Table Of ContentDevice Implications of the Theory
of Cylindrical Magnetic Domaix
By A, A THIELE,
‘anne evind Osos , 170)
“This paper ane the theory of eatndical magnetic domains! to
aliudreat dota devios, The etbiity conditions are ezxminal at
Dun to tha reyinn of poriba device operation and i i found Chet
the plate thickness 4 ~ sh end the oma diameter, = Al, where 1
fs the ratio of the wall eur per wat area to fines the eaturation
Inspnetcation suc ae preferred value, Whew te ofets of wall
eres end mutsly ave szcnsned, it fou! tha the peefrred
nd dunain dereter ae eve more strongly preferred,
Pad be ees Ton ane pecent of he
tinh the raturation maguetieutio, end Oat a damaceeoere iby atat
ny be defn Conieratin a the tet temparoson ad
tie abit
onditions shee ft demein warts having von antifeeramnyastic
Phometer and induced wniacit wsitouopy ave preferred. These ap
priate, oman wathote for smeneiring wnlriad gacametere are
deserted,
‘The eppliation of erlncreal maymctie domains or “bubbles” Wo
rmeniory nl loge doviees hae reewly ceerive consdessble atten
tien Such domain devine may aperste is a eontinuuzn of modes
ening fom the wall motion coeeivaty dominator to tbe
brebble" roe. [ysl caonivity dominates moe, atid fells deler-
tine the den configuration bith chen maine
In toe Fac bible mode the caer ey rust i sullen!
cle domains have wea eae el se ering the mow
grin id Rope Mabe alee hs Tonos, ae Noe 3a,
akser a)
726 umn AYSMRM-ernyEan oUTAD, ARE 6h
nn of individ demins eo dive enti, A previous paper
Aeveleped the theory of atic stahiity af eylad:ieal domaine in
atecile having ze coercivity: The ree work applies shi cory
to the determination of the pietrred conditions for corarseon af
Aevieesnperating ithe ard hail we Ts Zour that posting
fir erating donsir Enmeter Zetonsines prefreee yelues tor the
Plate thence, nagnetisation, enzctropy ronetent end the maxim
Conuble vai of ve wall lion exe
‘igure 1 own fhe el Ter tie donut straetate frees which the
static stabiliy theory wus developed. "Tae cooedinsta syrton ene
tymbols used ere axe the came asia Ret 1 except forthe adsiten
n'a fow sya such ns pp the well mobility; H,, the wall motion
ooercivity: Yo the domain velosity; abd amy fhe varistion in wall
‘ong. Thee tuaie) reprvents « single ieoluted domain in a plate of
muwgretc material of uniform theese, and. an inne exten, in
ho plang 7) = =. Bverywhore within the materia! the tagnenation
Tous uniform fontunuion) maugrtid, AP crested slong che upmend
ols nora ite = drcetion} within the domain and ala the casas
vend plese norma coher within tae matereh ‘The dancin wll ie
ews wo have aeglgble wlth snd the cerain wal energy ens,
ois iitielly cases to be indepenlent of ln vel enenthtion aad
(erveture. Tae doncain wall Fslvcieal in “he sen thab it every
‘whore eaaine a Tine parla t9 che plate normal. Urler ese nes
ig. 1—Dowesi enigersny an scan tt,
tions omy Ue onimnent of che apaially uniform applied fel whieh
lies along the plate normal wl itasuet wth the domain so chet only
this emmporeat is eons‘evel, This eamporent is denoted by IF en i
taken vovlive when cltsted wpa, the divert lending to ella
the doriin,
he nesnptions plist fm thie <ovuein model are mob all inde
persrs The nfeelaion of the essursrignt ie the dependence of
fhe essnmpticas exe comeia geonstry and the material paramere
fr disso AU2-3918 9f et 1, The valdty of che essary
tions ll "ot be diseased further Eero exeep towel that in exder
at domi of the ye lo be cotsiered hers om exis, demain zuslas~
tion and sal-wideh solrsons weuine roughly thet
1 Del o
re Ky isthe sania anisotropy ransinnl nd tha the valiiey of
Tie appmosinasions generally imnoves Ke ie neta shove this
‘The domain eudiasfuneton, (8) whol ie expanded inthe erie
=a
Arseribee tke Gosisit eae iv lve pono. Tes ary and 38, desribe
‘erties. im sdomain sae al stage feo «cian owas of redios
sade = ny and she 5 deseice she eet evar. (The 48,
Trav cign'teance only for seca variations.) Sines only near circular
flomaine are of Sstaevs hore Fe wand 0
PAS) m rab ryt SE Ay oe ald =
bl> lanl + Ea ioe ®
valued and amont
‘The first-aid sesond wantin o Ihe toca? domain eaersy with
pect to the Af. end deterrine dhe domain equibetun wa
ability eonition. Te total darian seg
By = Lip + Bt Be 6
is the sum of rive cern, Phe wi? evergy, Bn [2 the proctet af Te
ral cncray dene and the wall Te applied fel Seraeien
TretEs, Bu; is projmitional tothe poo: 0° the tamale values on
he vaverme Held elton sresy. The last term, By, the caters,
Iognevatavie energy af tae danagin, he everps watation hae the
FS nun as svetmae cmon. roman, aso 16H
”
where in the energy deivaven tae independent werables, tke a,
sd Af, ave bun writes ae ry sd 4 Toe eampetnery the aero
Seript indicate that the darvatives ara to be evaluate for see
sceeular domein, 10) ~ 5 and Oy indicates terms of order three or
hither. The fire docvatives aro the nemativs of the generalized Fores,
CaP p/ary wal — (OF 2/00), boing roseotvely the ruil at sgt
drneratsed frees of matntignn! poradviy m.‘The senond derivatives
{orm the stim mats of the eeazem wits (7 =/By ar being the
{nym} slomcone of the dal sulvnattix ad G@Ey/dty 860) ane
G@E,/a8 a8. le being the eormpmmnig element of the mixed ax
angie submateons aeapcetively. Ta Tot. 1 the dexvstives o? tho
total eneay with renpont In the ae, a ad, were ebtcined by cif
tating ts intel whi em lets guation (8) az eval
ing che resting integna tor the ease of eietly ereular domain
Tn the present woe it is onzvenient to waite the enengy vurintion
‘prosson in a mortage form in wai: enengy ie menaured in aie of
fea) (oh), the equilibrium condition #44 been sod to eliinnte
the svplied fll from the sseord verations (the nif mutrs ie of
Interest hore only for a domain equilibrium), tho Gin wl eceuad-
tilor variations whieh are iderteglly wor and delete, ad the mow
ro goneraline fares snd stress matrix elements. any sites
‘anetions of cienensionless vurtblo, the apoliod fe mcasurd in
Unite ot the waynetiation, Fite, unl rai of che plate tars,
ee id the haunting 08 the
he iain eter,
bo oseat 6
"hie somal fost: of Ue suey vasatcr i |Ref. 1, equntion (09)
aetteas [ii ir Cl]
“DLE feet
+ Bok 0]
‘where F and the S.aze eal tae forse and stability functions respoe-
ively and arp wvtten in terms of the complete elliptic integral of
the rw kind
tm) = [7 U= sot ta, 6
he comet etic inkl ef Uneven kin
eo)
= west hao ®
fond che fy Sanctions which ane suxdary Sanctions introluee for
Lae)
boa) = ale + YEO +
(ave)
<a HKG EAP cont)
Beahed = yg} [Lande + NELLY On Ef)
ioe KO BT GON,
‘Phe force yr sabiityPonetions ae
Poy = 2a ee BA +e = ay
my — 2 Dre om)
= de ie® — nor, 2)
snd
S03 Bere) tsk) = 140) + 10.
‘igure af Rol pote PUA) and § (4/2) for fh = 6 and a
70 summa srraae mews, JOORVAL, WaRER 1a7
Angurtinoy A and B of Ref 7 ge methods of computasion td eres
‘siren of Shee Sineions hile een TY aiaeueos tbo piysioal
inmeroroation ofthe terme in the enegy variation exprrsio [sons
Haw (6) be)
‘Beeiion TE of this paper diseunses the domtsin ane aod stalicy
implictions of the eneres vanation expansion (7). This deewsion
sinks the oonitons for the existence of eylindeen] Sarains nthe
oral absence of eissipetive proeeses ars these exitenee eadtion in
‘um place ceveralseetrctonr upon device design. ‘The scetion cen
‘cnder with discussion of domain sae nd comin elipticty im the
prsrenca of enso-ropic wall encray. Section IIT eonsiavs the efoe's
fof diasiparive poooceese (wal. peon coesivty and mbility? on
Aleaxin euetente ar moverncch, (re lon 0 main rel yt
tral miley, end the listing conitions under whieh the Yue ibe
‘node may br achieved, Sesion 1V sombinee the rests abteined here
find in Ref, Lt obsain several realms betwwon tnerial parameters
and devie performance.
‘This seeion entinines the arin equim aad sehiity candi
‘ions and suns a? le npie
24 Thr Keun Com
Tae domain i in equibrican shen all of the tst-ordr variations
of the tatal energy with repeat co the ay, and am are noo. Tn the
‘variation expansion (7), all the fhstoeder energy variations ext
the variation with respect to ar) (representing «variation in domein
fre wit nn variation in bye) are Mentivaliy ner, Sete somal
i ilially anid ea eiea'aeslinder and ere ace ng forte
tending ta tose it she domain cout rium shen it a eter
ferander Reving a dlansoese whieh Sea solution ta fro norte “ans
facing,
Lye mw _ 42)
it tea) -0 _
‘The (command) gncalne free apeatng in this oymtion have
‘ona name correavndenee wil che terms of the energy nan (1
{he int tem being produaeé hy the wall enesgs, the second by the
poled ek aml th “hid y te feral sagutetaie enon The
ome val fore, {fh ~ an tv¥h le tends to walyse
evLiscAL sa6enctHe Went we ™
Ie domass fou snd are postive) an iv ndeperdent of domain di
fnetonHaoh of Hiv nenmabaed gonerazad ono nny be ennverted
torem cyaivalent fel! by analtplsing by Muff. The eouivalent
allied is oj'cM, a Fens te wll eli proputional toche prod
1 the wal onnegy sl Che wall eurvatoxe. The nurmaized wl! ld
fore, SUA} eM.) vent ty eollaee the deacin for positive
plod flde sal its tagnitade i propesions. ta the demain diam
ler Toisas th equine ce apie al Te nore
Fao ntensal magnetetace ures ie + BCR, The fore fmetion, Pik
ceil nilh possve fist devvctivn ard argative sow sition
fF all prone, It approves a foe smal doin diameter and
1 tn] sein for bane dunes iineters ln Tet 1 ee estinn (38)
Teal 15g Bi Bice i's postive, the ineeal auc ero aay
Tia ts expt the domi. ‘Th nee! magaetcstatie fares an she
foal forse at hn oppaicely aretal. ‘Therriene equibiam 16 a
fais “or any ven diemerer hy suring che spplied field ty 1 se
‘lie enmpeneates for she difereann 'n angnitude of tess (ww Toren,
Tigh Ler fil depending om ahs Fae ie omnia ‘The ea:
Tat field. Zor cho. gase af the Hncemalmiagoetasoie form, Rl,
Aas oe the + wvorngel s oanpee of te tera demagnetis=
ing fe or th decor miszrelototie sear poseasal dterenes bee
thee typ foe hott of he gata diddy the pau ik
ness thie Bel apptoneher Set, for sen cain lamers and
AM, Uhye tel | or barge domain Slametets
So'vtnns to tho sepubauc. prebier mee be aeueed either Jr
lenin it Ihe easvaleae elds gf ww dewe by AL UL Moosel? or i
len af te gouwealge Soros svi FH) vith tae prefer? ech
Alepesing on the specie beatin Te the recent cu, ch ymeraze
free egutin ol he wre sino re general propesis ot sluone
ff tar cqaibbesum onvlew ay be ogsly ebtome ty straigat Live
conazeetions ou a plu".
Tigers 2 shove comics uel cme‘onntioms (along ith ability
fonaimcotione ities rl bo axphived ter) ke oyun cote
Servet consis ef fit neu pint mn che verte sx hase
rdinue 5m, then densing eight ine throug Che eat hore
aie Flop ie Ind, Th Eze 9 eativelne Ua eprecens
Tn! no totnn noite LL m0 that ibe ilorvetione wits GE wn)
‘tr Ihe wuifninn pointe. he dashed Tie sus uptotic to the Free
fangien ot the ongin kaa raanered hu ove ata erefare jroo
{ sefonc anf atiating the mete of mtr ll. In eal "the
Fost in Fig 2 fend ane this pia va La wortion aia i
Aeucted by "8. The symbol 7 as used snes 17h i che wall uray
rer init rer. fret of tice the magnotantatieseleonnng pst eit
Sooo ane ef ac iene plate gir hime hy ib i
‘ifort musrotined in 9 reckon verzediculr tote wane of Ue
ate
Fre the Bast exile, coir ve Tne WE far wha YR = 0,
HPs, — 0. Thi owe niet nile Pek OL ~ 378)
Since Ue slope of F is everywhere postive, thee wil ix geuerl be
nly one inereesem wach fae commnueton lin fr cay azo oF mez
welue of 11/4 insependent a? the tages of (8, ‘hot thie
single elution is unstable may be apstocstes: by considering ell
‘ration in tho selation ameter fra ita value at ¢: Inroesng the
ameter ofthe dom fm the darth ate easeanen eh magastude
1 the itor! roagretesttic free tars: hile the ether tar rena
teonelunt. The wt forge tnd to expe the domain ns ierot
‘hom noo as thy dranin espn from tee whstion dismieter- Te Seman
ie thar nemahle sil regen tn onus. An tly ila ag
rant alinor Cha Ti dem i nae with poet Uo vllonee Tin
‘easly agen in gerersl shit fot fixed elution diacotor te aolucion
‘ill boon ure ure (dhe destabilrng foro noresing bat with
Inarewing dinmetos) ws eld fe mard> toe nowntive, Str ble wale
3
sieos te the omilbvinn:pooblem asst only foe Salis aiing tha wall
Fil emg Lo cela the dna igh re saa enon
visi for any magni made of neeativenppted Held
Tosoave of she bigtime basaror of tho force Sanation sé hoge
lose motets, a eacally stable solusion to cho foee equstion wil
Sena? Gi viion tothe sadly atthe slubon) wher n vanish
tly sual pussve bigs ald i apples. Taal seit eden
sn tae come ay am Ue poet yosayrwch. The eaislenee of Pail
abner dace ich weir tal Sirti ray an oi be ae.) Se
the F function ren mrs snand Aervnive roeepabeze, the et
Dron solctigns sn saldien Uo a single "elily noatabie and 9 singe
tility stable sation, Tis situate Uutmwed the Fe
fr Hig. 2. The five aopnesnte UE ~ DHX, — O28 for whieh
Ue mle eatin, denoted By, opeurs st a — THT dhe
alle suai, enor Pe & nena ck ah — 2M. Debox Ie be
ferved saperimertaly the snstrole cehrion ane elas ennditions
This permis the exitones of she sbvble selon. TWA war done By
soplvisie short daration Minn Ge suse hin reed tho domain
Aiameter from the stable soliton arnt st 8 tn the unetsble ale
tion dlarsatr t cn thon rr th is Fld ite ogi vale
js, os fae onstable lution ws attained, The derisin tas'ng the
Sle selon digslen we Ven staid iio hy ecneetye
Tiago po.
TT th ali! iki icrzeaod fan vale for whic re ae iro
elution, the, the alucions spproact. each othor ail T 1 san fed
Yau thes censor. Tis point at whieh the solar exalare ie a
‘ins 0: anal raenaeaility sce at spit, Hho eanatruct.on 9b
tanger tof ‘Uva theo siaw fe ilor starve Uestae te lowest
andes by the sal vislation qraumenl wT cawnrling pereerupis.
here ato nn woutons for acpi Hila areaer tha the Hele 0 ra wl
jrasability Sires iv this ans the iawurd fs dominate at a
dlsmcter. the dma ctlloers ‘Sinee P remy wate Bee brow the
Asimpluse By the domains ange elapse fox appli Hekis wren
than ef») Tw Fig. 2 ioe i =, te ocigae point Cae
gi Hi4eA, — 8.989, 1.18 The seuenoe of he type of alae
tious whies acon ak is waded depends erly on the fs ant Fas
ff ths slaps nn emrantan fF, Taszefne fe axed wos! 2 Taka
fayoner eileen she ke of {7k [som t39 Hope we wary
tare oropetie af F tee aaioaonees of che eolagee dceler und Said
fa a giotn ves a hoy eis 3s eho, However, the essa 0
behets mnganess ed tae arzsea boharins of the downi dizeter
7S Tae RELL saan meu MioN Jonmyan, ROR
saa funetion of applied Feld wre eonconiont'y pratponed wail geuere
bly lance,
“The deve ineplieations ofthis autncetiar my be atmatanl by
soling thal eslincia! domains enet oly the preemner of aa. applied
‘sid appied in a direction ending to ellyse The demain ed having
f magelfade leas heat te,
Sane Sted
hp age of the stead variates of Us etal domain eneeey pro
‘avd gy ¢ teak vasiaticn in alage eaenctrise the aablsty of &
fy ori snmin, Snes poly desing: eu’ yin are af incest
bose, attantion is ceetretes to rirnsices af mich doneine fis ther
egitim sie age rot newer tape, Sinee ap arbisrare wal
asiation iv dheztihuble Uy Cis expen (21 sith sandiicn
the stebility probs! us 4 say Je goetaena of te
ferme in the eng emannas whieh 4c quadratic I the ary ead Ay
"Theee soebeiete ere level i: I sre sonwceat feat) 12 the
seifnee matin slemence ore snian, wed Mew ailfrse metab ole
rena" lnely leaped as ther nace sol, oF angUae et
ree rei wert
Th the enim soesey exanson (7) ube wily manera quadeace
eTrienta ane chr ceofiiente of the Lar") % 7-1, Ae requted hy the
a ommpleelywlesice
tlinrin” serie te sytem, 2 dora
We ih eegnet on angle, Thin ie incie.ted “ormal’y (21 by the
baunge eb ry nervy ler AMS, oF Aad. (The cule ane
Iu ines eubsvaticas are isenton| 7 avr.) Sito: tn ostine
‘Sraat fir om ange im onuation (7), (E22 ra litre il
ie gaa) ie cwiatiae anapitles azo quas-nectenl-ndee of the
sium, Nersor tems fonder thee rl ign Uh inti ay
vie wun ales never. the Heaton aepliudes bg lee nore
radon Fival's, be the energy et & dexaix it un tnfite pane i
indepen, of tae éorsan poston, the tnira ation. sills of the
Alovna'a is enn sd therefore sizes the vacation Se, wurramacrse te
Tun ote tn Iuenelaline, foe ez Lo, 4242. the eaeoent of
Sint the aceondandes exergy variation wis usp uy an ae teary
feuall lite ea variation a singly whe aun of the euotey
‘rior from cay venral eorrpaneat 0° the easation, tho aly ©
The alabiily ot wveulnregli-ieal maine rodnee to‘the study of
tho seabilcy of the domains wil esp to alam ral abe, ty
teste The spn o® he waronponding afi Satie etc deter
