Theory of the Spin Reorientation Transition of Ultra-Thin Ferromagnetic Films

The reorientation transition of themagnetization of ferromagnetic films is studied on a microscopic basis within Heisenberg spin models. Analytic expressions for the temperature dependent anisotropy are derived from which it is seen that the reduced magnetization in the film surface at finite temperatures plays a crucial role for this transition. Detailed phase diagrams in the temperature-thickness plane are calculated. The direction of the magnetization of thin ferromagnetic films depends on various anisotropic energy contributions like surface anisotropy fields, dipole interaction, and eventually anisotropy fields in the inner layers. These competing effects may lead to a film thickness and temperature driven spin reorientation transition (SRT) from an out-of-plane ordered state at low temperatures to an in-plane ordered state at higher temperatures at appropriate chosen film thicknesses. Experimentally, this transition has been studied in detail for various ultra-thin magnetic films [1, 2, 3]. Recently, it was found by Farle et al. [4] that ultra-thin Ni-films grown on Cu(001) show an opposite behavior: the magnetization is oriented inplane for low temperatures and perpendicular at high temperatures. Journal of Magn. and Magn. Mater., accepted for publication (1998) 2 It has been shown by various authors that the mechanism responsible for the temperature driven transition can be understood within the framework of statistical spin models. While Moschel et al. [5] showed that the temperature dependence of the SRT is well described qualitatively within a quantum mechanical mean field approach, most other authors focused on classical spin models. Results obtained by extended Monte Carlo simulations [6, 7] as well as mean field calculations of classical Heisenberg models [8, 9, 10], although different in detail, agree in the sense that a temperature driven SRT is obtained. Very recently, the perturbative approach [9, 10] has been extended to itinerant systems [11]. Note, however, that due to an unsufficient treatment of the spin-orbit induced anisotropy in this work no SRT is possible for monoand bilayers in disagreement with the above mentioned theories and with experiments. While the SRT from a out-of-plane state at low temperatures to an in-plane state at high temperature is found to be due to a competition of a positive surface anisotropy and the dipole interaction the interesting new result for ultra-thin Ni-films is argued [4] to have its origin in a stress-induced uniaxial anisotropy energy in the inner layers with its easy axis perpendicular to the film. This anisotropy is in competition with the dipole interaction and a negative surface anisotropy. Theory confirms this picture [9, 10]. In this paper a systematic investigation of the parameter and thickness dependence of the reorientation transition and in particular a calculation of the corresponding phase diagrams is presented. The calculations are done in the framework of a classical ferromagnetic Heisenberg model consisting ofL two-dimensional layers on a simple cubic (001) or face centered cubic (001) lattice. The Hamiltonian reads H = J2 Xhiji si sj Xi D i(szi )2 + !2 Xij r 3 ij si sj 3r 5 ij (si rij)(rij sj); (1) where si = (sxi ; syi ; szi ) are spin vectors of unit length at position ri = (rx i ; ry i ; rz i ) in layer i and rij = ri rj . J is the nearest-neighbor exchange coupling constant, D is the uniaxial anisotropy which depends on the Journal of Magn. and Magn. Mater., accepted for publication (1998) 3 layer index = 1 : : : L, and ! = 0 2=4 a3 is the strength of the long range dipole interaction on a lattice with nearest-neighbor distance a ( 0 is the magnetic permeability and is the effective magnetic moment of one spin). Note that only second order uniaxial anisotropies D enter the Hamiltonian (1). In our calculations we will restrict ourself to the case that all anisotropies are equal to Dv except at one surface, where the value is Ds, as this scenario is sufficient for explaining the basic physics of the temperature driven reorientation transition. TheHamiltonian (1) is handled in amolecular-field approximation, the reader is referred to [10] for further details. In the following we assume translational invariance within the layers and furthermore that the magnetization is homogeneous inside the film and only deviates at the surfaces, which is justified by numerical simulations [10]. Therefore we can set hsii = hs i if si is a spin in a volume layer ( = v) or in a surface layer ( = s). Now we must distinguish two cases: For L 2 we get a system with only one (surface) layer with the effective interactions ~ xss = x0 + 2x1(1 L 1 s ), where x is either the coordination number, z, or the dipole sum, , respectively. In this case we have Ls = L and Lv = 0. The effective uniaxial anisotropy is ~ Ds = Ds+(Ls 1)Dv and the Curie temperature of the unperturbated system is Tc(L)=J = ~ zss=3. For L > 2we get a system with two layers with the effective interactions ~ xss = x0, ~ xvs = x1, ~ xvv = x0 + 2x1(1 L 1 v ), and ~ xsv = 2x1L 1 v . Here we have Ls = 2 and Lv = L 2. Note that the coupling between the volume spin and the surface spin is asymmetric. The effective uniaxial anisotropies are ~ Ds = Ds +Dv and ~ Dv = LvDv, respectively. The critical temperature of the unperturbated system is given by Tc(L) J = z0 3 + z1 3 1 L 1 v +q1 + L 2 v : In this approach we are not restricted to films with integer values of the thickness. The number of next neighbors z and the constants containJournal of Magn. and Magn. Mater., accepted for publication (1998) 4 ing the effective dipole interaction between the layers for different lattice types and orientations are listed in table 1. lattice z0 z1 z >1 0 1 >1 sc(001) 4 1 0 9.034 -0.327 O(e 2 ( 1)) fcc(001) 4 4 0 9.034 1.429 O(e p2 ( 1)) Table 1: Numerical constants z and for sc(001) and fcc(001) lattices. A perturbation theory to the mean field Hamiltonian can be applied considering the terms proportional to ! and D as small perturbations which is justified in view of the smallness of these parameters [9, 10]. In this frameworkwe derived an analytical expression for the total anisotropy K(T ) of the system which is defined as the difference of the free energies of the in-plane state and the out-of-plane state. K(T ) depends on temperature only through the absolute value of the subsystem magnetizations m (T ) calculated with the unperturbed Hamiltonian and is given by K(T ) =X ~ D 1 3T m (T ) h (T ) ! 3! 4 X ; 0 m (T )~ 0L 0m 0(T ): (2) h = JP 0 ~ z 0m 0 is the mean field in subsystem . If we neglect the narrow canted phase, the reorientation temperature Tr is given by the condition K(Tr) = 0. For the homogeneous case (L 2) the magnetization is approximately given by m2(T ) 1 T=Tc 1 2T=5Tc : This approximation deviates less than 1% from the values obtained numerically. From (2) we then get K(T ) 1 T Tc " ~ Ds 3!L~ ss 4(1 2T=5Tc)# ; Journal of Magn. and Magn. Mater., accepted for publication (1998) 5 from which the reorientation temperature is determined as Tr Tc 52 15! 8 0L + 2 1(L 1) Ds + (L 1)Dv The phase diagram of the fcc(001) system calculated within this approach is shown in figure 1. One can distinguish three different reorientation schemes: For uniaxial volume anisotropies Dv ! < 3 4( 0 + 2 1) 8:92 (fcc) we find a normal SRT from perpendicular to in-plane magnetization with increasing temperature and film thickness (solid lines in figure 1). If Dv is large, we find a reversed SRT from in-plane to perpendicular magnetization with increasing temperature and film thickness (dashed lines in figure 1). In the intermediate range we obtain a third type of SRT where the magnetization switches from perpendicular to in-plane directionwith increasing temperature, but with decreasing film thickness. (dashdotted lines in figure 1). This workwas supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 166. One of us (A.H.) would like to thank M. Farle for fruitful discussions. Journal of Magn. and Magn. Mater., accepted for publication (1998) 6