Continuous spin-reorientation phase transition in the surface region of an inhomogeneous magnetic film

Journal of Experimental and Theoretical Physics, Vol. 101, No. 5, 2005, pp. 830–843.Translated from Zhurnal Éksperimental’no

œ

i Teoretichesko

œ

Fiziki, Vol. 128, No. 5, 2005, pp. 958–973.Original Russian Text Copyright © 2005 by Anisimov, Popov.

ORDER, DISORDER, AND PHASE TRANSITIONSIN CONDENSED SYSTEMS

Continuous Spin-Reorientation Phase Transition in the Surface Region of an Inhomogeneous Magnetic Film

A. V. Anisimov and A. P. PopovMoscow State Engineering Physics Institute, Kashirskoe sh. 31, Moscow, 115409 Russia

e-mail: [email protected] February 21, 2005

Abstract—A continuous spin-reorientation transition from a uniform magnetic state with the in-plane orienta-tion of the moments of all atomic layers to a nonuniform canted state in the surface region is considered. Thistransition was discovered in experiments on the divergence of magnetic susceptibility in a perpendicular mag-netic field at a temperature of about 240 K, which is lower than the Curie point of gadolinium, equal to 292.5 K.These experiments were carried out on an ultrathin iron magnetic film deposited on the (0001) surface of a thingadolinium film. It is shown that, in the vicinity of the spin-reorientation transition, the thermodynamic poten-tial has a form characteristic of the Landau theory of second-order phase transitions. The orientation angle ofthe moment of the surface atomic layer with respect to the plane of the film, which is chosen as an order param-eter, exhibits anomalous behavior and increases with temperature. Expressions are derived for the magnetic sus-ceptibility of each atomic layer. It is shown that, in the vicinity of the transition, the irregular part of the mag-netic susceptibility of each atomic layer exhibits behavior characteristic of the susceptibility in the Landau the-ory: it is less by a factor of two in the low-symmetry phase and diverges at the transition point. The regular partof the magnetic susceptibility of each atomic layer makes an additional contribution to the asymmetry of thetotal susceptibility in the vicinity of the transition point; this result follows from the fact that the inhomogeneousmagnetic system considered is semi-infinite. © 2005 Pleiades Publishing, Inc.

1. INTRODUCTION

Magnetic properties of films consisting of layers ofdifferent materials are of considerable fundamental andapplied interest. The phenomenon of giant magnetore-sistance [1], spin-reorientation transitions, etc., havebeen observed in these films. Although there are a largenumber of experimental studies on the magnetizationprocesses and spin-reorientation transitions in theseprocesses (see, for example, [2, 3]), theoretical modelsthat have been used until recently either deal with aninfinite medium or rely on numerical analysis [4];therefore, these models do not give a complete pictureof the behavior of a magnetic structure in an externalmagnetic field. The main difficulty associated with thetheoretical analysis of these systems lies in the consid-eration of the inhomogeneity of the magnetic structurethat is formed near the interface between differentmedia on the scale on the order of the lattice constantdue to the change in the chemical composition of thefilms. As a result, the parameters such as magnetizationand magnetic susceptibility, which characterize differ-ent magnetic states and phase transitions between them,become essentially dependent on the number of anatomic layer at a distance on the order of the lattice con-stant. Therefore, considerable attention has recentlybeen paid to the study of the surface and bulk anisotro-pies and to finding out the role of the exterior surface inthe process of reorientation analyzed by the models that

1063-7761/05/10105- $26.000830

take into account the real geometry of a magnet [5–7].In the present paper, we apply a method analogous tothat used in [5–7] to analyze the effect of an externalmagnetic field perpendicular to the surface of a Fe/Gdtwo-layer magnetic film and a continuous spin-reorien-tation transition discovered in this film.

In experiments with a two-layer film consisting ofone and a half atomic layers of Fe deposited on the sur-face of a Gd(0001) thin film, it was established that, astemperature increases, two successive spin-reorienta-tion transitions occur in this film at temperatures belowthe Curie point of bulk Gd, which is equal to 292.5 K[8]. At low temperatures, a magnetic structure is real-ized in which the moments of all atomic layers lie in theplane of the film; in this case, the surface moment isantiparallel to those of lower lying Gd layers. Anincrease in temperature to about 240 K leads to a con-tinuous spin-reorientation transition to a state that wecall here a Néel-domain-wall-like canted state. In thisstate, the surface moment and the moments of lowerlying Gd layers deviate from the in-plane orientation;deeper into the crystal, the orientation of the momentsof Gd atomic layers gradually approaches the in-planeorientation. A further increase in temperature leads to aslow increase in the deviation angle of the surfacemoment from the in-plane orientation. At a temperatureas high as 280 K, a discontinuous spin-reorientationtransition occurs from one canted state to another, in

© 2005 Pleiades Publishing, Inc.

CONTINUOUS SPIN-REORIENTATION PHASE TRANSITION 831

which the surface moment is virtually perpendicular tothe plane of the film.

Both transitions were discovered and investigatedby means of the spin-polarized secondary electronemission spectroscopy method. This method is sensi-tive to the magnetic state of only a few surface atomiclayers. Later, this system, denoted here as1.5Fe/Gd(0001), was investigated by a method basedon the magnetooptic Kerr effect [9]. This methodallows one to analyze the magnetic state of deeper bulklayers of Gd. In these experiments, it was establishedthat the moments of many Gd layers deviate from thein-plane orientation under spin-reorientation transi-tions; thus, many Gd layers take part in each spin-reori-entation transition.

A continuous transition from the state with a uni-form orientation of the moments of atomic layers to adomain-wall-like canted state is characteristic of thefilms that consist of layers with different chemical com-positions. In this sense, this type of transition is unique.Indeed, it does not occur in any magnetic film withhomogeneous chemical composition. In 1954, Néelshowed that the anisotropy of a surface layer may differfrom that of internal layers even in chemically homoge-neous ferromagnets such as Fe, Co, Ni [10]. Therefore,one may expect that, when, say, easy-axis anisotropy isrealized in the surface layer, while easy-plane anisot-ropy is realized in the bulk layers, a domain-wall-likecanted magnetic state can be formed in the subsurfaceregion. However, the exchange-interaction energy inthese ferromagnets is much greater than the anisotropyenergy; therefore, the surface moment cannot deviatefrom the in-plane orientation [11]. Naturally, the redis-tribution of the electron density between atomic layersin the surface region occurs even in chemically homo-geneous films; i.e., the Friedel oscillations of the elec-tron density and the associated phenomenon of inter-layer relaxation occur. These factors, together with thedifference between the environment symmetry ofatoms on the surface and in the bulk of a crystal, leadnot only to a difference between the surface and bulkanisotropies but also to a difference between theexchange interactions in the surface region and in thebulk of the crystal. However, experiments with Fe, Co,and Ni films show that the renormalization of theseparameters is not sufficient for the formation of acanted domain-wall-like structure in the surface region.A different situation is realized in the two-layer systemof 1.5Fe/Gd(0001). Here, the deposition of an ultrathinFe layer onto the surface of a thin Gd(0001) film witheasy plane anisotropy leads to the formation of anamorphous Fe/Gd film in the surface region of a sampleup to the Curie point of gadolinium [12]. It is wellknown that amorphous Fe/Gd films are characterizedby easy axis anisotropy, which favors a perpendicularorientation of the magnetic moment with respect to theplane of the film [13]. As a result, an ultrathin Fe/Gdfilm with anisotropy different from the bulk anisotropy

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHY

and enhanced exchange interaction is formed on thesurface of a Gd film. Due to the latter fact, the Curiepoint of the surface layer in the 1.5Fe/Gd(0001) sys-tem, 350 K, turns out to be higher than the Curie pointof Gd, which is 292.5 K [8].

The so-called one-layer approximation provides thesimplest explanation for the transition to a canted statein the surface region. Within this approach, it isassumed that only the magnetic state of the topmostatomic layer is different from the magnetic state of bulklayers. In the vicinity of the Curie point of Gd, theenergy of exchange interaction JSBMSMB between thesurface atomic layer and the adjacent subsurface layer,which is assumed to be a bulk layer in the one-layerapproximation, decreases to zero, because the magneti-zation MB of the bulk layers tends to zero. At the same

time, the surface anisotropy energy KS remainsfinite in this narrow interval of temperatures and is vir-tually independent of temperature. Therefore, at a cer-tain temperature, the anisotropy energy of the surfacebecomes comparable to the energy of exchange interac-tion in the surface region. Hence, a spin-reorientationtransition from a state with a uniform orientation of themoments of atomic layers to a canted domain-wall-likestate occurs in the 1.5Fe/Gd(0001) system. Naturally,the interpretation of the spin-reorientation transition toa nonuniform canted domain-wall-like magnetic statewithin the one-layer approximation is not quite correct.In this approximation, the thickness of a domain wall ison the order of the lattice constant, which contradictsboth the experimental data of [9] and the available dataon the relation between the anisotropy energy and theexchange-interaction energy in Gd. However, this inter-pretation at least provides a qualitative explanation forthe deviation of the surface moment from the in-planeorientation. Below, we describe a continuous spin-reorientation transition to a canted domain-wall-likestate in the surface region with regard to the deviationof the magnetic moment in many atomic layers of thefilm.

In this paper, we consider only the first of the twospin-reorientation transitions: a continuous spin-reori-entation transition from a state with a uniform orienta-tion of the moments of atomic layers to a canteddomain-wall-like state. This state was discovered byexamining a peak of magnetic susceptibility in a per-pendicular magnetic field [8]. According to the Landautheory of second-order phase transitions, the diver-gence of susceptibility at a certain temperature impliesthat a continuous second-order phase transition occursat this temperature [14].

The description and the physical interpretation ofthe continuous spin-reorientation transition to a cantedstate in the surface region require that one should takeinto account the deviation of the moments of manyatomic layers from the in-plane orientation. This is aconsequence of the fact that the value of the bulk

MS2

SICS Vol. 101 No. 5 2005

832

ANISIMOV, POPOV

anisotropy is small compared with the energy of inter-layer exchange interaction. The consideration of manylayers substantially complicates the description of thecontinuous phase transition due to the complexity ofthe thermodynamic potential that describes this system,

(1)

Nevertheless, the problem of the description of this tran-sition with regard to many layers, i.e., within model (1),was largely solved. First, based on stability theory, acriterion for the transition form a magnetic state with auniform orientation of moments to a nonuniformcanted state was derived for a semi-infinite ferromag-net. The effect of an external field parallel to the planeof the film on this criterion, the finiteness of the filmthickness, the influence of the anisotropy of the sub-strate onto which the film was deposited, and the inho-mogeneity of the chemical composition of the film onthis criterion was studied [15]. However, the effect of afield perpendicular to the film surface was investigatedonly in the limiting case of the infinite constant of bulkanisotropy KB ∞. Moreover, a (kS, kB)-phase dia-gram was constructed that indicates domains where auniform magnetic state and a nonuniform canted mag-netic state in the surface region are realized [15]; kS andkB are dimensionless surface and bulk reduced ani-sotropy constants, respectively, which are defined asfollows:

(2)

The parameter γ takes into account, in the simplestapproximation, the inhomogeneity of the film, i.e., thedifference between the surface and bulk values of theexchange interaction, which is associated with theinhomogeneity of the chemical composition of the1.5Fe/Gd(0001) film. Later, a criterion for the transitionto a canted state in the surface region was againobtained by the method of nonlinear area-preservingmaps [16].

In [17], along with a continuous transition to acanted state, which occurs at relatively low tempera-tures, we also described a discontinuous transition,occurring at a higher temperature, from one canted

Φ0 JSBMSMB θ1 θ2–( )cos–=

– JBBMB2 θn θn 1+–( )cos

n 2=

∞

∑

+ KSMS2 θ1sin

2KBMB

2 θn.sin2

n 2=

∞

∑+

kS

2KSMS2

JSBMSMB------------------------, kB

2KBMB2

JBBMB2

------------------,= =

γJSBMSMB

JBBMB2

------------------------.=

JOURNAL OF EXPERIMENTAL A

state to another canted state in which the surfacemoment is nearly perpendicular to the film plane. Inthat paper, we actually calculated the moment of anatom in each atomic layer and its orientation as a func-tion of temperature for any set of the model parameters.Then, using the results obtained, we calculated a signalrecorded in the experiment based on the magnetoopticKerr effect with regard to the exponential decrease inthe contribution of each atomic layer as the index of alayer increases; the curve obtained was compared withthe experimental curve and showed good agreement.

In spite of these achievements, there still remain anumber of questions concerning the description of acontinuous spin-reorientation phase transition from auniform magnetic state to a nonuniform canted state inthe surface region. First, the derivation of the criterionfor the transition to a canted state is based on the expan-sion of thermodynamic potential (1) only up to qua-dratic terms in the orientation angles θn of atomic lay-ers, regardless of which method is used, [15] or [16].This approximation allows one to derive the criterionitself but does not allow one to unambiguously judgethe kind of the spin-reorientation phase transitiondescribed by model (1). Indeed, according to the Lan-dau theory of second-order phase transitions, the ther-modynamic potential can be expanded in a series withrespect to the order parameter η at the transition pointT = TC:

(3)

The transition at T = TC from a low-symmetry state to ahigh-symmetry state is a second-order phase transitionif the coefficient of η2 changes its sign at the transitionpoint and the coefficient B of η4 is positive. Intuitively,it is obvious that a smooth increase in the surfaceanisotropy constant, which favors the perpendicularorientation of the surface moment, should lead to a con-tinuous spin-reorientation transition to a canted state inthe surface region. However, based on expression (1)for the thermodynamic potential of a semi-infinite crys-tal, it is rather difficult to judge the sign of the coeffi-cient multiplying the fourth power of the order param-eter. Moreover, it is not quite clear what physicalparameter can serve as the order parameter under a con-tinuous transition to a canted state and whether thermo-dynamic potential (1) in the vicinity of the phase transi-tion has a form characteristic of the Landau thermody-namic potential (3). Therefore, in [17], we determinedthe kind of the phase transition only for particular val-ues of the model parameters by numerical simulationon a computer. We also established that, in the limitingcase of the infinite value of the bulk anisotropy con-stant, KB ∞, the boundary that separates thedomain with a canted state from the domain with the in-plane orientation of the moments of all layers in the(kS, kB) phase diagram corresponds to a second-order

∆Φ a T TC–( )η2 Bη4.+=

ND THEORETICAL PHYSICS Vol. 101 No. 5 2005


spin-reorientation phase transition. The first problemsolved in the present paper is the derivation of anexpression for the thermodynamic potential in the formcharacteristic of the Landau theory (3) on the basis ofexpression (1) for the thermodynamic potential of asemi-infinite inhomogeneous crystal in the vicinity ofthe transition point.

The solution of this problem will allow one to makean unambiguous conclusion that the spin-reorientationtransition from a magnetic state with a uniform orienta-tion of the moments of atomic layers to a canted state inthe surface region, which corresponds to the intersec-tion of a line that separates the relevant domains in theearlier constructed (kS, kB) phase diagram, is a second-order phase transition for any values of the modelparameters kS, kB, and γ. Moreover, the Landau theoryimplies that the order parameter η that appears inexpression (3) for the thermodynamic potential is equalto zero in a high-symmetry phase and is different fromzero in a low-symmetry phase. The solution of the firstproblem formulated above allows one to answer thequestion of what is the order parameter in the continu-ous spin-reorientation transition considered and howdoes it behave with temperature. Finally, the solution ofthis problem is necessary for solving the next problem,which is the main problem of the present paper: the der-ivation of an expression for the magnetic susceptibilityof an inhomogeneous magnetic film described by ther-modynamic potential (1) in a perpendicular field in thevicinity of a continuous spin-reorientation transition.The magnetic susceptibility of a two-layer1.5Fe/Gd(0001) film is the basic physical quantity thatis measured in the experiment. However, the tempera-ture dependence of the magnetic susceptibility of aninhomogeneous 1.5Fe/Gd(0001) film has not beeninvestigated theoretically or discussed. According tothe Landau theory, the magnetic susceptibility divergesin the vicinity of a second-order phase transition, andits value in a low-symmetry phase is twice that in ahigh-symmetry phase. A question of whether theseproperties follow from model (1), which describes aninhomogeneous magnetic film, is the question that canbe answered by solving the second problem consideredin this paper. It is also of interest to consider how thechemical inhomogeneity and the bounded geometry ofthe film influence the above-mentioned features in thebehavior of the magnetic susceptibility in the vicinityof a continuous phase transition to a nonuniformdomain-wall-like magnetic state in the surface region.The solution of the second problem considered in thepresent paper will also help to answer the question as towhether it is correct, based on the divergence of themagnetic susceptibility, to classify the discovered tran-sition to a canted state as a second-order phase tran-sition.

One should bear in mind that, in a chemically inho-mogeneous semi-infinite film, the magnetic suscepti-bility is a physical parameter that depends on the index


of an atomic layer. Therefore, the derivation of anexpression for the magnetic susceptibility meets math-ematical difficulties associated with the fact that oneshould derive a separate expression for the magneticsusceptibility of each atomic layer. To our knowledge,presently there is only one publication, [18], in whichthe authors consider the influence of a perpendicularfield on an inhomogeneous semi-infinite ferromagnetdescribed by model (1). However, these authors inves-tigated only the evolution of magnetization profileswith a perpendicular field within the nonlinear mappingformulation of the mean-field theory; they did not con-sider the magnetic susceptibility in a perpendicularmagnetic field.

In Section 1, we establish a relation between the ori-entation angles of the moments of atomic layers in acanted state in the surface region of an inhomogeneoussemi-infinite ferromagnet and the orientation angles inan imaginary infinite domain wall in a homogeneousmagnet (γ = 1) obtained by an artificial continuation ofthe semi-infinite magnet beyond its surface. The solu-tion of this technical problem has been stimulated bythe fact that, to prove that thermodynamic potential (1)in the vicinity of a continuous spin-reorientation transi-tion has a form characteristic of the Landau theory (3)(which is demonstrated in Section 2), one shouldexpress each orientation angle θn in terms of the orien-tation angle θ1 of the surface layer using the so-calledequilibrium conditions. Then, one should substitute theexpressions obtained into the expression for thermody-namic potential (1). It turns out that this procedure issignificantly simplified if one uses the above-men-tioned concept of an imaginary domain wall of an infi-nite magnet and expresses the orientation angles θn ofthe moments of atomic layers in a real canted magneticstructure in the surface region in terms of fictitious ori-entation angles Θ1 of the moments of atomic layers inthe imaginary domain wall of an infinite homogeneousmagnet.

In Section 3, we derive expressions for the regularand irregular components of magnetic susceptibility ina perpendicular magnetic field as a function of theindex of an atomic layer in the system described bymodel (1). We show that, according to the Landau the-ory of second-order phase transitions, the magnetic sus-ceptibility of each atomic layer diverges at the transi-tion point and its value in a low-symmetry phase is halfthat in a high-symmetry phase. The theoretical resultsobtained are compared with experimental data. Theregular component of the magnetic susceptibility isanalyzed similarly. To conclude this section, we stressthat a direct calculation of the magnetic susceptibilityin a perpendicular field proves to be rather difficult. Atthe same time, the derivation of expressions for themagnetic susceptibility on the basis of the Landau the-ory and involving the concept of an order parametersubstantially simplifies the solution of this problem.

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834 ANISIMOV, POPOV

To conclude the Introduction, we make the follow-ing two remarks concerning formula (1) for the thermo-dynamic potential. The transition observed in theexperiment occurs at a sufficiently low temperature (atabout 230–240 K, which is by 50–60 K below the Curiepoint of Gd, equal to 292.5 K, and by 110–120 K belowthe Curie point of the surface, equal to 350 K), the mea-sured value of the magnetization of Gd layers still beingclose to the saturation value. Therefore, there are fewmagnons in the system. Thus, one can write out a mean-field thermodynamic potential by introducing the mag-netization amplitudes MB(T). The only possible alterna-tive state for the collinear states to which we restrict ourconsideration here is the Néel surface wall. Moreover,in view of the statement of the first problem (seeabove), we only focus on the very beginning of the tran-sition, when the deviation of moments from the in-plane orientation is arbitrarily small. Taking intoaccount that the transition occurs at a low temperature,we can neglect the possible angular dependence of themoment of Gd to a good accuracy. In addition, thermo-dynamic potential (1) is expressed in a discrete layer-by-layer approximation. The gradient term that entersthe well-known Ginzburg–Landau functional, which isused, in particular, for describing domain walls, arisesin (1) in the explicit form when passing to a continuumapproximation by the formula

where a is the interlayer distance. Then, we pass fromsummation to integration in (1). It can easily be shownthat the shape of a wall is described in this approxima-tion by the equation

Thus, there is no need to include the gradient term in (1)because it is already contained there in the implicitform. The continuum method yields approximateexpressions for the second- and fourth-order derivativesof the thermodynamic potential with respect to the ori-entation angle of the surface layer; these expressionsrepresent the limiting expressions for the exact formu-las obtained in this paper by a discrete method. We

mean the passage to the limit λ 1 – . Theparameter λ is introduced exclusively for convenience,to parameterize the bulk anisotropy constant kB by theformula kB = (1 – λ)2/λ. When λ is varied from zero tounity, kB varies from zero to infinity. The applicability

condition of the continuum method is given by ! 1

θn θn 1+–( )cos 1a2

2----- dθ x( )

dx--------------

x na=

2,–≈

d2θdx2--------

kB

2a2-------- 2θ x( )( ).sin=

kB

kB


and λ ≈ 1. In this case, if we pass again to the discretevariables, the shape of the domain wall is described bythe expression

As λ decreases, the shape of the wall becomes substan-tially different from that described by this expression.Therefore, it is essential to verify that the fourth-orderderivative at the transition point is positive for any pos-sible value of λ in the interval 0 < λ < 1, i.e., for anyvalue of the reduced bulk anisotropy constant kB fromzero to infinity. Thus, the results obtained below (seeSection 2) are of more general character compared tothose that would be obtained if one used the continuummethod. In spite of this fact, note that the application ofthe continuum approximation turned out to be ratherfruitful, for example, for the description and qualitativeexplanation of surface magnetism [19–21]. At the sametime, the construction of the (kS, kB) diagram of themagnetic states of a semi-infinite magnet within thecontinuum approximation, which was performedin [22, 23], proved to be qualitatively incorrect, whichwas pointed out in [15]. Here, we focus on finding outthe kind and the physical nature of a continuous spin-reorientation phase transition precisely from the view-point of the correct (kS, kB) phase diagram, which wasobtained earlier within the discrete method [15]. This isanother reason why we apply a discrete approximationin the present study.

2. DERIVATION OF EXPRESSIONSFOR THE ORIENTATION ANGLES

OF THE MOMENTS OF ATOMIC LAYERSIN TERMS OF THE ORIENTATION ANGLEOF THE MOMENT OF THE FIRST LAYER

Using the reduced anisotropy constants kS and kB (2)introduced in the previous section, we can write out thefollowing expression for the thermodynamic potentialreduced to the dimensionless form:

(4)

Searching for the state of the system described bythis thermodynamic potential, i.e., the minimization ofthis potential over each orientation angle θn , leads to

θn 2Θ1

2------

kB n 1–( )( )exptan

.arctan=

ϕ0

Φ0

JBBMB2

-----------------≡γkS

2-------- θ1 γ θ1 θ2–( )cos–sin

2=

+kB

2----- θnsin

2 θn θn 1+–( )cos– .

n 2=

∞

∑



the following infinite set of equations for the orienta-tion angles θn:

(5.1)

(5.2)

(5.3)

In the Landau theory, thermodynamic potential (3)corresponds to a nonequilibrium state until the orderparameter η reaches its equilibrium value; i.e., theorder parameter η is a variable quantity. Below, we willshow that the orientation angle θ1 of the surface atomiclayer may serve as an order parameter for a continuousspin-reorientation transition to a canted magnetic statein the surface region described by model (4). Therefore,the orientation angle θ1 of the surface is interpreted hereas a variable parameter; i.e., it must not be equal to itsequilibrium value. Hence, we will not use the equilib-rium condition (5.1). The orientation angles of all theother layers depend on the orientation angle θ1 of thesurface via the recurrence equations (5.2) and (5.3).

In accordance with the formulation of the first prob-lem of this paper, we must express each orientationangle θn in terms of the order parameter θ1 and thensubstitute the expressions obtained for these angles intothe formula for thermodynamic potential (4). However,due to the inhomogeneity of the chemical compositionof the film, γ ≠ 1, the equilibrium condition obtained bythe differentiation with respect to the orientation angleof the second layer (5.2) does not coincide with similarequilibrium conditions for all the other orientationangles (5.3). This fact substantially complicates thesolution of the first problem of the present paper. Toavoid this difficulty, we introduce the concept of animaginary domain wall in an infinite homogeneouscrystal in which the orientation angle of each atomiclayer satisfies the equilibrium conditions (5.3). In thepresent section, we find relations between real orienta-tion angles in a canted magnetic state in the surfaceregion and fictitious orientation angles in the imaginarydomain wall of an infinite homogeneous crystal, as wellas their expressions in terms of the order parameter θ1

∂ϕ0

∂θ1---------

γkS

2-------- 2θ1( ) γ θ1 θ2–( )sin 0,=+sin=

n 1,=

∂ϕ0

∂θ2---------

kB

2----- 2θ2( )sin θ2 θ3–( )sin+=

+ γ θ2 θ1–( )sin 0,=

n 2,=

∂ϕn

∂θn

---------kB

2----- 2θn( )sin θn θn 1+–( )sin+=

+ θn θn 1––( )sin 0,=

n 2.>


in the vicinity of a continuous spin-reorientation transi-tion to a canted magnetic state.

For γ ≠ 1, the profile of magnetization represents apart of a domain wall for all atomic layers except for thefirst. The equilibrium value of the orientation angle θ1

of the first layer differs from the corresponding orienta-tion angle in the imaginary domain wall, which isdenoted by Θ1. If the exchange interaction between thesurface and the adjacent subsurface layers is less thanthe exchange interaction between adjacent atomic lay-ers in the bulk of the magnet, γ < 1, then the real orien-tation angle θ1 of the surface is greater than the ficti-tious orientation angle Θ1 in the domain wall. If theexchange interaction between the surface and the adja-cent subsurface layers is greater than the exchangeinteraction between adjacent atomic layers in the bulkof the magnet, γ > 1, then the real orientation angle θ1

of the surface is less than the fictitious orientation angleΘ1 in the domain wall. Thus, in an inhomogeneousmagnetic film (γ ≠ 1), there is a jump in the equilibriumvalue of the orientation angle θ1 of the surface layerwith respect to the corresponding value of the fictitiousangle Θ1 in the domain wall of a homogeneous magnet,as illustrated in the figure. Therefore, we use the fol-lowing notation here: Θn are fictitious angles in adomain wall in an infinite homogeneous magnet, and θn

are real orientation angles in a canted magnetic state inthe surface region. For n > 1, we have Θn = θn , while for

γ < 1, θ1

γ > 1, θ1

Θ1

θn, Θn

γ = 1, θ1

–4 –3 –2 –1 0 1 2 3 4 n

Fictitious orientation angles of moments (circles) in animaginary domain wall and real orientation angles ofmoments (crosses) in a canted magnetic structure in the sur-face region as functions of the number n of an atomic layer.When γ > 1, the orientation angle θ1 of the surface layer isless than the corresponding value of the fictitious angle Θ1in the imaginary domain wall. When γ < 1, the orientationangle θ1 of the surface layer is greater than the correspond-ing value of the fictitious angle Θ1 in the imaginary domainwall. When γ = 1, the angle θ1 coincides with Θ1. For n = 2,3, 4, …, the real orientation angles θn of moments coincidewith the fictitious orientation angles Θn for any value of theparameter γ.

SICS Vol. 101 No. 5 2005

836 ANISIMOV, POPOV

n = 1, the relation between Θ1 and θ1 will be establishedbelow (see (7), (18), and (20)).

The fictitious orientation angles in the imaginarydomain wall satisfy the recurrence equation (5.3). Sincethe angle Θ1 also belongs to this domain wall, it mustobviously satisfy a similar equation

(6)

A comparison of this equation with Eq. (5.2) shows thatthe relation between the real orientation angle θ1 andthe corresponding fictitious orientation angle Θ1 in thedomain wall is given by the formula

(7)

A transition from the real orientation angle θ1 of thesurface atomic layer to the fictitious orientation angleΘ1 in the imaginary domain wall allows us to apply,instead of Eq. (5.2), Eq. (6), which is analogous toEq. (5.3); i.e., the system of recurrence equations (5.2),(5.3) becomes in a sense homogeneous.

To obtain an expansion of thermodynamic poten-tial (4) up to terms of the fourth order in each orienta-tion angle, we must first express the fictitious orienta-tion angles Θn in terms of the fictitious orientationangle Θ1 of the surface in the imaginary domain wall upto the cubic terms. To this end, we must expand the left-hand side of the recurrence equation in a series in termsof fictitious orientation angles in the imaginary domainwall,

(8)

up to the cubic terms in each orientation angle Θn . Forn ≥ 2, we will seek a solution in the form

(9)

Obviously, for n = 1, we should set

α1 = 1, β1 = 0 (10)

in this formula. Upon substituting (9) into the left-hand

side of Eq. (8) expanded up to and setting the coef-ficients at equal powers of Θ1 to zero, we obtain the fol-lowing recurrence equations for the coefficients αn

and βn:

(11.1)

(11.2)

Hereinafter, for convenience, we use the parameter λinstead of the reduced bulk anisotropy constant kB. The

kB

2----- 2θ2sin θ2 Θ1–( )sin θ2 θ3–( )sin+ + 0.=

γ θ2 θ1–( )sin θ2 Θ1–( ).sin=

kB

2----- 2Θnsin Θn Θn 1––( )sin+

+ Θn Θn 1+–( )sin 0,=

n 2,≥

Θn αnΘ1

βnΘ13

6------------.+=

Θ13

kB 2+( )αn αn 1–– αn 1+– 0, n 2,≥=

kB 2+( )βn βn 1+– βn 1–– η0αn3, n 2.≥=


relation between λ and kB is given by

(12)

As λ is increased from 0 to 1, the reduced bulk anisot-ropy constant kB decreases from +∞ to 0. Then, thecoefficient η0 can be expressed as

(13)

A solution to homogeneous equation (11.1) is given by

αn = λn – 1. (14)

By analogy with the case of inhomogeneous differen-tial equations, we will seek a solution to the inhomoge-neous recurrence equation (11.2) as a sum of the solu-tion to the corresponding homogeneous recurrenceequation and the partial solution to the inhomogeneousequation (11.2):

(15)

Substituting this expression for βn into the recurrenceequation (11.2), we obtain the following formula for

the coefficient :

(16)

It follows from boundary conditions (10) that = – .As a result, we obtain the following expression for anarbitrary fictitious orientation angle Θn in the imaginarydomain wall as a function of the first fictitious orienta-tion angle Θ1:

(17)

For small deviations of the moment vector of thesurface atomic layer from the in-plane orientation,there exists a one-to-one correspondence between thereal orientation angle θ1 and the fictitious orientationangle Θ1 introduced by formula (7). Let us find the rela-tion between Θ1 and θ1 up to the cubic terms. To thisend, we must expand both sides of Eq. (7) up to thecubic terms and apply the expression for Θ2 in theform (17). A solution is sought in the form

(18)

kB1 λ–( )2

λ-------------------.=

η0 4kB λ 1–( )3– 1 1λ---–

3

+=

= 1 λ2–( ) λ2 3λ– 1+( )

λ3---------------------------------------------------.–

βn β̃αn βαn3.+=

β

β 1 3λ– λ2+

1 λ2+---------------------------.=

β̃ β

Θn λn 1– Θ1λ2 3λ– 1+

λ2 1+--------------------------- λ3 n 1–( ) λn 1––[ ]

Θ13

6------.+=

θ1 aΘ1

cΘ13

6---------.+=



Expressions for the coefficients a and c can be obtainedafter equating the coefficients at equal powers of Θ1:

(19)

The inverse formula, which expresses the fictitiousangle Θ1 in terms of the real orientation angle θ1, isobtained analogously:

(20)

Finally, we express the fictitious angles Θn for n ≥ 2 interms of the real orientation angle θ1 of the surfaceatomic layer up to the cubic terms with the use of for-mulas (17) and (20):

(21)

3. DERIVATION OF EXPRESSION (3) FOR THE THERMODYNAMIC POTENTIAL

IN A FORM CHARACTERISTICOF THE LANDAU THEORY

According to the Landau theory of second-orderphase transitions, for a phase transition to be of the sec-ond order, it is necessary that the second derivative ofthe thermodynamic potential (3) with respect to theorder parameter η change its sign at the transition pointand that the fourth derivative (the coefficient B) be pos-itive. In this section, we will show that the thermody-namic potential (4) of an inhomogeneous magnetic filmsatisfies these conditions in the vicinity of a spin-reori-entation transition from a uniform magnetic state withthe in-plane orientation of the moments of all layers toa domain-wall-like canted state in the surface region.

Based on the results of the previous section, we canconclude that, for arbitrary values of the model param-eters λ and γ there exists a finite interval of values of thesurface orientation angle θ1 in which there exists a one-to-one correspondence between the orientation anglesθn and θ1 in a canted state in the surface region. The factthat the system is discrete is insignificant. Then, in thecanted magnetic state, one can expand the thermody-

namic potential (4) in a series in θ1 up to the terms .

a1 1 γ–( )λ–

γ-----------------------------,=

c1 λ–( )3

γ------------------- 1

γ2----- 1–

β2 1 1γ---–

,+=

β2 λ2 3λ– 1+( )λ 1 λ2–

1 λ2+--------------.–=

Θ1 ξθ1 εθ1

3

6-----, ξ+

1a---, ε c

a4-----.–= = =

Θn α̂nθ1β̃n

6-----θ1

3, α̂n+1a---λn 1– ,= =

β̂nβa3-----λ3 n 1–( ) c

a4----- β

a3-----+

λn 1– .–=

θ14


For simplicity and convenience, we decompose theexpression for the thermodynamic potential (4) into thebulk and surface parts:

(22)

(23)

These formulas show that the bulk part ∆ , whichcorresponds to a domain wall in a homogeneous mag-net, does not contain the parameter γ, which takes intoaccount the inhomogeneity of the exchange interactionin an inhomogeneous magnetic film. It neither containsthe reduced anisotropy constant kS of the surface atomiclayer, which is different from the reduced anisotropyconstant kB of the bulk layers of the magnet. Thesemodel parameters are contained in the surface part ∆ϕ0S

of the thermodynamic potential. Expanding ∆ in aseries with respect to each fictitious orientation angleΘn , substituting the expressions of Θn in terms of theangle Θ1 for Θn by formula (17), and summing the geo-metric progressions obtained, we derive the followingexpression for the bulk part of the thermodynamicpotential ∆ :

(24)

When deriving (24), we omitted all the terms that donot depend on the orientation angles. Formula (24)shows that the expression for ∆ does not containquadratic terms in Θ1. The increment of the total ther-modynamic potential can be expanded as follows. First,we should expand the surface part of the thermody-namic potential ∆ϕ0S in a series with respect to each ori-entation angle, both real and fictitious, up to thefourth-order terms. Second, we should substituteexpression (17) for the fictitious angle Θ2 in terms ofΘ1. Third, we should substitute expression (20) for thefictitious angle Θ1 in terms of the real orientation angleθ1 of the surface atomic layer into (23) and (24). As aresult, we obtain the following expression for the incre-ment of the total thermodynamic potential ∆ϕ0 as a

ϕ0 ∆ϕ̃0 ∆ϕ0S,+=

∆ϕ̃0λ 1–

2------------ Θ1sin

2 Θ1 Θ2–( )cos–=

+kB

2----- Θnsin

2

n 2=

∑ Θn Θn 1+–( ),cosn 2=

∑–

∆ϕ0S

γ kS kSC–( )2

--------------------------- θ1sin2

=

+γkSC

2----------- θ1sin

2 λ 1–2

------------ Θ1sin2

–

– γ θ1 Θ2–( )cos Θ1 Θ2–( ).cos+

ϕ̃0

ϕ̃0

ϕ̃0

∆ϕ̃0 31 λ2–

1 λ2+--------------

Θ14

4!------.≈

ϕ̃0

SICS Vol. 101 No. 5 2005

838 ANISIMOV, POPOV

function of the real orientation angle θ1 of the surfaceatomic layer:

(25)

Here, kSC is the critical value of the reduced constantof anisotropy of the surface atomic layer, whichdepends on the reduced bulk anisotropy constant kBand γ (formula (42) in [15]). The graph of the functionkSC(kB) determines the boundary, on the phase diagram(kS, kB), between the domain corresponding to a homo-geneous magnetic state with the in-plane orientation ofall moments and the domain corresponding to a cantedstate in the surface region. It follows from (25) that thesecond derivative of the thermodynamic potential withrespect to the order parameter θ1 changes its sign whencrossing the boundary between these domains on the(kS, kB) phase diagram. If the condition kS < kSC holds,then the second derivative of the thermodynamic poten-tial is negative; i.e., a canted state in the surface regionbecomes energetically favorable. When deriving for-mula (25), we used the following expression for thecritical value of the constant kSC of surface anisotropythat enters the criterion for the spin-reorientation tran-sition to a canted state kS < kSC:

(26)

The expression in square brackets in (25) is positive.Indeed, at the point of spin-reorientation transition, i.e.,at kS = kSC, the fourth derivative of the thermodynamicpotential with respect to the order parameter θ1 can berewritten as

(27)

For γ = 1, the expression for the increment of the ther-modynamic potential has an especially simple form:

(28)

If the criterion kS < kSC for the transition to a cantedstate is satisfied, then we can derive an equilibrium

∆ϕ0γ2--- kS kSC–( )θ1

2 Bθ1

4

4!-----,+≈

B ∂4ϕ0

∂θ14

-----------θ1 0=

≡ 1

a4----- 4γ kSC kS–( )a4 3

1 λ2–

1 λ2+--------------+=

+ 4 1 λ–( ) a3 1–( ) 1 λ–( )4 1 1

γ3-----–

.+

kSC1 λ–

1 1 γ–( )λ–-----------------------------.–=

B d4ϕ0

dθ14

-----------kS kSC=θ1 0=

3

a4----- λ4 1 λ2–( )

1 λ2+------------------------= =

+1 λ 1 2γ–( )–( )2 1 λ–( )2

γ3----------------------------------------------------------- 0.>

∆ϕ0 kS kSC–( )θ1

2

2!----- 3

1 λ2–

1 λ2+--------------

θ14

4!-----.+≈


value of the orientation angle of the surface atomiclayer from Eq. (25):

(29)

One can verify that this value of the order parameter θ1satisfies the equilibrium condition (5.1) up to the cubicterms. Based on the equilibrium condition (5.1) for thesurface atomic layer, one can derive an expression forthe coefficient B that coincides with the expression forthis coefficient defined by formula (25).

Thus, in the vicinity of a continuous spin-reorienta-tion transition to a canted state in the surface region, thethermodynamic potential (4) of a semi-infinite inhomo-geneous magnet has a form characteristic of the Landau

theory (3). The positiveness of the coefficient B at shows that this transition is of the second order. Thus,the spin-reorientation transition from a uniform mag-netic state with the in-plane orientation of the momentsof all layers to a canted state in the surface regiondescribed by model (4) is a physical realization of thesecond-order phase transitions described by the Landautheory.

Note that, at low temperatures, i.e., for kSC < kS, theorder parameter θ1 vanishes; an increase in temperatureleads to a continuous increase in the order parameter.Such behavior of the order parameter with temperaturecontradicts the vast majority of experimental data onsecond-order phase transitions. This makes the contin-uous spin-reorientation transition observed in the1.5Fe/Gd(0001) system a unique phenomenon. How-ever, according to the Landau theory, a decrease in theorder parameter with temperature is not a law of nature[14], because entropy increases all the same.

To conclude this section, we note that one canchoose the orientation angle of any atomic layer as anorder parameter because the orientation angles of allatomic layers are interdependent. However, the mostnatural choice of the order parameter is the orientationangle of the surface atomic layer. Obviously, any oddfunction of the surface orientation angle, say sinθ1,may also serve as the order parameter. Similar to theorientation angles, all values of sinθn are also interde-pendent; therefore, one can choose the total projectionof the moments of all atomic layers onto the normal tothe film surface as the order parameter.

4. MAGNETIC SUSCEPTIBILITY OF ATOMIC LAYERS

IN AN INHOMOGENEOUS MAGNETIC FILMIN A PERPENDICULAR MAGNETIC FIELD

In the presence of an external magnetic field perpen-dicular to the plane of the film, the expression for the

θ10( )

θ10( ) 6γ kSC kS–( )

B------------------------------.=

θ14



thermodynamic potential contains additional termscompared to expression (4) for ϕ0:

(30)

Here, h⊥ is the reduced value of the external perpendic-ular magnetic field, which is defined as

(31)

According to this definition of the parameter h⊥ , thecoefficient µ in the last term in (30) must be set equal tounity. However, it is more convenient to preserve thiscoefficient because it helps to follow up the origin ofdifferent terms when deriving the final formulas for thesusceptibility of each atomic layer in a perpendicularmagnetic field. Therefore, below we treat the parameterµ1 as the moment of the surface atomic layer and µ, asthe moment of a bulk atomic layer. Of course, in all theformulas below, we must set µ = 1.

With regard to the notations introduced, the equi-librium conditions for each orientation angle are rewrit-ten as

(32.1)

(32.2)

(32.3)

The switching on of the external field changes the ori-entation angle of the moment of each atomic layer com-pared to the equilibrium value at h⊥ = 0. In the bulk ofthe film, far from the surface, the deviation of themoment from the in-plane orientation is defined by theangle ΘB, which is independent of the index of anatomic layer:

(33)

Then, the expression for the magnetic susceptibility χBof a bulk atomic layer in a perpendicular field is given

ϕ ϕ 0 µ1h⊥ θ1sin– µh⊥ Θn.sinn 2=

∞

∑–=

h⊥MBH ⊥

JBBMB2

-----------------, µ1

MS

MB--------.= =

∂ϕ∂θ1--------

γkS

2-------- 2θ1sin γ θ1 Θ2–( )sin+=

– µ1h⊥ θ1cos 0,=

n 1,=

∂ϕ∂θ2--------

kB

2----- 2Θ2sin Θ2 Θ3–( )sin γ Θ2 θ1–( )sin+ +=

– µh⊥ Θ2cos 0,=

n 2,=

∂ϕ∂θn

--------kB

2----- 2Θnsin Θn Θn 1+–( )sin+=

+ Θn Θn 1––( )sin µh⊥ Θncos– 0,=

n 2.>

ΘBsinµh⊥

kB---------.=


by the formula

(34)

Now, we will find an expression for the magnetic sus-ceptibility of an arbitrary atomic layer in the surfaceregion near the point of a continuous spin-reorientationtransition to a canted state. In the canted state and ath⊥ = 0, the equilibrium profile with respect to the orien-tation angles represents a part of a Néel wall. Under theapplication of an external perpendicular field, the pro-file of the orientation angles represents a part of a mod-ified Néel wall. Thus, the nonequilibrium thermody-namic potential of the canted state depends not only onthe variable orientation angle θ1 of the surface, theorder parameter, but also on the external magnetic fieldh⊥ . In addition, the orientation angles of other layers,n = 2, 3, …, are determined by Eqs. (32.2) and (32.3)for a fixed value of the orientation angle of the surfaceatomic layer:

(35)

Let us show that the first derivative of the total thermo-dynamic potential ∆ϕ0 with respect to the external fieldh⊥ vanishes as the field value tends to zero. Indeed, afterthe rearrangement of terms, the expression for thisderivative for a fixed value of the orientation angle θ1 ofthe surface moment has the form

(36)

χBµ2

kB-----.=

∆ϕ θ1 h,( ) ∆ϕ0 θ1 h⊥,( ) µ1h⊥ θ1sin–=

– µh⊥ Θn,sinn 2=

∞

∑Θn Θn θ1 h⊥,( ).=

∂∆ϕ0

∂h⊥-------------

0

γ Θ20( ) θ1–( )∂Θ2

∂h⊥---------

0

sin=

+kB

2----- 2Θn

0( )( )∂Θn

∂h⊥---------

0

sinn 2=

∞

∑

+ Θn0( ) Θn 1+

0( )–( )∂ Θn Θn 1+–( )∂h⊥

---------------------------------0

sin

= γ Θ20( ) θ1–( )sin

kb

2---- 2Θ2

0( )( )sin+

+ Θ20( ) Θ3

0( )–( )sin ∂Θ2

∂h⊥---------

0

+kb

2---- 2Θn

0( )( )sin Θn0( ) Θn 1–

0( )–( )sin+n 3=

∞

∑

-+ Θn0( ) Θn 1+

0( )–( )sin ∂Θn

∂h⊥---------

0

0.=

SICS Vol. 101 No. 5 2005

840 ANISIMOV, POPOV

Here, are the equilibrium values of orientationangles for a fixed value of the orientation angle of thesurface moment in zero external field for n ≥ 2. Each setof square brackets in (36) contains the left-hand side ofthe equation for the equilibrium values of the orienta-tion angles given by (5.2) and (5.3). Hence, the switch-ing on of a weak perpendicular external field for a fixedvalue of the orientation angle of the surface momentdoes not change the total energy of the system, i.e., thesum of the exchange-interaction and anisotropy ener-gies. This means that the first term ∆ϕ0(θ1, h⊥ ) in thethermodynamic potential (35), being expanded in aseries with respect to the field h⊥ , does not contain aterm linear in h⊥ .

The magnetic susceptibility of each atomic layer ina canted state in the surface region represents a Laurentseries in the parameter (kSC – kS) ≡ kS:

. (37)

Below, we determine only the first two terms in thisseries: the irregular term An/∆kS and the constant Bn . Inthe previous section, we obtained an expression for theexpansion of the thermodynamic potential ϕ0(θ1, 0) inthe orientation angle θ1 of the surface moment up to thefourth-order terms (25). Here, we must generalize thisexpansion to the case of a nonzero perpendicular exter-nal magnetic field h⊥ . To find an expression for the sus-ceptibility in a canted state (a low-symmetry phase)with regard to the constant Bn in series (37), we shouldexpand the second and third terms in (35) with respectto the orientation angle θ1 of the surface moment up tothe cubic terms in θ1. This can be done by formula (17),in which the orientation angles Θn ≡ θn for n > 1 areexpressed in terms of the fictitious orientation angle Θ1of the surface in the imaginary domain wall, as well asby formula (20), in which the fictitious orientationangle Θ1 of the surface moment is expressed in terms ofthe real orientation angle θ1 of the surface moment in acanted state:

(38)

As a result, a generalized expression for the expansionof the thermodynamic potential in the orientation angleθ1 of the surface moment can be represented as

(39)

Θn0( )

χn

An

∆kS--------- Bn Cn∆kS Dn∆kS

2 …+ + + +=

h⊥ θn0( )( )sin

n 2≥∑ γkSC

kb-----------θ1h⊥– D

θ13

6-----h⊥ ,+≈

Dβ 1–( )λ3

1 λ3–( )a3------------------------

λ1 λ–( )a3

---------------------- ca--- β+

.–=

∆ϕ θ1 h⊥,( )γ kS kSC–( )θ1

2

2--------------------------------

Bθ14

4!---------+≈

– µ1h⊥ 1µγkSC

µ1kb---------------–

θ1 µ1h⊥θ1

3

6----- µ

µ1-----D 1–

.–


The perpendicular component of the moment of thesurface atomic layer can be expressed as µ1⊥ = µ1sinθ1.This formula implies that the susceptibility χ1 of thesurface atomic layer can be represented as

(40)

The differentiation of thermodynamic potential (39)with respect to the orientation angle θ1 of the surfacemoment followed by the differentiation with respect tothe field h⊥ leads to an equation in the parameter η1 thatgives an expression for the susceptibility χ1 of the sur-face atomic layer. The solution of this equation and thesubstitution of the obtained expression for η1 into theformula (40) for χ1 leads to the following expression forthe susceptibility of the surface atomic layer in thevicinity of the spin-reorientation transition to a cantedstate:

(41.1)

(41.2)

When deriving this expression, we used expression (29)for the equilibrium orientation angle of the moment ofthe surface atomic layer in zero external magnetic fieldh⊥ = 0.

Now, we pass to the determination of the suscepti-bilities χn of other layers, n ≥ 2. According to the defi-nition of the perpendicular component of the momentof the nth atomic layer, µn⊥ = µsinΘn , the expression forthe susceptibility of the nth atomic layer in a perpendic-ular field h⊥ can be represented as

(42)

According to the Landau approach, which underlies thepresent study, the orientation angle Θn (n = 2, 3, 4, …)of the moment of an arbitrary atomic layer is a functionof the order parameter, i.e., the orientation angle θ1 ofthe surface moment, and the external perpendicular

χ1∂µ1⊥

∂h⊥-----------

h⊥ 0=

µ1 θ10( )( )η1,cos= =

η1∂θ1

0( )

∂h⊥-----------

h⊥ 0=

.=

χ1

µ12 1 µγkSC/µ1kB–( )

γ kS kSC–( )------------------------------------------------, kS kSC,>=

χ1

µ12 1 µγkSC/µ1kB–( )

2γ kSC kS–( )------------------------------------------------=

+32---

µ12

B----- µ

µ1----- D

γkSC

kB-----------+

2– ,

kS kSC.<

χn µ Θn h⊥ 0=dΘn

dh⊥----------

h⊥ 0=

cos=

≡ µ Θn h⊥ 0= ηn.cos



field h⊥ : Θn = Θn(θ1(h⊥ ), h⊥ ). According to thisapproach, the derivative of the orientation angle Θn

with respect to the field for h⊥ = 0 and the equilibrium

value of the order parameter can be expressed as

(43)

According to expansion (21), the parameter νn isdefined by the formula

(44)

The partial derivatives satisfy the system of recur-rence equations obtained by differentiating Eqs. (32.2)and (32.3) (n ≥ 2) with respect to the field h⊥ at h⊥ = 0

and θ1 = . The superscript in points to the factthat this partial derivative is calculated at the equilib-rium value of the order parameter, the orientation angleof the moment of the surface atomic layer:

(45.1)

(45.2)

It is important that, when calculating the partial deriva-tive of the orientation angle Θn = Θn(θ1(h⊥ ), h⊥ ) withrespect to the field h⊥ , one should assume that the sec-ond term in (43), the order parameter θ1, is constant.This is why the system of equations (45) for the partial

derivatives does not contain the parameter η1 in thepenultimate term in (45.1). For the same reason, thesystem of equations (45) does not contain the partialderivative of the first equation (32.1) with respect to thefield h⊥ .

When determining the susceptibility χn in a uniformstate with the in-plane orientation of the moments of alllayers, kS > kSC, the orientation angle of each atomiclayer, including the surface layer, must be set equal to

θ10( )

ηndΘn θ1 h⊥( ) h⊥,( )

dh⊥---------------------------------------- h⊥ 0=

θ1 θ10( )=

=

= ∂Θn

∂θ1---------

∂θ1

∂h⊥--------- h⊥ 0=

θ1 θ10( )=

∂Θn

∂h⊥--------- h⊥ 0=

θ1 θ10( )=

+ νnη1 un0( ).+=

νn∂Θn

∂θ1--------- h⊥ 0=

θ1 θ10( )=

α̂n

β̂n θ10( )( )2

2---------------------.+= =

un0( )

θ10( ) un

0( )

kb 2Θ2u20( ) Θ2 Θ3–( ) u2

0( ) u30( )–( )cos+cos

+ γ Θ2 θ10( )–( )u2

0( ) µ Θ2cos–cos 0,=

n 2,=

kb 2Θnun0( ) Θn Θn 1+–( ) un

0( ) un 1+0( )–( )cos+cos

+ Θn Θn 1––( ) un0( ) un 1–

0( )–( )cos µ Θncos– 0,=

n 2.>

un0( )


zero, because it is these values of the orientation anglesthat correspond to the ground state under the conditionkS > kSC . In this particular case, the system of equa-tions (45) takes the form

(46)

A solution to this system of recurrence equations isgiven by

(47)

When determining the susceptibility χn in a cantedstate in the surface region (kS < kSC), the orientationangle Θn (n ≥ 2) of the moment of each atomic layer in

the system of equations for the parameter can beexpressed in terms of the orientation angle of themoment of the surface atomic layer by (21). As a result,the solution to system of equations (45) can beexpanded in a series in θ1. Since Eq. (45) contains onlycosines, this expansion contains only even powersof θ1:

.

According to (42) and (43), taking into account the qua-

dratic term bn( )2 in this expansion implies that thisterm appears in the expression for the susceptibility.

However, in view of (29), ( )2 ∝ ∆kS. Thus, takinginto account the quadratic term in the expansion

implies that the term proportional to ∆kS in expres-sion (37) for the susceptibility χn should be taken intoaccount. However, as pointed out above, in the presentpaper, we determine only the first two terms in the Lau-rent series expansion in ∆kS (37). Therefore, whendetermining the susceptibility in a canted state in thesurface region (kS < kSC), system of equations (45) for

the parameters is again transformed to system of

equations (46) with a similar solution for definedby formula (47). In other words, in both cases, kS < kSCand kS > kSC, system of equations (45) is reduced to sys-tem of equations (46).

Using (42)–(47) and (29), we obtain the followingexpressions for the susceptibility of the nth atomic

kb 1 γ+ +( )u20( ) u3

0( )– µ, n 2,= =

kb 2+( )un0( ) un 1+

0( ) un 1–0( )+( )– µ, n 3.≥=

un0( ) µ

kb---- 1 α̂n–( ), n 2.≥=

un0( )

un0( ) an bn θ1

0( )( )2 …+ +≈

θ10( )

θ10( )

un0( ) an bn θ1

0( )( )2 …+ +≈

un0( )

un0( )

SICS Vol. 101 No. 5 2005

842 ANISIMOV, POPOV

layer to the left and right of the transition point, respec-tively:

(48.1)

(48.2)

The first term in (48.1) and (48.2) represents theintrinsic susceptibility of the homogeneous substrate(n ≥ 2) perturbed by the presence of the surface atomiclayer of different nature. The summation of the firstterm over the layers (n = 2, 3, …, N) yields the follow-ing result for this part of the susceptibility of the sub-strate:

(49)

It follows from formula (49) that, in the case of γ = 0,which means the absence of the surface atomic layer,the susceptibility of each layer is equal to its bulk value( = 0). Conversely, the presence of the surfaceatomic layer (γ ≠ 0) with the negative anisotropy con-stant (kS < 0) reduces the susceptibilities of layers in thesubsurface region, because a surface atomic layer withnegative anisotropy constant favors the perpendicularorientation of the moments of atomic layers withrespect to the plane of the film. This effect is analogousto the application of a local perpendicular magneticfield to the second atomic layer.

The second term in (48.1) and (48.2) is the so-calledirregular part of the susceptibility, which diverges at thepoint of the spin-reorientation transition. The expres-sion for this term shows that the irregular part of sus-ceptibility in a canted state is half the irregular part ofsusceptibility in the uniform magnetic state with the in-plane orientation of the moments of all atomic layers.This result agrees with the results of the Landau theory,in which the susceptibility of a low-symmetry phase ishalf the susceptibility of a high-symmetry phase. Inboth (48.1) and (48.2), the irregular part of susceptibil-ity decreases to zero as λn (0 < λ < 1) as the index ofatomic layer increases; this result points to the fact thata continuous spin-reorientation phase transition occursprecisely in the surface region. The summation of the

χnµ2

kB----- 1 α̂n–( )

µµ1 1 µγkSC/µ1kB–( )γ kS kSC–( )

----------------------------------------------------α̂n,+=

kS kSC,>

χnµ2

kB----- 1 α̂n–( )

µµ1 1 µγkSC/µ1kB–( )2γ kSC kS–( )

----------------------------------------------------α̂n+=

+3µµ1

2B------------ µ

µ1-----D 1–

α̂n

+ 1µγkSC

µ1kB---------------–

β̂n α̂n3–[ ] ,

kS kSC.<

χsubµ2

kb-----N

µ2

kB-----

γkSC

kb-----------.+=

α̂n


irregular parts of the susceptibility over all layers,including the surface layer, yields the following result:

(50.1)

(50.2)

In a real experiment with a 1.5Fe/Gd(0001) film, themeasurement of the susceptibility in a perpendicularfield actually reduces to the measurement of the spinpolarization of secondary electrons that are knockedout from the surface region in a short period afterswitching off the external field. In the simplest approx-imation, we can assume that the contribution of individ-ual atomic layers to the resulting signal exponentiallydecreases with increasing layer number. It is obviousthat the summation of the irregular parts of susceptibil-ity (48.1) and (48.2) with an exponential factor givesrise to identical factors on the right-hand sides of (50.1)and (50.2). In other words, this procedure does notremove the divergence at the transition point; neitherdoes it lead to other significant changes in the structureof formulas (50.1) and (50.2). Therefore, the identifica-tion of a continuous spin-reorientation transition to acanted state as a second-order phase transition by thedivergence of the susceptibility at the transition point[8] is justified.

Finally, the summation of the last terms in (48.2)over the indices of atomic layers of the substrate withregard to a similar term in (41.2) for kS < kSC yields thefollowing expression for this component of the regularpart of susceptibility:

(51)

This term, just like the subsequent terms in expansion (37)

that are proportional to ∆kS, ∆ , …, makes an addi-tional contribution to the asymmetry in the behavior ofthe susceptibility in the vicinity of the spin-reorienta-tion transition point, in addition to the well-knownasymmetry of the irregular part of susceptibility, whichis less by a factor of 2 in the low-symmetry case.

χ irreg kS kSC>( ) χn irreg, kS kSC>( )n 1=

∞

∑=

= µ1

2 1 µγkSC/µ1kB–( )2

γ kS kSC–( )--------------------------------------------------,

χ irreg kS kSC<( ) χn irreg, kS kSC<( )n 1=

∞

∑=

= µ1

2 1 µγkSC/µ1kB–( )2

2γ kSC kC–( )--------------------------------------------------.

χreg

3µ12

B-------- µ

µ1-----D 1–

1µγkSC

µ1kb---------------–

,=

kS kSC.<

kS2



ACKNOWLEDGMENTSThis work was supported by NATO, project

no. PST.CLG.979374.We thank D.P. Pappas for help in formulating the

problem analyzed in this study and discussion ofresults.

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Translated by I. Nikitin

SICS Vol. 101 No. 5 2005

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Continuous spin-reorientation phase transition in the surface region of an inhomogeneous magnetic film