Physica B 426 (2013) 144–149 Contents lists available at SciVerse ScienceDirect Physica B journal homepage: www.elsevier.com/locate/physb Thickness dependent properties of magnetic ultrathin films Bach Thanh Cong a, Pham Huong Thao a,b,n a b Faculty of Physics, Hanoi University of Science, VNU, 334 Nguyen Trai, Hanoi, Viet Nam Faculty of Physics, Hue College of Education, 32 Le Loi, Thua Thien Hue, Viet Nam art ic l e i nf o a b s t r a c t Article history: Received 27 March 2013 Received in revised form 24 May 2013 Accepted 30 May 2013 Available online 20 June 2013 The dependence of magnetic properties on the thickness of few-layer thin films is investigated at finite temperature using the functional integral method for solving the Heisenberg spin model The temperature dependence of the ultra-thin film's magnetization and Curie temperature are calculated in terms of the mean field theory and of the Gaussian spin fluctuation approximations It has been shown that both Curie temperature and temperature interval, where the magnetization is non-zero, are strongly reduced with the thickness reduction by using the spin fluctuation approximations in comparison with the mean field results Curie temperature dependence on the film thickness calculated numerically well agrees with the experimental data for Ni/Cu(1 0) and Ni/Cu(1 1) ultrathin films & 2013 Elsevier B.V All rights reserved Keywords: Thin film Functional integral method Magnetic properties Introduction Size dependent effects of ferroelectric and ferromagnetic thin films have attracted a great interest due to its scientific and industrial importance The most important problem is to determine the size intervals where the long range order parameters of thin films like the Curie temperature (TC) or the magnetization considerably change, compared with those of the bulk one It was shown experimentally that TC is strongly decreased when the film thickness is reduced to several atomic layers, which has been observed in the ferromagnetic (FM) transition [1,2], and rare earth metal films [3] Several theories have been developed to explain the sizedependence of the thin film's macroscopic properties In the scaling theory approach [4,5], TC was shown to be shifted to a lower temperature than that of the bulk when the film thickness is smaller than the spin–spin correlation length Another group of researchers has used self-consistent spin wave concept [6] to show a sharp decrease of TC for few-layer FM films The Curie temperature in the Heisenberg ferromagnetic thin films was also calculated by using the standard mean field (MF) methods [7] including the Weiss mean field, the Oguchi cluster and the constant coupling approximations The analytical methods using famous Heisenberg, Ising and s–d models provide additional understanding of the electronic density functional theory (DFT) results for thin films [8,9] n Corresponding author at: Faculty of Physics, Hanoi University of Science, VNU, 334 Nguyen Trai, Hanoi, Viet Nam Tel.: +84 976693644 E-mail address: hthao82@gmail.com (P.H Thao) 0921-4526/$ - see front matter & 2013 Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.physb.2013.05.043 In the present paper, we study the magnetic properties of the ultra-thin films using the functional integral method (FIM) for Heisenberg spin systems in which the specific surface effects have been taken into account The FIM has been developed successfully for different bulk spin models [10,11]; hence it would be intrigued to apply for quasi-two-dimensional spin systems like thin films This paper is organized as follows: the general FIM for magnetic films is described in the next section Numerical results produced by the FIM theory are presented in Section Section shows the comparison between numerical and experimental results Functional integral method for magnetic thin films We consider here the magnetic film of a cubic lattice consisting of n spin layers, where the spin numbers N in a layer are practically infinite (Fig 1) The coordinate system is chosen as follows: Oz is perpendicular to the film surfaces and the spin layers are parallel to xOy plane Spin positions in the lattice are defined by the indexes νj, where ν is the layer (spin plane) index ðν ¼ 1; …; nÞ and Rj the two-dimensional vector denoting the position of the jth spin in this layer The Heisenberg Hamiltonian for the spin films is written as H ¼ −gμh∑Szνj − vj ∑ J ðR −R ÞS S νj;ν′j′ νν′ j j′ νj ν′j′ ð1Þ where the first term is the spin energy in an external homogeneous magnetic field h oriented along the z direction and the second term is the usual Heisenberg exchange interaction J νν′ ðR j −R j′ Þ denotes the exchange integral between Sνj and Sν′j′ spins, which is considered as the matrix element of the matrix JðRj −R j′ Þ Including the internal molecular field (Weiss molecular field) in B.T Cong, P.H Thao / Physica B 426 (2013) 144–149 145 The functional formalism for magnetic films is introduced in a more convenient way by using the well-known Stratonovich– Hubbard integral transformation (see Refs [10,11]), which is " # ( ) Zỵ dy i pAexp y2i þ ∑yi A1=2 exp ∑xi Aij xj ¼ @∏ x ð5Þ j ij i;j i 2π i i;j −∞ ^ 1=2 , which is square root is the matrix element of matrix A here ^ Applying an integral transformation (5), one can of the matrix A obtain the functional integral representation for the free energy, " # Z 1 6aị F ẳ F ln ðdφÞexp − ∑ φν ðqÞφν ð−qÞ−F int ðφÞ β ν;α;q 1=2 Aij Fig A spin position is defined in a spin lattice by a plane index νðν ¼ 1; 2; :::; nÞ and by a two-dimensional lattice vector Rj (denoted shortly by νj) Js is the nearest neighbor (NN) in-plane exchange integral of the surface layers; J are the NN in-plane exchange integral of interior layers and the NN out of plane exchange integral of adjacent layers N N shðS ỵ 1=2ịy F ẳ lnSpeH0 ị ẳ J νν′ ð0Þ〈Szν 〉〈Szν′ 〉− ln β νν′ β shðyv =2ị y ẳ gh ỵ J ð0Þ〈Szν′ 〉 ð6bÞ ð6cÞ ν0 the first term, we have here yν is the total field measured in the temperature unit and acting on spins in the v-layer q ¼ ðk; ωÞ is the four components wave vector and φαν qị is the auxiliary eld Fourier component ! H ẳ gh ỵ J R j R j ịSzj Szj j ỵ j;j vj J R j −R j′ Þ〈Szνj 〉〈Szν′j′ 〉− ∑ νj;ν′j′ φαν qị ẳ J R j R j ịSj Sj Z 1=2 k; ịexp ẵid; ω ¼ 2πm=β; m ¼ 0; 1; 2; … ð7Þ α in which the components of the spin fluctuation operators are defined as δSzνj ¼ Szνj −〈Szνj 〉; δSyνj ¼ Syνj ; δSxνj ¼ Sxνj ; α ¼ x; y; z here ðdφÞ is the measure of the functional integration, which is dened by [10] Z Z ỵ Z ỵ ;c d 0ị d qị ỵ d;s qị p p p 8ị dị ẳ ∏ π π 2π q≠0 −∞ αν −∞ −∞ Brackets ẳ SpeH ị=SpeH ị mean the thermodynamic average and β−1 ¼ kB T The formula (1) is rewritten by taking Fourier transformations for spin operators, which is The real and the imaginary parts of the complex field variable α;s φαν ðqÞ are denoted by φα;c ν ðqÞ and φν qị respectively H ẳ H ỵ H int ;s qị ẳ ;c qị ỵ i qị H0 ¼ ð2aÞ N ∑ J ð0Þ〈Szν 〉〈Szν′ 〉−∑gμhν Szνj ; vj H int ẳ 2bị ∑ J νν′ ðkÞδSαν ðkÞδSαν′ ð−kÞ ν; ν′; k α ð2cÞ The second term in the exponential part of Eq (6a) is the interacting functional which has an explicit form as * ( )+ 1=2 ^ ∑ β1=2 J νν′ kịl qịSl0 qị 10ị F int ị ẳ ln Texp ν;ν′;l;q The symbols in Eq (10) mean as given below: ẳ SpeH0 ị=SpeH0 ị; x qịiy qị qị ẳ where gh ẳ gh ỵ J 0ịSz l ẳ ; z; S7 qị ẳ S7 qị ẳ ẵSx qị iSy qị J kị ẳ J Rj R j ịexp ẵikRj R j ị Sz qị ¼ Szν ðqÞ−δðqÞN1=2 〈Szν 〉; j−j′ δSαν ðkÞ ¼ Sαν kị ẳ p1N Sj exp ẵikR j qị ẳ δðkÞδðωÞ ð11Þ The logarithmic functional in Eq (10) is readily presented in the series form which is and for α ¼ x; y ∞ −F int ðφÞ ¼ ∑ j Sz kị ẳ p1N Szj exp j 9ị ẵikRj ẳ Sz kịkịN1=2 Sz for ẳz 3ị In Eqs (2) and (3), kðkx ; ky Þ is the two-component wave vector relating to the translational symmetry in spin layers All spin sites in the same plane are equivalent due to this translational symmetry, then 〈Szνj 〉 ¼ 〈Szν 〉 The statistical operator of the system is given in the interaction representation using the temperature dependent scattering matrix sðβÞ, the non-interacting H0 and the interacting Hint Hamiltonians (see Eqs (2b) and (2c)): & Z β ' ^ − H int ịd 4ị expHị ẳ expH ịsị; sị ẳ Texp ∑ m ¼ m! ν1 ;ν′1 ;l1 ;q1 ;…νm ;ν′m ;lm ;qm 1=2 1=2 1=2 J ν ν′ ðkm Þ ′ ðk1 Þ⋯β m m ν1 β1=2 J ν ^ l10 ðq Þ⋯δSlm′ ðq Þ〉ir Âφlν11 ðq1 Þ⋯φlνmm ðqm Þ〈TδS m ν ν m ð12Þ The symbol 〈…〉ir means an irreducible average of spin component operators over non-interacting equilibrium statistical ensemble The graphical representation of the series (12) has been shown in Fig Here we use the Gaussian approximation, where the functional F int ðφÞ has the simple form as F int ị ẳ f z k; ω1 Þψ zν ð−k; ω2 Þb′ðyν Þδðω1 Þδðω2 Þ ;k;1 ;2 ỵ41 ỵ ịK y ịby ị k; ị ỵ k; ịg ð13aÞ 146 B.T Cong, P.H Thao / Physica B 426 (2013) 144–149 Fig Diagram representation for the series (12) Here wiggly lines mean the longitudinal field component φzν ðqÞ, cross lines (++++) and dashed lines (——) correspond to − ỵ qị and qị respectively The other graphical elements (ovals, arrows etc.) are described in the Izuymov's monograph [12] 1=2 l qị ẳ J kịl qị; K y ị ẳ y i 13bị where by ị ẳ S ỵ 1=2ịịcthS ỵ 1=2ịịy 1=2ịcthy =2Þ and b′ðyν Þ mean the Brillouin function and its derivative Calculating the functional integral presented in Eq (6a) by using Gaussian interacting functional Eq (13a), we find an expression for the free energy of the magnetic film: F¼ N N shS ỵ 1=2ịy J 0ịSz Sz ln ; shy =2ị ỵ 1 ^ ^ AðkÞj ^ Bðk; ^ ∑ lndet j1− þ ∑ lndet j1− ωÞj 2β k β k;ω ð14Þ βb′ðy1 ÞJ 11 ðkÞ B βb′ðy ÞJ ðkÞ B 21 B ^ Akị ẳB B by3 ịJ 31 kị B ^ @ βb′ðyn ÞJ n1 ðkÞ βb′ðy1 ÞJ 12 ðkÞ βb′ðy1 ÞJ 13 ðkÞ ⋯ βb′ðy2 ÞJ 22 ðkÞ βb′ðy2 ÞJ 23 ðkÞ ⋯ βb′ðy3 ÞJ 32 ðkÞ ^ βb′ðy3 ÞJ 33 ðkÞ ^ ⋯ ^ βb′ðyn ÞJ n2 ðkÞ βb′ðyn ÞJ n3 ðkÞ ⋯ βb′ðy1 ÞJ 1n ðkÞ βb′ðy2 ÞJ 2n ðkÞ C C C βb′ðy3 ÞJ 3n ðkÞ C C C ^ A βb′ðyn ÞJ nn ðkÞ ð15Þ and βbðy ÞJ 11 ðkÞ y1 −iβω B B βbðy2 ÞJ 21 ðkÞ B y2 −iβω B B ^ Bk; ị ẳ B by3 ịJ 31 ðkÞ B y3 −iβω B ^ B @ βbðyn ÞJ n1 ðkÞ yn −iβω βbðy1 ÞJ 12 ðkÞ y1 −iβω βbðy1 ÞJ 13 ðkÞ y1 −iβω ⋯ βbðy2 ÞJ 22 ðkÞ y2 −iβω βbðy2 ÞJ 23 ðkÞ y2 −iβω ⋯ βbðy3 ÞJ 32 ðkÞ y3 −iβω βbðy3 ÞJ 33 ðkÞ y3 −iβω ⋯ ^ ^ ^ βbðyn ÞJ n2 ðkÞ yn −iβω βbðyn ÞJ n3 ðkÞ yn −iβω ⋯ βbðy1 ÞJ 1n ðkÞ y1 −iβω βbðy3 ÞJ 3n ðkÞ C: C y3 −iβω C βbðyn ÞJ nn ðkÞ yn −iβω ð16Þ C C A χ zν1 j ;ν2 j2 ^ z ðτ1 Þ−〈Sz 〉ÞðSz ðτ2 Þ−〈Sz 〉Þ〉 ðτ1 ; ị ẳ z1 Rj1 R j2 ; ị ẳ TS j1 j2 ν2 χ xν1 j ;ν2 j2 ^ x ịSx ị ; ị ẳ x1 Rj1 R j2 ; ị ẳ 〈TS ν1 j ν2 j2 ;ν2 j2 ^ y ịSy ị ; ị ẳ χ yν1 ν2 ðRj1 −R j2 ; τ1 −τ2 Þ ¼ 〈TS ν1 j ν2 j ð17Þ The Fourier transformations of the spin correlation functions have the following form: R; ị ẳ ~ k; ịexp i ỵ ikR N k; ~ k; ị ẳ Z R; ịexpẵiikRd R α ∑ χ~ α ðk; ωÞ N α;k;ω νν ð19Þ Within the Gaussian approximation Eqs (13a) and (13b), the Fourier image of the spin correlation function becomes ! 20ị ~ k; ị ẳ J −1 ν1 ν2 ðkÞ−∑ J ν1 ν ðkÞCνν2 ðk; ωÞ β ν α where Cανν′ ðk; ωÞ is the element of the matrix C^ ðk; ωÞ and xðyÞ ^ Bk; ^ C^ kị ẳ ịị1 21ị Sz as a relative magnetization per site of the If we define mν ¼ νth spin layer, then in the MF approximation, we have z mMF ị ẳ S ¼ bðyν Þ ð22Þ In the spin fluctuation theory, when the Gaussian approximation is taken into account, we obtain from Eqs (17), (18a) and (18b) that ( )1=2 z ~ mSF ị ẳ S ẳ SS þ 1Þ− ð χ ðk; ωÞÞ ð23Þ ν ν N α;k;ω νν In order to compare with experiments, an average relative (or dimensionless) magnetization per thin film site is given as C βbðy2 ÞJ 2n ðkÞ C y2 −iβω C C Furthermore, one can introduce the correlation functions between the fluctuations of the spin components (or timeordered spin Green functions): χ yν1 j 〈δS2ν 〉 ¼ ∑ χ ανν ð0; 0ị ẳ z ^ ^ Akịị C^ k; ị ¼ ð1− ; ^ ^ AðkÞ and Bðk; ωÞ are n  n matrices In Eqs (18a) and (18b), τ is a time variable, τ ¼ τ1 −τ2 and R ¼ R j1 −R j2 The local spin fluctuation is easily derived from Eqs (17) and (18a) when ẳ 0; R ẳ 0, which is 18aị 18bị mị ẳ M n ẳ m ị Nn nẳ1 24ị M is the magnetic moment of the thin film Average magnetizations in the MF and SF approximations are denoted by mMF and mSF , respectively Numerical calculations In this part, we use the nearest neighbor (NN) exchange approximation where only NN in-plane exchange integral J νν ðΔÞ (Δ ¼ R j −R j′ means in-plane NN spin lattice vector), and NN out-ofplane exchange integral J ;ỵ1 ị ẳ J ỵ1; ị are nonzero In the framework of NN exchange approximation, the matrices (15) and (16) have the following explicit forms: βb′ðy1 ÞJ s ðkÞ βb′ðy1 ÞJðkÞ ::: n B βb′ðy ÞJ ðkÞ βb′ðy ÞJ ðkÞ βb′ðy ÞJðkÞ ⋯ C B C 2 22 B C n ^ C ÞJ ðkÞ by ịJ kị 0 by Akị ẳB 3 33 B C B C ^ ^ ^ ^ ^ @ A 0 ⋯ βb′ðyn ÞJ s ðkÞ ð25Þ B.T Cong, P.H Thao / Physica B 426 (2013) 144–149 ÞJ s ðkÞ y −iβω B n B βbðy2 ÞJ ðkÞ B y2 −iβω B B ^ Bk; ị ẳ B B B B ^ @ βbðy1 ÞJðkÞ y1 −iβω ⋯ βbðy2 ÞJ 22 ðkÞ y2 −iβω βbðy2 ÞJðkÞ y2 −iβω ⋯ βbðy3 ÞJ n ðkÞ y3 −iβω βbðy3 ÞJ 33 ðkÞ y3 −iβω ⋯ ^ ^ ^ ^ ⋯ βbðyn ÞJ s ðkÞ yn −iβω 0 1.0 C C C C C C C C C A 0.8 ð26Þ MF βbðy mν and 147 0.6 0.4 0.2 0.0 0.2 0.4 τ 0.6 The Fourier components of exchange integrals are easily derived from the geometrical structure of spin lattices: 27ị J ;ỵ1 kị ẳ Jkị ẳ Jexpikz aị Fig Temperature dependence of the magnetization of the FCC films with different thicknesses The surface exchange parameters are chosen as ξ ¼ 1, and S¼ 28ị n J ỵ1; kị ẳ J ;ỵ1 kị ¼ J ðkÞ ¼ Jexp ð−ikz aÞ 0.95 ð29Þ 0.90 b for a face centered cubic (FCC) thin film lattice k a J s kị ẳ 4J s cos k2x a cos 2y for surface layers ν ¼ 1; n k a J kị ẳ 4J cos k2x a cos 2y for interior layers ν ¼ 2; …; n−1 0.85 0.80 ξ=0.8 ð30Þ ! ky a kx a ikz a exp ỵ cos J ;ỵ1 kị ¼ JðkÞ ¼ 2J cos 2 ð31Þ 0.70 ð32Þ where a is a lattice constant of the thin film In the following parts, a relative dimensionless temperature τ ¼ T=T bC is used T bC is the Curie temperature of the corresponding bulk spin lattice, in which T bC ẳ SS ỵ 1ịZJị=3kB (Z is the number of NN spins, for FCC spin lattice, Z¼ 12) ξ ¼ J s =J is the ratio characterizing the difference of the surface and the interior exchange interactions of the thin films The Curie temperature within the MF (SF) theory is obtained from a solution of an equation mMF C ị ẳ (mSF C ị ẳ 0), where τC ¼ T C =T bC In the next parts, numerical calculations are performed for the FCC thin film lattice at zero external magnetic field h¼0 The magnetization is calculated in the approach of the MF approximation or the Weiss molecular field approximation for FCC spin lattice film using Eq (22) Fig shows the temperature dependence of the magnetizations of ultrathin films with different thicknesses n and parameter ξ ¼ It is clearly seen that the temperature interval, where the magnetization is nonzero, is shorter when the film is thinner However, layer magnetizations vanish at the same temperature, which is known as the Curie temperature of the thin films At a given temperature, magnetizations of layers decrease when position of the layer varies from the center to the film surfaces (Fig 4) for the cases of no-surface effect (ξ ¼ 1) and the surface exchanges are smaller than the interior ones (ξ o 1) The opposite tendency happens when the surface exchange is sufficiently large (see the ξ ¼ 2:2 curve in Fig 4) There are two factors originating this behavior: (i) the number of NN spins at surfaces is 1.0 0.8 0.6 0.4 ξ=0.8 ξ=1 ξ=2 0.2 0.0 0.0 3.1 Mean field approximation n (layer) Fig Dependence of magnetizations on layer position in the thin films with n ¼5, S¼ at τ ¼ 0:53 MF mV ξ=1 0.75 ξ=2.2 ! ky a kx a ikz a exp J ỵ1; kị ẳ J n;ỵ1 kị ẳ J n kị ẳ 2J cos ỵ cos 2 surface layer center layer mMF ν n 1.0 J s kị ẳ 2J s ẵ cos kx aị ỵ cos ky aị for surface layers ẳ 1; n J kị ẳ 2Jẵ cos kx aị ỵ cos ky aị for interior layers ẳ 2; …; n−1 n 0.8 a for a simple cubic thin film lattice er) (lay 0.2 0.4 0.6 τ 0.8 1.0 1.2 Fig Temperature dependence of the magnetization of a two-layer FCC film MF (mMF ¼ m2 ) calculated within the MF approximation for different surface exchange parameters ξ, and S ¼ smaller than that of the film center and (ii) the surface and internal exchanges are generally different Below the Curie temperature, the magnetization increases with the enhancement of the surface exchange parameter ξ (see Fig for n ¼2 case) The Curie temperatures of ultra-thin films obtained in MF approximation with different n and ξ are plotted in Fig Our MF calculation based on FIM is in good agreement with the results of the MF approximation [7] or of the spin wave theory [6], where the Curie temperature is subtracted with the deduction of the thin film thickness The analytic expression for TC was also given in our previous work [13] The mean field theory predicts a finite TC even in the monolayer film; however TC should be zero according to the 148 B.T Cong, P.H Thao / Physica B 426 (2013) 144–149 1.0 1.0 0.8 0.8 0.6 τC τC 1.2 0.4 0.6 ξ=1 (MF) ξ=1 (SF) ξ=0.8 ξ=1 0.4 0.2 ξ=0.5 (MF) ξ=0.5 (SF) ξ=2 0.0 0.2 n (layer) Fig The Curie temperatures obtained in the MF approximation versus the film thickness with different surface exchange parameters ξ Here S ¼ 1 n (layer) Fig The Curie temperature as a function of layer thickness n obtained within the SF (solid line) and the MF (dash line) approximations Here S ¼ 1.4 1.5 1.2 1.0 mνSF 1.2 0.9 0.8 mSF 0.6 n=1 0.6 0.0 n=5 0.0 0.1 0.2 0.3 0.4 τ ξ=2 0.0 n=4 0.0 ξ=1 0.2 n=3 0.3 ξ=0.8 0.4 n=2 0.5 0.6 0.7 Fig The magnetizations of the FCC films versus the temperature with various thicknesses, in which the surface exchange and spin parameters are set as ξ ¼ 1, and S ¼ Mermin and Wagner's theorem for one- or two-dimensional isotropic Heisenberg models with finite range interactions [14] For improving our MF results, we examine the temperature dependence of the magnetization and of the Curie temperature taking into account the SF, which is not treated in the usual MF theory but it is quite obvious in the FIM 0.1 0.2 τ 0.3 0.4 SF Fig The magnetization of a two-layer FCC film (mSF ¼ m2 ) calculated within the SF theory Parameters are the same as given in Fig components are involved The higher order terms in the series (12) must be included in order to increase the accuracy of the calculation for the monolayer thin film with the isotropic exchange Fig shows the temperature dependence of the magnetization for the symmetrical two-layer thin film in comparison with the results given in the MF theory (Fig 5) We can see that both the Curie temperature and the magnetization strongly decrease when the SF is considered, for example at τ ¼ 0:15, ξ ¼ 0:8, mSF ¼ 0:881; and mMF ¼ 0:998 3.2 Spin fluctuations in the Gaussian functional approximation Numerical calculations for the magnetizations and the Curie temperatures are carried out within the Gaussian functional approximation using Eqs (20)–(24) Fig shows the temperature dependence of magnetization with different thicknesses and without the surface exchange effect (ξ ¼ 1) It is noticed that the magnetization at zero temperature in the SF approximation is p SS ỵ 1ị1:4 for S ¼ (Fig 7), which is different from obtained by the MF approximation (Fig 3), where only the longitudinal z spin component is treated It is clearly seen that temperature ranges of nonzero magnetizations and the Curie temperatures obtained in the SF theory are essentially smaller than those of the MF method (Fig 8) This strong reduction proves the crucial role of the SF in the quasi-two dimensional systems For the monolayer film with the given parameters n ¼ S ¼ ξ ¼ 1, the MF method gives τMF C ≈0:35 but the SF theory with the Gaussian fluctuations gives τSF ≈0:05, which C is much improved in accordance with the Mermin–Wagner's theorem This enhanced accuracy is due to the Gaussian approximation, where the lowest order quadratic auxiliary field Comparisons with experimental results for nickel ultra-thin films The experimental investigations on the magnetic order of ultrathin nickel and nickel alloy films with several types of substrates are carried out in many works [1,15,16] It was observed that the Curie temperature for a given thickness is rather lower for Ni on Cu(1 0) than for Ni on Cu(1 1) [1,16] With an attempt to validate the SF approximation, the Curie temperature dependence on the thickness of FCC Ni ultra-thin films is plotted in Fig 10, in which the experimental data of Ni films grown on Cu(1 0) and Cu(1 1) surfaces (denoted as Ni/Cu(1 0) and Ni/Cu(1 1)) are taken from Ref [1] The Ni/Cu(1 0) and Ni/Cu(1 1) ultra-thin film cases are well described in the framework of Heisenberg model when the surface effects are included (ξ≈0:75 and ξ≈2:2 for the first and second cases, respectively) This chosen parameter values quite agree with the DFT calculations [8], which showed that NN exchange magnitude in the surface of Ni/Cu(1 0) films is smaller compared with the interior one Our results also imply B.T Cong, P.H Thao / Physica B 426 (2013) 144–149 149 exchange interactions The SF strongly reduces the Curie temperature with the decrease of the film thickness in comparison with the MF results, which is also consistent with the experimental observations for the ultra-thin nickel films 1.0 0.8 τC 0.6 Acknowledgment 0.4 0.2 0.0 n (layer) Ni/Cu(100) ξ=0.75 Ni/Cu(111) ξ=2.2 The authors thank the NAFOSTED Program 103.02.2012.37 and the fund for scientific research of Hue University, Vietnam Support from VNU University of Science is also acknowledged References Fig 10 The Curie temperature dependence on the FCC Ni ultra-thin film thickness The square (triangular) points are the theoretical results with ξ≈0:75 (ξ≈2:2), S ¼ 2, and h ¼0 The solid (dash) curve through square (triangular) points is guide to the eyes The experimental data of Ni/Cu(1 0) and Ni/Cu(1 1) films are taken from Ref [1] a lower Curie temperature with regard to smaller surface exchange parameter ξ (Fig 6) Conclusion The functional integral method is successfully applied for studying the magnetic properties of the Heisenberg thin films at finite temperature It is shown that the finite size effect experimentally observed in thin films (the dependence of the magnetization and the Curie temperature on the film thickness) can be illustrated by the Heisenberg model with different NN spin [1] F Huang, M.T Kief, G.J Mankey, R.F Willis, Phys Rev B 49 (1994) 3962 [2] 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Phys Conf Ser 200 (2012) 72020 [14] N.D Mermin, H Wagner, Phys Rev B 17 (1966) 1133 [15] Y Li, K Baberschke, Phys Rev Lett 68 (1992) 1208 [16] L.H Tjeng, Y.U Izderda, P Rudoff, F Sette, C.T Chen, J Magn Magn Mater 109 (1992) 288 ... layer magnetizations vanish at the same temperature, which is known as the Curie temperature of the thin films At a given temperature, magnetizations of layers decrease when position of the layer... investigations on the magnetic order of ultrathin nickel and nickel alloy films with several types of substrates are carried out in many works [1,15,16] It was observed that the Curie temperature... magnetization is calculated in the approach of the MF approximation or the Weiss molecular field approximation for FCC spin lattice film using Eq (22) Fig shows the temperature dependence of the