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Physica A 388 009 35 33 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.

Timisoara, Romania c Vojvodina Academy of Science Arts, Novi Sad, Serbia a r t i c l e i n f o a b s t r a c t Article history: Received 9 April 008 Received in revised form 7 October 008 Available

1 Physica A Contents lists available at ScienceDirect Physica A journal homepage: Low-temperature magnetization in nanofilms Vjekoslav Sajfert a,, Dušan Popov b, Bratislav Tošić c a Technical Faculty Mihajlo Pupin Zrenjanin, Serbia b Universitatea Politechnica, Timisoara, Romania c Vojvodina Academy of Science Arts, Novi Sad, Serbia a r t i c l e i n f o a b s t r a c t Article history: Received 9 April 008 Received in revised form 7 October 008 Available online November 008 PACS: i Ak a w The low-temperature magnetization of a film was analyzed by the use of exact Bose representation of spin operators that does not suffer from the presence of unphysical states. The magnetization of thin films has exponentially small temperature correctness, so that Dyson s proof about exponentially small correction coming from two bosons at ideal lattice point cannot be used in film analyses. The main conclusions of this work are that magnetic lattice of a thin film is more rigid than the macroscopic lattice that the autoreduction process the three layer film divides into two layer subfilms takes place in the film. 008 Elsevier B.V. All rights reserved. Keywords: Ultrathin magnetic films Low-temperature magnetization Exact Bose representation Two small parameters autoreduction Qualitative difference nano macrostructures 0. Introduction The problem of operators with mixed kinematics arose 80 years ago. Bosons fermions have clean kinematics while spin operators ], Pauli operators quasi-pauli operators 6] have mixed kinematics their kinematics is a combination of Bose Fermi kinematics. Mixed kinematics leads to statistical problems the calculation of mean values of operator products correspondingly to thermodynamical problems. Since thermodynamics deals with measurable quantities, correct evaluation of statistical means is necessary. At first the problem of mixed kinematics corresponding mixed statistics arose in quantum theory of magnetism. All attempts to solve this problem were based on the idea of finding an adequate boson representation of spin operators which would give correct statistical means. Following the appearance of lasers the interaction between optical excitations became topical. The original operators in the Hamiltonian of optical excitations Pauli quasi-pauli operators had mixed kinematics statistics too, boson representations of these operators were necessary for the explanation of nonlinear optical effects, such as energy shift in optical spectra two- three-photon absorption 7,8]. A short review of attempts concerning the problem of mixed kinematics is presented in the first section. The second section is dealing with the problem of the low-temperature magnetization in thin films the results of the second section are summarized in the conclusion. Corresponding author. Tel.: ; fax: addresses: sajfertv@ptt.rs V. Sajfert, dusan_popov@yahoo.co.uk D. Popov, btosic@yahoo.com B. Tošić /$ see front matter 008 Elsevier B.V. All rights reserved. doi:0.06/j.physa

2 36 V. Sajfert et al. / Physica A A short review of problems concerned with mixed kinematics The problem of the low-temperature magnetization of Heisenberg ferromagnet was started by Bloch 9] who approximated Heisenberg isotropic spin Hamiltonian with quadratic boson form. The Hamiltonian of Heisenberg ferromagnet is given by: H = SJ 0 S S z I, S S+ I, S S z S S z,, S + = S x + is y, S + = S x + is y are spin operators, are exchange integrals J 0 = l I l. The spin operators obey the following commutation relations:. S +, S ] = S z ; S +, S z] = S + ; S, S z] = S.. Bloch used boson representation of spin operators: S + = SB ; S = SB + ; S Sz = B+ B..3 obtained equivalent boson Hamiltonian: H = SJ 0 B + B S I, B + B.,.4 By diagonalisation of Hamiltonian.4, he found the dispersion law E = S J 0 J k J k = I 0 e i k using it he obtained the following expression for magnetization: σ = Sz S = S ζ 3/τ 3/ ; τ = Θ which is called Bloch s 3/ law. The Riemann s ζ function is given by: πi ; Θ = k BT.5 ζ p = n= n p..6 The next step in the theory of low-temperature magnetization was done by Holstein Primakoff 4], who used the following boson representation for spin operators: S + = S S z = S B + B. B+ B S B; S = SB + B+ B S Using this representation of spin operators, they obtained the expression for low-temperature magnetization:.7 σ = σ h + σ ah.8 σ h = ζ 3/ τ 3/ 3π ζ 5/τ 5/ 33π 6 ζ 7/τ 7/ + O τ 9/,.9 σ ah = 6πζ 3/ ζ 5/ τ 4..0 The terms in formula.9 are coming from exping the dispersion law in powers of magnitude k of k up to k 6 while anharmonic correction is the consequence of the interaction between spin waves. The result.8.0 was correct but the boson representation of spin operators, which is correct only if number of bosons at one lattice point is maximally equal to S arose discussions concerning the contributions of so-called unphysical states, i.e. the states with more than S boson per lattice point. The completed theory of the low-temperature magnetization was given by Dyson 0,]. He proved that the states with more than one boson at one lattice point involve exponentially small contributions in the low-temperature magnetization. Using this proof, he obtained that Holstein Primakoff s result was correct.

3 V. Sajfert et al. / Physica A Finally, the discussions about unphysical states were put ad acta by the appearance of Agranovich Toshich 3] boson representation of spin operators for spin operators S = /: ] S + ν ] = + ν! B+ν B ν B; S = B + ν + ν! B+ν B ν ; ν=0 ν=0 ν Sz = B + ν+ B ν+. + ν! ν=0 by exact boson representation of quasi-pauli operators 5]. It should be noticed that boson infinite series. satisfy commutation rules. for an arbitrary number of bosons. Using the representation., the exact results for the lowtemperature ordering parameter of antiferromagnet was determined ] the exact result for magnetization in case S > / was found in 6]. The correct result of Holstein Primakoff s formula.8 was obtained in 3]. Using the formula. Agranovich Toshich predicted the possibility of Bose condensation of Frenkel excitons. The similar idea was applied to ferromagnet in strong magnetic field µ B H >> I 4] the Bose condensation of spin waves was predicted. This prediction has been recently proved experimentally for YIG ferromagnet films in 5]. The exact Bose representation of spin operators. will be applied for the calculation of magnetization of ultrathin ferromagnet films with S = /. Ending this review of the low-temperature problems in determining magnetization, we quote the leading correction which is given by Bloch s approach in three layer film. This correction is proportional to τ Z it is clear that it is exponentially small., Zp q = n=. e nq n p. Low-temperature magnetization in three layer magnetic films with spin S = / The interest in magnetic problems has increased recently 6 4], but no has the problem of broken symmetry magnetic structures been in the foreground. We shall analyze the three layer magnetic film with spin S = / cut off in z direction from simple cubic ideal structure. The more compact clearer formulae will be obtained if we substitute spin operators with Pauli operators in accordance with the following formulae: S + = P, S = P + ; Sz = P + P; P, P + ] = P + P ; P + = P = 0. Using. the nearest neighbour approximation for a simple cubic structure, we obtain the Hamiltonian of three layer film in the following form: H = I,+ a + I, a + I,+ a + I, a + I,+ a + I, a P + P P + I,+ a P + a + I, a P a + I,+ a P + a + I, a P a + I,+ a P + a + I, a P a P + P I,+ a P + + a P + a + I, a P + a P a + I,+ a P + + a P + a + I, a P + a P a + I,+ a P + + a P + a + I, a P + + a P a a connects the nearest neighbours. In formula. the boundary conditions must be taken into account on the surfaces 0. It means that boundary conditions are:. I nx,n y,0;n x,n y, = I nx,n y,n z ;n x,n y,n z + = 0..3 The other exchange integrals will be denoted by I. The analysis will be carried out by means of the Pauli Green function Γ, t = θ t P t, P + 0] P t P + 0,.4 θ t is a Heaviside step-function. Differentiating Γ, t with respect to t taking into account that Ṗ = i h P, H], we obtain the equation for Pauli Green function Γ, t: i h d dt P t P + 0 = i hδ t δ, P + P + P, H] t P

4 38 V. Sajfert et al. / Physica A Further steps in the analysis are the following:. In accordance with.., Pauli operators will be substituted with: P = B B + B B ; P + = B+ B+ B+ B ;.6 P + P = B + B B + B in the Green function P, H] t P + 0 in correlation function, while in the higher order Green functions of the type P + t a P a t P b t P + 0 will be substituted with Bose operators, i.e. it will be taken that P = B. c. Applying Wick s theorem 5], we can write down: P t P + 0 B t B + t B t B t B + 0 B+ 0 B+ 0 B 0 B t B + 0 B+ t B t B t B + 0 B t B + 0 B+ 0 B 0 B t B + 0 B+ B 0 B t B + 0 B+ B 0 B t B It should be noticed that the higher order Green functions of the type B + BB B + B + B are rejected, since they are proportional to the square of boson concentration B + B. Inserting the formulae obtained by the application of Wick s theorem for bosons into.5, applying the boundary conditions.3, using the transformation for the boson Green function G, t = B t B + 0 in which translatory invariance in x, y plane is taken into account: G nx,n y,n z ;m x,m y,m z t = B nx,n y,n z t B + m x,m y,m z 0 = e iakxnx mx+iakyny my + k x,k y we obtain the following system of difference equations for the function α nz,m z kx, k y, ω : dωe iωt α nz,m z kx, k y, ω..8 n z = J α,mz + α 0,mz + ρα,mz = i h + B + B 0 δ,mz ; π.9 n z = 0 J J α,mz + ρ + ρ α 0,mz = i h + B + B 0 δ 0,mz ; π.0 n z = J J α,mz + ρ + ρ α,mz = i h π The notations in upper equations are the following: ρ = E I sin ak x + ak sin y I + I F + 3F IΦ ; + B + B 0 δ,mz.. ρ = I I F + F 3 IΦ ; Φ = Φ Φ ;. J = I F ; J = 4 I F; F = F F the functions F Φ are given by: F λ = k x,k y Φ λ = = ak q x,q y e E0 kx,ky,λ M k x, k y, q x, q y e E0 qx,qy,λ πτ 3/ Z 3/ λ ak 3 4 πτ Z λ = τ Z λ 7 + ak 3 8 πτ 3/ Z 3/ λ 9 + π λ τ Z + O τ 3 Z ; ] λ 6 π τ 5/ Z 5/ ] λ 6 π τ 3 Z 3 95 λ 768 π τ 5/ Z 5/ λ I Θ = λ.3 ] + O τ 7/ Z.4

5 V. Sajfert et al. / Physica A E 0 k x,k y,λ = I sin ak x + sin ak y + sin λπ 6 ; λ =,.5 M k x, k y, q x, q y = cos akx + cos ak y + cos aq x + cos aq y cos a k x q x cos a k y q y..6 It can be easily shown that all Eqs..9. after the substitution α nz,m z = a mz,λs nz,λ, λ=.7 S nz,λ = sin n z + ϕ λ + Y λ sin n z ϕ λ ; ] Y λ = + + cos ϕ 0.8 λ F + 3 Φ reduce to the unique one λ= J cos ϕ λ + ρ a mz,λ kx, k y, ω S nz,λ = i h π δ n z,m z ; n z, m z {0,, }.9 if the parameters ϕ λ satisfy the following equation: sin 3ϕ λ sin 3ϕ λ cos ϕ λ + sin ϕ λ cos ϕ λ sin ϕ λ F Φ = 0..0 The Kronecker symbol in.9 will be taken in the following form: δ nz,m z = λ= S nz,λt mz,λ. This system of algebraic equations enables us to calculate all introduced functions T mz,λ. Substituting. in.9 taking that. a mz,λ we reduce.9 into kx, k y, ω = b kx,k y,λ ω T mz,λ b kx,k y,λ ω = i + B + B 0 π ω ω, k x,k y,λ..3 ω kx,k y,λ = h E k x,k y,λ ; E k x,k y,λ = I sin ak x + ak sin y + ϕ λ sin IF IF 3 4 cos λπ 3 + IΦ. Combining.8,.7,., inversing Fourier s time-frequency transformation of the Green function.8, we obtain: G nx,n y,n z ;m x,m y,m z ω = i + B + B 0 e iak xn x m x+iak yn y m y S nz,λt mz,λ π k x,k y λ= ω ω 0 k x,k y,λ in this formula the unknown functions T mz,λ figure. They can be found from. putting δ nz,m z = for n z = m z δ nz,m z = 0 for n z m z. Having formula.5, we can find the spectral intensity, correlation function boson concentrations in the first approximation. We shall quote the first order boson concentration for subfilm 0,, only: B + B 0 = + B + B 0 k x,k y λ= S 0 n z,λ T 0 n z,λ E kx,ky,λ e Θ

6 330 V. Sajfert et al. / Physica A Fig.. Schematic presentation of autoreduction of three layer film into two equivalent two layer films. Now we shall determine the ordering parameter estimate the remaining order of magnitude. The expression for σ is given by: σ = P + P = B + B + 4 B + B 0]..7 Taking into account.6, we can write down: σ = k x,k y λ= S nz,λt nz,λ E kx,ky,λ e Θ. From formula.8 the term proportional B + B 0 B + B B + B 0 was rejected since it is extremely small. The fact that Eq..0 has only two real solutions in interval 0, π is of great importance. Since the number of film layers has three values, the transformation from configuration space to momentum space is not isomorphic. It is of the type 3 this means that the autoreduction takes place in the film 6 30]. The spin wave excitations exist in two layers only. It can be 0,, sets of layers. The set 0, is incompatible with the nearest neighbour approximation which was applied. Geometrically, it is clear that sets 0,, are completely equivalent see Fig.. It means that it is enough to examine magnetization in one of the mentioned subfilms. It was done for subfilm 0, it was found that magnetizations of layers are approximately given by: a is.8 σ n z = 0 = σ h n z = 0 + σ a n z = 0.9 σ h n z = 0 = τ Z σ a n z = 0 = aτ a = 8π Z Z 9 3 8π Z 0 Z..3 The magnetization for layer n z = is given by σ n z = = σ h n z = + σ a n z =.33 σ h n z = = τ Z 3 σ a n z = bτ b = 8π Z Z π Z 0 Z..36 In formulae.3.34 only the leading terms of magnetization are quoted.

7 V. Sajfert et al. / Physica A Conclusion Concluding this analysis, we would like to emphasize the following:. Magnetizations of layers 0 differ. It is the consequence of broken symmetry.. The temperature corrections are exponentially small due to the presence of zeta functions Z λ p. 3. The autoreduction of the n n type takes place in the film reducing the three layer ferromagnet into two layer one. 4. In connection with experimental fact that Bose condensation takes place of spin waves in ferromagnet film YIG, our intention is to examine theoretically the behaviour of thin magnetic films in extremely strong magnetic fields. Acknowledgements This work was supported by the Serbian Ministry of Science Technology: Grant No 4044, by Vojvodina Academy of Sciences Arts. References ] S.V. Tyablikov, Methods in the Quantum Theory in Magnetism, Plenum Press, New York, 967. ] N.N. Bogolubov, S.V. Tyablikov, Dokl. Akad. Nauk USSR ] V.M. Agranovich, B.S. Toshich, JETP ] T. Holstein, H. Primakoff, Phys.Rev ] D.I. Lalović, B.S. Tošić, R.B. Zakula, Phys. Rev ] D.I. Lalovi c, B.S. Tošić, R.B. Zakula, Phys. Rev ] N. Blombergen, Nonlinear Optics, Mir, Moscow, ] R. Loudon, l Phys. Rev ] F. Bloch, Z. Physik ] F.J. Dyson, Phys. Rev ] M.J. Skrinjar, B.S. Tošić, S.D. Stojanović, Rev. Res. Sci ] M.L. Kulić,Tošić, Phys. Stat. Sol. b K79. 3] Group of authors, Jubilee of Bratislav To sić. SFIN ] V.E. Demidov, O. Dzyapko, S.O. Demokritov, G.A. Melkov, A.N. Slavin, Phys. Rev. Lett ] V. Antonov, Electronic Structure Magneto-optical Properties of Solids, Kluwer Academic Publishers, ] K.H.J. Buschow, F.R. De Boer, Physics of Magnetism Magnetic Materials, Kluwer, New York, Boston, ] M. Getzlaff, Fundamentals of Magnetism, Springer, Berlin, Heidelberg, New York, ] J. Stöhr, H.C. Siegmann, Magnetism from fundamentals to Nanoscale Dynamics, Springer, Berlin, Heidelberg, ] R.M. White, Quantum Theory of Magnetism, in: Magnetic Properties of Materials, Springer, Berlin, Heidelberg, ] S. Maekawa, T. Shinjo, Spin Dependent Transport in Magnetic Nanostructures, CRC Boca Raton, London, New York, Washington, D.C, 00. ] M. Pavkov, M. Skrinjar, D. Kapor, M. Pantić, S. Stojanović, Phys. Rev. B ] M. Manojlović, M. Pavkov, M. Škrinjar, M. Pantić, D. Kapor, S. Stojanović, Phys. Rev. B ] P. Fröbrich, P.J. Kuntz, Phys. Rep ] R. Radu, M. Etzkorn, R. Siebrecht, et al., Phys. Rev. B ] B.S. Tošić, Statistical Physics, Faculty of Science, Novi Sad, ] V. Sajfert, R. Ðajić, M. Ćetković, B. Tošić, Nanotechnology ] V. Sajfert, R. Ðajić, B.S. Tošić, J. Nanosci. Nanotech ] V. Sajfert, J. Setrajčić, B. Tošić, R. Ðajić, Czechoslovak J. Phys ] V. Sajfert, J.P. Setrajčić, S. Jaćimovski, B. Tošić, Physica E ] V. Sajfert, J. Setrajčić, D. Popov, B. Tošić, Physica A 353C

Behaviour of Thin Magnetic Films at Low Temperatures

Behaviour of Thin Magnetic Films at Low Temperatures Copyright 009 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Computational Theoretical anoscience Vol.6,, 009 Vjekoslav Sajfert, Dušan Popov, Ljiljana