Table Of ContentDipolar Interactions and Origin of Spin Ice in Ising Pyrochlore Magnets
Byron C. den Hertog and Michel J. P. Gingras†
Department of Physics, University of Waterloo, Ontario Canada N2L 3G1
†Canadian Institute for Advanced Research
(February 1, 2008)
0 Recent experiments suggest that the Ising pyrochlore magnets Ho2Ti2O7 and Dy2Ti2O7 display
0 qualitative properties of the spin ice model proposed by Harris et al. Phys. Rev. Lett. 79, 2554
0
(1997). We discuss the dipolar energy scale present in both these materials and consider how
2
they can display spin ice behavior despite the presence of long range interactions. Specifically, we
n present numericalsimulations and amean fieldanalysis of pyrochloreIsingsystemsin thepresence
a of nearest neighborexchangeandlong rangedipolar interactions. Wefindthattwopossible phases
J
canoccur,alongrangeorderedantiferromagneticoneandtheotherdominatedbyspinicefeatures.
5 Our quantitative theory is in very good agreement with experimental data on both Ho2Ti2O7 and
2 Dy2Ti2O7. We suggest that the nearest neighbor exchange in Dy2Ti2O7 is antiferromagnetic and
that spin ice behavior is induced by long range dipolar interactions.
]
n
n An exciting development has occurred in the last two which by itself should cause a phase transition to a long
- yearswith the discoveryofanapparentanalogybetween rangeorderedgroundstate. Thus,howthesesystemsac-
s
i the low temperature physics of the geometrically frus- tuallydisplayspinice-likebehaviorismostpuzzling. For
d
trated Ising pyrochlore compounds Ho Ti O [1] and example, one might naively expect that the long-range
2 2 7
.
t Dy Ti O [2] (so called ‘spin ice’ materials), and pro- and anisotropic spin-space nature of the dipolar interac-
a 2 2 7
m ton ordering in real ice [3]. The magnetic cations Ho3+ tions would introduce so many constraints that a large
and Dy3+ of these particular materials reside on the py- degree of the degeneracy present in the simple nearest-
-
d rochlore lattice of corner sharing tetrahedra. Single-ion neighbor ferromagnetic spin-ice model [1,4] would be re-
n effects conspire to make their magnetic moments almost moved, and induce long-rangeorder. It is this issue that
o
ideally Ising-like,but withtheir ownsetoflocalaxes. In we wish to address in this paper.
c
particular,eachmomenthasitslocalIsingaxisalongthe
[
line connecting its site to the middle of a tetrahedron to
1
which it belongs (see inset of Fig. 1).
v
In a simple model of nearest neighbor ferromagnetic
9
6 (FM) interactions,suchasystemhasthe same‘ice rules’
3 for the construction of its ground state as those for the
1 ground state of real ice [3,4]. In both cases, these rules
0
predictamacroscopicallydegenerategroundstate,afea-
0
ture that a number of geometrically frustrated systems
0
/ possess [5–8].
t
a In Ho2Ti2O7, µSR data indicates a lack of ordering
m down to ∼ 50 mK despite a Curie-Weiss temperature
- θcw ∼1.9 K, while single crystalneutron scattering data
d
suggeststhedevelopmentofshort-rangeFMcorrelations,
n
but the absence of ordering down to at least 0.35 K [1].
o
c Ho2Ti2O7 also displays field dependent behavior consis- FIG. 1. Phase diagram of Ising pyrochlore magnets with
: tentwitha spinicepicture [9]. Quite dramatically,ther- nearest neighbor exchange and long range dipolar interac-
v
i modynamic measurements on Dy2Ti2O7 [2] show a lack tions. Dnn and Jnn are the Ising parameters for the near-
X of any ordering feature in the specific heat data, with est neighbor dipolar and exchange energies respectively (see
r the measured ground state entropy within 5% of Paul- text). Inset: An ice rule configuration of a tetrahedron. The
a four local h111i Ising axes meet in themiddle of thetetrahe-
ing’s prediction for the entropy of ice [3].
dron [4].
However, both spin ice materials contain further in-
teractions additional to the nearest neighbor exchange.
Often, rare earth cations can have appreciable magnetic
A previous attempt to consider dipolar effects in Ising
moments and, consequently, magnetic dipole-dipole in-
pyrochloreswasmade bySiddharthanet al.[10]. Inthat
teractions of the same order as, if not larger than the
work,thedipole-dipoleinteractionwastruncatedbeyond
exchange coupling, can occur. Furthermore, it has been
five nearest neighbor distances, and a sharp transition
suggested that the nearest neighbor exchange interac-
between paramagnetism and a partially ordered phase
tioninHo Ti O isactuallyantiferromagnetic(AF)[10],
2 2 7 (where rapid freezing occurs) was observed for interac-
1
tion parameters believed appropriate for Ho Ti O [10]. andmustbeconsideredwithcare. Inordertoincludethe
2 2 7
However, we argue that the truncation of dipole-dipole importantlongrangenatureofthedipole-dipoleinterac-
interactions can be misleading, and may introduce spu- tion,wehaveimplementedthewellknownEwaldmethod
rious features in various thermodynamic properties. For withinoursimulationtechnique,inordertoderiveanef-
example, we find that the sharp feature observed in the fective[14]dipole-dipoleinteractionbetweenspinswithin
specific heat for truncation beyond five nearestneighbor our simulation cell [12]. Unlike in dipolar fluid simula-
distances [10]is softened and rounded for truncationbe- tions,ourlatticespinsallowtheeffectiveinteractionsbe-
yond the tenth nearest neighbor shell, and the observed tweenallmomentstobecalculatedonlyonce,afterwhich
dynamical freezing is pushed to lower temperatures. As a numerical simulation can proceed as normal.
we show below, in the limit of infinite range dipoles, the A standard Metropolis algorithm was used in our
interaction parameters of Siddharthan et al. [10] yield Monte Carlo simulations. We used a conventional cubic
spin ice. cellforthepyrochlorelattice,whichcontains16spins. In
In this work, we consider the interplay between near- general,we foundit sufficient to simulate up to 4×4×4
estneighborexchangeanddipolarinteractionsbytaking cubic cells (ie. L=4, and 1024 spins) with up to ∼ 106
into account the long range (out to infinity) nature of Monte Carlo steps per spin when necessary. Thermody-
the dipolar interactions through the use of Ewald sum- namic data were collected by starting the simulations at
mation techniques [11,12]. Our Monte Carlo simulations high temperatures and cooling very slowly.
and mean field results show that dipolar forces are re- ReferringtoFig. 1,thecharacterizationofasystemas
markablyadeptatproducingspinicephysicsoveralarge having spin ice behavior was carried out by determining
region of parameter space. theentropy,vianumericalintegrationofthespecificheat
Some of our main conclusions are shown in Fig. 1. dividedbytemperaturedata. Pauling’sargumentforthe
For Ising pyrochlores, the dipole-dipole interaction at entropyoficeyieldsR[ln2−(1/2)ln(3/2)]or4.07Jmol−1
nearestneighbor is FM, and therefore favorsfrustration. K−1 [2,3]. We find for J /D = −0.91, (our spin ice
nn nn
Beyond nearest neighbor, the dipole-dipole interactions data point closest to the phase boundary in Fig. 1), this
can be either FM or AF, or interestingly, even multiply value for the entropy to within 3%, using a system size
valued [13], depending on the neighbor distance. Defin- L=4.
ingthenearestneighbordipole-dipoleinteractionasD Our thermodynamic data indicates that when the
nn
and the nearest neighbor exchange as J , our Monte nearest neighbor exchange is AF and large compared to
nn
Carlo results indicate that spin ice behavior persists in the dipolar interactions, the system undergoes a second
the presence of AF exchange up to J /D ∼ −0.91. orderphasetransitiontoanallinoralloutQ=0ground
nn nn
ForJ /D <−0.91we findasecondorderphasetran- stateasalludedtoearlier. ThisAFphasepersistsslightly
nn nn
sition to the globally doubly degenerate Q = 0 phase of beyond the point where the nearest neighbor dipolar in-
thenearestneighborAFexchange-onlymodel[15],where teraction(FM) is strongerthan nearestneighbor AF ex-
allspinseitherpointin,oralloutofagiventetrahedron. change.
Our Hamiltoniandescribing the Ising pyrochloremag-
nets is as follows,
H =−JXSzii ·Szjj
hiji
Szi ·Szj 3(Szi ·r )(Szj ·r )
+Drn3nX i|r |3j − i |irj |5j ij , (1)
ij ij
hiji
wherethespinvectorSzi labelstheIsingmomentofmag-
i
nitude |S| = 1 at lattice site i and local Ising axis z .
i
Because the local Ising axes belong to the set of h111i
vectors, the nearest neighbor exchange energy between
two spins i and j is J ≡ J/3. The dipole-dipole in-
nn
teraction at nearest neighbor is Dnn ≡ 5D/3 where D FIG. 2. Dependence of the specific heat peak height and
is the usual estimate of the dipole energy scale, D = temperature location on exchange and dipole-dipole interac-
(µ0/4π)g2µ2/rn3n. For both Ho2Ti2O7 and Dy2Ti2O7, tion parameters.
D ∼2.35 K.
nn
Itiswellknowninthefieldofelectrostaticinteractions
that the dipole-dipole interaction is difficult to handle Inthespiniceregime,ourspecificheatdatahasanum-
due to its 1/r3 nature. In general, a lattice summation ber of interesting features which help shed light on the
of dipole-dipole interactions is conditionally convergent, effectoflongrangedipole-dipoleinteractions. Ingeneral,
2
each data set shows qualitatively the same broad peak short range truncation of the dipole-dipole interaction.
asobservedinthe nearestneighborFMexchangemodel, As the nearestneighbor exchange interactionbecomes
andvanishesathighandlowtemperatures[9,10]. Wede- AF,wefindthatthe approximate‘collapse’ontoasingle
tect very little system size dependence upon comparison energy scale becomes less accurate, with features in the
of data from L = 2,3,4 simulation sizes. In particular, specific heat becoming dependent on J /D in a more
nn nn
thereisnonoticeablesizedependenceofthespecificheat complicated manner. It is within this regime that we
maximum, nor its position. As the AF/spin ice phase believe that both Ho Ti O and Dy Ti O exist.
2 2 7 2 2 7
boundary is approached from the spin ice side, the spe- Since D is a quantity which can be calculated once
nn
cific heat peak begins to narrow and more importantly, the crystal field structure of the magnetic ion is known,
both the peak height and peak position begin to vary. the nearestneighborexchangeJ is the only adjustable
nn
Aswediscussbelow,thishasimportantramificationsfor parameterin ourtheory. Fig. 2 enables us to test in two
the interpretation of experimental data. independent ways the usefulness of our approach to the
In Fig. 2 we plot the dependence of the specific heat long range dipole problem in spin ice materials.
peak height (C ) and the temperature at which it oc- If we consider Dy Ti O for example, specific heat
peak 2 2 7
curs(T )ontheratioofthenearestneighborexchange measurements by Ramirez et al. [2] indicate a peak
peak
and dipole-dipole energies, Jnn/Dnn. Note that in the height, CpDeyak of ∼2.72 J mol−1K−1. Given that
regime of large nearest neighbor FM coupling, the peak D ≈2.35 K for this material, the left hand plot of
nn
height plateaus to the value one observes in the nearest Fig. 2 indicates a nearest neighbor exchange coupling
neighbor FM exchange-only model. J ∼−1.2 K. The same experimental specific heat
nn
Indeed our data suggests that when the exchange be- data shows that this peak occurs at a temperature of
comes FM, the nearestneighbor effective bond energy is T ∼ 1.25 K. Using the plot of T in Fig. 2, we
peakDy peak
large enough to dominate the excitations of the system. independently arrive at approximately the same conclu-
This can be more dramatically seen in Fig. 3, where we sionforthevalueofthenearestneighborexchange. Thus,
haverescaledthe specific heatcurvesfor anumber ofin- wepredictthatAFexchangeispresentinDy Ti O with
2 2 7
teractionparametervaluesintermsofthe effective near- J ∼ −1.2 K. If there was no AF exchange present in
nn
est neighbor interaction Jeff ≡ Jnn +Dnn in the regime this system, our results in Fig. 1 imply that there would
Jnn/Dnn >0. beapeakinthespecificheatatatemperatureofatleast
∼ 2.3 K, which is not observed experimentally [2]. Our
bestfitforthespecificheatdataofDy Ti O byRamirez
2 2 7
et al. [2] is shown in Fig. 4, where we find good agree-
ment between theory and experiment for J = −1.24
nn
K.
FIG. 3. Specific heat for system size L = 2 rescaled
into units of the effective nearest neighbor interaction Jeff.
Jnn/Dnn =0correspondstopurelydipolarinteractionswhile
Jnn/Dnn =∞ corresponds to nearest neighbor FM exchange
only.
This figure shows that in terms of an effective energy
scale,the medium to long range effects ofthe dipolar in-
teractionsare‘screened’by the system,andone recovers
qualitativelytheshortrangephysicsofthenearestneigh-
bor spin ice model. Remarkably, inclusion of long range
dipolar interactions appears to have the effect of remov-
ing a tendency towards ordering which can come from a
3
dueto narrowlyspacedcrystalfieldlevels[8]. While this
makes the interpretation of experimental data more dif-
ficult, initial estimates of the nearest neighbor exchange
and dipole moment [8] yield J /D ∼−1, placing this
nn nn
system very close to the phase boundary of Fig. 1. In-
deed, µSR measurements suggestthat Tb Ti O fails to
2 2 7
orderdownto70mK[7]andthusitwouldappearthata
spinice pictureforthis materialcannotbea priori ruled
out.
Whilewebelieveourapproachyieldsareasonablysuc-
cessful quantitative theory of spin ice behavior in Ising
pyrochlores,there still remains the question of why long
rangedipolarinteractionsdonotappeartoliftthemacro-
scopicdegeneracyassociatedwiththeicerules,andselect
an ordered state.
Mean-field theory provides a more quantitative basis
for examining this issue. Following the approach used
in Refs. [16,17], we find that the soft-modes for Ising
pyrochlore systems described by Eq. 1 consist of two
very weakly dispersive branches (less than 1% disper-
sion) over the whole Brillouin zone (except at Q ≡ 0).
Suchasetofquasi-dispersionlessbranchesisverysimilar
to the two completely dispersionless soft branches of the
nearest-neighborFMspinicemodel. Consequently,both
FIG. 4. Comparison of (a) specific heat and (b) entropy
nearest-neighborFM and dipolar spin ice behave almost
databetweenDy2Ti2O7 [2]andMonteCarlosimulationwith
identically over the whole temperature range spanning
Jnn =−1.24 K, Dnn =2.35 K and system size L=4.
O(w) ≤ T < ∞ where w is the bandwidth of the soft
Specific heat measurements on a powdered sample of branchofdipolarspinice. Wenotethatthisneardisper-
Dy Ti O in a magnetic field were also reported in Ref. sionless behavior is only recovered asymptotically as the
2 2 7
[2]. Three field independent peaks were observed at low longrangedipolesareincludedouttoinfinity. Thelifting
temperature. Fora largefieldin theh110idirection,two of degeneracy at the 1% level in the apparentabsence of
spins on each tetrahedron are pinned by the field, while any small parameters in the theory is at this point not
the other two remain free, since their Ising axes are per- completely understood. We do not believe it is due to
pendicular to the applied field. Due to the dipolar inter- numerical or computational error. A partial explanation
action, there will be a coupling between the fluctuating may be that (111) Ising anisotropy, the 1/r3 long-range
spins on these two sub-lattices. Our preliminary simu- nature of dipolar interactions, and the specific relation-
lations on small lattice sizes suggest a field independent ship between the topology of the pyrochlore lattice and
ordering at low temperature as observed in experiment. the anisotropic(spin−space) coupling of dipolar interac-
Considering Ho Ti O , experimental data on its ther- tions,combineinasubtlemannertoproduceaspectrum
2 2 7
modynamic properties is not so categorical. The specific ofsoftmodesthatapproximateverycloselythespectrum
heatdataofSiddharthanet al.[10]indicatesafeatureat of the nearest-neighbor ferromagnetic spin ice model of
∼ 0.8 K, although it has been suggested that this could Harrisetal.[1]. Inotherwords,theremustbeanalmost
be due to an additionalcontributionin this temperature exactsymmetry fullfilled in this systemwhen long-range
range from an anomalously large hyperfine coupling in dipolar interactions are taken into account. However,
Ho3+ [18]. Nevertheless, using the plot of T in Fig. the same long-range nature of these interactions renders
peak
2, we find substantial AF exchange coupling of the same it difficult to construct a simple and intuitive picture of
orderofmagnitudeasintheDycompound. Wenotethat their effects, and we have not been able to identify such
Siddharthanetal.[10]anddenHertogetal.[18]findfrom “almost exact” symmetry.
analysis of magnetization measurements a similar order Also,in these systems,the absenceofsoftfluctuations
of magnitude for J . Furthermore, our numerical simu- (Ising spins) combined with the macroscopic degeneracy
nn
lations within this region of parameter space indicate a associated with the ice rules may be such that correla-
Curie-Weisstemperatureof∼2K,inagreementwithan tions associated with a ‘true’ ground state are dynam-
experimentalestimatebyHarriset al. ofθ ∼1.9K[1]. ically inhibited from developing. For energetic reasons,
cw
Tb Ti O isanIsingpyrochloresystemofsimilartype statesobeyingtheicerulesarefavoreddowntolowtem-
2 2 7
to the Ho3+ and Dy3+ based materials [7,8], but the perature, by which time such large energy barriers ex-
Ising anisotropy is reduced to much lower temperature ist that evolution towards the true ground state is never
4
achieved. In simulation terms, at low temperature the
Boltzmannweightsforlocalspinflips‘towards’thisstate
are simply too small.
In conclusion, using Ewald summation techniques we
haveconsideredtheeffectsoflongrangedipole-dipolein-
teractions on the magnetic behavior of Ising pyrochlore
systems. Our results show that spin ice behavior is re-
covered over a large region of parameter space, and we
find quantitative agreement between our approach and
experimental data on spin ice materials.
We thank S. Bramwell, B. Canals, and P. Holdsworth
for useful discussions. We are grateful to A. Ramirez
for making available his specific heat data. M.G. ac-
knowledges financial support from NSERC of Canada,
Research Corporation for a Research Innovation Award
and a Cottrell Scholar Award, and the Province of On-
tario for a Premier Research Excellence Award.
[1] M. J. Harris et al.,Phys. Rev.Lett. 79, 2554 (1997).
[2] A.P. Ramirez et al., Nature399, 333 (1999).
[3] L. Pauling, The Nature of the Chemical Bond (Cornell
UniversityPress, Ithaca, 1945), p.301.
[4] For the particular type of Ising pyrochlores discussed
here, the ice rules imply that every tetrahedron must
havetwo spins pointing intoit, and two pointing out.
[5] J. Villain, Z. Phys.B 33, 31 (1979).
[6] R.MoessnerandJ.T.Chalker,Phys.Rev.Lett.80,2929
(1998).
[7] J. S. Gardner et al. Phys.Rev.Lett. 82, 1012 (1999).
[8] M. J. P. Gingras et al., cond-mat/0001317, unpublished
(1999).
[9] M. J. Harris et al.,Phys. Rev.Lett. 81, 4496 (1998).
[10] R.Siddharthan et al.,Phys. Rev.Lett. 83, 1854 (1999).
[11] M. Born and K. Huang, Dynamical Theory of Crystal
Lattices(OxfordUniversityPress,London,1968),p.228.
[12] S. W. de Leeuw, J. W. Perram and E. R. Smith, Proc.
R.Soc. Lond. A 373, 27 (1980).
[13] This is due to thenature of the local Ising axes. For ex-
ample, there are 12 third nearest neighbors to a spin i.
Six have displacement vectors rij which are perpendic-
ular to their local Ising axis, giving two values for the
dipolar interaction at this neighbor distance.
[14] Physically,thiscorrespondstohavinganinfinitenumber
of replicas of the simulation cell, and using the Ewald
method to sum convergently the dipolar interaction be-
tween say spins i and j (within the simulation cell) and
spin j’s images (outside the simulation cell). The result,
is theeffective Ising interaction of spin i with spin j.
[15] S.T.BramwellandM.J.Harris,J.Phys.Condens.Mat-
ter10, L215-L220 (1998).
[16] J.N.Reimers,A.J.BerlinskyandA.-C.Shi,Phys.Rev.
B 43, 865 (1991).
[17] N.P. Raju et al. Phys. Rev.B 59, 14489 (1999).
[18] B.C. den Hertog et al., cond-mat/9912220, unpublished
(1999).
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