UtiImpPac L. T1DepEngiPhys*Cor AbstCOMimpleequatCOMHowrequia muproceconsuimpleApplproviuser-demoApplimpleand LifshMaxwthis strucnano KeywnanoLifsh 1. In

Nrecenbecaupotene.g. matepropesmallmagntechnCarloestabstruc

ilization oplementackage witeich*1, A. Hüpartment of Eineering, Unisics, Thin Filrresponding au

tract: One MSOL Multipement user-tions (PDE)

MSOL’s predwever, using th

ires manual imultitude of probess is not just uming. As aementation onlication Progides an easy -defined simuonstrate the lication Progementation osimulation pa

hitz-Gilbert well’s equatioimplementatiotures with dimmeters can be

words: Microstructures, ma

hitz-Gilbert eq

ntroduction

Nanostructuredntly been focuuse of theirntial applicati

data storagrials. In ordeerties of nano-l number onetic molecuniques like qo and spin dynblished. Movtures like ma

of COMSation of ath a Custütten2, C. SchEngineering Siversity of Aplms and Nanouthor: Wilhelm

of the big physics is th-defined par

which can defined app

he tool’s standamplementationblem-specific error-prone b

an alternative ne can use Cgramming Iand efficient

mulation packusage of C

gramming Inf a micromaackage. Thereequation is

on for the magon the magnemensions in the analyzed.

magnetics, Javagnetic nanop

quation.

d magnetic used by manyr particular on in many i

ge, biomedicer to calcula-sized structurof magnetic ules, spin-discquantum and namics simulaving on to agnetic nanopa

SOL Mua Micromtomized Uhröder1 Sciences andpplied Scienostructures, Bm-Bertelsmann

advantages he possibilityrtial differen

be coupled lication modard user interf

n of the PDEs parameters. T

but also very tto this man

COMSOL’s JInterface wh

way to creatkage. Here, COMSOL’s Jnterface by agnetic modelein, the Lands coupled gnetic field. Wetic propertiese order of sev

va API, magnparticles, Land

materials hy research gro

properties interesting fiecine and smate the magnres hosting on

moments crete simulatclassical Mo

ations [1] are wlarger magnarticles or mu

ltiphysicmagnetic M

User Inte

d Mathematicces BielefeldBielefeld Unn-Str. 10, DE-

of y to ntial

to des. face and

This time nual Java hich te a we

Java our ling dau-

to With s of eral

netic dau-

have oups and

elds, mart netic nly a like tion onte well netic ulti-

lagbedmsso

2

mdecZdcs

2

Tmraaathbmothcmdoaadd

s’ JAVAModelingerface

cs, Computatd, Bielefeld, Giversity, Biel

-33602 Bielefe

ayer systems, growing dramabecomes larequations of mdifferential emagnetic mompace and timeimulation tech

of Micromagne

2. Micromag

The orientmoment withindetermined byenergy with contributions, Zeeman energdemagnetizatiocontributions ection.

2.1 Energy co

The exchangemagnetic momelative to each

adjacent magnanother, wherean antiparallelhe Zeeman en

between magmagnetic fieldobtained, whenhe external

contribution, magnetic momdirections in caorient perpendan easy-plane above describdemagnetizatiodetermined by

A API for g and Sim

ional MateriaGermany, 2Dlefeld, Germeld, lisa.teich@

the number oatically. If thege enough

motion can beequations whment as a cone. These contihniques are knetics.

gnetic Model

tation of the n a micromay minimizing

respect to such as the e

gy, the anisotron energy. are explaine

ntributions

e energy causments align h other. In ferrnetic momentseas in antiferrl orientation inergy accountgnetic momds. Minimal n magnetic mo

magnetic the anisotro

ments to align wase of an easydicular to that

anisotropy. bed energy on energy or y long-range

r the mulation

als Science &Department o

many @fh-bielefeld.

of atomic mome density of m

the spin-de replaced by hich describntinuous funcinuous modelnown under th

ling

continuous mgnetic descrip

g the systemcompeting

exchange enerropy energy, These d in the fo

ses that neighin a preferreromagnetic ma align parallelromagnetic mis preferred. Sts for the inte

ments and eZeeman ene

oments lie parfield. The

opy energy, with certain pr-axis anisotropt direction in . In contrast

contributionstray-field eninteractions.

& f

de

ments is moments

discrete partial

be the ction of ing and

he name

magnetic ption is

ms total energy

rgy, the and the energy

llowing

hboring ed way aterials, l to one

materials Second, eraction external ergy is rallel to

third causes

referred py or to case of to the

ns, the nergy is

. This

meancertadepenotherdependemamagn 2.2 M All eadd usystewell-depen(LLGpreceaccor

The ethe cthe anZeemto Mdemacan whicCOM 3. COM

Inmodesimumodethe mand weakapplidema 3.1 W In ormicrobe mthey problis us

ns that the dain point of nds on the enr energy contrnd on theiagnetization enetic domains.

Micromagneti

energy contribup to the totalem. Minimizin-known static ndent Land

G) Eq. (1) essional motrding to

2 ∆

effective fieldcontributions fnisotropy, the

man energy. ThMaxwell’s eqagnetization fibe transformh allows one

MSOL Multiph

MicromagMSOL Mult

n COMSOL es are needed

ulations. In the “PDE, weakmicromagnetica correspond

k formulation. ication modeagnetizing or s

Weak Form M

rder to obtainomagnetic equ

multiplied by ahave to be

lem domain. Sed to integrat

demagnetizatif the micromntire configuraributions explair local envenergy causes .

ic equations

butions mentil energy of th

ng this total enBrown equati

dau-Lifshitz-G[2]. The LL

tion of mag

2 〈

d Heff given byfrom the exch

e demagnetizathe coupling of

quations is rield Hdem This

med into a wto implement

hysics.

gnetic Motiphysics

Multiphysics,d to accomplishe first placek form” is usc equations, nding norm co

Second, the “e is used tstray field.

Modeling

n the weak fouations, the ea set of test fue integrated Subsequently,te by parts. A

ion energy amagnetic sysation whereas ained above ovironment. Tthe formation

oned in Sec. he micromagnnergy leads to on or to the tim

Gilbert equatLG describes gnetic mome

, 〉

y Eq. (2) contahange interactition field, andf the LLG Eq.realized via s set of equati

weak formulatt them directly

odeling w

, two applicatsh micromagn, the applicatsed to implemnamely the Londition in th“Magnetic fielo calculate

ormulation of equations haveunctions . Tover the en Green’s form

After implying

at a tem the

only The n of

2.1 netic

the me-tion the

ents

(1)

(2)

ains ion,

d the . (1) the

ions tion y in

with

tion netic tion

ment LLG heir lds” the

the e to

Then ntire mula

the

NfoTbthpimTuthth

3

AthmthapucthfibmloficthmswInmC

Neumann bouformulation ofThis formulatboundary condhe weak form

problem [3].mplemented

Therefore, theused. The test he test functiohe test functio

〈 ,2μ

μ

| | 13.2 Stray Field

As mentioned he magnetic

magnetic fieldhe stray field

approaches capredefined appused and forcomputed usinhe context of

field has to bebecause of thmagnetic momong-range nat

field is very tcalculation of he LLG Eqs.

methods for atray field are

we use the man Fig. 1 the

magnetized cuCOMSOL Mu

undary condf the original ption, togetherdition and a no

mulation of the. This for

in COMS so-called tesoperator may ons used abovons for the vari

〉

〈, ,

2μ 〈 ,

〈 ,

d Computatio

in Sec. 2.2, thstray field

d Heff Eq. (2). d is very elaan be used. plication moder that reasonng the magnetf Maxwell’s ee computed fohe bidirection

ment and the sture, the comptime consumithe other ene

. (1) and (2)a more efficie

commonly usagnetic vector

stray field oube as a result ltiphysics is sh

dition, a varproblem is ac

r with the Dorm condition

e micromagnetrmulation caSOL Multipst operator hanot be mistak

ve. It rather geiable it operat

,

〉〈 , 〉

〉

on

he LLG is couvia the e

The computaaborate and d In our ca

e “Magnetic fin the stray ftic vector poteequations. Thr every iterati

nal coupling stray field. Duputation of thing comparedergy contribut). Hence, altent calculation

sed. [4]. In thir potential apof a homogenof a simulatio

hown.

riational chieved. Dirichlet n, yields tic PDE an be physics. as to be ken with enerates es on.

〉

(3)

(4)

(5)

upled to ffective ation of different se, the ields” is field is ential in he stray ion step of the

ue to its he stray d to the tions of ernative n of the is work, pproach. neously on with

Figurcube

3.3 Cstray The smagnappli(7).

For used magnformThis conficompconfimom 3.4 S The sto beachieconficonficonfitime-handdepenthe sdepenreasoconddepen

re 1: Stray fieldcomputed with

Coupling of y field and the

stray field Hdenetic field H iication mode “

a bidirectionby COMSO

netic field, ham of the micr

way, the iguration of puted which iguration and

ment.

Simulation Pr

simulation of e performed ieve convergeiguration has iguration is tiguration, the -dependent si

dle, every simundent steps, estationary standent steps se

onable initiaditioning, standent simulati

d of a homogenCOMSOL

Maxwell’s ee micromagn

em in Eq. (2) cin the equation“Magnetic fiel

nal coupling, L for the coave to be buiromagnetic eq

stray fieldthe magne

in turn chad results in a

rocess

the micromagin a certain w

ence. First, to be generat

too far away problem cannmulations areulation starts

even if one is oate of the syerve as pre-col values. A

ationary as ions can be do

neously magnet

equations tonetic equation

corresponds tons of COMSOlds”, Eqs. (6)

the expressi

mponents of ilt into the wquations as wd for a gietic moment anges the aca new magn

gnetic system way, in ordera proper inited. If the inifrom a reali

not be solved.e much easierwith a few timonly interestedstem. The tim

onditioning to After this p

well as timone.

tized

the ns

the OL’s

and

(6) (7) ions the

weak well. iven

is tual

netic

has r to itial itial istic As r to me-d in me-get

pre-me-

4

mnCapopsInc

4

Inmpms

DoMmUmmracdmacfip

Fn

4. Validation

In order tomicromagneticnecessary to COMSOL to sanalytical resupackages. Theof NIST offerproblems. Fimulation resun our case,

chosen as a ref

4.1 µMag Stan

n µMag Stmagnetic nanoparameters imagnetic statetructure:

1. Flowemagn

2. Vortexstructu

Depending on one of these staMagnetic dommagnetization Usually, a magmagnetic dommaterial in domeduced. At

anisotropy enecases, splittingdomains resultmagnetic systeadditional stracontributions cfield energy.particles are fo

Figure 2: Posnanoelement, µM

n of the Simu

o check the imc equations i

compare rsome kind of ults or results e Micromagners a set of stor these ults can be reµMag Stand

ference.

ndard Proble

andard Proboelement witis considerees are expect

er state: etization strucx state: a c

ture of the magthe edge leng

ates forms themains are are

within mgnetic sampleains. By splittmains, the strathe same ti

ergy have to beg up the mats in a lower

em. In relativeay field andcannot be com. Consequen

ormed.

ssible magnetiMag Standard P

ulation Mod

mplementationin COMSOL

results achievreference resufrom other s

etic Modelingtandardized eproblems, deferenced onldard problem

em #3

lem #3, a th arbitrary med. Two dted for this k

a homogcture, curl- or vorgnetic momength of the nane stationary staeas of homogmagnetic mae consists of mting up the m

ay field energyime exchange considered.

agnetic materir total energyely small systed anisotropy

mpensated by thntly, single-

ic states of Problem #3 [5]

del

n of the L, it is ved in ults, i.e. oftware

g Group example detailed ine [5].

m #3 is

cuboid material different kind of

geneous

rtex-like nt. no cube, ate. geneous aterials. multiple

magnetic y can be ge- and In most ial into

y of the ems, the

energy he stray -domain

a cubic

4.2 V To microfollowaccor

The eof 30betw8 ∙lengtwhervorteFor tmodeadditsurrocalcucould 4.3 R Belowhomomomobserof30pre-cbeen resulof ththe mthe dseleccompmagnbeen magn

Whenfollowfounddirecmagn

Validation Mo

validate thomagnetic eqwing materialrding to [6]: 6.28 1.7 10 6.2 5.3 edge length of0 to 60 nano

ween the two m43 ∙ 10

th, the flowereas above thex state shouldthe simulationel the microtion to that ounding volumulation of the d use infinite e

Results - Flow

w the criticaogeneous con

ment, the so-rved, as show0 . After a conditioning,

performed. Flt. The surfacee magnetizatio

magnetic momdirection of

cted positioponents of thenetic volume,

determined netization. , , ,n compared towing discrepad. The averag

ction is very snetization. Mo

odel

he implemenquations in l parameters h

831583 ³,773528 ∙ 10

, 2831583 ∙ 10128 ∙ 10 ,

30 ∙ 10f the cube is vmeters, becaumagnetic state

. Below ther state shouhe critical e

d be obtained. n it is not oomagnetic syt one needsme which is stray field. A

elements..

wer State

al edge lengnfiguration o-called flowe

wn in Fig. 3 foa few time-dep

a stationary Fig. (3) depice plot shows on M, which

ment m. The athe magnetic

ons. By ie magnetizatio, the followi

in units of

0.001 0.001 0.985

o the official µancies and simge magnetizatsmall as compost of the µMa

ntation of COMSOL,

have been cho

,

,

, , …60 ∙ 10

aried in the rause the transites is expectedhis critical e

uld be observedge length, nly necessaryystem itself. s to specifyessential for

Alternatively,

gth of 8 ∙of the magner state can or an edge lenpendent steps

simulation cts the stationthe x-componis proportionaarrow plot shoc moment mintegrating

on over the whing values hf the saturat

µMag results, milarities cantion in x- andpared to the tag results do n

the the

osen

. ange tion d at edge ved, the

y to In

y a the one

a netic

be ngth

for has

nary nent al to ows

m in the

hole have tion

the n be d y-total not

FPCpfl

inwoWthadsorlec

4

Faewzddnrosmbp

4

iminin

Figure 3: ResulProblem #3 wCOMSOL. The parallel to the zflower state.

nclude these vwould expect of the magnetiWhen comparhe z-direction

average value odifference lies ubmitted µM

obtained orienesults for an eength can

conclusive.

4.4 Results – V

For an edge lena vortex-like stexpected. Fig. with an edge lz-components due to the vodifferences bnoticed betweesults. Again,

of all results suhows that

micromagneticbe used for siproblems.

4. JMicroma

For the mplemented mn COMSOL Mnterface “JMic

lt of the simulawith an edge

magnetic momz-axis of the c

values. Howefor a homogeization and thring the averan a differenceof all µMag rewithin the sta

Mag results. ntation of theedge length bebe considere

Vortex State

ngth above thetructure of the4 shows the r

length of 60of the magn

ortex structureetween 1.56een our resu this lies withiubmitted to M

the implemc equations is imulating arbi

ag

efficient usamicromagnetiMultiphysics, cromag” has b

ation of µMag Slength of 30

ment has an oricube, which is

ver, this is weneous config

he magnetic mage magnetizae of 1.34%esults is noticeatistical error

Together we magnetizatielow the criticed reasonabl

e critical edgee magnetic moresult for a sim

. Each, thenetization cane. In the y-di6% and 3.5ults and the in the statistic

Mag. So, the vamentation ocorrect and c

itrary microm

age of our ic modeling f

the customizbeen develope

Standard

nm in ientation called a

what one guration moment. ation in % to the

ed. This over all ith the on, the

cal edge le and

e length, oment is mulation e x- and ncel out irection, 59%are

µMag cal error lidation of the can now magnetic

newly features

zed user ed,

Figurlengthlengthobser

takinApplthis wPDEavoidmode 4.1 COn standsimucorreeveryor exCOMmodeautom 4.2 JIn “JMiwell and simufollownamegeomsimptypesmainFig. 5automparamis chthe u“COMeverysumm

re 4: Simulatioh of 60 nm, wh. As expectedrved.

ng advantage olication Prograway, the manus and problemded which saveling process.

COMSOL’s Jthe basis

dalone applicaulation file esponding Moy Model Javaxternally, can

MSOL. Conseeling procesmatized.

JMicromag Uour micromicromag” all as the correhidden from

ulation the uwing steps. Fie and the sto

metry has to bely chooses ons, e.g. a spheren dimensions 5). All necessmatically. Ameters have toecked formalluser can specMSOL Settingy step the usemary window

nal result for a which is aboved, a vortex-like

of COMSOL’samming Interfual implementa

m-specific paraves time and si

Java API of COMSO

ations can be dcan be e

odel Java-file a-file, generaten directly be equently sevess can be

User Interface magnetic mod

manual parasponding PDE

m the user. user has to irst, the user dorage locatione defined. In dne of the prede or a cube, anof the chose

ary input fieldAfterwards, o be provided.ly by “JMicrocify simulatiogs” page. In aer can check on the “Projec

cube with an e

e the critical ee structure can

s Java face (API). In ation of the

ameters is implifies the

L’s Java Adeveloped. Evexported to

and vice veed by COMSimplemented

eral steps in simplified

deling interfameter inputsEs are packa

To performgo through

defines the projn. After this, doing so, the udefined geomend then fills inen structure (ds are highligh

the mate Every user in

omag”. Moreovon details on addition to that

the input in ct Properties”

edge edge n be

API, very

a ersa, SOL d in

the or

face s as aged m a

the oject

the user etry

n the (see hted erial nput ver, the

t, at the

Find

pthsfiuthTmththsds

5

pocCimwmd“maCaaeththmuhwh

Figure 5: Geomnterface, geome

defined.

page. All setthere. After pimulation file

file can easily user. The Modhe geometry, p

The simulationmouse-click. Fhe possibility he micromagoftware packa

data that is uimulation file

5. Conclusion

We have crpackage by imof the Landaucoupling to COMSOL mplementation

with other simmicromagneticdeveloped th“JMicromag” micromagneticand user friendCOMSOL Muand evaluate arbitrary geomexisting implehis basic formhe implemen

magnetic anisuniaxial anisohigher order awithin the exhand, the use

metry settings petry type and m

tings and valuproviding all es are generatebe imported i

del Java-file coparameters ann can then be Furthermore, w

to generate thgnetic modelage Nmag [7],used to genes.

ns and Outlo

reated a micromplementing thu-Lifshitz-Gilb

Maxwell’s Multiphysics

n, we obtainemulation resultsc problem. he specializwhich allows

c simulations dly way. With ultiphysics can

micromagnmetries and maementation difmalism can b

ntation of difsotropy. At otropies are

anisotropies caxisting framewer interface “

age of JMicrommain dimension

ues are collenecessary da

ed. The Modento COMSOLontains the eqund additional s

started with jwe have implehe simulation fing and sim, with the samerate the CO

ook

omagnetic simhe weak formbert equation

equations s. Using ed a good agrs based on a s

Furthermoreed user in the user to pin a very e

these enhancn be used to snetic problematerials. Basedfferent extensbe implementefferent types

the momentimplemente

an easily be iwork. On the

“JMicromag”

mag user s can be

ected in ata, the el Java-L by the uations, settings. just one emented files for

mulation me input OMSOL

mulation mulation

and its within

our reement standard e, we nterface perform efficient ements,

simulate ms for d on the sions to ed, e.g. of the

t, only ed, but inserted e other can be

enhanbe tris poCurrefiles COMfunctso thwithiMult 7. AThis “Nan 7. R 1. LcompMoleWinpSinga2. WNew 3. AparticProcCom4. H.for m(20085. TechComMicrhttp://6. ZMagnMicr(20047. T.to MicrMagn

nced. For exaansferred to thossible to raently “JMicrothat have to

MSOL Multiphtionality of “Jhat import anin “JMicromatiphysics.

cknowledgework was sup

nomagnets”).

eferences

L. Engelhardputationally coecular Clustepenny). Woapore (2011)

W. F. Brown,York (1963)

. Weddemanncles by

ceedings of themsol, Milan (20

. Szambolics micromagneti8) National In

hnology, Cenmputational Mromagnetic M/www.ctcms.ni. Zhao, Modnetic Materromagnetics, 4) Fischbacher Multiphysics

romagnetic Simn., 43, 6, 2896

ample changeshe user interfa

aise the levelomag” generato be importehysics. It is imJMicromag” cnd simulationag” and not w

ment pported by th

dt, C. Schröomplex magneer Magnets orld Scienti

, Jr., Microm

n et al., Detecmagnetoresi

e 2009 COMS009) et al., Finite eism, COMPE

nstitute of nter for T

Materials SciModelling Ast.gov/~rdm/mu

deling Shape rials with University o

et al., A Syst of Finitmulations: Nm6-2898 (2007)

s of the PDEs face. Moreovel of automattes the simulatd manually i

maginable that can be extendn can be starwithin COMS

he DFG (FOR

öder, Simulatetic molecules(Ed: R. E.

ific Publish

magnetics. Wi

ction of magnistive sens

MSOL Conferen

element approEL, 27, 266-

Standards Theoretical ience, µMagActivity Groumag.org.htmlEffects in N

Web Baof New Orle

tematic Approte-Element-Bamag, IEEE Tra

can er, it ion. tion into the

ded, rted

SOL

R945

ting s, in

P. hers,

iley,

netic ors, nce.

oach 276

and and

g – oup.

Nano ased eans

oach ased ans.

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice