2008:054 CIV
E X A M E N S A R B E T E
A Monte Carlo study of
Ferrimagnetic Heisenberg models
Thomas Falk
Luleå tekniska universitet Civilingenjörsprogrammet
Teknisk fysik
Institutionen för Tillämpad fysik, maskin- och materialteknik Avdelningen för Fysik
Thisreportistoinvestigate,throughMonteCarlosimulations,theferri-
magneti behaviourof threedierent latti esonthe anisotropi Heisen-
bergmodel.
In the rstmodel, with a layered simple ubi stru ture, onditions
fortheexisten eof ompensationpointsareestablished.
The se ond and third models are set up to resemble the real sub-
stan es,with knownmaterial parameters, ofyttriumirongarnet (YIG)
andgadoliniumirongarnet(GdIG).Thesimulationsaimto he kthe or-
re tnessofthetwomodelsintermsofferrimagneti and riti albehaviour,
su hasthe ompensationpointTcompandthe riti altemperatureTc.
I Introdu tion 6
1 Earlier work in the eld 6
2 Ferrimagnetismand CompensationTemperatures 6
3 Te hnologi al Interestsin Ferrimagnets 7
3.1 Opti alisolators . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Magneto-Opti al(MO)re ording . . . . . . . . . . . . . . . . . . 8
II Basi s on Magnetism 9 4 Quantum theoryof Magneti Ordering 9 4.1 OriginoftheMagneti Moment. . . . . . . . . . . . . . . . . . . 9
4.2 TheEx hangeIntera tion . . . . . . . . . . . . . . . . . . . . . . 9
4.2.1 Antisymmetrizationofwavefun tions . . . . . . . . . . . 10
4.2.2 Two-ele tronwavefun tions. . . . . . . . . . . . . . . . . 11
4.2.3 TheHeisenbergHamiltonian . . . . . . . . . . . . . . . . 11
4.2.4 Superex hange . . . . . . . . . . . . . . . . . . . . . . . . 12
5 Magnetismasa Colle tionof Atoms 13 5.1 EquilibriumstatesandtheBoltzmanndistribution . . . . . . . . 13
5.2 TheIronGarnets . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2.1 YttriumIronGarnet(YIG) . . . . . . . . . . . . . . . . . 14
5.2.2 GadoliniumIronGarnet(GdIG) . . . . . . . . . . . . . . 15
5.3 Mole ular eld modeloftheGarnets . . . . . . . . . . . . . . . . 15
III Latti e Representation & Simulational Methods 17 6 Modelling 17 6.1 SpinmodelsofFerrimagnetism . . . . . . . . . . . . . . . . . . . 17
6.2 TheAnisotropi Heisenbergmodel . . . . . . . . . . . . . . . . . 18
6.2.1 Model1: TheLayeredSimpleCubi latti e . . . . . . . . 18
6.3 TheGarnetlatti e . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.3.1 Model2: YttriumIronGarnet . . . . . . . . . . . . . . . 21
6.3.2 Model3: GadoliniumIronGarnet . . . . . . . . . . . . . 22
7 EquilibriumThermalMC-Simulations 22 7.1 Cal ulatingObservables . . . . . . . . . . . . . . . . . . . . . . . 22
7.2 TheMasterEquation. . . . . . . . . . . . . . . . . . . . . . . . . 23
7.3 DetailedBalan e andA eptan eRates . . . . . . . . . . . . . . 23
7.4 Importan e Sampling. . . . . . . . . . . . . . . . . . . . . . . . . 24
7.5 MarkovChains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.6 TheSingleSpin-FlipMetropolisAlgorithm . . . . . . . . . . . . 25
7.7 Te hniques atlowtemperatures. . . . . . . . . . . . . . . . . . . 26
8 MonteCarlo simulations 28
8.1 Model 1: TheLayeredSimpleCubi stru ture. . . . . . . . . . . 28
8.2 Model 2: YttriumIron Garnet . . . . . . . . . . . . . . . . . . . 30
8.3 Model 3: GadoliniumIronGarnet . . . . . . . . . . . . . . . . . 31
V Con luding remarks 35
9 Con lusions 35
10 Futuredire tions 36
11 A knowledgements 38
2.1 Demonstrationof ompensationtemperature . . . . . . . . . . . 7
4.1 Symmetri wavefun tion . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Antisymmetri wavefun tion. . . . . . . . . . . . . . . . . . . . . 10
4.3 Superex hange . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.1 Heisenbergintera tion . . . . . . . . . . . . . . . . . . . . . . . . 17
6.2 Latti egeometries: Honey omb andtriangular . . . . . . . . . . 18
6.3 MixedspinHeisenbergmodel . . . . . . . . . . . . . . . . . . . . 19
6.4 Numberings hemefor imple ubi latti e. . . . . . . . . . . . . 20
6.5 Thegarnetunit ell . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7.1 TheRedu ed onemethod. . . . . . . . . . . . . . . . . . . . . . 27
8.1 32x32x32versus8x8x8 . . . . . . . . . . . . . . . . . . . . . . . . 29
8.2 JA/JB phaseplane . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.3 JAB/DA phaseplane . . . . . . . . . . . . . . . . . . . . . . . . 31
8.4 JAB/Tcomp phaseplane . . . . . . . . . . . . . . . . . . . . . . 32
8.5 YIGmagnetizationandspe i heat measurements. . . . . . . . 33
8.6 GdIGmagnetizationandspe i heatmeasurements . . . . . . . 33
8.7 YIGmagnetization urves . . . . . . . . . . . . . . . . . . . . . . 34
8.8 GdIGmagnetization urves . . . . . . . . . . . . . . . . . . . . . 34
Introdu tion
1 Earlier work in the eld
There hasbeenmu h eortonstudyingferrimagneti modelsre entlyandthe
rst model studied in this report is intended to go one step further from [1℄,
where alayeredmixed spin(12, 1)Ising model is studied. Ising models donot
realisti ally represent the ve tor spinsin real materials and this is why some
hangesaremadeas anattempt toimprovethemodel.
As for the garnets, extensive resear h has been arried out in the experi-
mental area. Themostinterestingpartseemstobethemagneto-opti al(MO)
ee tsofthevariousrare-earth(RE)andrare-earth-transition-metal(RE-TM)
garnetsandtheirpotentialappli ationsasmediumsinthin- lmmanufa turing.
Other kindsof magneti properties arestudied as well, usuallywith means of
Raman-,infrared-andphotolumines en espe tros opy,X-raydira tion,et .
IntheareaofMonteCarlosimulations,theinterestingarnetsis onsiderably
less. TherearesomeworkontheHeisenbergmodelonthegarnetlatti e,though,
su has in[2℄. Therearealsoa oupleofpapersontheIsingmodelwithgarnet
stru ture,[3℄and[4℄,butthese arefo usedondierentthings.
2 Ferrimagnetism and the on ept of a Compen-
sation Temperature
Aferrimagneti substan eisamaterialthat onsistsofmorethanonesublatti e
of ions with unequal net magneti moments, pointing in dierent dire tions
in spa e. In the simplest ase it is two anti-parallell sublatti es where the
numberofionsononeofthelatti esex eedstheother. Ferrimagnetsdierfrom
antiferromagnetsinthattheirtwoopposingsublatti esdonotexa tly an eland
thereforetheyexhibitaresultantmagnetization. Contrarytoferromagnetstheir
magnetization arises due to non-parallell alignment of their atomi magneti
moments.
Inthe groundstate aferrimagneti substan e hasa spontaneousmagneti-
zationdue to unequal an ellationof spins,as dis ussedabove. Thismagneti-
zation willde rease in magnitude within reasing temperaturedue to thermal
motion. Whenthe temperaturerea hesthe riti alCurie temperatureTc the
magnetizationdropsoveryrapidlytozero.
Sin etheopposingsublatti esinthematerialare,ingeneral,dierentfun -
tions of temperature there may be some point below the Curie temperature
wheretheyexa tly an el. Thisisknownas the ompensationpointTcomp and
hasimportantte hnologi alappli ations.
The two most ommon types of materials that exhibit ferrimagneti be-
haviourarethespinels andthegarnets. Thesegroupshaveafairly ompli ated
rystal stru ture with two and three magneti sublatti es, respe tively. The
ions in the spinel stru ture order into tetrahedraland o tahedral sites, where
the garnetshaveone further sublatti e in dode ahedral sites. The most om-
mon ompositionsforthespinelsandgarnetsarewhenironandoxygeno upy
thetetrahedralando tahedralsites,e.g. asin magnetite(Fe3O4)andtheiron
garnets.
Far from allthespinels and garnetshavea ompensationtemperature. Of
those thata tually haveone,themostnotablearesomeof therare-earthiron
garnets 1
(RIG). Two of them are of parti ular interest and will be studied
in some detail, the yttrium iron garnet (YIG) and Gadolinium Iron Garnet
(GdIG).
0 0.5 1 1.5 2 2.5 3 3.5 4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Temperature
|Magnetization|
T comp M
M
A
B
Figure2.1:Thisishowthemagnetization urveswouldlooklikeifa ompensationtemper-
aturewaspresent. Sin ethe absolutevalue oftheBlatti emagnetizationisplottedrather than its realvalue the ompensationpointwillappearinthe interse tion between the two
graphs.
3 Te hnologi al Interests in Ferrimagnets
Yttriumandgadoliniumirongarnetshavenumerouste hnologi alappli ations,
su has in MO omponents. Thereare threepropertiesthat amaterialhasto
owetobesuitablefortheseappli ations;ithastoshowlowmagneti losses,low
opti al losses(absorbtion)and highMO ee t. TheRIGs (in ludingyttrium)
havethis.
3.1 Opti al isolators
Thepurposeofanopti alisolatoristo makesurethatlightonlygoesthrough
inonedire tion. Thisisessentialine.g. opti al breswherethelight anbere-
e tedba kandforthanddisturbthesignal. The ommon hoi eoflightsour e
used in ommuni ationsis 1.30µm and 1.55µm. Sin e YIG is transparentto wavelengthslongerthan1.10µmtherewill benoabsorbtionanditistherefore
suitableasanisolatormaterial 2
.
1
Wherethedode ahedralsitesinthegarnetsareo upiedbyarare-earthion.
2
Itisunusual,however,thatpuresubstan es,su hasYIG,areusedforanyappli ations.
Mu hbetterproperties anbea hievedifthe materialisdopedwithanotherelement. For
example, inthe YIG ase, the MOee t an beimprovedifdopedwithbismuthand the
temperaturedependen eofthesameee tisloweredifdopedwithterbium.
polarisedwiththeuseofapolariser,ledthroughtheYIG rystalandthrough
anotherpolariser(analyzer)whi h istilted45◦. Bygivingthe rystala ertain
length, the Faraday ee t 3
will rotate the angle of polarisation exa tly 45◦.
When entering the bre, some of the light will bere e ted ba kthrough the
rystalandthereforerotatedanother45◦. Sin ethelightnowisperpendi ular to the original polarisation it will be blo ked and unable to disturb the laser
signal.
3.2 Magneto-Opti al (MO) re ording
Another use for theRIGs (and YIG) is as mediums in MO drivesbe ause of
their out-of-plane anisotropy and high MO ee ts. The prin iple works like
this. Thedata is storedin the(memory) material with theuseof alaser. To
re ordsomethingon thematerial thelaserheats aspot and an,with theaid
ofamagnet, ipbetweenupanddownstates(where,say up orrespondstoa
re orded mark). When theinformation is erased, the spot is heated to above
thematerialsCuriepointwhere nomagnetismexists.
Ifamaterial hasa ompensation point, however,it isadvantageousto use
this temperaturetogo betweentheup,downandzerostates. Sin e theGdIG
hasa ompensation pointjust belowroomtemperatureit haspotentialusein
thisarea. Thereasonwhythe ompensationpointisadvantageoustotheCurie
temperatureis that theinformation stored hereis morestable due to the low
temperature. Thereisalsoa onsiderablylessamountofenergythathastobe
suppliedtothematerialin ordertoraisethetemperature,whi hwilllowerthe
needoflaserpoweraswell.
Wheninformation is read one usesthe samelaser to shine on thesurfa e.
Some ofthelightisre e ted from themagneti surfa ewitharotationof po-
larizationdue to the Kerr ee t (whi h is thesameas the Faradayee t but
nowreferto there e tedlight insteadofthetransmitted). Depending onthe
dire tionofrotationitispossibletode idethedire tion ofmagnetization,and
onsequentlywhi hareasthatarere ordedandwhi harenot.
TheMO re ordingis prefered to onventional re ording due to the longer
durabilityofthedis s. Sin emagnetizationreversaldoesnotinvolveanymove-
mentofatomsthedis swillnotdegradewiththenumberofreadingandwriting
pro esses. One analso a hieveahigherresolution. More detailsonthis wide
area anbefoundin [6℄.
3
Thelightwillintera twiththematerialsmagnetizationand anexternalmagneti eld,
whi hwill ausethepolarisationtorotate. Theangleofrotationwilldependonthemagneti
eld strengthB,the length of dof the rystal and the so alledVerdet onstant ν. This
onstantisveryhighfordopedYIGsubstan es. MoreonMOee ts anbefoundine.g. [5℄.
Basi s on Magnetism
Thispartwillgivesomeoverviewofthephenomenonofmagnetism. Itisnotin-
tendedtogivea ompletedes riptionofmagnetism,asitwouldbeverylengthy,
but rather somebasi understanding. Many books havebeen written on this
hugeareaofphysi s,su has [7℄or themorere ent[8℄.
Thereare twowaysof explainingmagnetism; Onthemi ros opi s ale us-
ingquantum me hani s or onthema ros opi s alewithstatisti al me hani s.
Thesetwoapproa heswillbe overedinsomedetailbutitwillbebene ialto
havesomeknowledgeonthesetwoareas.
4 Quantum theory of Magneti Ordering
Thisse tionwillgiveabriefba kgroundtothephenomenonofmagnetismfrom
a mi ros opi point of view. It is assumed that the reader has some basi
knowledge about quantum me hani s, su h as the S hrödinger equation, the
Pauli ex lusionprin iple, et . Seee.g. [9℄or [10℄.
4.1 Origin of the Magneti Moment
Atomi magneti momentsarise due toele tronsorbitingthenu leus (ele tri
urrents) and the ele tronsown built-in property, alled spin. The spingives
thebiggest ontributiontothemagneti momentofthetwoandisthereforethe
mostimportantsour e,at leastwhendis ussing magnetism.
Whentwo (or several) atoms are su iently lose to ea h other, like in a
solid,their orrespondingwavefun tionswilloverlapandamagneti statemight
be reated. Theele tronsmagneti momentswill eitheralignor anti-alignand
there is a ertain property of the wave fun tions that is responsible for this.
This isbrie ywhat magnetismis about;A ooperativebehaviourof magneti
moments 4
.
Ifa solidis magneti or not depends on various things ( rystaltype, ele -
troni stru ture,interatomi separation,et .) anditisimpossibletoinvestigate
it in anydetailex ept for somesimple ases. There are howeversomesimple
prin iples tostartwith,andtheyareoutlinedin somedetailbelow.
4.2 The Ex hange Intera tion
An atomi magneti moment in a solid that is ae ted by the dipole eld of
aneighbouringatom isfar from enoughto ause any magneti ordering. The
largestmagneti intera tionand theoriginofthemagneti momentalignment
is something alled ex hange intera tion. It originates from the fa t that the
ele tronwavefun tionshavetobeantisymmetri withrespe ttoparti leinter-
hange. It isessentialformagnetismandisthereforereviewedbelow.
4
Thespinandorbitalmomentsalso oupleinternally,whi his alledspin-orbit oupling.
Thisisalsoimportantformagnetismbuthasnothingtodowithitintermsofalignedmagneti
moments.
Suppose that there is a system of two identi al ele trons with position oor-
dinates r1 and r2 and spin oordinates s1 and s2. The positions are mea-
sured in spheri al polar oordinates and the spinis along the z axis, so that r1= r2= (r, θ, φ) and s1= s2= (sz). There are four plausible forms of the
wavefun tionbut onlyone ofthemisphysi allyvalid, namely
Ψ(a, b) = 1
√2[ψα(a)ψβ(b) − ψα(b)ψβ(a)] . (4.1)
Here ψα(a), ψα(b), ψβ(a) and ψβ(b)are thewavefun tions of ele tron1 with
energy eigenvalue Eα, ele tron 2 with energy eigenvalue Eα, ele tron 1 with
energyeigenvalueEβ, ele tron2 with energyeigenvalueEβ, respe tively. The
(a, b)isashorthandfor(r1, s1; r2, s2)and1/√
2 isjust theretogetthe orre t
normalization.
Eqn(4.1)istheonlyformtoobeythePauliex lusionprin iple. Ifthewave
fun tionwouldhavebeenneithersymmetri norantisymmetri ,theparti ledis-
tribution|ψ|2wouldbealteredduringinter hangeoftheparti les. Asymmetri wavefun tion,however,wouldsuggestthattheindividualwavefun tionsofthe
parti lesare identi al,whi h isnotaloud. Therefore,thewavefun tion hasto
beantisymmetri .
Figure4.1: Symmetri wavefun tion. Sour e: [7℄
Figure4.2: Antisymmetri wavefun tion.Sour e: [7℄
Theantisymmetrization of wavefun tions is easily extended to N parti le
systemsbytheSlaterdeterminant
