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non-magnetic films: Monte Carlo approach
Applied Surface Science 255 (2009) 8695–8700
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Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Dependence of the magnetization on the interface morphology in ultra-thin magnetic/non-magnetic films: Monte Carlo approach A. Razouk, M. Sahlaoui *, M. Sajieddine Laboratoire de Physique et Me´canique des Mate´riaux, Faculte´ des Sciences et Techniques, Universite´ Sultan Moulay Slimane, BP 523, 23000 Be´ni-Mellal, Morocco
A R T I C L E I N F O
A B S T R A C T
Article history: Received 31 March 2009 Received in revised form 11 June 2009 Accepted 13 June 2009 Available online 21 June 2009
Using Monte Carlo simulations, we have studied the dependence of magnetic properties on interface morphology in magnetic/non-magnetic (M/NM) multilayers. Our aim is to relate macroscopic magnetic properties of the multilayers to their concentration profile at the interface. Our model consists of an alternate staking of magnetic and non-magnetic layers with disordered interfaces. We have considered different concentration and the existence of local magnetic domains at the interface. The results indicate the crucial dependence of magnetization amplitude with interface multilayers atomic composition and the spatial arrangement of magnetic atoms. In particular, we show that isolated islands at the interface leads to the apparition of super-paramagnetic behavior. ß 2009 Elsevier B.V. All rights reserved.
PACS: 87.55.K 75.70.i 67.30.er 67.30.hp 75.10.Jm Keywords: Monte Carlo simulation Heisenberg model Interface morphology Multilayers Curie temperature
1. Introduction Studies of metallic magnetic multilayers are not only of fundamental interest but are also essential for development of new magnetic multilayers [1]. Recent experimental techniques have been associated with analytical tools (Mo¨ssbauer spectrometry, squid magnetometry, polarized neutron reflectivity, etc.) to determine magnetic properties of multilayers and the morphology of interfaces in relation with the substrate temperature and the deposition process [2]. Since the discovery of giant magnetoresistance, novel problems have been raised up concerning magnetism of ultra-thin films and interfaces. It is believed that the spin-dependent scattering occurs mainly at interface sites, but the relation between interface morphology (e.g. surface reconstruction, roughness or interdiffusion) and the spin-dependent scattering probability is not yet clear. On the theoretical side, study has revealed that the spindependent scattering is dependent largely on the electronic
* Corresponding author. E-mail address: [email protected] (M. Sahlaoui). 0169-4332/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2009.06.061
structure of magnetic layers, especially when they are adjacent to non-magnetic layers [3]. The interface order is treated in the mean of ab initio calculation [4] within it is shown a depend spin with interface concentration. The theoretically predicted effects will be suppressed by interface roughness and interdiffusion, which cannot be avoided in many multilayers systems such as the Fe/Cu(0 0 1) system; where intermixing is known to occur [5]. On the other hand Monte Carlo simulations have been widely used in recent years to study the magnetic properties of multilayers. Various effects such as surface anisotropy, spin dynamics and magnetic semiconductor have been investigated [6–9]. The aim of this paper is to study the dependence of magnetic properties on the interface morphology in M/NM multilayers by Monte Carlo (MC) method with Heisenberg Hamiltonian. The MC method is one of the powerful techniques to simulate complex models and present the advantage that the mean-field approximation is not made. The additional complications are introduced to simulate really the structural dependence of the magnetic properties. Furthermore, in the case of Fe/Cu multilayers, structural imperfections can lead to the formation of an island structure with sub-multilayers Fe layers at the interface. This may lead to the formation of nanometer sized grains embedded in the Cu medium.
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Fig. 1. Schematic representation of two-dimensional cross-section through the multilayers system consisting of magnetic (M) and non-magnetic (NM) materials with uniform distribution of atoms at the interface ((MkNM1k)(Mk0 NM1k0 ) (a), k and k0 are the mixture parameter in the interface 0 k 0.5 and 0.5 k0 1), mono-domain (MD) configuration (b), decoupled cluster (DC) (c), in this case a magnetic cluster is localized within the non-magnetic layer.
In the following section, we describe the model and the simulation technique. The numerical results are presented in Section 3. Our conclusion and perspectives are given in Section 4. 2. Model 2.1. Atomic model We describe our multilayer system with a real interface in the sense that it is not really sharp but exhibits a certain degree of disorder. We simulate the magnetization for interfaces with controlled morphologies at different temperatures, by introducing an interface mixture in a systematic manner. The system (Fig. 1) consists of ferromagnetic Heisenberg layers with different materials M (magnetic) and NM (nonmagnetic) and disordered interface in between. This study could be carried out on multilayers formed by two different magnetic layers [10]. The interface atomic model is characterized by some arrangement of M and NM atoms so that (M1xNMx) is a three-dimensional alloy with 0 x 1 (x = 0 or x = 1 abrupt interface, x 6¼ 0 interface alloys). In this work, we have limited the interdiffusion to two atomic layers; this is due to the weak miscibility between transition metals as iron and copper. We consider the interface made up of two areas; an atomic layer beside the layer NM rich in element NM and poor in element M (MkNM1k) with 0 k 0.5 and the other layer adjacent to the layer M rich in element M (Mk0 NM1k0 ) with 0.5 k0 1 (Fig. 1a). k (or k0 ) is related to the abundance of element M in each of the two areas for a fixed composition x. For two monolayers at the interface, we can write k + k0 = 2x. Such interface configurations are predicted in real
samples (Fe/Cu) analyzed experimentally, using Mo¨ssbauer spectroscopy [2,11]. Furthermore, at fixed concentration x, k and k0 values, two extremes situations of rough interfaces are probable. The first one corresponds to uniform distribution (UD) of atoms at the interface (Fig. 1a). About all the atoms are surrounded in the same manner. In the second limit, the interface atoms are clustered to form two different mono-domains (MD) (M and NM). The magnetic domain is coupled to the magnetic layer (Fig. 1b). In each cluster, atoms are surrounded by the maximum of atoms of the same species (M or NM). In our simulation, cluster distribution at the interface can be modified systematically and different configurations between UD and MD cases would be considered. Also, decoupled clusters (DC) within the non-magnetic area without interactions with magnetic layer are considered (Fig. 1c). 2.2. Energetic model We describe the system by an isotropic Heisenberg model in absence of effects of anisotropy; a simple Hamiltonian H was used for the simulation as shown below: X X H¼ J i j Si S j þ B Si hi; ji
i
where Si is a classical Heisenberg spin and the sum is taken over nearest-neighbour pairs of spins. Jij denotes nearest-neighbour exchange interaction and is assumed to be JM–M between different magnetic atoms. The exchange parameters are defined in temperature units. B represents a weak magnetic field; applied parallel to the film plane to activate spin fluctuations along the layers and is about 50 Oe.
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In order to relate our results to real systems, such as M/NM multilayers, we make the assumption that M element is Curie metal (Fe); the NM element will be copper. According to heavy M– M interaction, the JMM interaction is considered to be positive (ferromagnetic material). JMM have been estimated by Monte Carlo simulations of one pure monatomic M systems, so that the maximum at the specific heat as the magnetic susceptibility is located at Curie temperature T C amorphous 202 K [12], we found Fe JMM 110 K in the case of amorphous iron. The choice of the amorphous iron is arbitrary; this study is easily extrapolated with crystalline iron or others multilayer systems by including the parameter JMM in an adequate way. We found JMM 580 K, for crystalline iron. The applicability of this model to the present system permits us to reduce the number of external parameters to the minimum and can be justified by noticing that: (i) Only nearest-neighbour interactions should be taken into account since magnetic studies of such systems in particular in amorphous alloys show evidence that the value of the nextnearest-neighbour interactions is one order of magnitude smaller [13]. Indeed, the exchange interaction are very strong in transition metal but at short range. (ii) Other type of interactions that we did not consider here (e.g. anisotropy of the magnetic domains, dipolar interactions between them and interlayer coupling) can be omitted in a first approximation, this is due to the almost absence of perpendicular anisotropy in Fe/Cu systems and the weak value of the interlayer coupling strength in comparison to the exchange interaction parameter [2]. The validity of the above approximation is suitable because the interlayers antiferromagnetic coupling is not treated in this study. The dipolar interactions (often used for dilute limit) are taken into account indirectly in our model by the adjustment of the exchange parameter at the critical temperature of the system and by the alignment of spins to the layers by the application of weak magnetic field paralleling to the layers. This reduces the dipolar interactions to the minimum. (iii) The concentration of magnetic atoms at the interface (x = 0.5) permits us to consider a constant value of the spin at the interface as justified experimentally [12]. 2.3. Numerical simulation Numerical simulations were performed by using the importance Monte Carlo procedure at each temperature based on the standard Metropolis algorithms based on random trial step [14,15]. Our results were obtained by a slow decrease of the temperature starting from the paramagnetic state. At each temperature, 5 103 Monte Carlo steps (MCS) have been used for discarded for equilibration before averaging the physical quantities over the following 2 104 MCS. This procedure allows reaching equilibrium at each temperature when the Hamiltonian of the system does not display too much frustration, which is the case here. For our simulations, we have employed two magnetic layers, two layers at the interface and two non-magnetic layers. In the lateral dimensions, we have considered several linear sizes ranging from 10 10 to 20 20 atoms. Periodic boundary conditions have been applied along the three-directions. Since we are not interested in critical behavior, finite size effects have non-significant influence on the physical properties under consideration. The thermodynamic magnetization per atom and the susceptDhP P y 2 x 2 ibility were calculated from mðTÞ ¼ þ þ i mi i mi P z 2 1=2 2 2 i=N, and x(T) = N (hm i hmi )/kBT, respectively, i mi where N is the number of atomic systems, kB is the Boltzmann
Fig. 2. Concentration dependences of magnetization per atom of (M1xNMx) interface with 0 x 1 at T = 25 K. (D) The experimental data of the Fe1xCux alloys given by Ushida et al. [16]. (&) The value simulated by MC.
constant and h i is the average over the MCS, i.e. thermal average at temperature T. 3. Results and discussion Firstly, we take the interface in the form of a homogeneous amorphous M1xNMx films alloy with 0 x 1 as reported in Fig. 1. The interface spin exchange (Ji) is set equal to the bulk spin exchange (Jb). We plotted the variation of interface magnetization per atom as function of concentration x at low temperature (25 K) (Fig. 2). For comparison, we have reported the experimental data obtained for Fe1xCux disordered alloys [16]. It can be clearly seen
Fig. 3. Thermal variation of the reduced magnetization per atom (m/ms) of the interface in the case of MkNM1k (x = 0,5) for k = 0.1, 0.3, and 0.5. ms and TC are the bulk values of the saturated magnetization and Curie temperature of the magnetic layers.
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Fig. 4. Thermal variation of the reduced magnetization per atom of M/NM multilayers with abrupt interface (x = 0) and disordered interface ((MkNM1k) (Mk0 NM1k0 ), k and k0 are the mixture parameters in the interface, 0 k 0.5 and 0.5 k0 1; x = 0,5) (a) and the magnetization per atom versus the concentration k at T = 25 K (b). ms and TC are the bulk values of the saturated magnetization and Curie temperature of the magnetic layers.
that over the entire range of concentration, the Monte Carlo simulation results for the average magnetization per atom for this alloys are in good agreement with the experimental data. The average magnetization per atom of M1xNMx films alloys decrease as x increases. Similar tendency was also found previously in NixPd1x system [17]. The analysis of the result shown in Fig. 2 suggests that the gradual diminution of the magnetization is, most likely, associated with local environmental effects. In particular, this means a first neighbours number of magnetic atoms decreasing with x. At fixed concentration x, the interface morphology presents several configurations with different atomic arrangement. Such configurations will affect the magnetic properties of the system. In the following, we propose a rough interface with a particular distribution of different atoms with the same concentration x as described below (Fig. 1a). The particular case treated in the
Fig. 5. The number of spin flips in whole system during 5000 MCS at reduced temperature T/TC = 0.8 in the case of alloying interface.
Fig. 6. Thermal variation of the reduced magnetization in the whole system for different interface morphologies.
following is corresponding to x = 0.5 at the interface. Then, we consider the variation of the concentration at each region of the interface represented by the parameter k (k + k0 = 1 for x = 0.5). Fig. 3 shows the effect of concentration k on the thermal evolution of the magnetization per atom of the interface. For the reason of clarity, only average curves of some values of k are drawn. The interface magnetization decreases with increasing interface mixture for all temperatures. It decreases as much as 50% at some temperatures for T/TC about 0.5, as the degree of mixture increases towards the formation of a perfect alloy (M0.5NM0.5) with uniform mixture. This variation affects the thermal of reduced magnetization of multilayers M/NM as presented in Fig. 4. It decreases with increasing interface mixture. This variation reaches as much 20% at some temperatures when the parameter k increases from 0.0 to 0.5. To represent dynamical fluctuations of spins present in the sample, the variation of spin flips at a fixed temperature in function of magnetization distribution in the sample is reported in Fig. 5. The peaks are attributed to different sub-layers present in the sample what are magnetic layers and the two areas at the interface.
Fig. 7. Variation of magnetic susceptibility for the whole system versus reduced temperature for (a) mono-domain (MD) and (b) disconnected clusters (DC) at the interface.
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Fig. 8. Schematic representation of the configuration of decoupled clusters linked by a magnetic chain (a), two magnetic chains (b) and magnetic atoms randomly distributed (c).
Multilayer magnetization configuration is treated with a presence of different distribution clusters at the interface. Fig. 6 shows an example of the thermal dependence of the reduced magnetization for MD (Fig. 1b) and DC states at the interface (Fig. 1c). It can be seen that the magnetization in MD configuration, is very close with that in abrupt interface case. The magnetic layer involves the spins configuration to MD cluster which behaves like an extension of the magnetic layer. In DC case, the thermal magnetization is lower than that of the MD cluster, but exhibits an inhomogeneous variation for temperatures inferior than some temperature TB 0.25TC. This behavior is related to the spins order in decoupled cluster. The spins are organized independently of the magnetic layer, can be reoriented along others easy magnetic axes and would be ‘‘frozen’’ in this state. But, with a temperature relatively higher than TB, these irregularities are non-existent. In fact thermal fluctuations are dominating, so the average calculated quantities are taken on many several spins configurations. The temperature TB, which separates these two states, is called in the literature ‘‘blocking temperature’’. TB is function of the anisotropy energy barrier and the observing time. In our simulations the origin of the anisotropy is reported to the islands shape at the mixed interface. The spins are preferably oriented in parallel to the layers according to the longest side of the cluster. The observing time corresponds to the number of Monte Carlo steps. The temperature variation of the susceptibility (Fig. 7) exhibits a maximum around Curie temperature T = TC in MD case where occurs a paramagnetic–ferromagnetic transition. But it presents two maximums in DC case for TC and TB temperatures. Similar super-paramagnetism behavior of Fe/Cu multilayers is found experimentally [18]. The change in magnetic behavior in this system with decreasing Fe thickness is attributed to an evolution from multilayer to island structure rather to the formation of a non-magnetic phase.
Considering other clusters distributions at the interface in our simulations, it is shown that an isolated cluster present a blocked state when it have a minimum of interaction with magnetic layer. This leads us to determine the critical condition where two adjacent linked clusters behave like single domain. Two ferromagnetic domains are considered separately in our simulation and brought in contact as represented in Fig. 8. It is shown in Fig. 9 that
Fig. 9. Thermal variation of the reduced magnetization for mono-domain cluster (a), two clusters linked by two magnetic chains (b), two clusters linked by one magnetic chain (c), and two clusters linked by magnetic atoms randomly distributed in between (d).
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separated clusters form one magnetic domain when they are jointed by at least one mono-atomic chain. Consequently cluster configurations at the interface influenced the magnetic behavior of the sample. The simulations confirm that domains with a common mono-atomic chain with the magnetic layer do not show a superparamagnetic behavior. 4. Conclusion In summary, we have studied the dependence of magnetic properties on the interface morphology in the Heisenberg multilayers M/NM by using Monte Carlo simulations. Our simulations confirm that the form of distribution of the atoms in interface, i.e. morphology interface modifies its magnetic properties and consequently the magnetic properties of the multilayers. In particular the existence of the clusters within the interface leads to the apparition of super-paramagnetic state. In near future, we project to investigate the influence of more complicated situations at the interface as the diffusion of magnetic atoms in non-magnetic layers, or reactivity at the interface. Taking into account different parameters like layer thicknesses, stepped interface, real exchange parameter and interlayer coupling is necessary to compare Carlo Monte data with experimental results M/NM multilayers, such as Fe/Cu or FeCoB/MgO systems.
References [1] P. Bruno, G. Bayureuther, P. Beauvillain, C. Chappert, G. Lugert, D. Renard, J.P. Renard, J. Seiden, J. Appl. Phys. 68 (1990) 5759. [2] S. El Khiraoui, M. Sajieddine, M. Hehn, S. Robert, O. Lenoble, C. Bellouard, M. Sahlaoui, K. Benkirane, Physica B 403 (2008) 2509. [3] C. Prados, D.V. Dimitrov, J. Magn. Magn. Mater. 192 (1999) 19. [4] J.-T. Wang, L. Zhou, Yoshiyuki Kawazoe, D.-S. Wang, Phys. Rev. B60 (1999) 3025. [5] Th. Detzel, N. Memmel, Th. Fauster, Surf. Sci. 293 (1993) 227. [6] E. Talbot, P.E. Berche, D. Ledue, J. Magn. Magn. Mater. 320 (2008) 2814. [7] D. Zhao, F. Liu, D.L. Huber, M.G. Lagally, Phys. Rev. B62 (2000) 11316. [8] Q.Y. Wu, Z.G. Chen, R. Wu, G.G. Xu, Z.G. Huang, F.M. Zhang, Y.W. Du, Solid State Commun. 142 (2007) 242. [9] Terrence C. Gibb, J. Mater. Chem. 9 (1999) 2851. [10] A. Razouk, M. Sahlaoui, M. Sajieddine, E. Agouriane, Ferroelectrics 372 (2008) 31. [11] J.A. Hutchings, K. Newstead, M.F. Thomas, G. Sinclair, D.E. Joyce, P.J. Grundy, J. Phys. Condens. Matter 11 (1999) 3449. [12] P. Hansen, C. Clausen, B. Much, M. Rosenkranz, K. Witter, J. Appl. Phys. 66 (7) (1989) 56. [13] S.N. Kaul, Phys. Rev. B 27 (1983) 5761. [14] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, M. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [15] P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, New York, 2000. [16] M. Ushida, K. Tanak, K. Sumiyama, Y. Nakamura, J. Phys. Soc. Jpn. 58 (1989) 1725; K. Sumiyama, T. Yoshitake, Y. Nakamura, J. Phys. Soc. Jpn. 53 (1984) 3160. [17] Z.B. Tang, C.S. Tian, L.F. Yin, G.S. Dong, J. Xiaofeng, J. Magn. Magn. Mater. 310 (2007) 1804. [18] D.W. Lee, D.H. Ryan, Z. Altoun, A. Kuprin, Phys. Rev. B 59 (1999) 7001.
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Dependence of the magnetization on the interface morphology in ultra-thin magnetic/non-magnetic films: Monte Carlo approach A. Razouk, M. Sahlaoui *, M. Sajieddine Laboratoire de Physique et Me´canique des Mate´riaux, Faculte´ des Sciences et Techniques, Universite´ Sultan Moulay Slimane, BP 523, 23000 Be´ni-Mellal, Morocco
A R T I C L E I N F O
A B S T R A C T
Article history: Received 31 March 2009 Received in revised form 11 June 2009 Accepted 13 June 2009 Available online 21 June 2009
Using Monte Carlo simulations, we have studied the dependence of magnetic properties on interface morphology in magnetic/non-magnetic (M/NM) multilayers. Our aim is to relate macroscopic magnetic properties of the multilayers to their concentration profile at the interface. Our model consists of an alternate staking of magnetic and non-magnetic layers with disordered interfaces. We have considered different concentration and the existence of local magnetic domains at the interface. The results indicate the crucial dependence of magnetization amplitude with interface multilayers atomic composition and the spatial arrangement of magnetic atoms. In particular, we show that isolated islands at the interface leads to the apparition of super-paramagnetic behavior. ß 2009 Elsevier B.V. All rights reserved.
PACS: 87.55.K 75.70.i 67.30.er 67.30.hp 75.10.Jm Keywords: Monte Carlo simulation Heisenberg model Interface morphology Multilayers Curie temperature
1. Introduction Studies of metallic magnetic multilayers are not only of fundamental interest but are also essential for development of new magnetic multilayers [1]. Recent experimental techniques have been associated with analytical tools (Mo¨ssbauer spectrometry, squid magnetometry, polarized neutron reflectivity, etc.) to determine magnetic properties of multilayers and the morphology of interfaces in relation with the substrate temperature and the deposition process [2]. Since the discovery of giant magnetoresistance, novel problems have been raised up concerning magnetism of ultra-thin films and interfaces. It is believed that the spin-dependent scattering occurs mainly at interface sites, but the relation between interface morphology (e.g. surface reconstruction, roughness or interdiffusion) and the spin-dependent scattering probability is not yet clear. On the theoretical side, study has revealed that the spindependent scattering is dependent largely on the electronic
* Corresponding author. E-mail address: [email protected] (M. Sahlaoui). 0169-4332/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2009.06.061
structure of magnetic layers, especially when they are adjacent to non-magnetic layers [3]. The interface order is treated in the mean of ab initio calculation [4] within it is shown a depend spin with interface concentration. The theoretically predicted effects will be suppressed by interface roughness and interdiffusion, which cannot be avoided in many multilayers systems such as the Fe/Cu(0 0 1) system; where intermixing is known to occur [5]. On the other hand Monte Carlo simulations have been widely used in recent years to study the magnetic properties of multilayers. Various effects such as surface anisotropy, spin dynamics and magnetic semiconductor have been investigated [6–9]. The aim of this paper is to study the dependence of magnetic properties on the interface morphology in M/NM multilayers by Monte Carlo (MC) method with Heisenberg Hamiltonian. The MC method is one of the powerful techniques to simulate complex models and present the advantage that the mean-field approximation is not made. The additional complications are introduced to simulate really the structural dependence of the magnetic properties. Furthermore, in the case of Fe/Cu multilayers, structural imperfections can lead to the formation of an island structure with sub-multilayers Fe layers at the interface. This may lead to the formation of nanometer sized grains embedded in the Cu medium.
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Fig. 1. Schematic representation of two-dimensional cross-section through the multilayers system consisting of magnetic (M) and non-magnetic (NM) materials with uniform distribution of atoms at the interface ((MkNM1k)(Mk0 NM1k0 ) (a), k and k0 are the mixture parameter in the interface 0 k 0.5 and 0.5 k0 1), mono-domain (MD) configuration (b), decoupled cluster (DC) (c), in this case a magnetic cluster is localized within the non-magnetic layer.
In the following section, we describe the model and the simulation technique. The numerical results are presented in Section 3. Our conclusion and perspectives are given in Section 4. 2. Model 2.1. Atomic model We describe our multilayer system with a real interface in the sense that it is not really sharp but exhibits a certain degree of disorder. We simulate the magnetization for interfaces with controlled morphologies at different temperatures, by introducing an interface mixture in a systematic manner. The system (Fig. 1) consists of ferromagnetic Heisenberg layers with different materials M (magnetic) and NM (nonmagnetic) and disordered interface in between. This study could be carried out on multilayers formed by two different magnetic layers [10]. The interface atomic model is characterized by some arrangement of M and NM atoms so that (M1xNMx) is a three-dimensional alloy with 0 x 1 (x = 0 or x = 1 abrupt interface, x 6¼ 0 interface alloys). In this work, we have limited the interdiffusion to two atomic layers; this is due to the weak miscibility between transition metals as iron and copper. We consider the interface made up of two areas; an atomic layer beside the layer NM rich in element NM and poor in element M (MkNM1k) with 0 k 0.5 and the other layer adjacent to the layer M rich in element M (Mk0 NM1k0 ) with 0.5 k0 1 (Fig. 1a). k (or k0 ) is related to the abundance of element M in each of the two areas for a fixed composition x. For two monolayers at the interface, we can write k + k0 = 2x. Such interface configurations are predicted in real
samples (Fe/Cu) analyzed experimentally, using Mo¨ssbauer spectroscopy [2,11]. Furthermore, at fixed concentration x, k and k0 values, two extremes situations of rough interfaces are probable. The first one corresponds to uniform distribution (UD) of atoms at the interface (Fig. 1a). About all the atoms are surrounded in the same manner. In the second limit, the interface atoms are clustered to form two different mono-domains (MD) (M and NM). The magnetic domain is coupled to the magnetic layer (Fig. 1b). In each cluster, atoms are surrounded by the maximum of atoms of the same species (M or NM). In our simulation, cluster distribution at the interface can be modified systematically and different configurations between UD and MD cases would be considered. Also, decoupled clusters (DC) within the non-magnetic area without interactions with magnetic layer are considered (Fig. 1c). 2.2. Energetic model We describe the system by an isotropic Heisenberg model in absence of effects of anisotropy; a simple Hamiltonian H was used for the simulation as shown below: X X H¼ J i j Si S j þ B Si hi; ji
i
where Si is a classical Heisenberg spin and the sum is taken over nearest-neighbour pairs of spins. Jij denotes nearest-neighbour exchange interaction and is assumed to be JM–M between different magnetic atoms. The exchange parameters are defined in temperature units. B represents a weak magnetic field; applied parallel to the film plane to activate spin fluctuations along the layers and is about 50 Oe.
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In order to relate our results to real systems, such as M/NM multilayers, we make the assumption that M element is Curie metal (Fe); the NM element will be copper. According to heavy M– M interaction, the JMM interaction is considered to be positive (ferromagnetic material). JMM have been estimated by Monte Carlo simulations of one pure monatomic M systems, so that the maximum at the specific heat as the magnetic susceptibility is located at Curie temperature T C amorphous 202 K [12], we found Fe JMM 110 K in the case of amorphous iron. The choice of the amorphous iron is arbitrary; this study is easily extrapolated with crystalline iron or others multilayer systems by including the parameter JMM in an adequate way. We found JMM 580 K, for crystalline iron. The applicability of this model to the present system permits us to reduce the number of external parameters to the minimum and can be justified by noticing that: (i) Only nearest-neighbour interactions should be taken into account since magnetic studies of such systems in particular in amorphous alloys show evidence that the value of the nextnearest-neighbour interactions is one order of magnitude smaller [13]. Indeed, the exchange interaction are very strong in transition metal but at short range. (ii) Other type of interactions that we did not consider here (e.g. anisotropy of the magnetic domains, dipolar interactions between them and interlayer coupling) can be omitted in a first approximation, this is due to the almost absence of perpendicular anisotropy in Fe/Cu systems and the weak value of the interlayer coupling strength in comparison to the exchange interaction parameter [2]. The validity of the above approximation is suitable because the interlayers antiferromagnetic coupling is not treated in this study. The dipolar interactions (often used for dilute limit) are taken into account indirectly in our model by the adjustment of the exchange parameter at the critical temperature of the system and by the alignment of spins to the layers by the application of weak magnetic field paralleling to the layers. This reduces the dipolar interactions to the minimum. (iii) The concentration of magnetic atoms at the interface (x = 0.5) permits us to consider a constant value of the spin at the interface as justified experimentally [12]. 2.3. Numerical simulation Numerical simulations were performed by using the importance Monte Carlo procedure at each temperature based on the standard Metropolis algorithms based on random trial step [14,15]. Our results were obtained by a slow decrease of the temperature starting from the paramagnetic state. At each temperature, 5 103 Monte Carlo steps (MCS) have been used for discarded for equilibration before averaging the physical quantities over the following 2 104 MCS. This procedure allows reaching equilibrium at each temperature when the Hamiltonian of the system does not display too much frustration, which is the case here. For our simulations, we have employed two magnetic layers, two layers at the interface and two non-magnetic layers. In the lateral dimensions, we have considered several linear sizes ranging from 10 10 to 20 20 atoms. Periodic boundary conditions have been applied along the three-directions. Since we are not interested in critical behavior, finite size effects have non-significant influence on the physical properties under consideration. The thermodynamic magnetization per atom and the susceptDhP P y 2 x 2 ibility were calculated from mðTÞ ¼ þ þ i mi i mi P z 2 1=2 2 2 i=N, and x(T) = N (hm i hmi )/kBT, respectively, i mi where N is the number of atomic systems, kB is the Boltzmann
Fig. 2. Concentration dependences of magnetization per atom of (M1xNMx) interface with 0 x 1 at T = 25 K. (D) The experimental data of the Fe1xCux alloys given by Ushida et al. [16]. (&) The value simulated by MC.
constant and h i is the average over the MCS, i.e. thermal average at temperature T. 3. Results and discussion Firstly, we take the interface in the form of a homogeneous amorphous M1xNMx films alloy with 0 x 1 as reported in Fig. 1. The interface spin exchange (Ji) is set equal to the bulk spin exchange (Jb). We plotted the variation of interface magnetization per atom as function of concentration x at low temperature (25 K) (Fig. 2). For comparison, we have reported the experimental data obtained for Fe1xCux disordered alloys [16]. It can be clearly seen
Fig. 3. Thermal variation of the reduced magnetization per atom (m/ms) of the interface in the case of MkNM1k (x = 0,5) for k = 0.1, 0.3, and 0.5. ms and TC are the bulk values of the saturated magnetization and Curie temperature of the magnetic layers.
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Fig. 4. Thermal variation of the reduced magnetization per atom of M/NM multilayers with abrupt interface (x = 0) and disordered interface ((MkNM1k) (Mk0 NM1k0 ), k and k0 are the mixture parameters in the interface, 0 k 0.5 and 0.5 k0 1; x = 0,5) (a) and the magnetization per atom versus the concentration k at T = 25 K (b). ms and TC are the bulk values of the saturated magnetization and Curie temperature of the magnetic layers.
that over the entire range of concentration, the Monte Carlo simulation results for the average magnetization per atom for this alloys are in good agreement with the experimental data. The average magnetization per atom of M1xNMx films alloys decrease as x increases. Similar tendency was also found previously in NixPd1x system [17]. The analysis of the result shown in Fig. 2 suggests that the gradual diminution of the magnetization is, most likely, associated with local environmental effects. In particular, this means a first neighbours number of magnetic atoms decreasing with x. At fixed concentration x, the interface morphology presents several configurations with different atomic arrangement. Such configurations will affect the magnetic properties of the system. In the following, we propose a rough interface with a particular distribution of different atoms with the same concentration x as described below (Fig. 1a). The particular case treated in the
Fig. 5. The number of spin flips in whole system during 5000 MCS at reduced temperature T/TC = 0.8 in the case of alloying interface.
Fig. 6. Thermal variation of the reduced magnetization in the whole system for different interface morphologies.
following is corresponding to x = 0.5 at the interface. Then, we consider the variation of the concentration at each region of the interface represented by the parameter k (k + k0 = 1 for x = 0.5). Fig. 3 shows the effect of concentration k on the thermal evolution of the magnetization per atom of the interface. For the reason of clarity, only average curves of some values of k are drawn. The interface magnetization decreases with increasing interface mixture for all temperatures. It decreases as much as 50% at some temperatures for T/TC about 0.5, as the degree of mixture increases towards the formation of a perfect alloy (M0.5NM0.5) with uniform mixture. This variation affects the thermal of reduced magnetization of multilayers M/NM as presented in Fig. 4. It decreases with increasing interface mixture. This variation reaches as much 20% at some temperatures when the parameter k increases from 0.0 to 0.5. To represent dynamical fluctuations of spins present in the sample, the variation of spin flips at a fixed temperature in function of magnetization distribution in the sample is reported in Fig. 5. The peaks are attributed to different sub-layers present in the sample what are magnetic layers and the two areas at the interface.
Fig. 7. Variation of magnetic susceptibility for the whole system versus reduced temperature for (a) mono-domain (MD) and (b) disconnected clusters (DC) at the interface.
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Fig. 8. Schematic representation of the configuration of decoupled clusters linked by a magnetic chain (a), two magnetic chains (b) and magnetic atoms randomly distributed (c).
Multilayer magnetization configuration is treated with a presence of different distribution clusters at the interface. Fig. 6 shows an example of the thermal dependence of the reduced magnetization for MD (Fig. 1b) and DC states at the interface (Fig. 1c). It can be seen that the magnetization in MD configuration, is very close with that in abrupt interface case. The magnetic layer involves the spins configuration to MD cluster which behaves like an extension of the magnetic layer. In DC case, the thermal magnetization is lower than that of the MD cluster, but exhibits an inhomogeneous variation for temperatures inferior than some temperature TB 0.25TC. This behavior is related to the spins order in decoupled cluster. The spins are organized independently of the magnetic layer, can be reoriented along others easy magnetic axes and would be ‘‘frozen’’ in this state. But, with a temperature relatively higher than TB, these irregularities are non-existent. In fact thermal fluctuations are dominating, so the average calculated quantities are taken on many several spins configurations. The temperature TB, which separates these two states, is called in the literature ‘‘blocking temperature’’. TB is function of the anisotropy energy barrier and the observing time. In our simulations the origin of the anisotropy is reported to the islands shape at the mixed interface. The spins are preferably oriented in parallel to the layers according to the longest side of the cluster. The observing time corresponds to the number of Monte Carlo steps. The temperature variation of the susceptibility (Fig. 7) exhibits a maximum around Curie temperature T = TC in MD case where occurs a paramagnetic–ferromagnetic transition. But it presents two maximums in DC case for TC and TB temperatures. Similar super-paramagnetism behavior of Fe/Cu multilayers is found experimentally [18]. The change in magnetic behavior in this system with decreasing Fe thickness is attributed to an evolution from multilayer to island structure rather to the formation of a non-magnetic phase.
Considering other clusters distributions at the interface in our simulations, it is shown that an isolated cluster present a blocked state when it have a minimum of interaction with magnetic layer. This leads us to determine the critical condition where two adjacent linked clusters behave like single domain. Two ferromagnetic domains are considered separately in our simulation and brought in contact as represented in Fig. 8. It is shown in Fig. 9 that
Fig. 9. Thermal variation of the reduced magnetization for mono-domain cluster (a), two clusters linked by two magnetic chains (b), two clusters linked by one magnetic chain (c), and two clusters linked by magnetic atoms randomly distributed in between (d).
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separated clusters form one magnetic domain when they are jointed by at least one mono-atomic chain. Consequently cluster configurations at the interface influenced the magnetic behavior of the sample. The simulations confirm that domains with a common mono-atomic chain with the magnetic layer do not show a superparamagnetic behavior. 4. Conclusion In summary, we have studied the dependence of magnetic properties on the interface morphology in the Heisenberg multilayers M/NM by using Monte Carlo simulations. Our simulations confirm that the form of distribution of the atoms in interface, i.e. morphology interface modifies its magnetic properties and consequently the magnetic properties of the multilayers. In particular the existence of the clusters within the interface leads to the apparition of super-paramagnetic state. In near future, we project to investigate the influence of more complicated situations at the interface as the diffusion of magnetic atoms in non-magnetic layers, or reactivity at the interface. Taking into account different parameters like layer thicknesses, stepped interface, real exchange parameter and interlayer coupling is necessary to compare Carlo Monte data with experimental results M/NM multilayers, such as Fe/Cu or FeCoB/MgO systems.
References [1] P. Bruno, G. Bayureuther, P. Beauvillain, C. Chappert, G. Lugert, D. Renard, J.P. Renard, J. Seiden, J. Appl. Phys. 68 (1990) 5759. [2] S. El Khiraoui, M. Sajieddine, M. Hehn, S. Robert, O. Lenoble, C. Bellouard, M. Sahlaoui, K. Benkirane, Physica B 403 (2008) 2509. [3] C. Prados, D.V. Dimitrov, J. Magn. Magn. Mater. 192 (1999) 19. [4] J.-T. Wang, L. Zhou, Yoshiyuki Kawazoe, D.-S. Wang, Phys. Rev. B60 (1999) 3025. [5] Th. Detzel, N. Memmel, Th. Fauster, Surf. Sci. 293 (1993) 227. [6] E. Talbot, P.E. Berche, D. Ledue, J. Magn. Magn. Mater. 320 (2008) 2814. [7] D. Zhao, F. Liu, D.L. Huber, M.G. Lagally, Phys. Rev. B62 (2000) 11316. [8] Q.Y. Wu, Z.G. Chen, R. Wu, G.G. Xu, Z.G. Huang, F.M. Zhang, Y.W. Du, Solid State Commun. 142 (2007) 242. [9] Terrence C. Gibb, J. Mater. Chem. 9 (1999) 2851. [10] A. Razouk, M. Sahlaoui, M. Sajieddine, E. Agouriane, Ferroelectrics 372 (2008) 31. [11] J.A. Hutchings, K. Newstead, M.F. Thomas, G. Sinclair, D.E. Joyce, P.J. Grundy, J. Phys. Condens. Matter 11 (1999) 3449. [12] P. Hansen, C. Clausen, B. Much, M. Rosenkranz, K. Witter, J. Appl. Phys. 66 (7) (1989) 56. [13] S.N. Kaul, Phys. Rev. B 27 (1983) 5761. [14] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, M. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [15] P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, New York, 2000. [16] M. Ushida, K. Tanak, K. Sumiyama, Y. Nakamura, J. Phys. Soc. Jpn. 58 (1989) 1725; K. Sumiyama, T. Yoshitake, Y. Nakamura, J. Phys. Soc. Jpn. 53 (1984) 3160. [17] Z.B. Tang, C.S. Tian, L.F. Yin, G.S. Dong, J. Xiaofeng, J. Magn. Magn. Mater. 310 (2007) 1804. [18] D.W. Lee, D.H. Ryan, Z. Altoun, A. Kuprin, Phys. Rev. B 59 (1999) 7001.