Micromagnetics : Successor to domain theory ?

HAL Id: jpa-00235996

https://hal.archives-ouvertes.fr/jpa-00235996

Submitted on 1 Jan 1959

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Micromagnetics : Successor to domain theory ?

William Fuller Brown

To cite this version:

William Fuller Brown. Micromagnetics : Successor to domain theory ?. J. Phys. Radium, 1959, 20

(2-3), pp.101-104. �10.1051/jphysrad:01959002002-3010100�. �jpa-00235996�

MICROMAGNETICS : SUCCESSOR TO DOMAIN THEORY ?

By ^{WILLIAM FULLER} BROWN, Jr,

University ^of Minnesota, Minneapolis ^{14, U S A.}

Résumé. 2014 Dans la théorie des domaines, les notions de paroi ^et de domaine sont considérées

comme fondamentales quand l’épaisseur ^et l’énergie ^d’une paroi ^{ont été} calculés. La présente

communication concerne un calcul tridimensionnel complet ^et cohérent basé sur la notion de l’aimantation continue, calcul dans lequel ^on ne fait pas appel ^a priori ^{aux notions de domaine}

et de paroi. ^{Si ces} derniers s’avèrent necessaires, ils devront découler naturellement de la théorie.

Les états d’équilibre ^{stables sont} les solutions d’un problème ^non linéaire à données aux limites.

Ce problème ^a été resolu dans le cas à une dimension de la théorie classique ^des domaines, dans le cas où l’aimantation étant sensiblement constante, ^{il est} possible ^de faire ^une approximation linéaire, ainsi que dans certains autres cas ^{non linéaires} pouvant ^être résolus ^{soit par une approxi-}

mation de Ritz, ^soit numériquement par ^{une machine à calculer} électronique.

Abstract.

In domain theory, ^the domain and wall concepts ^{are treated as} fundamental as

soon ^as the thickness and energy of ^a wall have been calculated. The alternative considered here is a complete, self-consistent, three-dimensional calculation based on the continuous-magneti-

zation concept, in which domains and walls ^{are not} postulated ; ^when they ^{are valid} concepts, they

must emerge automatically ^{from the} theory. ^{The stable} equilibrium ^{states are} solutions of a

nonlinear boundary-value problem. ^This problem has been solved in the one-dimensional case of traditional domain theory, in linearizable cases of nearly ^uniform magnetization, ^{and in} special

nonlinear cases amenable to a Ritz approximation ^{or to numerical solution} by high-speed ^com- puters.

PHYSIQUE 20, 1959,

In domain theory, ^{we work with " domains " and}

domain " walls ". In micromagnetics, ^{the dis-}

tance scale is smaller ; ^the working concept ^{is a} magnetization ^J

^{7g v whose} magnitude 7g ^is constant, ^but whose direction v( = ce 1 + p j + y k)

’varies continuously ^with position. ^{On the still}

smaller atomic scale, ^we speak of lattice sites and electron spins. ^At first, ^{domains were} merely postulate ; later the concept ^was refined ^and partly justified by calculations at the micro-

magnetic level. The resulting theôry ^{is full of} logical gaps and contradictions. I wish ^to examine

an alternative : direct operation ^{ait. the micro-} magnetic ^level.

At this level we do not postulate ^{domains ; we} just ^{assume a} spatially ^{variable v. The} problem

of micromagnetostatics ^{is to determine a function v}

that minimizes the potential ^energy [1], [2]

Here Ho ^{and H’ are the} parts ^{of the} magnetizing

force H due respectively ^to external and ^to internal sources ; dr is a volume element of the specimen.

In an atomic interpretation, ^{the C term is} predomi- nantly exchange energy, the wi term crystalline anisotropy energy, and the Ho ^{and H’ terms}

external and internal magnetic energy. H’ must be calculated from J by potential theory.

Landau and Lifshitz [1] miniïîlized W under constraints. They ^{assumed a} one-dimensional solenoidal variation of v, ^with Ho

0, ^{and with} prescribed ^values v = + 1 at z - + ^oo. To pres- cribe these values is to postulate ^{a domain struc-}

ture ; therefore such a calculation does not explain

the origin

of domains. Minimization without constraints for a finite specimen [2], ^{leads to the} partial differential equations

and the boundary ^conditions

or in vector form

Since the minimization procedure generates ^its

own boundary conditions, ^{we are not at} liberty ^to prescribe boundary ^{values as was done} by ^Landau

and Lifshitz. Any ^{solution of} (4) ^{makes W sta-} tionary, ^{but not} necessarily ^{a minimum ; a stabi-} lity ^test is also necessary.

By including dynamic terms, ^one gets ^a dynamic generalization ^of (4) [3]. ^{Workers in} ferromagnetic

resonance might ^benefit by studying previous

work in micromagnetostatics.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:01959002002-3010100

102

There are several possible procedures for attack-

. ing the nonlinear boundary-value problem ^(4).

(a) ^Patch together different one-dimensional ^solu- tions in différent regions. ^{This is} the method of traditional domain theory [4]. ( b) Study ^{cases in}

which v is almost uniform because of a large ^field

or a large anisotropy ; ^{then the} transverse compo- nents of v are small, ^and Eqs. (4) ^{become linear} [2, 5]. (c) ^{Assume a form for the function} v(x, y, z)

with undetermined parameters, ^and adjust ^them

for minimum W. This is tbe Ritz method, ^used

’

most systematically by Kondorskij [6]. (d) ^Attack

the nonlinear problem directly by ^numerical methods; ^this requires high-speed computers [7].

In _one important case, the linear equations ^of ( b)

are rigorous. ^{This is the case of a} perfect ellip-

soidal crystal, initially magnetized along ^a principal

axis coincident with a direction of easy magne-

tization ; ^{one seeks to find how} large ^{a reversed}

field is necessary ^to upset ^the stability. My ¹⁹⁴⁵

calculation by ^{this method} [8] predicted stability

up ^{to a} rather large ^.reversed field, ⁱⁿ apparent contradiction to experiment. To resolve this para-

dox, ^Stoner [9] ^invoked imperfections ; ^{but his} suggestion ^{has never been made} quantitative. ^The paradox disappears ^{in the case of fine} particles ;

here the theory ^{fits in well with the} concept ^of

"

single-domain " particles. Recently ^Shtrikman

and I, working independently ^but cooperatively,

have obtained precise stability ^{criteria and calcu-}

lated the nonlinear behavior after instability ^sets

in. Thus we can now supplement the Stoner- Wohlfarth theory ^of uniformly magnetized par- ticles [10] ^{with a} theory that says just ^{how fine a}

FIG. 1.

Theoretical magnetization ^{curve of an infinite} cylinder just ^too large

for Stoner-Wohlfarth behavior. For discussion, ^see second reference 7.

particle ^{must be to} stay uniformly magnetized,

and how it will behave if it is not quite ^{that fine}

(cf. fig. 1) [7]..

But what about a specimen o f ordinary size, ^for

which there is still a paradox ? I believe that the

theory correctly describes the behavior of a speci-

men of the type assumed-one with ^no imperfec- tions. (However, ^{the case of a} perfect crystal

with magnetostrictive properties requires ^intro-

duction of surface energy ^terms that have never

been completely investigated [11].) ^{Let us there-}

fore consider a specimen that contains imper- fections o f random character.

In the micromagnetics ^{of fine} particles, ^{we made}

substantial _progress when ^we separated ^our pro- blem into two parts : ^first study ^{the linear} equa-

tions, ^{then use the results of this} study ^{to select a} special ^situation fornonlinear calculation. Let us

try ^{a similar} teÇhnique ^here.

If the transverse deviation u

oc 1 + p j ^is small, ^{it is} sufficient to express W ^to the second order in u. The anisotropy Energy density wl(oc, P, y)

w(ce, z3) ^{can then be written}

where the g’s ^{now include} termes that vary ^ran,

domly ^with position. ^{The terms linear in a} and B represent ^the transverse deviating ^forces

g

91 ’ + g2 j ^{treated in} references 2 ; ^{the qua-}

dratic _terms determine whether these forces increase or decrease with amount of deviation.

The 1945 stability argument hinged ^{on the fact}

that the quadratic ^form

is positive ^{definite as} long ^as

^H is smaller than ^a certain value (#à gij 1 la). ^{If the} gij "s include random terms, there may be points ^{in the} specimen

where the positive definiteness fails at much smaller values of

H ; ^such points may ^{serve as} centers for nucleation of magnetization ^reversal.

In this linear approximation,

where _x. is the n th eigenvalue, ^and un the ^corres-

ponding normalized vector eigenfunction ^{of the} homogeneous boundary-value problem ^obtained by setting ^g

0, ^{and where}

’

(ut

complex conjugate ^of un). The solution (7)

is unique ^as long ^{as T is} positive ^{definite. Insta-} bility ^{occurs when - H reaches} the smallest eigen-

value xo ; the deviation that then occurs starts as a

small deviation of the _{form uo.}

But this calculation is subject ^{to an error not} present ^{in the} perfect-crystal ^{case. There g}

0,

and therefore u

0 until instability ^{sets in. Here}

g # 0 ; ^{and before} instability ^sets in, ^the

transverse deviation u will in general have attained such a size that the linear approximation ^{is no} longer ^{valid. In} fact, ^{for H}

xo Eq. (7) gives

u = oo unless _g is orthogonal ^to uo ( Go = 0).

A safer application ^{of the} linear theory is ^{the follow-} ing. ^{Given random} imperfections ^{of a} specified

sort, ^{we can use} (7) ^to find which component ^G,, produces ^the largest ^{effect within the} linear range.

We can then assume a disturbance that corres-

ponds ^{to this} component alone, ^and carry ^{out a} nonlinear calculation by ^numerical methods. If

we neglect ^{the random} components of gll, g22, and g12 and ^assume periodic boundary ^conditions

the Gn ^’s become Fourier transforms. Thus in effect we are examining the spatial ^noise spectrum, selecting ^the part ^{of it to which} the small devia- tions are most sensitive, ^and retaining ^this part alone when we go ^{on to} consider large deviations.

In this way (tho only by considerable labor), ^we

may hope ^to determine the conditions for nuclea- tion of magnetization ^{reversal. Such a} theory regards nucleation as the essential irreversible process. Conventional coercivity ^theories bypass

this problem and discuss hypothetical irreversible

jumps of " walls ", ^{assume to be} already ^nucleat-

ed. In the proposed theory, the existence of such walls and jumps either will follow from the ^non- linear calculation or will be rejected ^{as a} myth.

Clearly micromagnetics ^{is not} yet ready ^to eject

domain theory ^{from the} position ^{that it now holds} by default. But micromagnetics ^{can at least for-}

mulate explicitly, ^{and attack} honestly, problems

that domain theory ^{evades. With the new} digital

calculators ^to help ^{us, there is little excuse for} resorting ^any longer ^to such evasion.

REFERENCES

[1] ^{LANDAU (L.)} and LIFSHITZ (E.), Phys. ^Z. Sowjetunion, 1935, 8, 153. ELMORE (W. C.), Phys. ^{Rev., 1938,} 53, 757.

[2] ^BROWN (W. F., Jr.), Phys. Rev., 1940, 58, 736 ; 1941, 60, 139.

[3] ^BROWN (W. F., Jr.), Phys. Rev., 1949, 75, ^1959.

MACDONALD (J. R.), ^Proc. Phys. ^Soc. (London), 1951, ^A 64, 968. ^GILBERT (T. L.), Phys. Rev., 1955, 100, ¹²⁴³ and Armour Res. Foundation Proj.

n° A059, Suppl. Report, 1 May 1956.

[4] For reviews at three successive stages, ^{see : BROWN} (W. F., Jr.), ^J. Appl. Phys., ^{1940, 11, 160-172.}

KITTEL (C.), ^{Rev. Mod.} Physics, 1949, 21, ^541-583 (see also McKEEHAN (L. W.), Phys. Rev., 1950, 79, 745). ^KITTEL (C.) ^{and GALT} (J. K.), Solid State

Physics, ^{vol. 3} (Acad. Press, ^New York, 1956),

pp. 437-564.

[5] ^HOLSTEIN (T.) ^{and PRIMAKOFF} (H.), Phys. Rev., 1941, 59, ^{388. NÉEL} (L.), ^{C. R. Acad.} Sc., Paris, 1945, 220, ^{738 and 814 and J.} Physique Rad., 1948, 9,

184 and 193.

[6] KONDORSKIJ (E.), ^{Dokl. Akad.} Nauk, S. S. S. R., 1952, 82, 365 ; ^{Izvest. Akad.} Nauk, ^{S. S. S.} R. ;

Ser. Fiz., 1952, 16, 398.

[7] ^BROWN (W. F., Jr.), Phys. Rev., 1957, 105, 1479 ; J. Appl. Phys., 1958, 29, ^470. FREI, ^SHTRIKMAN and TREVES, Phys. Rev., 1957, 106, ^{446. AHARONI} (A.) and SHTRIKMAN (S.), Phys. Rev., 1958, 109,

1522.

[8] ^BROWN (W. F., Jr.), ^{Rev. Mod.} Physics, 1945, 17, 15.

[9] ^STONER (E. C.), Repts. Progr. Phys., 1950, 13, 83, esp. p. 116.

[10] ^STONER (E. C.) and WOHLFARTH (E. P.), Phil. Trans.

Roy. ^Soc. (London), 1948, ^A 240, ^{599. For some} generalizations ^{see BROWN} (W. F., Jr.) and MORRISH

(A. H.), Phys. Rev., 1957, 105, 1198.

[11] ^HAYASHI (T.), ^Z. Physik, 1931, 72, 177. ^POWELL (F. C.), Proc. Camb. Phil. Soc., 1931, 27, 561 ; BROWN (W. F.), Jr.), ^{Rev. Mod.} Physics, 1953, 25,

131.

DISCUSSION

Mr. Nagamiya.

^Without knowing your inte-

resting calculation, ^{I have been} trying ^{to find the}

nucleation field for a spherical specimen ^(aniso- tropy ^axis parallel ^{to the} applied ^{field). The}

variational equation ^was linearized ^{and a} parti-

cular case where the magnetization ^{deviates as a}

function of the position ^{in a} plane containing ^the

applied ^{field was} examined. A more general ^case

is being investigated.

104

Mr. Brown.

The difficulty ^{with the} sphere ^is

that the partial differential equations ^{are awkward-} ly anisotropic. ^{I found solutions} only ^{for a uni-}

form deviation and for the magnetization-curling mode ; other modes _{may also} be important.

Results of a more complete investigation ^{would be} interesting.

Mr. Dreyfus.

In variational problems ^one generally ^{deals with} continuous and continuously

derivative functions (class ^C functions). ^Don’t

you think it is impossible ^to get any information ^on discontinuous functions (which ^are supposed ^to represent domains) by starting only from conti-

nuous functions as you tried in your determination of coercitive field in an ellipsoid ?

Mr. Brown.

It is true that nonlinear partial

differential equations ^{sometimes have discon-}

tinuous solutions. In our problem ^the physical requirement ^{of finite} exchange energy precludes ^an

actual discontinuity ^of direction ; ^{the " discon-}

tinuities " of domain theory ^{are therefore} regarded

in the Landau-Lifshitz interpretation ^as approxi-

mations. In a numerical calculation by ^{a finite-}

difference approximation, the mathematical diffi- culties you mention do ^not occur ; there, is, ^of

course, uncertainty ^how well the finite-difference

approximation represents ^the continuous-range problem, ^{but there is also} uncertainty ^{how well the}

latter represents ^the physically ^correct description

at the atomic level. From this point of view thé finite-difference approximation may ^even be a

better one. 1 agree that there are many ^unans- wered questions ^{here. In the linear} stability ^cal- culation, ^{1 think the} situation is essentially ^the

same as in a calculation of elastic stability.

Mr. Jacobs.

I would like to call to the atten- tion of the Congress ^{an abstract of a} paper pre- sented to the Amer. Phys. ^Soc. (June 1958) by my

colleagues R. W. de Blois and C. P. Bean, ^entitled

"

Nucleation of Ferromagnetic ^{Domains in Iron}

Whiskers ". They observe nucleation field up to 480 Oe for the reversal of magnetization ⁱⁿ regions ^of nearly perfect ^iron whiskers about 5 _{jL thick. This is ^near the theoretical limit 2K1/M2 (m ^{510 Oe for} iron) ^{for the coercive force due to} magnetocrystalline anisotropy.

Mr. Brown.

It is a satisfaction to see expe- riment finally verifying theory, especially ^{as the}

outcome of my 1945 calculation was at the time rather startling.

Mr. Shtrikman (Remark).

^Our group ^at Rehovoth has studied the influence on the nuclea- tion field of certain defects, ^as for instance a cylin-

drical inclusion whose axis is perpendicular ^{to the}

easy axis, ^{or a} region ^{of zero} anisotropy. ^The larger ^defects possible ⁱⁿ larger particles ^could explain the decrease of coercive force with increas-

ing size.