Spin Hamiltonian, Competing Small Energy Scales and Incommensurate Long Range Order in the Highly Frustrated Gd3Ga5O12 Garnet Antiferromagnet

0.22 MB·English

by  Taras Yavors'kii

Spin Hamiltonian, Competing Small Energy Scales and Incommensurate Long Range
Order in the Highly Frustrated Gd Ga O Garnet Antiferromagnet
3 5 12
Taras Yavors’kii,1 Matthew Enjalran,2 and Michel J.P. Gingras1,3
1 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
2 Department of Physics, Southern Connecticut State University, New Haven, CT 06515, USA
7 3 Department of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
0 (Dated: February 2, 2008)
0
2 Despite the availability of a spin Hamiltonian for theGd3Ga5O12 garnet (GGG) for overtwenty
fiveyears,therehassofarbeenlittletheoreticalinsightregardingthemanyunusuallowtemperature
n
properties of GGG. Here we investigate GGG in zero magnetic field using mean-field theory. We
a
J reproducethespinliquid-likecorrelations and,mostimportantly,explainthepositionsofthesharp
peaksseeninpowderneutrondiffractionexperiments. Weshowthatitiscrucialtotreataccurately
2
thelong-range natureofthemagneticdipolar interactionstoallow foradeterminationofthesmall
2
exchangeenergyscalesinvolvedintheselectionoftheexperimentalorderingwavevector. Ourresults

showthattheincommensurateorderinGGGisclassical innature,intrinsictothemicroscopicspin
]
i Hamiltonian and not caused by weak disorder.
c
s
-
rl The diversity of empirical data collected over the past tions that develop below 200 mK [9, 10]. This could be
t fifteen years has demonstrated that geometrically frus- interpreted as evidence that exotic mechanisms involv-
m
trated triangular and tetrahedral arrangements of an- ing either quantumfluctuations ordisordereffects areat
.
t tiferromagnetically coupled spins are highly partial to- play in GGG. For example, it has been suggested that
a
m wards the realizationof exotic correlatedphases in mag- disorder, such as Gd3+ on Ga3+ sites, is responsible for
neticmaterials[1,2,3]. Thereasonfortherichandtypi- nucleating regions with the spin order observed in neu-
-
d callymaterialspecificpropertiesoffrustratedmagnetsis tron scattering in zero magnetic field [9, 10].
n understood. It stems from their sensitivity to perturba- In this letter we address the adequacy of the Hamilto-
o
tions beyond the frustrating nearest-neighbor antiferro- nian H of Ref. 11 to explain the low temperature prop-
c
[ magnetic (AFM) exchange which, on its own, leads to a erties of GGG. It has been demonstrated [9, 12] that H
macroscopic number of exactly degenerate and compet- with only a nearest-neighbor exchange interaction can
2
ing, hence fragile, classical ground states. In this paper describe liquid-like spin correlations [9, 10]. We show
v
3 we show, through a careful theoretical analysis of neu- below that H can also describe well the incommensurate
0 tronscatteringexperiments,thatthe extensivelystudied spin-spincorrelationsandthatthereisnoobviousreason
4 Gd3Ga5O12 garnet (GGG) is precisely such a system, to extend H. Rather, it is the correctprocedure to treat
1 though evidence for this fact emergesfrom a perspective itsmanycompetingsmallenergyscales,inparticularthe
1
on the problem that has heretofore escaped scrutiny. longrangedipolarinteractions,andtheunusuallyprecise
5
GGG displays a gamut of complex and interesting specification of weak second and third nearest-neighbor
0
/ lowtemperature magnetic phenomena. Inzeromagnetic exchange couplings, that are foremost needed, and that
t
a field, the behavior of GGG is uniquely rich. The nonlin- have been missed so far.
m earmagneticsusceptibilityχ3 peaksatTg ∼180mK[4], Inorderto correctlyparameterizeH andthe resulting
- indicatingaspinglasstransition[5]. However,muonspin spin correlations we rely on neutron scattering experi-
d
relaxation[6,7]andMo¨ssbauerspectroscopy[8]findper- ments[9,10],whicharethemostdirectandunambiguous
n
o sistentspindynamicsdowntoT ≪Tg. Meanwhile,pow- probe of magnetic correlations. Our basic hypothesis is
c der neutron scattering data [9, 10] indicate that GGG is that conventional criticality controls the development of
v: on the verge of developing true incommensurate long- the sharppeaks observedin neutronexperiments [9, 10],
i range magnetic order with a correlation length (ξ ≈100 andthat,asinconventionalmagnets,amean-fieldtheory
X ˚A) extending over 8 cubic unit cells below 140 mK. (MFT) treatment of the pertinent Hamiltonian should
r
A Hamiltonian H describing GGG, that we shall ex- be able to capture the Gaussian regime in the approach
a
plicitly define below, has long been available [11]. It as- to criticality. Our finding of a unique ordering wave
sumes classical Gd3+ spins, is parameterized as a sum vector q shows that GGG, unlike the AFM nearest-
ord
of empirical exchange contributions up to third nearest- neighbor Heisenberg pyrochlore, is not pathological and
neighborsaswellasamagneticdipolarcontribution,and is amenable to such a soft-mode determination and cal-
ignores potentially important quantum fluctuations or culation of the neutron scattering intensity profile. We
disorderinherenttoGGG[4]. Previousnumericalstudies show that GGG is very similar to the now well under-
based on H have been unable to provide a quantitative stood dipolar spin ice materials [13, 14], where a MFT
explanationforthebulk [4]anddynamical[6,7,8]prop- thattakesintoaccountthelong-rangenatureofthedipo-
erties of GGG or the incommensurate spin-spin correla- larinteraction[13](i)identifiesthecorrectorderingwave
2
We use a GaussianMFT [16] to obtain the softmodes
48
units qan-gdricda,lcwuleatceaIn(Qca)locfutlahteemIo(dqe)lH(in. Uthsiengthaesremlfo-addyanpatmiviec
b.
36 ar limit)atanarbitrarypositivedimensionlesstemperature
Q), τ = T/TMFT−1, where TMFT ≈ 2 K and depends on
I( c c
Ewald J2 andJ3. Due tothe log|τ| growthofthe powderMFT
24 Bragg peak intensities, as opposed to the 1/τ growth of
R=1000
c
theq-dependentintensities,τ mustbesetrathersmallin
R=10
c
order to work with the experimental profiles in the criti-
12 Rc=5 cal regime (cf. Fig. 4). We consider a finite q-space grid
Rc=4 with 323 points in the first Brillouinzone, and construct
0 Rc=3 Q, A o-1 a three dimensional cubic interpolation scheme to sepa-
ratelyevaluatethenumeratorandthedenominatorofthe
0.4 0.6 0.8 1 1.2 1.4 1.6
MFT I(q) (cf. Eq. A37 in Ref. 16). For q ≈ q +G,
ord
FIG.1: MFTI(Q)ofGGGwithJ2 =−0.003K,J3 =0.010 where Bragg peaks occur (G is a FCC reciprocal lat-
K [11] and τ = 10−4 for Rc = 3,4,5,10,1000. The Rc = ∞ tice vector), we back up the interpolation by calculating
profile is obtained using the Ewald method. The profiles are I(q) exactly [16]. This ensures q-grid size independent
uniformlyshiftedtoemphasizetheirstrongdependenceonRc results [16]. I(Q) is computed by spherically averaging
for small values, as well as their saturation to a well defined
I(q) numerically.
limit at large Rc.
WefindthatHwithdipolarinteractionsandexchange
interactions beyond nearest-neighbor possesses a unique
q . Our numerical study suggests that the soft mode
vector [14], and (ii) correctly predicts the symmetry of ord
selection in the full H is highly sensitive to the values
the scattering close to the critical temperature.
of J and J . This finding is in general agreement with
We examine the spin correlations in GGG by cal- 2 3
resultsfromMonteCarlosimulationsofGGG[9,10](see
culating the powder neutron scattering intensity I(Q)
alsoRef.[11]foracommentonthesensitivityoftheGGG
and comparing it with experimental data [9]. The
specific heat on J ). However, our crucial new observa-
scattering q-vector-dependent intensity is given by 3
tionisthatq isalsoverysensitivetoanad-hocchoice
I(q) = (F(|q|)2/N) hS⊥ · S⊥ieıq·rij, obtained for ord
the model HamiltonPianij [1i1] Hj = H + H with of the cut-off distance Rc in Hdip. We demonstrate this
e dip byplottinginFig.1I(Q)forvariousR forJ =−0.003
He = Pi>jnP3n=1Jnδrij,rnSi·Sjo and Hdip = KandJ3 =0.010K(valuesfrom[11]). cForRc2<∼O(102),
Pi>j[D(r1/rij)3][Si·Sj −3(Si·rˆij)(Sj ·rˆij)]. Labels i notonlyis qord anon-monotonousfunctionofRc,asthe
and j span the N sites of a garnet lattice (c.f. Fig. 2 positions of the peaks indicate, but so is the unit cell
in Ref. [11]), r and rˆ are the length and direction magnetic form factor, as is evident from the significant
ij ij
of the vector separation between them, and r enumer- changesinthe overallshapeofthediffuse scatteringpat-
n
ate distances between the sites starting from the near- ternsnearcriticality(τ ≪1). Fig.1demonstratesthatH
est neighbors. The size of the GGG conventional cu- withdifferentlowRc givescompletely differentspin-spin
bic unit cell is a = 12.349 ˚A [9]. The isotropic vectors correlations in the critical regime. Only for rather large
Si havelength S(S+1)toclassicallyapproximatethe cut-offs does qord become insensitive to Rc, approaching
S = 7/2 Gd3+pspins. S⊥i are the spin components per- the Rc = ∞ limit only when Rc >∼ 103. Figure 1 sug-
pendiculartoq,F(|q|)istheGd3+ magneticform-factor gests that it is essential to set Rc to its true Rc = ∞
[15], and h...i denotes a thermal average. The spin cou- limittoavoidinducingspuriousorderedphasesbytreat-
plings in H include dipolar interactions Hdip of strength ing the lattice sum in Hdip by a shortcut-off method, as
D, cut-off at a distance R , and exchangeinteractionsof was done previously [9, 10, 11, 12, 17]. We henceforth
c
empirical strength J , n ≤ 3. We show below that R acknowledge this and implement the Ewald method [16]
n c
must be set to the true long-range limit Rc = ∞ in or- to process Hdip with Rc =∞.
der to obtain physically meaningful results. Throughout The scattering profiles in Fig. 1 and the q positions of
the paper, we keep D = 0.0457 K [12] and the nearest- the sharp peaks, even in the R = ∞ limit, are incom-
c
neighborexchangeJ =0.107K[11]fixed. However,be- patible with the experimental data [9] (cf. Fig. 4, upper
1
cause of the large uncertainties on J and J estimated panel). However,theuncertaintyinpreviouslyestimated
2 3
from high-temperature analysis (−0.015 ≤ J ≤ 0.009 values of J and J [11] allows fine-tuning them in or-
2 2 3
K, −0.035≤J ≤0.100 K) [11], and the unreliability of der to obtain a better match with the low temperature
3
theMonteCarloprocedure[11]torefinethembecauseof data[9]. Toproceed,weareguidedbythegeneralunder-
poor statistics (see also discussion in Ref. [10]), we con- standing of disordered systems near the critical bound-
sider these as free parameters to be adjusted by fitting arybetweenlong-rangeorderedandspinglassphases[5].
the experimental I(Q) [9]. Specifically, we make the reasonable assumption that,
3
J=-0.01K, J= 0K,
10 J32= 0.0015K 30 J32=-0.0285K
20
5
10
I(Q), arb. units 500 JJ02==.5--00..0000135KK, 1 1.5 200 JJ02==.5- 00..0010K7K, 1 1.5
40 3 15 3
30
10
20
10 5
0 0
0.5 1 1.5 0.5 1 1.5
Q, Å-1
FIG. 3: Scattering profile for regions of the J2 −J3 plane
where the positions Qi of the sharp peaks have been opti-
FIG. 2: The optimization neutron-scattering penalty func- mized; τ = 1.6×10−6. Upper left and bottom right panels
tion, P, in the J2 −J3 plane behaves discontinuously and represent correspondingly located black domains in Fig. 2.
shows minima in the black domains. Only for the couplings The other two panels represent small domains in the central
demarcated by a dotted trapezoid does the theoretical scat- and bottom-left part of Fig. 2. The overall shape of I(Q)
teringprofilereconstituteboththelocationofthesharppeaks varies significantly, but all the profiles display sharp peaks
and the overall shape of the broad and diffuse experimental agreeing with theexperiment (shown by vertical lines).
scattering [9, 10]. The circle corresponds to the values of J2
andJ3 suggestedinRef.11. Theupper-leftandbottom-right
white regions correspond to different qord =0 structures.
Figure2showsthatevensmall(atthe≈1%levelofJ )
1
changesin J andJ show up aslarge changesinthe or-
2 3
given the sizeable and saturated ξ ∼ 100 ˚A correlated deringwavevectorand,correspondingly,inP too. There
existonlyrelativelysmalldomains(showninblack)that
regions below 140 mK [9], the weak disorder in GGG
minimize P. Itturnsout,however,thatforeachofthese
freezes-in the q-dependence of quasi-critical incommen-
domains, the overall shape of the MFT powder scatter-
surate correlated regions as T is crossed upon cooling.
g
ing profile changes noticeably even though q remains
We therefore use a two-step fitting procedure to deter- ord
almostconstantat the value of 2π/a[0.29,0.29,0]. Simi-
mine J and J . First, we establish whether the model
2 3
larly to the standard experimental procedure to analyze
Hallowsforaq compatiblewiththeexperiment. Sec-
ord
powderneutronscatteringdata,onceq isdetermined,
ond,weexaminewhetherthemodelwithagiven{J ,J } ord
2 3
one needs to solve for the magnetic structure. The sen-
set is able to reproduce the overall structure of correla-
sitivity of the unit cell magnetic form factor to {J ,J }
tions both in the paramagnetic and “frozen-in critical” 2 3
leadstoasignificantqualitativechangeinthediffusescat-
regimes [9].
tering pattern near criticality, hence offering a second
We find that q for R =∞ belongs to the (hhl) re-
ord c
optimization channel for the determination of {J ,J }.
ciprocal plane, with its location within the plane highly 2 3
dependent on J and J . To determine the optimum J Figure3 showsthe dependence ofthe theoreticalI(Q)
2 3 2
and J3, we generate a sequence of data points {Q˜} ≡ on {J2,J3} within the black domains of Fig. 2. A com-
|q (J , J )+G|, and then match these points to the parisonofsuchtheoreticalprofileswiththeexperimental
ord 2 3
experimental sequence of sharp peaks in powder GGG one reveals that only in the upper left black domain are
data [9]. To do so, we introduce a penalty function, thetheoreticalandexperimentalprofilessimilar;astrong
P(J , J ), thatprovidesa measureofmismatchbetween discrepancy in other domains allows us to exclude them
2 3
the first three clearly discernible and strongest experi- from further consideration. Within the upper left do-
mental peaks at Q∗ = 0.64 ˚A−1, Q∗ = 0.85 ˚A−1 and main, the match is visually better for rightmost values
1 2
Q∗3 = 1.07 ˚A−1 of Ref. 9 (see 43mK experimental data of J2 with the restriction that its value cannot cross the
in top panel of Fig. 4) and the closest theoretically domain boundary at −0.004K.
determined peaks: P = 100 × maxi=1,2,3∆Qi, where We support and further refine the above analysis by
∆Q =min |Q˜−Q∗|withthe numericalfactor100 fitting the MFT scattering profiles to the experimental
i Q˜∈{Q˜} i
chosento setthe scale, P ≡1, fora maximum mismatch I(Q) in the paramagnetic regime. For each value of J
2
of0.01˚A−1. Amap ofP in the J −J plane is shownin and J we consider a χ2-type penalty function for the
2 3 3
Fig. 2. A good match between theory and experiment is overall shape of I(Q), which is adjusted by parameters
denoted in black. The discrepancy between experiment responsible for a uniform and linear in |Q| background
andtheoryincreasesasthe gradationofgrayvariesfrom terms, as well as by the overall scale factor. With τ as
black to white. a free fitting parameter, we do the fits in the interval
4
Toconclude,we havefoundthat, asinspinices[1,13,
14], an ad-hoc R cut-off of the dipolar interactions at
8000 c
less than a few hundrednearest-neighbordistances leads
7000 to spurious long-range ordered phases. Only a proper
unts treatment of an infinite Rc allows one to get a handle
6000 Q), co on the very small exchange interactions beyond near-
I( estneighborswhichdictatetheincommensurateordering
5000
wave vector q . With a reasonably well parameterized
ord
Hamiltonian now in hand, further theoretical studies to
4000
explore the origins of the complex phenomena displayed
Q, A o-1
3000 by GGG in zero and nonzero magnetic field may now be
0.4 0.6 0.8 1 1.2 1.4 1.6 possible. Perhaps most importantly, we have identified
thelikelyreciprocalplanethatcontainsq ,withq ≈
ord ord
17
2π/a[0.29,0.29,0]. Withthispredictionavailable,itmay
now be possible to perform single crystal neutron scat-
14
11 units t1e57riGndgmabesaosruprteimonenptrsoibnleamreinflehcitgiohnqgueaolmityetsriyngtolebceraytsttahles
b.
Q), ar with natural Gd abundance [18], and allow for a quan-
7 I( titative investigation of the development of correlations
in GGG at low temperatures. This may prove a fruitful
4 endeavor to help shed light on the mysteries of GGG.
1 Q, A o-1 We acknowledge the contributions of M. Dion and B.
denHertogtothe earlierstageofthis project. We thank
0.4 0.6 0.8 1 1.2 1.4 1.6
A. Del Maestro, B. Gaulin, R. Moessner, O. Petrenko,
FIG.4: Positionsofthefirstthreestrongestpeaksoftheex- A. Ramirez and P. Schiffer for useful discussions. We
perimentalscatteringprofile(pointedtobyarrows intheup- thankO.Petrenkoforprodivingneutrondata. Thiswork
perpanel)aswellastheoverallshapeofscatteringprofileare was funded by NSERC of Canada, the Canada Research
usedascriteria fortheoptimization procedurein theJ2−J3 Chair Program (Tier I) (M.G), Research Corporation
plane. The resulting mean-field theory profile at dimension-
less temperature τ =1.6×10−6 (lower panel) compares well (M.G.,M.E.),theCanadafoundationforInnovation,the
OntarioInnovationTrust,andtheCanadianInstitutefor
with the experimental one. The positions of three experi-
mentalpeaksaredenotedbyverticallines. Theexperimental Advancedresearch. M.G.acknowledgestheU.ofCanter-
scattering profile has been obtained by digitizing Fig. 2 of bury (UC) for an Erskine Fellowship and the hospitality
Ref. 9. The theoretical profile is shown for J2 = −0.005K, oftheDept. ofPhysicsandAstronomyatUCwherepart
J3 =0.010K, denoted by a box in Fig. 2. of this work was completed.
Q=0.26−1.57˚A−1 whichallowsfor the complete shape
determination of the first experimental broad diffusive [1] Frustrated SpinSystems,ed.H.T.Diep,WorldScientific,
scattering peak at 1.05˚A−1 without a |Q|2 background Singapore (2004).
[2] A.P.Ramirez,inHandbookofMagneticMaterials,ed.K.
correction. Most importantly, an analysis of the (para-
H.J.Buschow(ElsevierScience,Amsterdam,2001),Vol.
magnetic) data at 175 mK [9] allows us to confirm the
13.
outcomeoftheanalysisofthecriticaldataat43mKand [3] J.E.Greedan,J.ofAlloysandCompounds408-412,444
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2 2
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3 [6] S.R. Dunsiger et al.,Phys. Rev.Lett. 85, 3504 (2000).
J ≤0.012 K, see Fig. 2.
3 [7] I.M. Marshall et al.,J. Phys. Condens. Matter 14, L157
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the three strongest experimental peaks leads to a theo- [10] O.A. Petrenkoet al.,Physica B 266, 41 (1999).
[11] W.I. Kinney and W.P. Wolf, J. Appl. Phys. 50, 2115
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[12] O.A. Petrenko and D. McK. Paul, Phys. Rev. B 63,
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of the correlations can be described by MFT. 1339 (2001).
5
[14] R.G. Melko and M. J. P. Gingras, J. Phys.: Condens. [16] M. Enjalran and M. J. P. Gingras, Phys. Rev. B 70,
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