D, a collinear 8-SL region is indicated in Fig.
ij
actions(showninFig. 1). Thenearest-neighbordistance 1. Two cycloids are also obtained: in the upper left,
has been set to 1. The SW frequencies of the collinear cycloid II with wave-vector 4πx/3; in the lower right,
phases are obtained by performing a Holstein-Primakoff cycloid I with the variable wave-vector indicated in the
1/S expansion about the classical limit. Above the crit- figure [15]. Finally, a CNC phase appears just below the
ical anisotropy Dc, a collinear phase is locally stable if 4-SL phase and disappears above J1 /D 20. Regions
| | ≈
the SW frequencies ω(k) are positive and real for every of local stability for the collinear phases are indicated in
momentum k. Fig. 3 by the dashed black lines [14]. The results in Fig.
MC simulations were used to find the non-collinear 2canbeobtainedfromFig. 3bydrawingalinefromthe
magnetic phases of the TLA. The simulations were origin with slope 2.28 (gray) (so that J2/J1 = 0.44),
| | −
started at a high-enough temperature to rule out whichpassesfromthe4-SLphasethroughtheCNCphase
metastable states. To mimic the process of thermal an- into cycloid I.
nealing,the systemwasslowlycooledtoafinaltempera- The classical energies of each of these phases can be
ture(inunitsof J1 S2)rangingfrom4 10−3to1 10−4. written as E/S2 = A1J1 +A2J2 +A3J3 ADD. The
| | × × −
Lowering the final temperature further did not signifi- coefficients for each phase are given in Table I. Only a
cantly change the resulting non-collinear phase. Using non-collinear phase with 0.5 < A < 1 can intercede
D
lattices of varying sizes with periodic boundary condi- between a collinear phase with A = 1 and a cycloid
D
tions, we found that there was no substantial change for with A =0.5. For the CNC phase with A 0.71,the
D D
≈
lattices greater than 16 16. errorbarsindicatetherangeofparametersobtainedfrom
×
In Fig. 1, the 4-SL phase is separated into regions I MC simulations near J1 /D = 5.7 and J2 /D = 2.5.
| | | |
andIIbythecurveJ1/J2 2=J2/J3. Inregion4-SLII, Thisphaseischaracterizedbyratherweaknext-neighbor
−
the instabilitywave-vectorsaregivenbyk=(π π/3)x, correlations with small A2 . The CNC phase space in
± | |
independent of the exchange parameters; in region 4-SL Fig. 3 is obtained by using the results of Table I to
I, the instability wave-vectors depend on the exchange evaluate the energies of the MC spin configurations as
3
gion can be related by a symmetry operation to one of
25 these three. We find that the instability at k1 always
has a larger intensity than the “twins” at k2 or k3 or
20 Cycloid II: q= 4!/3 thanany of the otherwave-vectorsrelatedby symmetry.
Correspondingly, cycloid I along any diagonal in Fig. 3
J = 2.28J
1 2 has the same wave-vector q as the dominant instability
15
D of the 4-SL phase.
J| | / 110 4-SL phase CNC phase whSicimhilsawritccohnecslutsoiocnysloairdeIreaalcohnegdafnoyr dthiaego8n-SaLl inphFasige.,
3. Although the SW instability of the 8-SL phase oc-
curs simultaneously at several wave-vectors, the domi-
5 Cycloid I: qvaries
nant wave-vector instability of the 8-SL phase coincides
8-SL phase withthe wave-vectorqofcycloidI alonganydiagonalin
q = 0.684!
0 Fig. 3.
0 2 4|J | / D 6 8 10 However, the CNC phase that intercedes between the
2
FIG.3: (Color online)Phasediagram fortheTLAasafunc-
tion of |J1|/D and |J2|/D with J3 = 1.3J2 containing five
regions: 4-SL (blue), 8-SL (green), CNC (violet), cycloid I
8
(variablegreen-orange),andcycloidII(maroon). Thedashed (a)
(white) lineseparates regions 4-SLI and4-SLII.Thedotted
(black)curvesdenotethemetastableboundariesforthe4-SL 6
and 8-SL regions. Cycloid I has wave-vectors q that range
S
from 0.684π to 0.923π in intervals of 0.016π. J|14
ω/|
functions of J1 /D and J2 /D. Hence, the CNC region 2
may be unde|res|timated. | |
The ordering wave-vector q = qx of cycloid I is eval- 50
uated by minimizing E with respect to q. So q depends (b) q ~ 0.83π q = 4π/3
I II
only on the ratios J2/J1 and J3/J2, as indicated by the 40 |J2|/D = 1.30
diagonal lines in Fig. 3. Cycloid II with q = 4π/3 cor- |J2|/D = 1.50
responds to the 120o N´eelstate found in a classicalTLA 30 |J2|/D = 2.00
|J|/D = 2.50
withD =0 andnearestandnext-nearestneighborinter- s)20 |J2|/D = 2.67
satcattioen[1s5[1]6is]wsthaebnle|Jfo2r/Jn1o|n<zer1o/8D. oAveslrigahrtalyngdeisotfoertxecdhaNn´egeel Unit10 2
parameters with |J3/J2|>1/2, so that the diagonalline Arb.
in Fig. 1 passes through the 4-SL and 8-SL phases. MC 10
sIimanudlaItIioinnsFwige.re3u.sedtoconfirmthe stabilityofcycloids nsity ( 8 (c) Sx kx ~ 0.87π
With decreasing D or moving away from the origin of nte Sy
I 6 S
Fig. 3alongadiagonal,the4-SLphasebecomesunstable z
either to cycloid II or to the CNC phase. The white line 4
bisecting region the 4-SL strip in Fig. 3 corresponds to k ~ 1.13π
x
the white pointinFig. 1 atthe borderbetweenthe 4-SL 2 k ~ 1.38π
x
I and 4-SL II regions with J2/J1 = 0.36. In region
| | − 0
4-SL II or above the white diagonal line, the 4-SL phase 0 0.5 1 1.5 2
hasinstabilitiesatthewave-vectors(π π/3)x. TheSW k /π
± x
intensity at the larger of these two wave-vectors always
dominatesandthe4-SLphaseevolvesintocycloidIIwith
wavector 4πx/3.
FIG. 4: (Color online)(a-b) SW frequency and intensity ver-
In region 4-SL I, the 4-SL phase has three unique SW sus wave-vector kx for the 4-SL phase with |J1|/D = 5.5,
instabilities: one at wave-vector k1 along the x axis, J3 = 1.3J2, where |J2|/D varies from 1.30 to 2.67. (c)
another at k2 rotated by π/3, and a third at k3 ro- Fourier transform for the Sx, Sy, and Sz components of the
tated by π/3. All three have the same magnitude with CNCphasewiththesameexchangeparametersasaboveand
π/2 < k −< π. Other SW instabilities in the 4-SL I re- |J2|/D=2.5.
i
4
TABLE I:Energy Coefficients for Collinear, Cycloid, and CNC Phases
Phase A1 A2 A3 AD
4-SL 1 -1 1 1
8-SL 0 1 1 1
Cycloid I -`cos(q)+2cos(q/2)´ -`1+2cos(3q/2)´ -`cos(2q)+2cos(q)´ 1/2
Cycloid II 3/2 -3 3/2 1/2
CNC 0.595±0.001 -0.097±0.001 1.161±0.001 0.712±0.001
4-SLandcycloidIphasesischaracterizedbyseveralelas- in Ref.[17].
tic peaksshowninFig. 4(c). Withinthe precisionofour
MCsimulations,thedominantwave-vectork 0.87πof Tosummarize,wehaveshownthatthedominantwave-
x
the CNC phase coincides with the dominant≈instability vector of a non-collinear phase in a frustrated TLA cor-
wave-vectorof the 4-SL phase that preceeds it. responds to the dominant instability wave-vector of a
Todemonstratehowthemagneticgroundstateevolves collinearphaseastheanisotropyisloweredandspinfluc-
fromcycloidII intothe 4-SLphaseandthenintocycloid tuationsbecomesofter. TheCNC phasesketchedinFig.
I, we plot in Fig. 4(a) and (b) the SW frequencies and 2 is a more reasonable candidate for the multi-ferroic
intensities versus wave-vector for the 4-SL phase with phase observedin Al-doped CuFeO2 than the one previ-
J1/D = 5.5, J3/J2 = 1.3, and J2 /D varying from ously proposed.
− | |
1.30to2.67. As J2 /Dapproachesthelowerlimitforthe
| | This research was sponsored by the Laboratory Di-
stability of the 4-SL phase, the SW intensity dominates
rectedResearchandDevelopmentProgramofOakRidge
at the cycloid II wave-vector q = 4πx/3. At the upper
National Laboratory, managed by UT-Battelle, LLC for
limit, the SW intensity dominates at the cylcoid I wave-
theU.S.DepartmentofEnergyunderContractNo. DE-
vector q 0.83πx. Hence, the wave-vector of the SW
≈ AC05-00OR22725 and by the Division of Materials Sci-
instabilities for the collinear phases correspond to the
ence and Engineering and the Division of Scientific User
ordering wave-vectorsof the non-collinear phases.
Facilities of the U.S. DOE.
Bear in mind that the transition between magnetic
groundstatesisnotalwayssignaledbythesofteningofa
SW mode. In fact, the transition between the 4-SL and
8-SL phases in Fig. 1 occurs even at D = , when the
∞
spinsareIsingvariablesandtheSWgapsforbothphases [1] Z. Fang et al., Phys. Rev. Lett. 84, 3169 (2000).
diverge. [2] W. Zheng et al., Phys. Rev. B 75, 184418 (2007).
The CNC phase may be related to the multi-ferroic [3] N. Terada et al., J. Magn. Magn. Mat. 272-276, e997
phase observed in Al-doped CuFeO2 [3], which was re- (2004);N.Teradaetal.,Phys.Rev.B70,174412(2004);
cently investigated by Nakajima et al. [17]. Based N.Teradaetal.,J.Phys.: Cond.Mat.19,145241(2007).
[4] See,forexample,Frustrated Spin Systems (WorldScien-
on neutron-scattering measurements, those authors con-
tific, New Jersey, 2004), edited by H.T. Diep.
cluded that the ground state is a modified cycloid with
[5] T. Takagi and M. Mekata, J. Phys. Soc. Jpn. 64, 4609
the same spin on sites R and R′ (see above). This (1995).
phase has peaks at wave-vectors on either side of πx, [6] S. Mitsuda et al., J. Phys. Soc. Jpn. 60, 1885 (1991).
in agreement with the neutron measurements. However, [7] M. Mekata et al., J. Phys. Soc. Jpn. 12, 4474 (1993).
amodifiedcycloidcannotbestabilizedbyaHamiltonian [8] F. Yeet al., Phys. Rev. Lett. 99, 157201 (2007).
with the form of Eq.(1), regardless of the exchange and [9] R. S.Fishman et al., Phys. Rev. B 78, 140407 (2008).
[10] R. S.Fishman, J. Appl. Phys. 103, 07B109 (2008).
anisotropy parameters. With an additional phase slip δ
[11] T. Kimuraet al., Phys. Rev. B 73, 220401(R) (2006).
for the spins at sites R′, a pure cycloidwith δ =0 and a
[12] S. Kanetsuki et al., J. Phys.: Cond. Mat. 19, 145244
singleelasticpeakalwayshaslowerenergythanthephase
(2007).
proposedin Ref.[17] with δ = q/2. This conclusionhas [13] S. Sekiet al., Phys. Rev. B 75, 100403(R) (2007).
−
been verified by MC simulations. [14] M.Swanson,J.T.Haraldsen,andR.S.Fishman,(unpub-
Like the non-collinear phase proposed earlier [17], the lished).
CNC phase also contains FM correlations between sites [15] Due to the anisotropy, the cycloids will be slightly dis-
R and R′ or R′′. So the CNC phase also has elastic torted by the tilting of the spins towards the ±z direc-
tions with an energy gain of order D2S2/z|J1| (z = 6 is
peaks on either side of πx at k 0.87π and 1.13π, as
x ≈ thenumberofnearestneighbors).Withintheprecisionof
shownin Fig. 4(c). Because the FM correlationsarenot our MC simulations, this distortion is undetectable and
perfect and vary along the x direction, the CNC phase a pure cycloid is obtained.
contains several other elastic peaks that may allow it to [16] Th. Jolicoeur et al., Phys. Rev. B42, 4800 (1990).
beexperimentallydistinguishedfromthephaseproposed [17] T.Nakajimaetal.,J.Phys.Soc.Jpn.76,047309(2007).