Stripe magnetic domains and lyotropic liquid crystals

D. Sornette

To cite this version:

D. Sornette. Stripe magnetic domains and lyotropic liquid crystals. Journal de Physique, 1987,48 (9), pp.1413-1417. <10.1051/jphys:019870048090141300>. <jpa-00210570>

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Stripe magnetic domains and lyotropic liquid crystals

D. Sornette

Laboratoire de Physique de la Matière Condensée, CNRS UA 190, Université des Sciences, Parc Valrose,06034 Nice Cedex, France

(Reçu Ie 22 dicembre 1986, accepti sous forme définitive le 9 juillet 1987)

Résumé.- On étend le domaine de validité d’une analogic récemment proposée entre les domaines magnétiquesen ruban dans les films de grenat uniaxial et les cristaux liquides smectiques, par des arguments énergétiques etdes considérations de symétries. On montre de plus que la croissance des domaines sous champ magnétique peutêtre vue comme un gonflement, dans une analogie avec des phases lamellaires lyotropes. Ceci donne une nouvelle

interprétation à la transition "ruban~bulle" qui s’interprète alors comme résultant d’une instabilité péristaltique.Quelques conséquences pour les expériences sont discutées.

Abstract.- A recent analogy between stripe domain structures of uniaxial magnetic garnet films and smectic liquidcrystals is extended, using general energetic and symmetry considerations. The analogy is furthermore strengthenedby remarking that the growth of magnetized stripes in a magnetic film is similar to the swelling of lyotropic smecticliquid crystals occurring under changes of water or oil concentrations. This sheds new light on the stripe~bubbletransition occurring as the magnetic field is raised, which appears as a "peristaltic" unstability in the smectic language.Several consequences, relevant for experiments, are discussed.

J. Physique 48 (1987) 1413-1417 SEPTEMBRE 1987,

Classification

Physics Abstracts75-70K - 75-60E - 64.70M

1. Introduction

Thin uniaxial magnetic films are used for a vari-ety of technological applications, which among othersare magnetic computer memory, data-flow data pro-cessing, optical image processing, magneto-optic dis-play devices [1]... The typical application uses singlecylindrical Weiss domains of negative magnetization,the so-called (two-dimensional) "bubble", surroundedby a positively magnetized "sea".

Structures composed of many "bubbles" in dif-ferent lattice configurations are stable under suffi-

ciently high magnetic field. They minimize an energyF, which is the sum of the dipolar (or demagneti-zation) energy, the Bloch wall surface energy contri-bution (r denotes the Bloch wall surface energy) andthe magnetization-magnetic field interaction term [2].For magnetic fields slightly less than the saturationfield down to zero fields, the "bubble" phase is unsta-

ble with respect to the stripe domains phase whichconsists in straight parallel stripes of constant widthd(H)/2 of alternate magnetization (...+/2013/+/2013...).d(H) is the period of the 1D stripe structure whichdepends on H. These different structures have beenobserved long ago and are essentially understood intheir undeformed state [3,4].

It is a trivial observation that these stripe do-mains bear some gross resemblance to 2D-smectic liq-uid crystals : however, this statement has recently[5,6] been made quantitative by remarking that, with-in a Landau-Ginzburg expansion in terms of the mag-netization order parameter, the deformation energy ofthe magnetic stripe domains structure (MSDS) canbe cast under the smectic form :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048090141300

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u(x, y) denotes the displacement of the nth stripe atposition x = nd(H) counted positively along theaxis as a function of the distance y parallel to thestripes. B and K are the compression and curvatureelastic moduli respectively. The first term is a con-

stant reference energy, the second term is the com-

pression energy associated with a local change in thestripe period, and the third term describes the energycost for bending the stripes.

This deformation energy was obtained within atruncation approximation of the Fourier decomposi-tion of the magnetization distortion M(x,y), whichkept only the first Fourier component cos(27rx/d) andused the "phase" approximation : M(x,y) = Ms.m(x - u(x, y)) with m(x) = cos(21rx/d).The truncation approximation is valid near the Curietemperature Tc for which the width £ m r/Ku of theBloch walls is of the order of the period d and themagnetization profile is therefore smooth (Ku is theanisotropy energy favoring a uniaxial magnetizationperpendicular to the film).

This analogy (1) allowed us to explain the ap-pearance [7] of undulations of the stripes, observedin MSDS under magnetic field cycles 0 --; Hoax -0, in terms of an undulation instability [6] analo-

gous to that of smectics occurring under extensionalstrains [8]. However, many experimental systems donot verify the condition (T -- Tc) which is necessaryto validate the Fourier decomposition truncation ofthe magnetization. In many cases, the magnetizationprofile of an MSDS at low temperatures consits ofalternate plateau of saturated +Ma and -M$ magne-tizations of extensions d+ and d- respectively, whichis therefore incompatible with the single Fourier com-ponent truncation.

The aim of the present letter is to cure this weak-ness of the "smectic" theory, using general energeticand symmetry considerations. The analogy is fur-thermore strengthened by showing that the growthof magnetized stripes’ under a magnetic film is similarto the swelling of lyotropic smectic liquid crystals oc-curring under changes of water or oil concentrations.This sheds new light on the stripe-;bubble transi-tion occurring as the magnetic field is raised, whichappears as a "peristaltic" unstability in the smecticlanguage. Along with the development, we discussseveral consequences for experiments.

2. The deformation energy

Here, we derive expression (1) for any undis-torted 1D-magnetization profile M(x, y) = Msm(x),where m(a:) can be any periodic function of period

d(£Q. The argument follows closely that developedfor smectics [9] and we recall it briefly to stress thatit relies on very general assumptions which can applyto a variety of systems [10] once the following twoconditions are satisfied :

i) existence of an optimal period d(H) aroundwhich the energy is quadratic,

ii) invariance of the description of the system un-der global rotation.

It is useful to distinguish between two types of defor-mations : compression/extension and bending. Thefirst condition i) allows us to write the compression/extension part of the energy density f8 {u} = fs (0)=(1/2)B{2 (Un+l - un) /d}2 , where u,, (y) = u(x ==nd,y). This is the generic form of the energy associ-ated with a local change of the stripe period d’ = d+2(un+l - un)/d, away from the optimal period d[4].In the continuous description limit, 2(un+l - ’Un) /d -- 9tt/9:c and one recovers the second term of ther.h.s. of (1).Let us now consider the deformations dependent upony. One also expects an expansion of fs in terms ofsuccessive derivative and powers of u. However, it isimportant to recognize that similarly of the case ofsmectics, terms like 9M/9y or (au/ay)2 cannot exist.Indeed, a constant 9u/ay represents a uniform rota-tion around the z-axis perpendicular to the film andcannot change the MSDS energy. Therefore, to lowestorder, the first term is proportional to (a2u/ay2)2which cannot appear at the first power due to the

symmetry u --+ -u. This terminates the derivation ofthe phenomenological expression for the deformationenergy (1), and justifies the use of (1) for explain-ing the undulation instability appearing in saturatedmagnetic stripes [6]. Of course, the relevant expres-sion for the elastic constants are no more given byB ;e, (2/1r2) (T, - T) JD/ad2Tc, K = )..2B,).. = d/47r,where J is the exchange energy, D the film thicknessand a the lattice length.

3. Effect of a magnetic field

Let us start from an undistorted MSDS of perioddo, prepared in the absence of magnetic field. Let usapply progressively a uniform positive magnetic fieldH perpendicular to the film. As H increases, thewidth d+ of the (+) stripes increases at the expenseof the width d- of the (-) stripes which decreases.For weak H, d++ d- remains constant, equal to theperiod do, but for higher fields, d- decreases slowlyand attains a limit at high H whereas d+ increasesmore and more rapidly and diverges near completesaturation, occurring for H = H8 N (0.75)47r Ma.

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Correspondingly, the period d(H) = d+ + d- divergesas H - H8. These observations are explained by acomputation of the total energy of the MSDS sum ofthe dipolar Bloch wall surface and interaction ener-gies [4]. Since, at any H, the topology remains thesame and the period d(H) corresponds to the mini-mum of the energy, the previous reasoning of §2 ap-plies : the deformation energy of the MSDS under

field H also takes the form (1) with field dependentelastic modulus B(H) and K(H).

During the growth of the period d(H) as H in-creases, some stripe must disappear. Indeed, for afilm of width L, one has Lx = Ndo = N(H) d(H)where N (respectively N(H) = Ndo/d(H)) is the to-tal number of stripe pairs (+/2013). Suppose that acycle 0 --+ Hmax - 0 is described. At the value H =Hmax, a number N - N (Hn1ax) of stripe pairs havedisappeared (see Ref.[6] for a discussion on differentmechanisms of their disappearance). When the fieldis decreased back to zero, the period d(H) decreasesand the system favours the appearance of new stripes.However, this involves the nucleation of pairs of Blochwalls which is in general forbidden or in any case ex-tremely slow. Therefore, the MSDS remains with

N(Hmax) stripe pairs. For H Hmax, the equilib-rium period becomes d(H) d(Hn,ax). In the smecticlangage, the system suffers from a global extensionalstrain 8(H, Hn,ax) = N(H1nax). {d(H1nax - d(H)} =L.. {d(Hl11ax - d(H)} /d(Hmax), or a reduced strain

With this definition (2) of the strain, it is straight-forward to apply the analysis of reference [6] to thecase of MSDS under a magnetic field smaller than thepeak Hmax of the cycle. This should provide the bestexperimental conditions for testing quantitatively thesmectic theory. In particular ; the continuous controlof X(H, Hmax) as a function of the decreasing fieldH enables us to test the undulation instability, andits corresponding characteristic exponents [6].

4. The deformation energy with peristalticmodes

4.1. THE DEFORMATION ENERGY REVISITED.- When

considering deformations of the MSDS with very largeperiods d(H), it is important to recognize the ex-istence of another deformation mode involving in-homogeneous variations of the stripe thicknesses :the so-called peristaltic modes also coined "buckling"modes in the magnetic literature. They can be de-scribed in the smectic language by observing the sim-ilarity between the change of d+ and d- with H andthe swelling of lyotropic liquid crystals made of wa-

ter/surfact ants/ oil [11]. A variation of magnetic fieldH corresponds to a change in, say, the chemical po-tential of the water component, resulting in a growthof the water layer thickness.

Beyond this similarity in morphology, the twoconditions i) and ii) of §2 allow us to translate thetreatment developed in reference [12] to the case of"swollen" MSDS. Let us sketch the derivation and theresults. One introduces the displacements Vn (Wn)of the Bloch walls (-/+) (respectively (+/-)) awayfrom their equilibrium positions. The total defor-mation energy as a function of Vn, Wn, d+ and d-,reads :

For small Yn and Wn, one writes, as usual, thatd+ and d- correspond to the minima of the energy,which leads to a quadratic form for Fl (d+ -t- Vn+ 1 -

These expressions allow a "mode" decomposition.One finds [12] an "acoustic" band uq (so coined be-cause of the absence of a gap) and an "optical" bandçq (so coined because of the presence of the gap A) :

u corresponds to the stripe displacement at constantthickness and ç describes the thickness variations ofthe stripe pair (+) and (-) at fixed period d. u de-scribes nothing else than the undulation and com-pression/extension modes (note that the first termof the r.h.s. of (5) is the Fourier transform of theharmonic part of (1)) and ç corresponds to the peri-staltic modes. Note that the terminology "acoustic"of "optical" band may be misleading since the en-ergy (5) only describes static deformations and notany dynamic of the stripes which would involve thedifferent propagating modes of the system (see Ref.[3] for information on the propagating modes). Also,the above derivation of F does not suppose what the

origin of these deformations is. For lyotropic liquidcrystals, the origin comes from thermal coupling witha bath whereas for the stripe magnetic domains thedeformations stem from the existence of an appliedexternal field as, already stressed in reference [6].Within the local harmonic approximation, A and B

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verify

A is the "peristaltic" elastic modulus of the deforma-tion of d+ and d- at fixed d, in the absence of anyother deformation ( bending or compression / exten-sion ).

For reasonably swollen MSDS (this is specifiedbelow, A involves the same energy contributions asB. Since lc = r/,uo M.2, the ratio of the Bloch wallsurface energy to the magnetic energy, is the onlyrelevant length scale, one expects A m B / t;. In thiscase, A - Bq 2 remains positive insuring the stabilityof the stripes.

The moduli A, B and K depend on the value ofthe applied field as we discuss below.

4.2 "STRIPE-BUBBLE" TRANSITION : A PERISTAL-

TIC INSTABILITY.-

However, for higher "swelling", the expressions(6) for A and B are no longer valid due to the pres-ence of the long range dipolar magnetic interaction.A and B acquire a dependence on H which is esti-

mated below. As a consequence, the demagnetizationenergy may lead to a negative value of A - Bqx , forsufficiently high qz .

Let us estimate this phenomenon more quantita-tive.

’

Paramount to the stability of "bubbles" or stri-pes is the fact that the demagnetization field pene-trates inside the layers only up to a thickness of theorder of the film thickness D. As a consequence, a

variation of d+ with H (for d+ » D) does not costany additional energy. Only the change of d- with Hcontributes. One therefore expects A(H) m (d- (H) /d(H)). V (H) /12, where d- Id represents the volumefraction of a period of the MSDS which contributesto the peristaltic deformation.

A peristaltic instability may appear for small val-ues of qy, if the elastic coefficient controlling the peri-staltic mode ç becomes negative, i.e.

Since the maximum value of q.,,, is m 7r/d-, conditions(7) yields

For H - H3’ it is known that d- becomes of theorder of do/4 [4]. Taking typically 1, -- do / 15 [1],one finds dp -- 3.5 do roughly independent of H. Thismeans that, when the MSDS period d(H) increasesbeyond dp, the stripe domains become unstable withrespect to the variation of their thicknesses. This cor-responds to the appearance of pinches of the layerswhich can be naturally interpreted as the start-off ofthe "stripe-;bubble" transition. Note that this the-ory predicts that the pinches occur at isolated points(the instability occurs for qy --+ 0). This indicates thatthe "stripe--+ bubble" transition proceeds via the local"rupture" of layers, which, in a second step, collapseto form "bubbles". Notice that the stripe-bubblytransition is unrelated to the finger formation appear-ing at decreasing fields after the undulation instabil-ity [6].

5. Concluding remarks.

The smectic theory of thin uniaxial magneticfilms has been extended to a large temperature rangea.nd magnetization profiles. This analogy providesuseful guides for reasoning and interpreting experi-ments. Furthermore, a more correct description oftlie magnetic stripe domain structure involves a dif-ferent kind of smectic liquid crystal namely lyotropoiclamellar phases. The influence of a magnetic field onthe magnetic stripe structure corresponds to a changein chemical potential of one component of the ly-otropic smectic. The "stripe-bubble" transition oc-cu.rring as the magnetic field increases beyond somethreshold corresponds to a peristaltic instability inthe smectic language. The main message presentedin §4.2 is that the d(H) law constitutes the essen-tial ingredient for understanding the stripes-bubbletransition.

Precise experimental tests of the qualitative andquantitative predictions of the smectic theory are stilllacking. We holpe that the present note will encour-age experimental works in this respect.

References

[1] O’DELL, T.H., Rep. Prog. Phys. 49 (1986) 589.

[2] KITTEL, C., Rev. Mod. Phys. 21 (1949) 541.

[3] see for example MALOZEMOFF, A.P. and SLONC-

ZEWSKI, J.C., Magnetic domain walls in bubblematerials (Academic Press, New York) 1979.

[4] KOOY, C., and ENZ, U., Philips Res. Repts 15(1960) 7.

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[5] GAREL, T., and DONIACH, S., Phys. Rev. B26(1982) 325.

[6] SORNETTE, D., J. Physique 48 (1987) 151.[7] MOLHO, P., GOUZERH, J., LEVY, J.C.S., and

PORTESEIL, J.L., J. Magn. Magn. Mat. 54-57(1986) 857.MOHLO, P.H., PhD Thesis, Grenoble (1985).

[8] RIBOTTA, R., and DURAND, G., J. Physique 38(1977) 179.

[9] DE GENNES, P.G., The physics of liquid crystals(Clarendon Press) 1974.

[10] see for example Cellular structures in instabili-

ties, Lec. Notes Phys. 210, ed. WESFREID, J.E.and ZALESKI, S., (Springer-Verlag) 1984.

[11] EKWALL, P., Composition, Properties and struc-tures of Liquid Crystalline phases in systems ofAmphiphilic Compounds in Adv. Liq. Crys. 1

Amphiphilic mesophases made of defects, 1, ed.BROWN, G.H., and HELFRICH, W. ; (1975), LesHouches, XXXV, ed. BALLIAN, R., e t al. (1980).

[12] PARODI, O., Microémulsions et phases lamellai-res in Colloides et Interfaces, Aussois France

p.355 (1983).