… aν z rν j Spin wave in ultrathin magnetic film Le Ngan, Bach Thanh Cong Computing materials Science Laboratory, Faculty of Physics, VNU University of Science, Rj 334 Nguyen Trai, Hanoi, ν Viet nam Abstract In this work the temperature dependent dispersive law for spin wave in ultrathin magnetic films consisting from several atomic spin layers is calculated using double time Green function technique and anisotropic exchange Heisenberg model The chain of Green functions is decoupled within Tyablikov-Bogoliubov approximation Dependence of spin wave spectra on the number of layers and temperature is analyzed Keywords: Magnetic thin film, Heisenberg model, Double times Green function Introduction At present time magnetism of low dimensional spin systems involves much interest of researchers In [1] the ground magnetic state of perovskitenanoclusters is studied using density functional theory (DFT) method Thermodynamic properties at finite temperature of ultra- thin magnetic films (quasi- two dimensional case)is proposed to calculate by the functional integral method [2] The spin waves theory in magnetic single layer with dipole and isotropic exchangeinteractions is developed in [3].The aim of this research is to investigate the spin wave in the case of anisotropic exchange interaction in untrathin magnetic films using the popular double – time Green’s function method [4] We will restrict our treatment to ferromagnets Model and formalism We consider a magnetic thin film having cubic structure and consisting of n atomic spin layers.Each atom has a magnetic ma = − g µ B S Fig 1: Arrangement of thin film atomic spin lattice relative to coordinate system moment The Oz axis of coordinates system is chosen perpendicular to the surfaces of the thin film and atomic spin planes are parallel to xOyplane (see figure 1).There is a translational symmetry of spin arrangement in xOy plane andthe number of spin in every plane N is to be very large (N ~ ∞) rν j is the position vector of spin Sν j ( rν j = R j + aν zˆ ) Here a is the lattice constant, ν is an Rj index of layer and aν zˆ is the two-component vector describing position of the spin on the xOy plane, is the component of the position vector on the Oz axis.Heisenberg Hamiltoniandescribed theinteracting spin system in the thin film is written as: ν= H =− z ∑ Jν j ,ν ' j ' Sν j Sν ' j ' − g µB B0 ∑ Sν j νj ν j ,ν ' j ' (1) rν j The first term in (1) is an exchange interaction between two spins at sites and Jν j ,ν ' j ' = Jνν ' ( R j − R j ' ) depends only on the distance, rν ' j ' , which This exchange integral is a periodical Rj − Rj ' function of two dimensionallattice vector The second term in (1) is the energy of the spin system in the external field orientated along the Oz axis Normally, we use the up, down Sν±j = Sνx j ± iSνyj , Sνz j and Z components of spin operator, And Hamiltonian (1) is rewritten in Sν±j , Sνzj terms of H =− as + − z z z ∑ Jν ,ν ' ( R j − R j ' ) Sν j Sν ' j ' + Sν j Sν ' j ' − g µ B B0 ∑ Sν j νj ν j ,ν ' j ' ( ) (2) In order to study the kinetics of the system at finite temperature, we introduce the following retarded double times Green function: Gν j ,ν ' j ' ( t , t ') = Sν+j ( t ) ; Sν−' j ' ( t ' ) = −iθ ( t − t ' )  Sν+j ( t ) , Sν−' j ' ( t ' )  − (3) An equation of motion for Green function (3) in the energy representation has been obtained as: r r EGν jν ' j ' ( E ) = Sνz j δνν 'δ jj ' − ∑ Jν1ν R j1 − R j ν1 j1 − Sνz1 j1 Sν+j ; Sν−' j ' E + g µ B B0 ( Sν+j ; Sν−' j ' E } ){ Sνz j Sν+1 j1 ; Sν−' j ' E (4) The lower symbol E in (4) denotes the energy representation In what follows we drop this symbol for clarity The chain for Green function (4) can be decoupled using Bogolyubov Tyablikov’s procedure, where: Sνz j Sν+1 j1 ; Sν−' j ' → Sνz j Sν+1 j1 ; Sν−' j ' Sνz1 j1 Sν+j ; Sν−' j ' → Sνz1 j1 Sν+j ; Sν−' j ' (5) Because of translational symmetry in the thin film plane, the average value of the projection Sνz j = Sνz of moment spin onto the z-axis does not depend on j and j 1, i.e .The formula (4) now becomes: r r  E − g µ B − J R − R ∑  B ν1ν j1 j ν1 j1  ( )  Sνz1  Gν jν ' j ' ( E ) = Sνz δνν 'δ jj '  ( − ∑ Jν1ν R j1 − R j ν1 j1 ) Sνz Gν1 j1ν ' j ' ( E ) (6) Due to the translational symmetry of the spin distribution in the thin film planes, Green k functioncan be expanded in the Fourier series of wave vector space Gν jν ' j ' ( E ) = ik ( R j − R j ' ) ∑ Gνν ' ( k , E )e N k (7) One has from (6)  z  z z %  E − g µ B B0 − ∑ J% ν1ν (0) Sν1  Gνν ' ( k , E ) = Sν δνν ' − ∑ Jν 2ν ( k ) Sν Gν 2ν ' ( k , E ) ν1 ν2   (8) ikR j J% νν ' ( k ) = ∑ Jνν ' ( R j ) e j Where (9) We use the nearest neighbor approximation (n.n.) then Fourier image of exchange integral (9) becomes:   J s ( cos k x a + cos k y a ) when ν ' = ν J% νν ' ( k ) =  when ν ' = ν ±  J p Js J p ( (10) ) denotes the in-plane (next plane) n.n exchange Spin wave spectra in single and double layers thin films 3.1 Single layer film For single layer film with isotropic exchangewe obtain the following retarded Green function from (8): Sz Sz G11 ( E , k ) = = z z E − E( k) E − g µ B B0 − J% + J% 11 ( ) S 11 ( k ) S (11) The spin wave spectrum as a pole of the Green function is derived as: E (k , T ) = g µ B B0 +  − ( cos k x a + cos k y a )  mJ s (12) m = Sz / S In (12), is an average magnetization per sites which is temperature dependent and satisfies following equation: m = 1− ∑ NS k  ε ( k ,τ )  exp  −1  τ  (13a) Here the dimensionless energy and temperature are ε (k ,τ ) = k T E (k , T ) τ= B Js Js (13b) ; The isotropic exchange case is not interest because of absence of magnetization in monolayer spin film with isotropic Heisenberg exchange (Mermin-Wagner theorem) and it is not proper described in Tiablikov-Bogolivbov approximation Then we study the more interesting case J s = J s21 + J s22 J s1 J s of single layer film with anisotropic exchange: ( ( ) is n.n exchange along Ox (Oy) direction).We obtain the following spin wave spectrum formonolayer film: ε k,a = { [ ] } gµ B B0 + J s1 + J s − cos k x a − ρ cos k y a mS J s1 ρ = J s2 / J s1 (14) τ = k BT / J s1 characterizes anisotropy and is dimensionless temperature Figure2 and show the temperature dependence of the magnetization of the monolayer thin film with anisotropic exchange Figure show and show the spin wave spectra for the cases where magnetizations are described in figures 2, and One sees from figures 4, that intensity of spin wave reduces with increasing temperature enhancement of spin wave in single layer film τ and the anisotropy gives to 1 8 m m T 0 0 0 0 τ 0 α 0 5 T c 0 0 Fig 2:Temperature dependence of magnetization of single layer film with S=1 on temperature for ρ = 1.7 τ 0 0 α 0 c Fig 3: Temperature dependence of magnetization of single layer film with S=2 on temperature for ρ = 1.7 10 25 E ( τ= 0.01) E (τ= 0.01) k E ( τ= 0.02) k 20 k E ( τ= 0.05) E (τ= 0.03) k k 15 E (τ= 0.096) k E E k k 10 0 0 1 2 3 4 k Fig 4: Spin wave energy spectra in the first Brillouin zone of single layer cubic spin film at different temperatures for S=1, anisotropic ρ = 1.7 exchange parameter 0 1 k 3 4 Fig 5: Spin wave energy spectra in the first Brillouin zone of single layer cubic spin filmat different temperatures for S=2, anisotropic exchange ρ = 1.7 parameter 3.2 Double layers thin film Similarly, by symmetry we obtain the expression for two Green functions in case of two layers film:   1 ÷ G22 ( k , E ) = G11 ( k , E ) = m ( E − Ek ,m )  −  E − ( Ek , m + J p m ) E − ( E k , m − J p m ) ÷   (15)  1 G12 ( k , E ) = G21 ( k , E ) = J p m  −  E − ( Ek , m + J p m ) E − ( E k , m − J p m )   ÷ ÷  (16) Ek±,m We obtain the two branches for energy spectrum of spin waves ξ k+,m = Ek+,m Js = { g µ B B0 +  − ( cos k x a + cos k y a )  J s mS Js } (17) ξ − k ,m Ek−,m = = g µ B B0 +  − ( cos k x a + cos k y a ) + 2η  J s mS Js Js { } Parameter η =Jp/Js characterizes an anisotropy behavior of exchanges and magnetization obeys the following equation: m= Sz S ≅ 1− η m2 − + S S = − 2S 2N    ÷ α  ÷ exp ξ τ − k ,α =±  k ,m  ∑ ( ) Figure 6, and show the temperature dependence of the magnetization of the double layers thin film with anisotropic exchange between layers Figure show and show the spin wave spectra for the cases where magnetizations are described in figures 2, and 3.One sees from figures 7, and that intensity of spin wave reduces (increases) with increasing (reducing) temperature (anisotropy) like in monolayer case But differing with single layer case, there is an energy gap between two spin waves branches and the gap is wider when temperature is lower Crossing between branches at different temperatures in the Brillouin zone is also possible 10 (E )- (0 01) k 8 (E )+(0.01) gap τ = 0 k - 6 (E ) (0 09) k (E )+(0.09) k E m k gap τ = 0 T 0 0 0 0 τ 0 0 c 0 0 0 Fig 6: The dependence of magnetization on η = temperature for , S = 0 0.5 1.5 k 2.5 3.5 4.5 Fig 7: Dependence of the spin wave energy spectrum k on the wave vector 0.01 and τ = 0.09 for at different temperatures τ = η = ,S=1 7 10 0.9 (E )- (0.01) k (E )+(0.01) gap τ = 0 k (E ) (0.1) (E )+(0.1) - k E m k 0.8 0.7 τ = 0.6 0.5 gap k Tc 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 τ Fig 8: Temperature dependence of magnetization for double layer film for S = 1,and anisotropic exchange η = 0.2 0 0.5 1.5 k 2.5 3.5 4.5 Fig 9: Energy spectrum of spin waves in the first Brillouin zone of double layers film at different temperatures τ=0.01; τ=0.1 for S=1,and anisotropic exchange parameter between η = 0.2 parameter between layers layers Acknowledgements Authors thank the NAFOSTED grant 103.02.2012.37 for support References [1]Nguyen Thuy Trang, Bach Thanh Cong, Pham Huong Thao, Pham The Tan, Nguyen Duc Tho,Hoang Nam Nhat, Physica B 406 (2011) 3613 [2] Bach Thanh Cong, Pham Huong Thao, Physica B 426 (2013)144–149 [3] E Meloche, J I Mercer, J P Whitehead, T M Nguyen, and M.L Plumer, Phys Rev B 83 (2011) 174425 [4] S.V Tyablikov, Methods in the quantum theory of magnetism, Plennum press, New York, 1967  ... the spin wave spectra for the cases where magnetizations are described in figures 2, and One sees from figures 4, that intensity of spin wave reduces with increasing temperature enhancement of spin. .. Fig 4: Spin wave energy spectra in the first Brillouin zone of single layer cubic spin film at different temperatures for S=1, anisotropic ρ = 1.7 exchange parameter 0 1 k 3 4 Fig 5: Spin wave... (1) is the energy of the spin system in the external field orientated along the Oz axis Normally, we use the up, down Sν±j = Sνx j ± iSνyj , Sνz j and Z components of spin operator, And Hamiltonian