Journal of Magnetism and Magnetic Materials 183 (1998) 283—291On tilted magnetization in thin filmsL. Udvardi, R. Király, L. Szunyogh*, F. Denat, M.B. Taylor, B.L. Györffy,B. U jfalussy, C. Uiberacker Department of Theoretical Physics, Institute of Physics, Technical University of Budapest, Budafoki u& t 8, H-1521 Budapest, Hungary Institut fu( r Technische Elektrochemie, Technical University of Vienna, Getreidemarkt 9, A-1060 Vienna, Austria H.H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol BS8 1TL, UK Research Institute for Solid State Physics, Hungarian Academy of Sciences, PO Box 49, H-1525 Budapest, HungaryReceived 25 October 1995; received in revised form 18 July 1997AbstractWe suggest a new mechanism for explaining the tilt of the magnetization away from the surface normal in certainmagnetic ultra-thin films. Our arguments are based on a simple classical spin Hamiltonian in which the magnetocrystallinesurface anisotropy is described by H "! λ (s )# γ (s ), where λ and γ are non-negative phenomenologicalconstants, s denotes the z-component (normal to the surface) of the spin at the site labelled by i. In this paperwe study only the ground state. In contrast to the usual explanation which attributes the experimentally observed tiltedmagnetization to the fourth-order term involving γ , we show that the second-order term alone can lead to this interestingphenomenon. Our explanation implies that the magnetization of the successive layers are not collinear. As an illustrationof our arguments we discuss the experimentally observed orientational transition of the Co/Au(1 1 1) system inquantitative details. 1998 Elsevier Science B.V. All rights reserved.PACS: 75.10.Hk; 75.25.#z; 75.30.Gw; 75.30.Kz; 75.30.Pd; 75.50.Rr; 75.70.FrKeywords: Reorientation transition; Anisotropy — surface; Exchange coupling — dipolar; Non-collinear magnetism1. IntroductionAs is well known by now, ultra-thin films of Fe,Co and Ni on non-magnetic substrates or sandwichedbetween non-magnetic metals, such asCu, Ag or Au are frequently magnetized alongout-of-plane directions, whereas on the basis of* Corresponding author. Tel.: #36 1 463 4109; fax: #361 463 3567; e-mail: firstname.lastname@example.org arguments one would expect theirmagnetization to lie in-plane . Currently, thissurprising phenomenon is at the centre of muchexperimental [1—9] and theoretical [10—15] attention.In this paper we shall study two mechanismswhich can give rise to a ground-state magnetizationtilted at an angle θ with respect to the surfacenormal of the film.This problem is of interest because in manycases, such as Au/Co/Au or Cu/Fe/Cu sandwiches[4—6], where the ground state of a monolayer is0304-8853/98/$19.00 1998 Elsevier Science B.V. All rights reserved.PII S0304-8853(97)01084-6
284 L. Udvardi et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 283—291perpendicularly magnetized and the rotation inplane,as further layers are added, takes place graduallywith the tilt angle θ changing continuouslyfrom θ"0atN"1toθ"π/2 at the critical thicknessof N layers. In the face of it, this may bea textbook example of the well-known orientationaltransition from one magneto-crystalline easyaxis (perpendicular) to another one (in-plane) .Indeed, this is the view taken by all the authors whohave discussed the matter so far [4—7,12]. In whatfollows, we shall suggest an alternative explanationwhich implies a physical picture very different fromthat of the conventional argument.As the others [12,15] we shall study a model ofclassical vector spins μ "μ s (s "1) localized atthe lattice sites labelled by i. It is described by theHamiltonianH"! 1 2 J s ) s # 1 2 ω s ) s !3 (s ) r )(s ) r )r r ! λ (s)#γ (s), (1) where the first term is an exchange interactionenergy, the second one is due to the dipolar interactionbetween the spins while the third and thefourth terms describe the magneto-crystalline anisotropy.To simplify matters we shall assume thatJ "J for all nearest neighbor sites and zero otherwise.Similarly, we shall neglect the variation of themagnetic moment μ from site to site, even near thesurface, and adsorb it into the dipolar couplingconstant ω "ω"(μ /4π)μ/a, where a is the (in-plane) lattice constant. Consequently, the magnitudesof the position vectors r are measured inunits of a. The single-site magneto-crystalline anisotropyterms have been added to describe theeffects of spin—orbit coupling . Due to the reducedsymmetry at the surface (interface), the coefficientsλ are much larger on the surface (interface)than in the bulk . Therefore, we shall assignfinite values to them only on the surface (interface)layers. On the contrary, when not zero γ will be taken to be the same on all layers. An example ofthe geometries of interest is illustrated in Fig. 1.Fig. 1. Scheme of a thin magnetic film on a substrate displayingthe convention used for the orientations of magnetization.Of course, the above model is only a caricature ofthe real physical situation which concerns itinerantelectrons. Nevertheless, although the first principlesaccount of magneto-surface anisotropy by Szunyoghet al.  could address most of the problemswe shall investigate in this paper, for the point wewish to make, the above model will suffice.To illustrate the dilemma, let us assume that allthe spins are parallel and make an angle θ with thesurface normal. Furthermore, let us denote the energyof such a configuration for N (100) layers of anFCC lattice by E (θ) and seek the ground state ofthe classical Hamiltonian (1) — neglecting for themoment the last term — by minimizing E (θ) with respect to θ. For N"1E (θ)"!2J! Aω#(Aω!λ) cos(θ), (2) where A is the so-called dipolar Madelung constant. Evidently, for λ'Aω the ground state correspondsto θ"0, while for λ(Aω to θ"π/2.For N'1, E (θ) can also be readily determined,and — as it will be shown in Section 3 — yieldsa ground-state orientation θ"0 for N(N and θ"π/2 for N(N , with no intermediate tiltingangle θ in between. Thus the question arises: ‘‘whatdoes give rise to a tilt?’’As discussed by Landau and Lifshitz , forbulk magnets with two symmetry-determined easyaxis one can predict a magnetization which pointsat an intermediate direction between them if one
L. Udvardi et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 283—291 285includes a contribution to the spin Hamiltoniananalogous to the fourth-order term in Eq. (1). Indeed,it can be readily seen that for N"1 and γO0the minimum in the energy occurs at a tilted angleθ"arccos((λ! Aω)/2γ). Thus, it is natural thatChappert and Bruno  and Jensen and Bennemann explained the observed tilt for themagnetic thin films by invoking such higher-ordercontribution to the anisotropy energy. The principleaim of the present paper is to point out thatthis is not the only mechanism for the tilted magnetizationin the ground state. In fact, as we shallshow presently, the above solution of the energyminimization problem to find the ground state isincorrect and misleading. If we allow the spin orientationto vary from layer to layer there is anotherstationary state whose energy is lower than that ofthe uniform configuration discussed above and correspondsto an average orientation tilted with respectto the surface normal even when γ"0. Thepossibility of such non-collinear ground-state spinconfiguration in magnetic thin films was first notedby Mills [18,19]. Here we investigate in more detailthe circumstances when it can arise. To illustratethe mechanism at work, in Section 2 we study thecase of a bilayer. In Section 3 we investigate theorientational transition from perpendicular to inplanemagnetization with increasing N, while inSection 4 the effect of the fourth-order anisotropyterm is studied. Quantitative fits to experimentaldata for the Co/Au(1 1 1) system are performed inthe last section stressing the importance of noncollinearityeven in the presence of fourth-orderanisotropy (γO0).2. The case of a bilayerConfining ourselves to spin states in which thespins are parallel in a given layer but their orientationsmay differ from layer to layer, the energy ofN layers per 2D unit cell implied by Eq. (1) can bewritten asE (θ , θ ,2, θ )"! (Jn #A ω) sin(θ ) sin(θ )! (Jn !A ω) cos(θ ) cos(θ )! λ cos(θ ) # γ cos(θ ), (3) where n is the number of nearest neighbors ofa site in layer p within layer q and A is the dipolecoupling constant determined by the lattice geometry.In order to find the extremal values of theenergy, Eq. (3), as a function of the set of angles(θ , θ ,2, θ ) we seek the solutions of the Euler— Lagrange equations:E (θ , θ ,2, θ ) "! (Jnθ #A ω) cos(θ ) sin(θ )# (Jn !A ω) sin(θ ) cos(θ )#2λ sin(θ ) cos(θ ) !4γ sin(θ ) cos(θ )"0. (4)The aim of this and the forthcoming section is toelaborate on the observation that allowing differentspin orientations in different layers naturally yielda solution to the above equation which correspondsto an average magnetization tilted withrespect to the surface normal even without requiringthe presence of the fourth-order anisotropyterm in Eq. (3) [15,18,19]. Consequently, in thepresent (and also in the next) section γ is chosen tobe zero. Noting that many of our general conclusionsare also valid in the multilayer cases, wenow confine ourselves to the case of a bilayer. Inparticular, we will be looking for the set of anisotropyconstants (λ , λ ) at given J and ω for whichtilted average magnetization can occur.To begin with, we observe that the uniform perpendicularand in-plane magnetizations, θ "θ "0 and θ "θ "π/2, respectively, satisfy Eq. (4).Interestingly, along the line in the parameterspace (λ , λ ),λ #λ " (A #A )ω, (5)the energies of the two solutions are degenerate.Below this line the easy axis lies in the plane, whilst
286 L. Udvardi et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 283—291above it is perpendicular to it. Clearly, if the magnetizationis to change continuously across thisline, solutions corresponding to tilted magnetizationsshould exist in its vicinity. An efficient wayto investigate these is to choose a pair of anglesθ* , θ* and determine the parameters λ , λ by demandingthat θ* and θ* satisfy Eq. (4). Substitutingthese parameters into Eq. (3), we can easily expressthe energy of the tilted solutions as a function of theorientations in the two layers:E (θ* , θ* )"!(Jn # A ω)! (Jn # A ω) sin(θ* )sin(θ* ) #sin(θ* )sin(θ* ) . (6)Note that the position of the minimum, θ* and θ* ,as well as the minimum energy E (θ* , θ* ) are functionsof the parameters J, ω, λ and λ . Evidently,for θ* Oθ* the average magnetization is tilted withrespect to the surface normal. It is reassuring tonote that for no interaction between the two layers,i.e., for n "0 and A "0, and on substitutingn "4 for FCC (1 0 0) layers as well as abbreviatingA by A, Eq. (6) gives the double of the energyin Eq. (2) for θ"π/2.By using Eq. (6) we now can compare the energiesof the different types of solutions, i.e., those ofthe collinear (θ "θ "0 or π/2) and the noncollinearones. After straightforward algebra wefindE (θ "θ "0)!E (θ* , θ* )" (n J!A ω) cos(θ* )cos(θ* ) #cos(θ* )cos(θ* ) !2 *0,E θ "θ " !E (θ* , θ* )" (n J# A ω) sin(θ* )sin(θ* ) #sin(θ* )sin(θ* ) !2 *0.Remarkably, whenever they exist, the energy ofnon-collinear states is always below or equal to theenergies corresponding to the perpendicular or inplanemagnetizations.(7)(8)Of course, there are regions in the parameterspace where θ "θ "0orπ/2 are the only solutions.It is particularly interesting to investigate theboundaries of such regions in the λ versus λ planefor fixed values of the parameters J and ω. Itisstraightforward to show that the lower and upperbound of the tilt area are given by(2λ !Jn !( A # A )ω)(2λ !Jn !( A # A )ω)"(Jn !A ω),and(2λ #Jn !( A #A )ω)(2λ #Jn !( A #A )ω)"(Jn # A ω), (10)respectively. The derivation of the above equationswith a complete study of the magnetic ground stateof the model will be published elsewhere. Herewe restrict our considerations to the evolution ofthe tilt in the λ versus λ parameter space. Theground-state phase diagram is shown in Fig. 2 forω"0.1 as measured in units of J. We note that thearea of the tilt magnetization is getting larger asω is increasing. Moreover, as the dipolar couplinggoes to zero (ωP0) the upper and lower bound ofthe tilt zone tends to the line defined in Eq. (5) andwe get back the Ne` el model with ferromagneticground state. Since the exchange coupling J isgenerally much larger than the strength of thedipole—dipole interaction ω as well as the magnitudesof the anisotropy term λ , the orientations ofthe spins in the different layers tend to be almostparallel, i.e., the difference between the angles of thetilted solutions will be small, typically less than0.1 rad as was discovered by Mills et al. [18,19]. Itis also worthy to mention that collinear solutionother than those perpendicular or parallel to thelayer can exist only for λ "λ " (A #A )ωwhere, however, the orientations of the magnetizationare undefined (critical point).3. MultilayersAlthough the mathematics is somewhat morecomplicated, the above discussion can be repeated(9)
L. Udvardi et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 283—291 287Fig. 2. The ground-state phase diagram of a bilayer on the λ /ω versus λ /ω plane for ω"0.1 (in units of J) and γ "γ "0. Schematicpictures of the spin directions in the three characteristic regions of the phase diagram are shown on the right-hand side. The dashed lineis defined by Eq. (5).for N ('2) layers starting from Eq. (4). In particular,it can be shown that if a non-collinear solution(θ* , θ* ,2, θ* ) exists for the multilayersystem, its energy is always smaller than thosecorresponding to the ferromagnetic states beingmagnetized perpendicular or parallel to the surface.In order to dramatize the effect we are considering,the anisotropy constants of the inner layers(λ , λ ,2, λ ) have been chosen to be zero andas in Section 2 the ground-state phase diagrams arepresented in the λ versus λ parameter space. Thischoice is consistent with the results of first-principlescalculations [17,20] which generally predictedlarge magnetic anisotropy at the surface offerromagnetic films, or rather at the interface of thefilm and the substrate, while yielding small valuesin between. Note that in order to keep correspondencewith the previous section we continue to usethe notations λ and λ for the two non-vanishinganisotropy constants.Since the region where tilted solutions exist isexpected to separate those of the uniform in-planeand perpendicular magnetizations it has to be closeto the line along which the energy of these twosolutions are equal,λ #λ " A ω, (11) where the right-hand side is an obvious generalizationof that of Eq. (5). Since the dipolar Madelungconstants A , which depend only on the actuallayer geometry, rapidly decrease with an increasingvalue of p!q , the right-hand side of Eq. (11)increases monotonously with N. Therefore, fora given set of parameters λ , λ and ω Eq. (11) clearly determines the value of N where — as mentionedin Section 1 — within the subset of collinearmagnetic states a first-order transition from perpendicularto in-plane magnetization occurs. As inthe case of Eqs. (9) and (10) the phase boundaries inthe λ versus λ plane can be determined yielding a phase diagram for different multilayers as shownin Fig. 3 for ω"0.01. Evidently, the larger thenumber of layers (N) the larger the area of the tiltzone. As far as the N dependence of the orientationof the magnetization for a given set of (λ , λ )is concerned, there are two possibilities: the point(λ , λ ) is outside or inside one of the tilt zone. In
288 L. Udvardi et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 283—291Fig. 3. The ground-state phase diagram of multilayers with N"2—5 on the λ /ω versus λ /ω plane for ω"0.01 (in units of J) and γ "0 (p"1,2,N). The average angle versus N phase diagrams are depicted for the points A, B and C on the right-hand side (see in thetext).the first case (point A in Fig. 3) there is a discontinuousreorientation transition as the number oflayers exceeds a critical value N . In the second case(point B in Fig. 3) at the critical thickness, the point(λ , λ ) is lying in the tilt zone of which, the layeraveragedmagnetization will have an intermediateangle and mimic a continuous transition. Ofcourse, since the physical values of N form a discreteset the word ‘continuous’ in the present contextneeds to be interpreted with appropriate care.Experimentally, during the growth process partiallyfilled layers can be achieved representinga non-integer value for N, however, a theoreticaldescription of such cases is beyond the scope of thepresent work. Interestingly, the tilt regions correspondingto different values of N can overlap (seepoint C in Fig. 3). This implies that for a given setof parameters from such a region there exist (atleast) two different thicknesses of the multilayerwith tilted ground-state magnetization.4. The effect of the fourth-order termAbove, we have argued that the fourth-orderterm in Eq. (1) is, in principle, not necessary toexplain the presence of tilted magnetizations detectedin several experiments [4—6]. In what followswe reintroduce the fourth-order term into the theoryand study its effect on the ground-state phasediagram. This will allow us to make contact withthe conventional arguments of Refs. [4,12].An estimate, based on very general symmetryarguments, of the fourth-order anisotropy leads tothe conclusion that it is comparable on all layersand also not much different from its bulk value[4,12]. Therefore, to simplify the foregoing discussionwe let γ "γ for all layers p. At the sametime, as before, we take λ ’s to be non-zero only onthe top and the bottom layers. As it turns out, theextra term γ cos(θ )inE (θ ,θ ,2,θ ) doesnot complicate matters too much. For instance, theline along which the energies of the uniform perpendicularand parallel magnetizations are equal,i.e. the generalization of Eq. (11), is given byλ #λ " 3 A ω#Nγ. (12) 4 Whilst it is not surprising that the present generalizedtheory leads to a ground-state magnetizationwhich is tilted towards the surface normal, it is
L. Udvardi et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 283—291 289unexpected that, again, the absolute minima referto non-collinear solutions. To demonstrate thisstatement we generalize Eqs. (7) and (8) for a bilayerin the presence of fourth-order anisotropyterms to yieldE (θ "θ "0)!E (θ*, θ* ) "(n J!A ω) cos(θ* )) #cos(θ* )cos(θ* ) !2 #γ sin(θ* )#γ sin(θ* )*0, (13)E (θ "θ "π)!E (θ*, θ* ) "(n J#A ω) sin(θ* )) #sin(θ* )sin(θ* ) !2 #γ cos(θ* )#γ cos(θ* )*0. (14)Introducing the notation λ "λ #2γ cos(θ )we find the same boundaries for λ and λ as forλ and λ in Eqs. (9) and (10). Since at the boundarybetween the areas of the tilted and in-plane magnetizationson the ground-state phase diagramθ tends to π/2, consequently, λ tends to λ , thefourth-order anisotropy does not affect the shape ofthe lower boundary of the tilt region. However,at the upper boundary θ P0 and, therefore,λ Pλ #2γ. As a result, this boundary is simplyshifted by 2γ on the phase diagram as compared tothe case of γ"0. As mentioned in Section 1, ina monolayer the fourth-order anisotropy causesa tilting of the magnetization. In a multilayer this isnot necessarily the case. However, it makes the tiltarea broader relative to the case of γ"0. Thisbroadening of the tilt region is shown in Fig. 4 fora bilayer.5. Application to the Co/Au(1 1 1) systemand conclusionsHaving discussed the possibility of spin tilting ingeneral, the question arises whether measurementson real systems can be reproduced by the spinHamiltonian (1) with a reasonable set of parameters.The most appealing system for sucha study is the Co/Au(1 1 1) system where both thegeometrical and the magnetic structure is simple,and the intermediate transition region, where spintilting is found experimentally, is relatively broad.In Fig. 5 the experimentally measured mean orientationsof magnetization in the Co /Au(1 1 1)thin films [5,6] are recorded. Characteristically, upto a film thickness of 3 layers the magnetizationpoints normal to the surface, while above 6 layers itlies parallel with it. In between there is a continuousFig. 4. The ground-state phase diagram of a bilayer on the λ /ωversus λ /ω plane for ω"0.04 and γ"0.01 (both in units of J).As a comparison, the upper boundary corresponding to the caseof γ"0 is shown by a dashed line. The region where tiltedcollinear spin-states, although not necessarily ground states,exist is indicated by shading.Fig. 5. Mean angles of the magnetization in the Co/Au(1 1 1)system as deduced from Refs. [5,6]. The open circles indicatethat, most likely because of the lower Curie temperature, at300 K no magnetization for N(2 was found.
290 L. Udvardi et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 283—291transition in the angle of magnetization. Since thepresent formalism accounts merely for perfect,two-dimensional translation invariant layers, fromthe set of points we used in our numerical fit weexcluded the value for the non-integer coverage(N"4.5) also given in Refs. [5,6].Because of the small number of data points, wehad to decrease considerably the number ofparameters in the spin Hamiltonian. Some ofsuch restrictions were introduced before, namely,J "J, ω "ω, λ "0 for p"2,2,N and, in thepresence of fourth-order anisotropy, γ "γ for all p.Additionally, we choose λ at only one terminallayer to be non-zero. This approach is partiallysupplied by our experience of previous first-principlesstudies predicting generally much larger anisotropyat the interface between the magnetic filmand the non-magnetic substrate than at the surface[17,20]. Therefore, in total there remained fourindependent parameters which define our modelHamiltonian (1): J, ω, λ and γ. If, however, one isinterested only in the angles which characterize theground states, but not in the ground-state energies,it is sufficient to deal with three ratios J"J/ω, λ"λ/ω and γ"γ/ω. It should also be notedthat we calculated the dipole—dipole Madelungconstants A for an FCC(1 1 1) geometry.To highlight the role of various factors whichdetermine the tilt angle, we used three differentschemes to fit the experimental data: (A) withoutfourth-order anisotropy (γ"0), (B) with fourth-orderterm and also, (C) based on the collinear tiltedstate for γO0. It is noteworthy that whilst withinscheme C the parameters λ and γ unambiguouslydetermine the magnetic orientation, a numerical fitof the proper, non-collinear ground states of theHamiltonian (1) to a measurement of the magneticdirections also determines the Heisenberg exchangeparameter J.The parameters as obtained from the differentkinds of numerical fits are listed in the upper part ofTable 1. In all the three cases the mean anglespresented in Fig. 5 could be recovered with anaccuracy less than 0.02 rad. Apparently, the valueof J for case A is seven times smaller than that forcase B. In view of the previous sections, this in fact,is not surprising, since for γ"0 the only way tobroaden the tilting regions corresponding toTable 1Parameters fitted to the measured mean directions of magnetizationin the Co/Au(1 1 1) system. Three different models wereused: non-collinear ground states without (A) and with (B)fourth-order anisotropy term, and collinear tilted state in thepresence of fourth-order anisotropy term (C). Note that in caseC there is no information on the exchange parameter J. Thestrength of the dipolar interaction was fixed to ω"6.63 μeV (seein the text)A B CJ"J/ω 179 1250 —λ"λ/ω 37.3 43.0 44.3γ"γ/ω — 1.25 1.56J (meV) 1.19 8.29 —λ (meV) 0.247 0.285 0.294γ (μeV) — 8.29 10.34K (mJ/m) 0.554 0.638 0.657K (kJ/m) — 79.1 98.8a given thickness of the film — thus to achieveoverlap between different layer thicknesses (seeFig. 3) — is to increase the ratio ω/J or equivalentlydecrease J. Since for B and C the competition ofthe second and fourth-order anisotropies broadensthe tilting regions, the value of J can be considerablyincreased in these cases. A less pronounceddifference between schemes B and C is a decrease ofγ by about 20%. This trend is again clear, sinceby taking into account the proper non-collinearground states the tilting region broadens relative tothat of the collinear case (see Fig. 4).In order to provide parameters which can becompared to first-principles calculations, or evento other experiments, we made an estimate ofthe magnetic moment of Co to be 1.7 μ  andthen, by using the 2D lattice constant of the (1 1 1)facet of FCC gold (a"5.43 a.u.), we calculatedωK6.63 μeV. The corresponding values for J,λ and γ are seen in the middle part of Table 1. Thesurface and second hexagonal anisotropy constants,K and K , respectively, as computed fromthe parameters λ and γ are presented in the lowerpart of Table 1. While the surface anisotropy constantscompare reasonably well with that givenin Refs. [5,6] (0.62 mJ/m), our values for K are somewhat lower than the bulk value of
L. Udvardi et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 283—291 291K (143 kJ/m). A possible reason for that might bethat in our model we considered an FCC(1 1 1)layer geometry with the lattice constant of bulkgold rather than the hexagonal Co bulk structure.As pointed out above, an interesting feature ofthe non-collinearity of ground states is that theygive information on the exchange parameter J. Thefact, however, that the experiments were carried outat room temperature, implies that — at least forN'2 [5,6] — the Curie temperature, ¹ , for thethin film must be higher. From this point of view,J for case A seems to be rather low, while for caseB it displays a fairly reasonable value. This may betaken as an indication that fourth-order anisotropydoes play a role in the Co/Au(1 1 1) system. However,the connection between J and ¹ for thinfilms is not firmly established and we would preferto defer judgement on this matter pending furtherexperiments. A particularly useful one of thesewould be a study of the reorientation as a functionof λ —λ . Evidently, for the fourth-order (γ only)mechanism, a change in λ —λ would have a littleeffect, while the non-collinear state would be dramaticallyaltered. Perhaps the easiest experimentswould be to compare the behavior of a non-magneticsubstrate/magnetic film/open surface systemwith that of a magnetic film sandwiched betweentwo equivalent non-magnetic covers. As Fig. 4 indicates,in the latter case, λ "λ , the transitionregion would be dominated by γ. On the otherhand, in the asymmetric system the non-collinearmagnetism could produce a significantly broadertransition.In summary, we have presented a careful investigationof the ground states of a simple spin-Hamiltoniancontaining magnetostatic dipole—dipoleinteraction, as well as second- and fourth-ordersurface magneto-crystalline anisotropies. We haveshown that continuous reorientation transitionwith respect to the thickness of an ultra-thin magneticfilm is a direct consequence of the model evenin the absence of fourth-order anisotropy. Interestingly,we find that the spins corresponding to thetilted average magnetization are necessarily noncollinear.Fits to the measured directions in theCo/Au(1 1 1) thin films show that at present thereare not enough experimental data to distinguishbetween alternative explanations for the phenomenon.Finally, we proposed some new experimentswhich could do so.AcknowledgementsThis paper resulted from a collaboration within,and partially funded by the TMR Network on‘Ab initio calculations of magnetic properties ofsurfaces, interfaces and multilayers’ (ContractNo. ERB4061PL951423). Financial support of theHungarian National Scientific Research Foundation(OTKA Nos. T024137 and T021228) and ofthe Austrian Ministry of Science (GZ 308.941/2-IV/3/95) as well are kindly acknowledged.References U. Gradmann, J. Magn. Magn. Mater. 54—57 (1986) 733. B. Heinrich, K.B. Urquhart, A.S. Arrott, J.F. Cochran, K.Myrtle, S.T. Purcell, Phys. Rev. Lett. 59 (1987) 1756. N.C. Koon, B.T. Jonker, F.A. Volkening, J.J. Krebs, G.A.Prinz, Phys. Rev. Lett. 59 (1987) 2463. C. Chappert, P. Bruno, J. Appl. Phys. 64 (1988) 5736. R. Allenspach, J. Magn. Magn. Mater. 129 (1994) 160. R. Allenspach, M. Stampanoni, A. Bischof, Phys. Rev. Lett.26 (1990) 3344. H. Fritzsche, J. Kohlhepp, H.J. Elmers, U. Gradmann,Phys. Rev. B 49 (1994) 15665. F. Baudelet, M.-T. Lin, W. Kuch, K. Meinel, B. Choi, C.M.Schneider, J. Kirschner, Phys. Rev. B 51 (1995) 12563. D.E. Fowler, J.V. Barth, Phys. Rev. B 53 (1996) 5563. J.G. Gay, R. Richter, Phys. Rev. Lett. 56 (1986) 2728. J.G. Gay, R. Richter, J. Appl. Phys. 61 (1987) 3362. P.J. Jensen, K.H. Bennemann, Phys. Rev. B 42 (1990) 849. D. Pescia, V.L. Pokrovsky, Phys. Rev. Lett. 65 (1990) 2599. M.B. Taylor, B.L. Gyo¨ rffy, J. Phys.: Condens. Matter5 (1993) 4527. A. Moschel, K.D. Usadel, Phys. Rev. B 49 (1994) 12868. L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Electrodynamicsof Continuous Media, 2nd ed., Pergamon Press,New York, 1984, p. 159. L. Szunyogh, B. U jfalussy, P. Weinberger, Phys. Rev. B 51(1995) 9552. D.L. Mills, Phys. Rev. B 39 (1989) 12306. R.P. Erickson, D.L. Mills, Phys. Rev. B 43 (1991) 10715. B. U jfalussy, L. Szunyogh, P. Weinberger, Phys. Rev. B 54(1996) 9883. D. Weller, J. Stöhr, R. Nakajima, A. Carl, M.G. Samant, C.Chappert, R. Mégy, P. Beauvillain, P. Veillet, G.A. Held,Phys. Rev. Lett. 75 (1995) 3752.