Effect of surface orientation and thickness on the magnetization of anisotropic FCC ferromagnetic films

Journal of Magnetism and Magnetic Materials 231 (2001) 98–107

Effect of surface orientation and thickness on the magnetization of anisotropic FCC ferromagnetic films C. Cuccia, M.G. Pinia, P. Politib,*, A. Rettorib,c a

Istituto di Elettronica Quantistica, Consiglio Nazionale delle Ricerche, Via Panciatichi 56/30, I-50127 Firenze, Italy b Istituto Nazionale per la Fisica della Materia, Unita" di Firenze, Largo E. Fermi 2, I-50125 Firenze, Italy c Dipartimento di Fisica, Universita" di Firenze, Largo E. Fermi 2, I-50125 Firenze, Italy Received 3 July 2000

Abstract The dependence on temperature of the layer magnetization of a Heisenberg ferromagnetic ultrathin film in presence of magnetocrystalline single-ion anisotropy was theoretically investigated in the framework of a Green’s function approach using the random phase approximation (RPA). The effect of surface orientation and of film thickness N on the Curie temperature TC was carefully investigated in the case of face centered cubic (FCC) films: the steepest increase of TC ðNÞ was found in the case of the FCC(1 1 1) orientation and the smoothest in the FCC(1 1 0) one. Our results for TC ðNÞ were successfully fitted by a finite-size scaling relation ½TC ð1Þ  TC ðNÞ=TC ðNÞ ¼ ðN=N0 Þl , giving a shift exponent l ’ 1:5, irrespectively of the surface orientation. Finally, the temperature evolution of the magnetization profile was analyzed, as well as its limiting shape at TC . # 2001 Elsevier Science B.V. All rights reserved. PACS: 75.70.Ak; 75.40.Cx; 75.40.s Keywords: Thin films; Magnetization; Curie temperature; Shift exponent

1. Introduction Magnetism in ultrathin films is now a fairly well established field of research: see, for example, the collection of papers edited by Pescia [1] and the more recent one by Heinz [2], as well as the reference books edited by Heinrich and Bland [3]. On the experimental side, the availability *Corresponding author. Tel.: +39-55-2307851; fax: +39-55229330. E-mail address: politi@fi.infn.it (P. Politi).

of sophisticated growth and characterization techniques has led to the discovery of new phenomena which opened the prospect of important technological applications [4]. On the theoretical side, ultrathin films represent a valuable testing ground for the comprehension of magnetism in two dimensions [5], as well as of the crossover to three dimensions when the film thickness is increased [6]. In systems with reduced dimensionality, magnetic anisotropies play a fundamental role in the stabilization of long range order: though smaller than exchange by

0304-8853/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 0 3 3 - 6

99

C. Cucci et al. / Journal of Magnetism and Magnetic Materials 231 (2001) 98–107

orders of magnitude, they are able to drastically modify the temperature dependence of the magnetization [7]. In this paper, we consider an ultrathin ferromagnetic film with face centered cubic (FCC) structure, subject to an easy-axis crystal field anisotropy. Using a Green’s function (GF) technique [8] and the random phase approximation (RPA) [9,10], which are expected to give good results in the whole temperature range, we calculated the temperature dependent layer magnetizations for the high symmetry orientations: (1 0 0), (1 1 0) and (1 1 1). To our knowledge, the case of an FCC(1 1 1) ultrathin film, which is potentially interesting in view of the interpretation of experimental results on epitaxial Ni films [11], as well as that of an FCC(1 1 0) one, have been investigated by this technique for the first time. In fact, in previous papers where a similar theoretical approach was developed, it was applied to simple cubic and FCC(1 0 0) ultrathin films only [12–15]. Within this framework, the Curie point TC is determined as the temperature at which the magnetizations of the different layers vanish simultaneously. In particular, we investigated the sensitivity of TC to surface orientation. We analyzed the dependence of the Curie temperature on film thickness using a finite-size scaling formula which gives the same shift exponent (l ’ 1:5) for the three different surface orientations. Finally, we investigated how the magnetization profile evolves in temperature and its limiting shape at TC , which allows to throw new light on the exponent l. The theoretical framework is developed in the next section, while the results for the temperature dependence of the magnetization and the thickness dependence of the Curie temperature are presented and discussed in Section III. Some calculations and some lengthy expressions are relegated to the Appendices.

Heisenberg exchange interaction and easy-axis magnetocrystalline anisotropy is H ¼

N X J X Sm ðrÞ Sm0 ðr0 Þ 2 m;m0 ¼1 r;r0

D

N X X m¼1

z ½Sm ðrÞ2 :

ð1Þ

r

Here, m and m0 are the layer indexes running from 1 to N and r, r0 denote two-dimensional vectors which determine the position within the film plane; J > 0 denotes the ferromagnetic exchange constant and D > 0 is a uniaxial single-ion anisotropy favoring the magnetization to lie perpendicularly to the film plane. If anisotropy prevails on dipolar interaction, a gap appears in the spin-wave spectrum and the long range character of dipolar energy is not essential in the stabilization of longrange order [16–19]. Furthermore, if magnetic domains are neglected, the main effect of dipolar interaction is to reduce the effective value of D [18]. Recent Kerr and Brillouin experiments in epitaxial Cu/Ni/Cu/Si(1 1 1) heterostructures [20] revealed that indeed there is a thickness regime where the magnetization is perpendicular to the film. To calculate the temperature dependence of the z layer magnetization hSm ðrÞi, we use a Green’s function formalism [8] and an random phase approximation (RPA) decoupling scheme [9,10,12–15], which are expected to give consistent results in the whole temperature range, the critical region included. We first consider the equation of motion for the Green’s function 1 h½A; Bi þ <½A; H; B>E ; EE ¼ ð2Þ 2p þ where, following Callen [9], we choose A ¼ Sm ðrÞ z 0 ðaÞ aSm ðr Þ  0 0 and B ¼ e 0 Sm0 ðr Þ  Bm0 ðr Þ. Performing the first commutator on the r.h.s. of Eq. (2), one obtains z

0

þ  0 ðrÞ; eaSm0 ðr Þ Sm h½Sm 0 ðr Þi ¼ dm;m0 dr;r0 Ym ðaÞ;

ð3Þ

where Ym ðaÞ is defined as 2. Theory

z

Ym ðaÞ ¼ Sm ðSm þ 1Þðea  1ÞheaSm i z

The Hamiltonian of a ferromagnetic ultrathin film with N parallel planes and subject to

z i þ ðea þ 1ÞheaSm Sm z

z 2  ðea  1ÞheaSm ðSm Þ i:

ð4Þ

100

C. Cucci et al. / Journal of Magnetism and Magnetic Materials 231 (2001) 98–107

We observe that the r-dependence of spin-dependent thermodynamic averages can be dropped owing to the translational invariance within the film plane. In the following, the r-dependence will be restored whenever it allows to make the decoupling procedure clearer. Performing the second commutator of Eq. (2), one obtains higher order Green’s functions. Following Lines [10], their decoupling, within the RPA, has to be performed in different ways, depending on the site indexes of the operators. Green’s functions containing products of spin operators on different sites, which result from the commutator with the exchange term of the Hamiltonian, are decoupled in the usual way [8]

After the RPA decouplings, the equation of motion takes the form " # N X X z z þ hSj ðpÞi  DCm ðrÞhSm ðrÞi
þ 0 z þ E  hSj ðpÞi
1 X   ðrÞ ¼ S ðkk Þeikk r ; Sm Nk k m ð5Þ

while a different decoupling scheme [10] is required for the Green’s function resulting from the commutator with the uniaxial anisotropy term in Eq. (1) þ z z þ 0 E



ð6Þ

XX j

z hSm ðrÞi
p

1 0 dm;m0 dr;r0 Ym ðaÞ: BðaÞ m0 ðr Þ>E ¼ 2p

ð8Þ

The translational invariance within each layer allows us to perform the Fourier transformation of the spin operators 1 X þ þ Sm ðrÞ ¼ S ðkk Þeikk r ; Nk k m ð9Þ

where Nk denotes the total number of spins within each layer. Introducing the Fourier-transformed Green’s functions GðaÞ ðE; kk ; m; m0 Þ ¼ 1 X ikk ðrr0 Þ þ 0 e E ; Nk k

ð10Þ

k

the equation of motion can be rewritten as X ½Edm;n  T# m;n GðaÞ ðE; kk ; n; m0 Þ ¼ n

where Cm is defined as z 2 3hðSm Þ i  Sm ðSm þ 1Þ Cm ¼ : z i2 hSm

þJ

k

z 0 z þ E  hSm ðrÞi
z þ 0 Cm hSm ðrÞiE ;

p

k

0 BðaÞ m0 ðr Þ>E ;

0 BðaÞ m0 ðr Þ>E ;

j¼1 0 BðaÞ m0 ðr Þ>E

ð7Þ

This decoupling implies that the average of an even power of the magnetization is expressed as the product of two averages of (lower order) even powers, thus it is qualitatively sound up to TC [10]. Moreover, it should be noted that Lines’ decoupling scheme is basically valid in the limit of small anisotropy [10]. In particular, it breaks down completely D=J ! 1, where it gives TC ! 1 instead of the Ising limit. This is due to the fact that only the spin excitations between the ground state and the first excited states, which diverge in energy as D=J ! 1, were taken into account, whilst the excitations between jSi and j  Si, which do not diverge, were neglected.

1 dm;m0 Ym ðaÞ 2p

ðm; n ¼ 1; 2; ; NÞ;

ð11Þ

where in general T# is an N  N non-hermitian matrix. Denoting by om ðkk Þ its eigenvectors and # k Þ and P# 1 ðkk Þ the right and left eigenvecby Pðk tors, respectively, we obtain X Plm ðkk ÞP1 ðkk Þ 1 ml : ð12Þ GðaÞ ðE; kk ; l; lÞ ¼ Ym ðaÞ 2p E  om ðkk Þ m Now, we use the spectral theorem to express the þ spin-dependent thermal average hBðaÞ l Sl i in terms of the imaginary part of the corresponding Green’s function Z þ1 1 ðaÞ þ hBl ðrÞSl ðrÞi ¼ 2 ImE¼oþi0þ ¼ Yl ðaÞfl ;

ð13Þ

101

C. Cucci et al. / Journal of Magnetism and Magnetic Materials 231 (2001) 98–107

where we have defined fl 

In the case of a FCC film with (1 1 1) surface orientation the N  N matrix T# takes the form

N Plm ðkk ÞP1 1 XX ml ðkk Þ : Nk k m¼1 eb h om ðkk Þ  1

ð14Þ

k

Now, taking into account the relation, valid for any value of the spin Sl Slþ ¼ Sl ðSl þ 1Þ  Slz  ðSlz Þ2 ;

ð15Þ

we obtain the following expression for the thermal þ average hBðaÞ l Sl i in Eq. (13) he

aSlz

Sl Slþ i

¼ Sl ðSl þ 1ÞOl ðaÞ  DOl ðaÞ  D2 Ol ðaÞ;

ð16Þ

where by Ol ðaÞ, DOl ðaÞ and D2 Ol ðaÞ we denote, z respectively, the quantity heaSl i and its first and second derivative with respect to a. By equating the quantity Yl ðaÞfl in Eq. (13) and the r.h.s. of Eq. (16), one obtains, after some algebraic manipulations, the differential equation ð1 þ fl Þea þ fl D2 Ol ðaÞ þ DOl ðaÞ ð1 þ fl Þea  fl  Sl ðSl þ 1ÞOl ðaÞ ¼ 0:

ð17Þ

This can be solved, with two boundary conPthe Sl ditions Ol ð0Þ ¼ 1 and p¼Sl ðD  pÞOl ð0Þ ¼ 0, to give Ol ðaÞ ¼

fl2Sl þ1 eaSl  ð1 þ fl Þ2Sl þ1 eaðSl þ1Þ ½fl2Sl þ1  ð1 þ fl Þ2Sl þ1  ½ð1 þ fl Þea  fl 

Hence, we obtain for the thermal average of the magnetization hSlz i ¼ DOl ð0Þ hSlz i ¼ ð1 þ fl Þ2Sl þ1  fl2Sl þ1

z þ 3JhSm1 ið1  dm;1 Þ z þ 3JhSmþ1 ið1  dm;N Þg z  dm1;n f3JhSm ig ðkk Þð1  dm;1 Þg z  dmþ1;n f3JhSm igþ ðkk Þð1  dm;N Þg;

ð22Þ where the geometric factor related to the interactions between nearest neighbors within a layer is the two-dimensional (2d) structure factor for the triangular lattice  1 X ikk drk 1 ð2dÞ g ðkk Þ ¼ e ¼ cosðkx aÞ 6 dr 3 k pffiffiffi !#   3 kx a ky a ; þ 2 cos ð23Þ cos 2 2 and the geometric factors related to the interactions between nearest neighbors on adjacent layers are 1 X ikk drþ gþ ðkk Þ ¼ e 3 dr þ    pffiffi pffiffi 1 i= 3ky a kx a i=2 3ky a ¼ e þ 2 cos e ; 3 2

:

ð18Þ

ðSl  fl Þð1 þ fl Þ2Sl þ1 þ ðSl þ 1 þ fl Þfl2Sl þ1

z z i þ 6JhSm i½1  gð2dÞ ðkk Þ T# m;n ¼ dm;n fDCm hSm

:

ð24Þ and g ðkk Þ ¼ ½gþ ðkk Þ * . Thus, for the FCC(1 1 1) surface orientation, the matrix T# turns out to be complex, non-hermitian and tridiagonal. For the (1 0 0) and (1 1 0) surface orientations, the explicit form of the matrix is reported in Appendix A: in both cases, it turns out to be real and non-symmetric.

ð19Þ Moreover, from Eq. (17) we obtain the relation hðSlz Þ2 i ¼ Sl ðSl þ 1Þ  ð1 þ 2fl ÞhSlz i;

ð20Þ

so that the parameter Cm in Eq. (7) can be z rewritten in terms of hSm i as z 2Sm ðSm þ 1Þ  3ð1 þ 2fm ÞhSm i : ð21Þ Cm ¼ 2 z hSm i

3. Discussion and conclusions To calculate the temperature dependence of the magnetization hSlz i of each layer of the film, we first determined the eigenvalues and eigenvectors # Eq. (22), then solved selfof the matrix T, consistently the set of Eq. (19), (14), and (21), starting from the low temperature estimates

102

C. Cucci et al. / Journal of Magnetism and Magnetic Materials 231 (2001) 98–107

Fig. 1. Calculated layer-dependent magnetizations Mn  hSlz i=S of FCC(1 1 1) films as a function of reduced temperature T=TC ð1Þ for two different values of the film thickness N [J ¼ 1 K, D ¼ 0:01 K, S ¼ 1, TC ð1Þ ¼ 6:042 K].

hSlz i ¼ Sl , fl ¼ 0 and Cl ¼ ½2ðSl þ 1Þ  3=Sl , 8l ¼ 1; 2; . . . ; N. In Fig. 1, we report the normalized layerdependent magnetizations Mn  hSlz i=S as a function of temperature, for two different values of the film thickness, N, in the case of FCC(1 1 1) surface orientation. Clearly, in absence of any term breaking the up-down symmetry, the magnetization profile is symmetric with respect to the center of the film. Also, owing to the reduced coordination number of the surface spins, the magnetization increases from the surfaces to the bulk. Fig. 1 also shows how rapidly Mn ðTÞ goes to the ‘‘bulk’’ value (i.e. MN=2 ðTÞ). For sufficiently low temperatures, Mn ðTÞ, as calculated within the RPA approximation, is expected to coincide with that predicted by free spin-wave theory. For a film with a rather large number of planes (N ¼ 40), we have reported in Fig. 2 the calculated deviation of the surface (n ¼ 1) and ‘‘bulk’’ (n ¼ N=2 ¼ 20) magnetization from the saturation value as a function of the variable T 3=2 , in order to verify whether they follow the well-known laws: 1  Ms ðTÞ=Ms ð0Þ ¼ Bs ðT=JÞ3=2 and 1  Mb ðTÞ=Mb ð0Þ ¼ Bb ðT=JÞ3=2 , with Bs ¼ 2Bb , predicted by Mills and Maradudin [21] for a semi-infinite ferromagnetic film within free spin-wave theory. Indeed, we find Bs ’ 0:030 and Bb ’ 0:015 for the temperature ranges where such formulas are applicable: for T5D, an exponential departure of the magnetization from the saturation value is found, due to the anisotropy-induced gap in the film dispersion relation, and at high temperatures free spin-wave theory

Fig. 2. Calculated deviation of the surface (squares) and bulk (circles) magnetization from the saturation value as a function of T 3=2 for a film with N ¼ 40 and FCC(1 1 1) surface orientation. Values of the parameters as for Fig. 1. Fitting of the ‘‘T 3=2 ’’ regions give Bs ’ 0:030 and Bb ’ 0:015 (see the main text).

Fig. 3. Calculated temperature dependence of the mean magnetization of a film with N ¼ 10 planes, for three different surface orientation: FCC(1 1 0) (squares); FCC(1 0 0) (triangles); FCC(1 1 1) (circles). Lines are guides to the eye.

breaks down; our results show that for a surface spin it breaks before than for a bulk one. It is worth to report the temperature dependence P % of the mean magnetization MðTÞ ¼ N1 N n¼1 Mn , since usually this is the quantity which is % experimentally measured. In Fig. 3 MðTÞ is plotted for three fcc films with the same thickness, N ¼ 10, and the same anisotropy-to-exchange ratio, D=J ¼ 0:01, but differing for the orientation of the surface plane. Firstly, we observe that

C. Cucci et al. / Journal of Magnetism and Magnetic Materials 231 (2001) 98–107

Fig. 4. Reduced Curie temperature as a function of the thickness N of FCC(1 1 1), FCC(1 0 0) and FCC(1 1 0) films with the same Hamiltonian parameters (J ¼ 1 K, D ¼ 0:01 K, S ¼ 1). Lines are fits by the finite-size scaling formula: ½TC ð1ÞTC ðNÞ=TC ðNÞ ¼ ðN=N0 Þl , where TC ð1Þ ¼ 6:042 K is the result of RPA calculation for the bulk anisotropic system.

TC ð1 1 1Þ > TC ð1 0 0Þ > TC ð1 1 0Þ. This is just a trivial mean field effect, related to the different coordination number zs of a spin at the surface. In fact, denoting by z0 , z1 and z2 the number of nearest neighbouring spins lying in the same plane (z0 ), in adjacent planes (z1 ) and at a distance of two atomic planes (z2 ), we have the following. A spin on the FCC(1 1 1) surface has z0 ¼ 6, z1 ¼ 3 and z2 ¼ 0, so that zs ð1 1 1Þ ¼ 9; a spin on the FCC(1 0 0) surface has z0 ¼ 4, z1 ¼ 4, z2 ¼ 0, so that zs ð1 0 0Þ ¼ 8; finally, in the case of the FCC(1 1 0) surface, one has z0 ¼ 2, z1 ¼ 4 and z2 ¼ 1, giving zs ð1 1 0Þ ¼ 7. Therefore P the average cordination number zðNÞ  ð1=NÞ N n¼1 zðnÞ, with zðnÞ being the coordination number of a spin lying in the nth plane, is such that zðNÞjð1 1 1Þ > zðNÞjð1 0 0Þ > zðNÞjð1 1 0Þ . However, considerations based on the average coordination number only may be wrong, as shown in the following where we investigate (see Fig. 4) how TC ðNÞ approaches the bulk [10] value TC ð1Þ as the film thickness N is increased. Numerical results in the large N limit are generally explained by the finite-size scaling formula [22]  l TC ð1Þ  TC ðNÞ N ¼ : ð25Þ TC ð1Þ N0

103

Fig. 5. The shift of TC ðNÞ from TC ð1Þ versus film thickness N on a log–log scale. The fitted values of the critical exponent l are: 1.53 for FCC(1 1 1), 1.50 for FCC(1 0 0) and 1.49 for FCC(1 1 0) surface orientation, respectively.

Sometimes, for not too large values of the film thickness, better agreement is obtained using the modified expression [22]  l TC ð1Þ  TC ðNÞ N ¼ : ð26Þ TC ðNÞ N0 The quantity l is called shift exponent and, according to the standard finite-size theory, it is the inverse of the exponent n for the correlation length, diverging at TC ð1Þ in the bulk system: l ¼ 1=n. Still, this relation is not always fulfilled: e.g., for the spherical model it is true only if one imposes the spherical constraint on each separate plane of the film [23]. We mentioned the spherical model because in three dimensions it has the same critical exponents as the isotropic version of the model discussed in the present paper and studied with the same technique [23]. On the other side, we are not aware of any study of the critical exponents for our anisotropic model, Eq. (1). Even the mean field value of l is not free from misunderstandings. A na.ıve application of ideas based on the average coordination number zðNÞ would give l ¼ 1. In fact, the hypothesis that TC ðNÞ is proportional to zðNÞ gives (if z2 ¼ 0) 1 ½2zs þ ðN  2Þzb  N 2 ¼ zb  ðzb  zs Þ: N

TC ðNÞ / zðNÞ ¼

ð27Þ

104

C. Cucci et al. / Journal of Magnetism and Magnetic Materials 231 (2001) 98–107

Nonetheless this result is not correct even in a mean field approach because it does not properly take into account the magnetization profile in the film. In Appendix B we show that, if zðNÞ is replaced by an average hzðNÞi weighting each plane according to the magnetization on that plane PN n¼1 zðnÞMn hzðNÞi  P ; ð28Þ N n¼1 Mn then the relation TC ðNÞ  hzðNÞi does hold, leading to the correct mean field result l ¼ 2: see Eq. (B.8). In Fig. 5, we report our numerical RPA results for the Curie temperature of the film TC ðNÞ (with N ranging between 3 and 40) in a log–log scale in order to determine the shift exponent l by Eq. (26). For the three considered surface orientations, all the calculated points lie on straight lines with nearly the same slope. The bad alignment of the circle at N ¼ 40 is most likely due to the worse determination of TC ðNÞ for (1 1 1) orientation, which requires the diagonalization of a complex # More precisely, we obtain l ¼ 1:53, 1.50 matrix T. and 1.49 for the FCC(1 1 1), FCC(1 0 0) and FCC(1 1 0) orientations, respectively. Thus, taking into account that the error in the determination of TC is of the order of percent, one can conclude that, within RPA approximation, l ’ 1:5 is the common value of the shift exponent l, characterizing the approach of TC ðNÞ to the bulk value in an FCC film. Such a value is consistent with experimental data on Ni/Cu(1 1 1) films [24] and agrees with numerical studies on the Ising model (high temperature series expansion [25] and Monte Carlo simulations [22]). We might think to apply the ideas about hzðNÞi to understand the value l ’ 1:5 found in our RPA treatment. Fig. 6 gives the magnetization profile for a N ¼ 40 film at different temperatures and Fig. 7 shows the limiting profiles (T ! TC ðNÞ) for different values of N. If we use these profiles to evaluate numerically Eq. (28), we are able to determine the ‘‘shift’’ exponent lz for the average coordination number hzðNÞi (see Appendix B). We do not find exactly lz ’ 1:5, but a slightly greater value lz ’ 1:6=1:7. We have not been able to settle whether this discrepancy is real or it is a

Fig. 6. Evolution of the reduced magnetization profile with increasing temperature for a film of N ¼ 40 planes, from T ¼ 0 to T ¼ TC . Lines are guides to the eye.

Fig. 7. Reduced magnetization profiles at the critical temperature TC for FCC(1 0 0) films with N ¼ 10; 20; 30; 40. To make comparison easier the origin of the horizontal axis has been shifted to the center of the films. Lines are guides to the eye.

problem of numerical precision in the evaluation of expression (28). In conclusion, in this paper we developed a procedure for calculating the layer dependent magnetizations of a ferromagnetic Heisenberg film with single-ion anisotropy, in the case of fcc structure. Within this formalism and the random phase approximation it is possible to follow the temperature dependence of the magnetization in the whole ordered region. Particular care

C. Cucci et al. / Journal of Magnetism and Magnetic Materials 231 (2001) 98–107

was devoted to the study of the effect of surface orientation on the magnetic properties of the film. While for low thicknesses different results were found for TC depending on the surface orientation (as it could be expected even on the basis of a simple mean field argument), in the limit N ! 1 the same asymptotic behavior was found, with a shift exponent l ’ 1:5 independent of the orientation. Finally, we proposed a possible interpretation of the latter result in terms of the average coordination number hzðNÞi, weighting each plane according to its magnetization, even though the result lz ¼ l has been rigorously proven for mean field theory only.

Acknowledgements This work was supported by CNR under Integrated Project Surface and interface magnetic anisotropies in ultrathin magnetic films and their dependence on temperature. Many thanks are due to G. Carlotti and G. Gubbiotti for fruitful discussions and suggestions. Appendix A: Expressions of the T# matrix in the case of FCC(1 0 0) and FCC(1 1 0) surface orientations In the main text, Eq. (22), we reported the explicit expression for the non-symmetric N  N matrix T# in the case of the FCC(1 1 1) surface orientation. Here, we consider the other two possible surface orientations of the FCC lattice. For the FCC(1 0 0) orientation we have z0 ¼ z1 ¼ 4 and z2 ¼ 0 (see the main text), so that the matrix takes the tridiagonal form z z T# m;n ¼ dm;n fDCm hSm i þ 4JhSm i½1  gð2dÞ ðkk Þ þ 4JhSm1 ið1  dm;1 Þ

þ 4JhSmþ1 ið1  dm;N Þg z  dm1;n f4JhSm ig ðkk Þð1  dm;1 Þg 

z dmþ1;n f4JhSm igþ ðkk Þð1

105

where the geometric factor related to the interactions between nearest neighbors within a layer is     ky a 1 X ikk drk kx a e ¼ cos cos ; gð2dÞ ðkk Þ ¼ 4 dr 2 2 k

ðA:2Þ and the geometric factors related to the interactions between nearest neighbors on adjacent layers are 1 X ikk drþ e 4 dr þ      ky a 1 kx a ¼ cos þ cos ; 2 2 2

gþ ðkk Þ ¼

ðA:3Þ

and g ðkk Þ ¼ ½gþ ðkk Þ * . In conclusion, in the case of the FCC(1 0 0) surface orientation, the N  N matrix T# is real, non symmetric and tridiagonal. For the FCC(1 1 0) orientation, we have z0 ¼ 2, z1 ¼ 4 and z2 ¼ 1 so that the matrix takes the pentadiagonal form T# m;n ¼ z z dm;n fDCm hSm i þ 2JhSm i½1  gð2dÞ ðkk Þ

þ ½4JhSm1 i þ JhSm2 ið1  dm;2 Þð1  dm;1 Þ þ ½4JhSmþ1 i þ JhSmþ2 i  ð1  dm;N1 Þð1  dm;N Þg z  dm1;n f4JhSm ig ðkk Þð1  dm;1 Þg z  dmþ1;n f4JhSm igþ ðkk Þð1  dm;N Þg z  dm2;n fJhSm ið1  dm;2 Þð1  dm;1 Þg z  dmþ2;n fJhSm ið1  dm;N1 Þð1  dm;N Þg;

ðA:4Þ where the geometric factor related to the interactions between nearest neighbors within a layer is "pffiffiffi # 2 1 X ikk drk ð2dÞ g ðkk Þ ¼ ðkx aÞ e ¼ cos ðA:5Þ 2 dr 2 k

 dm;N Þg; ðA:1Þ

and the geometric factors related to the interactions between nearest neighbors on adjacent

106

C. Cucci et al. / Journal of Magnetism and Magnetic Materials 231 (2001) 98–107

the mean field Curie temperature TC ðLÞ of a film of thickness L is found to satisfy the relation   TB  TC ðLÞ z1 p2 1 ¼ : ðB:6Þ TB z0 þ 2z1 L2

layers are 1 X ikk drþ gþ ðkk Þ ¼ e 4 dr þ "pffiffiffi #   2 kx a kz a ¼ cos cos ; 2 2 2

ðA:6Þ

and g ðkk Þ ¼ ½gþ ðkk Þ * . In conclusion, in the case of the FCC(1 1 0) surface orientation, the N  N matrix T# is real, non-symmetric and pentadiagonal.

Appendix B: Shift exponent and magnetization profile in mean field approximation Let us consider a magnetic film with N planes labeled by n ¼ 1; . . . ; N. We will suppose here that z2 ¼ 0. In proximity of the Curie temperature TC ðNÞ the linearized mean field equations for Mn ðTÞ are: z0 M1 þ z1 M2 ¼ tM1 ;

ðB:1Þ

ðB:2Þ

z0 MN þ z1 MN1 ¼ tMN ;

ðB:3Þ

where t ¼ TC ðNÞ=J is the reduced Curie temperature and tB ¼ z0 þ 2z1 is the reduced bulk Curie temperature (t ! tB for N ! 1). In the continuum limit, valid at large N, Eq. (B.2) becomes the differential equation (for simplicity we take a ¼ 1) z1 q2x M þ DtMðxÞ ¼ 0;

Dt  tB  t;

ðB:4Þ

and the surface equations (B.1, B.2) give the boundary conditions qx M ¼ MðxÞ for x ¼ L=2;

where zðnÞ is the coordination number of a spin lying on the nth plane. By defining lz as the ‘‘shift exponent’’ of the weighted coordination number: ðzB  hzðNÞÞi=zB  Llz , in the continuum limit it is possible to obtain its analytic evaluation zB  hzðNÞi MðL=2Þ  R L=2 zB dxMðxÞ

z0 Mn þ z1 ðMnþ1 þ Mn1 Þ ¼ tMn ; n ¼ 2; . . . ; N  1

The shift exponent is therefore l ¼ 2, and not l ¼ 1 as derived from simple considerations based on the mean coordination number zðNÞ. The reason of this disagreement is that the nonuniform surface profile plays an important role in determining l. In fact, rather than considering zðNÞ (see Eq. (27)), let us perform a weighted average of the coordination number PN n¼1 zðnÞMn hzðNÞi  P ; ðB:7Þ N n¼1 Mn

ðB:5Þ

where the film has a thickness L ¼ N  1 and surfaces in x ¼ L=2. The even solution offfi pffiffiffiffiffiffiffiffiffiffiffi Eq. (B.4) is MðxÞ ¼ M0 cosðqxÞ with q ¼ Dt=z1 . The boundary condition determines the relation between q and L and therefore the L-dependence of t: q tanðqL=2Þ ¼ 1. The previous equation can be solved numerically and its limiting expression at large L can be found analytically: q ¼ ðp=L  2p=L2 Þ. Therefore

L=2

cosðqL=2Þ p2 ¼ R L=2  2: 2L 2 0 dx cosðqxÞ

ðB:8Þ

Therefore, in mean field theory, lz ¼ l ¼ 2.

References [1] D. Pescia (Ed.), Appl. Phys. A 49 (1989) No 5–6. [2] K. Heinz, (Ed.), J. Phys.: Condens. Matter 11 (1999) No 48. [3] B. Heinrich, J.A. Bland (Eds.), Ultrathin Magnetic Structures I and II, Springer, Berlin, 1994. [4] F.J. Himpsel, J.E. Ortega, G.J. Mankey, R.F. Willis, Adv. Phys. 47 (1998) 511. . [5] C.H. Back, Ch. Wursch, A. Vaterlaus, U. Ramsperger, U. Maier, D. Pescia, Nature 378 (1995) 597. [6] D.L. Mills, J. Magn. Magn. Mater. 100 (1991) 515. [7] M. Bander, D.L. Mills, Phys. Rev. B. 38 (1988) 12015. [8] R.A. Tahir Kheli, in: C. Domb, M.S. Green (Eds.), Phase Tansitions and Critical Phenomena, Vol. 5b, Academic Press, London, 1976, p. 259, and references therein. [9] H.B. Callen, Phys. Rev. 130 (1963) 890. [10] M.E. Lines, Phys. Rev. 156 (1967) 534. [11] See G. Carlotti, G. Gubbiotti, Riv. Nuovo Cimento 22 (1999) 1, and references therein.

C. Cucci et al. / Journal of Magnetism and Magnetic Materials 231 (2001) 98–107 [12] Diep-The-Hung, J.C.S. Levy, O. Nagai, Phys. Stat. Sol. b 93 (1979) 351. [13] P. Bruno, in: S.S.P. Parkin, H. Hopster, J.P. Renard, T. Shinjo, W. Zinn (Eds.), Magnetic Surfaces, Thin Films, and Multilayers, Material Research Society Symposium Proceeding, Vol. 231, 1992, p. 299. [14] L. Zhou, L. Hu, Z. Lin, Y. Kawazoe, R. Tao, Phys. Rev. B 57 (1998) 7863. [15] R. Schiller, W. Nolting, Solid State Commun. 110 (1999) 121. [16] S.V. Maleev, Sov. Phys. JETP 43 (1976) 1240. [17] Y. Yafet, J. Kwo, E.M. Gyorgy, Phys. Rev. B 33 (1986) 6519. [18] P. Bruno, Phys. Rev. B 43 (1991) 6015.

107

[19] P. Politi, A. Rettori, M.G. Pini, J. Magn. Magn. Mater. 113 (1992) 83. [20] G. Gubbiotti, G. Carlotti, G. Socino, F. D’Orazio, F. Lucari, R. Bernandini, M. De Crescenzi, Phys. Rev. B 56 (1997) 11 073. [21] D.L. Mills, A.A. Maradudin, J. Phys. Chem. Solids 28 (1967) 1855. [22] K. Binder, P.C. Hohenberg, Phys. Rev. B 9 (1974) 2194. [23] M.N. Barber, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena, Vol. 8, Academic Press, London, 1983, p. 145, and references therein. [24] C.A. Ballentine, R.L. Fink, J. Araya-Pocket, J.L. Erskine, Phys. Rev. B 41 (1990) 2631. [25] T.W. Capehart, M.E. Fisher, Phys. Rev. B 13 (1976) 5021.