Giant magnetoresistance in a three-dimensional lattice of dipolar interacting magnetic nanoparticles

10 December 2001

Physics Letters A 291 (2001) 325–332 www.elsevier.com/locate/pla

Giant magnetoresistance in a three-dimensional lattice of dipolar interacting magnetic nanoparticles Chen Xu a , Zhen-Ya Li b,a , I.E. Dikshtein c , V.G. Shavrov c,∗ , P.M. Hui d a Department of Physics, Suzhou University, Suzhou 215006, China b CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China c Institute of Radioengineering and Electronics, Russian Academy of Sciences, 103907 Moscow, Russia d Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received 10 September 2001; accepted 26 September 2001 Communicated by V.M. Agranovich

Abstract The magnetization and the giant magnetoresistance (GMR) of three-dimensional multilayered lattice of dipolar interacting fine anisotropic magnetic particles embedded in a nonmagnetic metallic matrix are numerically investigated. Using a Monte Carlo method the dependence of the magnetization curve and the GMR effect on the magnetic field H , temperature, the number of layers, the magnetic anisotropy, the separation between neighboring particles, and the particle-size distribution are examined systematically. We found that the enhanced dipolar interaction at high particle densities and for wide particle-size distributions reveals itself as a substantial suppression in the change in the negative MR with the applied field H . For out-of-plane magnetic field orientation, the negative MR is shown to increase more rapidly with H as the number of layers increases. This effect is attributed to the demagnetization field arising from the free magnetic poles forming at the sample surface. The relevance of the present results to the understanding of the magnetic and transport properties in granular composites is discussed.  2001 Elsevier Science B.V. All rights reserved. Keywords: Magnetic granular alloy; Magnetic nanoparticles; Giant magnetoresistance

1. Introduction Since the first observation of the giant magnetoresistance (GMR) effect in Fe/Cr multilayers [1], there has been a rapid growth in research activities on GMR and related phenomena due both to the fundamental interests and potential applications as magnetoresistance sensors. Recently, GMR has also been observed in magnetic granular alloys [2–5] consisting of an as-

* Corresponding author.

E-mail address: [email protected] (V.G. Shavrov).

sembly of fine magnetic particles or clusters (e.g., Co, Fe, Ni) embedded in a nonmagnetic conducting matrix (typically Cu, Ag, Au). Such granular materials are immiscible or almost immiscible alloys under equilibrium, implying that phase separation is essential to the GMR effect [6]. It is well known that GMR in multilayers and granular alloys have a common origin, namely the spin-dependent scattering of conduction electrons occurring at interfaces and/or in the bulk of the ferromagnetic constituent. Novel methods of sample preparation allow the tailor making of granular metals with specific magnetic and magneto-transport properties through controlled variation of the average

0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 6 8 7 - 9

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particle size and the volume fraction of the magnetic fine particles or clusters. Currently, ion-beam techniques has been applied to modify magnetic properties and GMR of granular CoAg and CoCu layers [4, 5,7]. GMR measurements [5] were found to be highly sensitive to small change in the granular structure. Technological requirements for high packing densities make the role of interparticle coupling increasingly important [8] as the coupling modifies significantly the magnetic properties of the system. Altbir et al. [9] pointed out that the dipolar interactions are dominant in Co–Cu systems. Previous works have also discussed the dependence of GMR on the particle-size distributions [8,10–12] and successfully explained some experimental findings. In the present Letter we focus on the study of the equilibrium magnetization and magnetoresistance of a three-dimensional (3D) multilayered assembly of fine ferromagnetic particles and their dependence on the growth parameters (separation between the neighboring particles, particle-size distribution, and the number of layers), the temperature and the applied magnetic field H . We expect that the geometry of the sample and the corresponding demagnetization energy strongly affect the MR. The simulations are performed using a Monte Carlo–Metropolis (MC) algorithm that is proved to be extremely helpful to understand the role of the dipole interactions and the magnetic anisotropy in the magnetic and transport properties of granular materials [13].

2. The model and simulation method

We model the system as a collection of 3D nanosized spherical particles located on the cells of a simple cubic lattice consisting of S = 8 × 8 × P cells with lattice constant r0 . Since the average size of the particles in such granular composites is typically around 3 nm [6,14], it seems natural to regard each particle as a saturated single domain. The magnetic moment and the direction of the uniaxial anisotropy axis are taken to be random in space. The diameter of the ith particle is di . Including the classical dipolar interaction and crystalline anisotropy energy, the total energy of the system for a particular configuration {µ  i } of the

magnetic moments is given by  S   
Eij + Ku Vi sin2 αi − µ  i · H , E {µi } = i=1 j >i

(1) where Ku is the effective anisotropy constant including surface, magnetocrystalline and shape contributions, Vi is the volume of the ith particle, αi is the angle between the direction of the crystalline axis and magnetic moment µ  i of the ith particle, and H is the applied magnetic field. The energy of the classical dipole–dipole interaction Eij is given by Eij =

µ i · µ  j − 3(µ  i · rˆij )(µ  j · rˆij ) rij3

,

(2)

where rij is the separation between the ith and the j th particles, and rˆij is the unit vector along the direction connecting µ  i and µ  j . For a given temperature, the reduced equilibrium magnetization m can be calculated by averaging  | iµ i| M m= (3) = Ms Nµ over configurations of dipole moments after thermal equilibrium has been reached. The crucial quantity for studying GMR in granular system is the average value cos θij , where θij is the relative angle between the magnetic moments at sites i and j [15–18]. It implies that the magneto-transport properties are primarily caused by the spin-dependent scattering process of conduction electrons from one magnetic particle to another. The spin-dependent scatterings at interfaces between the magnetic particles contribute more significantly to the MR than scatterings within the magnetic particles [19,20]. For situations in which the separation between neighboring particles does not far exceed the electronic mean free path λ [21], the change in the resistivity of a granular system with the degree of field-induced magnetic order may be simply represented by [17]  (λ)  ρ = ρ0 − k cos θij , (4) where ρ0 and k are constants. Assuming that there are no correlations between the magnetic moments of the particles, the magnetoresistance ∆ρ/ρ can be written according to Eq. (4) as k k ∆ρ (λ)  = − cos θij = − cos θi cos θj ρ ρ0 ρ0

C. Xu et al. / Physics Letters A 291 (2001) 325–332

=−

k 2 m . ρ0

(5)

Such a quadratic dependence of ∆ρ/ρ on m has actually been observed in some experiments [3,22]. However, other experiments [10,18,23] showed that ∆ρ/ρ does not vary quadratically with m, probably due to the effects, e.g., stronger correlations between particles, caused by a size distribution of the magnetic particles. For a system involving coupling between magnetic (λ) particles, cos θij 0 becomes nonvanishing even for H = 0 and the expression for the magnetoresistance becomes (λ) (λ) ∆ρ cos θij 0 − cos θij = , (λ) ρ Kt − cos θij 0

(6)

where Kt = ρ0 /k is a field-independent constant. The thermal averages in Eqs. (4)–(6) are obtained using the standard MC procedures and the Metropolis algorithm [24]. According to this algorithm, the orientations of the magnetic moments of the particles are initially chosen at random and the energy Ei is determined. The configuration of magnetic moments is then changed at random and the energy difference ∆E = Ef − Ei , with Ef being the energy of the new configuration, is calculated. If ∆E < 0, the new configuration is accepted. If ∆E > 0, the new configuration is accepted with probability exp(−∆E/kB T ). If the new configuration is rejected, the old configuration is kept. Typically thermal equilibrium is achieved after 104 Monte Carlo steps per spin (MCS). Further increase in the number of MCS to an order of magnitude higher (∼105 MCS) does not substantially change the results. Then, we evaluate the thermal averages as a simple arithmetic average over 500 accepted configurations in the calculations of the reduced magnetization and magnetoresistance. Free boundary conditions are used in the calculations.

3. Numerical results and discussion Data for our MC simulations are calculated as follows. Initially, each particle is assigned a random direction of its uniaxial anisotropy axis with Ku = 4.0 × 106 erg/cm3 and a random direction of its magnetic

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moment. The value of Ku is typical of magnetic materials, e.g., cobalt. The particles are placed in the cells of a 8 × 8 × P lattice, with the external magnetic field H applied along the [001]-direction. The separation r0 between the particles is taken to be 6 nm, unless specified otherwise. This value is comparable in magnitude with the electronic mean free path or spin diffusion length [21]. Néel pointed out that single-domain ferromagnetic small particles may exhibit superparamagnetism, i.e., the reversal of the magnetization through a thermally activated process over the anisotropy barrier even in the absence of an external magnetic field. In other words, above a certain temperature referred to as the blocking temperature Tb , an ensemble of singledomain particles behaves as a collection of paramagnetic molecules with giant magnetic moments (m ∼ 103µB ). According to the prediction of the Néel– Brown theory, the superparamagnetic relaxation time τ for a single-domain particle for H = 0 can be written as τ = τ0 exp(Ku V /kB T ), where τ0 is a characteristic time of the system. The blocking temperature Tb(0) is then defined as the temperature above which the particle has enough time, within the observation time τobs , to overcome the anisotropy barrier (Ku V ) and (0) to reverse the magnetization. Obviously, Tb depends on the method used to observe the relaxation [25] (0) and is given by Tb = Ku V /[kB ln(τobs /τ0 )]. Taking τobs = 100 s and τ0 = 10−9 [26], the blocking temperature Tb(0) of a particle of diameter d = 4 nm is about 38 K. In our calculations, we choose T = 40 K which is close to the blocking temperature. 3.1. Influence of uniaxial anisotropy and dipolar interaction For concreteness, we consider a 8 × 8 × 5 lattice of particles with a particle-size distribution given by a logarithmic-normal function [27] with standard deviation ∆ = 0.1 nm and average diameter dav = 3 nm. The choice refers to typical average size of particles in granular systems [14,21,28]. The separation r0 between neighboring particles is taken to be 6 nm. Fig. 1 shows the magnetization and the magnetoresistance ∆ρ/ρ as a function of the applied field H . The dependence of ∆ρ/ρ on H is calculated via Eq. (6) with Kt = 3.5 taking into account both the effects of anisotropy and interactions after the system reaches

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Fig. 1. Field dependence of (a) Mz /Ms and (b) MR for a 8 × 8 × 5 lattice of particles with a log-normal particle-size distribution with ∆ = 0.1 nm and dav = 3 nm, T = 40 K, r0 = 6 nm. Squares: noninteracting isotropic particles (Ku = 0). Circles: particles with random anisotropy only. Downward triangles: particles with dipolar interaction only. Upward triangles: particles with both dipole interaction and random anisotropy.

equilibrium. Four different sets of data are shown corresponding to a system of noninteracting isotropic particles (squares, Ku = 0), a system of particles with random anisotropy only (circles), a system with dipolar interaction only (downward triangles, Ku = 0), and a system of particles with both random anisotropy and dipolar interaction (upward triangles). We notice that the magnetization and magnetoresistance exhibit dif-

ferent field dependence, depending on the interplay of the thermal fluctuation energy, the single-particle anisotropy energy, and the dipolar interaction energy. In particular, the magnetic moments of a system of noninteracting isotropic particles subjected to an externally applied magnetic field are aligned so as to become parallel to H , with M(T , H ) in the form of the Langevin function and the saturation field Hs depending on the temperature T . For low temperatures the magnetization Mz and the modulus of magnetoresistance |∆ρ|/ρ increase rapidly with H . For a system of particles with random anisotropy only, which can be used to approximate an ensemble of well-separated particles, the magnetization reversal of an ensemble of single-domain particles is well described by both a coherent rotation of the magnetic moments of the particles from the easy axes to the direction of the applied magnetic field and by a thermally activated process over the anisotropy barrier. In this case the magnetization and the negative magnetoresistance in the system are saturated in an applied field Hs ≈ 2Ku /Ms . For H = 0, a system with dipolar interaction only (Ku = 0) possesses an interaction-induced anisotropy. For 8 × 8 × 5 lattice the demagnetization factor along the [001]-direction is larger than those along the [100]and [010]-directions. Thus, an out-of-(001)-plane orientation of the magnetization corresponds to a higher energy. Therefore, an in-(001)-plane configuration of magnetic moments is realized in this case, with the ground state of the lattice of dipoles depending crucially on the boundary conditions [29]. For free boundary conditions the ground state is antiferromagnetic due to the demagnetization effect of the lateral boundaries, which is not as strong compared with the demagnetization effect of the (001)-surface. When an external magnetic field is applied, the magnetic moments rotate so as to align with the magnetic field. In the case where both dipolar interaction and random uniaxial anisotropy of particles are involved, it is more difficult for the magnetization and for the modulus of magnetoresistance |∆ρ|/ρ to reach saturation since these effects impede the ordering of the magnetic moments and hence reduce the magnetization of the system. This leads to the inflectional dependence of the magnetoresistance on the applied field in low fields (see Fig. 1(b)), a behavior that have been observed in experiments [30,31].

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Fig. 2. Field dependence of MR for a 8 × 8 × 5 lattice of particles with dipole interaction. The other parameters are d = 3 nm, T = 40 K, r0 = 6 nm (squares), r0 = 4.5 nm (circles), and r0 = 3.5 nm (upward triangles).

The anisotropy induced by dipolar interactions is very sensitive to the spatial arrangement of the particles (average separation between the particles and their size) and it is enhanced considerably with decreasing r0 . The effects of varying r0 on MR are shown in Fig. 2. The magnetoresistance in a strongly dipolar system (upward triangles for r0 = 3.5 nm and circles for r0 = 4.5 nm) are harder to become saturated in comparison with a moderately dipolar system with larger r0 (squares for r0 = 6 nm). Fig. 3 shows |∆ρ|/ρ for calculations in a system of dipolarly coupled particles with fixed diameter d = 3 nm at T = 40 K (squares) and T = 300 K (circles). The difference is attributed to successful magnetization reversal in ferromagnetic particles with an increase in temperature. We note that the absolute value of MR at T = 300 K is higher than experimental values since we have not considered the corresponding change of Kt with temperature, which will affect the value of MR according to Eq. (6).

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Fig. 3. Field dependence of MR for a 8 × 8 × 5 lattice of particles with dipole interaction. The other parameters are d = 3 nm, r0 = 6 nm, T = 40 K (squares), and T = 300 K (circles).

Fig. 4. Field dependence of MR for a 8 × 8 × 5 lattice. T = 40 K, r0 = 6 nm. Squares: A log-normal particle-size distribution with dav = 3 nm and ∆ = 0.1 nm. Circles: a log-normal distribution with dav = 3 nm and ∆ = 0.3 nm. Upward triangles: a narrow particle-size distribution from 2 to 4 nm. Downward triangles: a wide particle-size distribution from 0 to 6 nm.

3.2. Influence of particle-size distribution In order to investigate the effects of a particlesize distribution on the field dependence of magne-

toresistance, four sets of data for the 8 × 8 × 5 lattice of magnetic particles with random anisotropy and dipolar interaction are shown in Fig. 4. The situa-

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tions correspond to log-normal distributions with average diameter dav = 3 nm and standard deviations ∆ = 0.1 nm (squares) and ∆ = 0.3 nm (circles), a narrow particle-size distribution from 2 to 4 nm (upward triangles), and a wide particle-size distribution from 0 to 6 nm (downward triangles). Fabrications of systems with narrow (after annealing the sample) and wide particle-size distributions have been reported in Refs. [15] and [23], respectively. For a narrow particlesize distribution in the range of 2–4 nm, the field dependences of magnetoresistance ∆ρ/ρ (upward triangles in Fig. 4) are similar to those for a very narrow log-normal particle-size distribution with ∆ = 0.1 nm (squares). In both cases the blocking temperature of the largest particle is Tb ≈ 38 K, and all the particles are superparamagnetic. For a wide particle-size distribution (downward triangles in Fig. 4) in the range of 0–6 nm, the dipolar coupling between the larger particles is enhanced. Some of the larger particles become blocked at T = 40 K, and the smaller ones are still in the superparamagnetic state. In this case both the dipolar interaction and the blocking of the larger particles lead to a substantial reduction in the average magnetic moment Mz and the modulus of |∆ρ|/ρ in the range of magnetic fields H < Hs relative to the case of a narrow particle-size distribution. The reduction of the negative GMR was observed, and explained by Hickey et al. [10] and Xu et al. [23] for systems with a wide particle-size distribution. A log-normal distribution with standard deviation ∆ = 0.3 nm (circles) represents a case which is intermediate between the cases of narrow (upward triangles) and wide particlesize distributions (downward triangles). Eq. (5) suggests a quadratic dependence of ∆ρ/ρ on the magnetization. In Fig. 5, we show the dependence of ∆ρ/ρ on M/Ms for all the cases studied. Except for the case of a wide particle-size distribution, the numerical data are well described by the theoretical prediction of Eq. (5), as shown by the solid line in the figure. The results indicate that the MR in a system with narrow particle size distribution is well described by a quadratic dependence on M/Ms . However, deviation from the (M/Ms )2 dependence of MR has also been observed [23]. In Ref. [23], the melt-spun Au–Fe samples consist of many small magnetic particles and also some large particles. This wide spread in particle sizes was also found in other melt-spun Co–Cu samples [32, 33]. Therefore, our numerical results for a wide parti-

Fig. 5. Dependence of MR on Mz /Ms for a 8 × 8 × 5 lattice. T = 40 K, r0 = 6 nm. The data represent results of a lattice of noninteracting isotropic particles (closed squares, Ku = 0), particles with random anisotropy only (open squares), dipolar coupled magnetic particles with random anisotropy and four types of particle-size distributions: log-normal distributions with dav = 3 nm and ∆ = 0.1 nm (closed circles) and ∆ = 0.3 nm (open circles), a narrow particle-size distribution from 2 to 4 nm (closed upward triangles), and a wide particle-size distribution from 0 to 6 nm (open upward triangles). The form of the parabolic line is given by Eq. (5) and the dashed line is given by Eq. (7).

cle size distribution may be used to model the situation in melt-spun samples. As suggested in Ref. [23], we fit our results to a modified form of Eq. (5) by including a term linear in m and a constant term, i.e., m2 ∆ρ =− + αm + β, ρ Kt

(7)

with α and β being the fitting parameters. The fitted line (dashed line) in Fig. 5 corresponds to Kt = 3.5, α = 0.094, and β = −0.013. The scattering process of conduction electrons between superparamagnetic and blocked particles [23] may lead to a term linear in m in ∆ρ/ρ. The constant term is due to the scattering process between saturated blocked particles in which the magnetic moments are aligned (or nearly aligned) to the applied field H and thus will not change when H is further increased. This constant term contributes to ∆ρ/ρ only after the blocked magnetic moments are aligned above a certain H . The contribution of the blocked but unaligned moments to ∆ρ/ρ was dis-

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Fig. 6. Field dependence of MR for a 8 × 8 × P lattice of dipolar coupled isotropic particles with fixed diameter d = 3 nm and H  [001]: P = 1 (circles), P = 2 (squares), P = 4 (diamonds), P = 6 (closed upward triangles), P = 8 (open upward triangles).

cussed in Ref. [34]. Eq. (7) fits the simulation data very well in the range of M/Ms > 0.2. Both effects become apparent in our calculations since both superparamagnetic and blocked particles coexist in systems with a wide distribution of particle sizes. Thus, a wide distribution of particle sizes may explain the deviation from the parabolic law for the MR as a function of the magnetization [10,11,23]. 3.3. Influence of the number of layers To illustrate the effects of film thickness and surface, we carried out calculations in 8 × 8 × P (P = 1, 2, 4, 6, 8) lattices of dipolarly coupled but isotropic particles with fixed diameter d = 3 nm. Fig. 6 shows that for H  [001] the negative magnetoresistance |∆ρ|/ρ increases more rapidly with the field H as the number of layers P increases. This is attributed to the decrease in energy of the demagnetization field, which arises from the free magnetic poles forming at the (001)-surfaces. The demagnetization factor along the [001]-direction and the anisotropy in geometry are maximal for the 2D array (8 × 8 × 1) and are minimal for a 8 × 8 × 8 lattice. Since the anisotropy in geometry favors an in-(001)-plane configuration of magnetic

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Fig. 7. Field dependence of MR for a 8 × 8 × P lattice of dipolar coupled isotropic particles with fixed diameter d = 3 nm and H  [100]: P = 1 (circles), P = 2 (squares), P = 4 (diamonds), P = 6 (upward triangles), P = 8 (downward triangles).

moments, the magnetization Mx and negative magnetoresistance |∆ρ|/ρ are reduced with an increasing number of layers for an external magnetic field H applied along the [100]-direction. The effect is illustrated in Fig. 7.

4. Summary We presented numerical simulation results for the field dependence of the magnetoresistance of a model of granular magnetic composites taking into account the effects of sample thickness. The system typically consists of randomly oriented nanosized ferromagnetic particles interacting through dipole interaction and embedded in a nonmagnetic metallic matrix. It is shown that the magneto-transport properties of such system are determined by the interplay of the singleparticle random anisotropy energy, the dipolar interaction energy, the thermal fluctuation energy, and the energy associated with the anisotropy in the geometry of the sample. For better understanding of experimental findings, the effects of particle-size distributions and the average separation between particles on the magnetic properties are studied. It is shown that the enhanced dipolar interaction at high particle densi-

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ties and for wide particle-size distributions manifests itself as a substantial suppression in the change of the magnetization and the negative MR as a function of the applied field H . For a wide distribution of particle sizes, the calculated curve ∆ρ(M/Ms )/ρ is found to be similar to that observed experimentally [10,11, 23] and may explain the deviation from the parabolic law for the MR as a function of the magnetization. We found that for a lattice of dipolar coupled particles, the energy of the demagnetization field, which arises from the free magnetic poles forming at the surfaces of a sample, depends sensitively on the number of layers and affects the magnetization and magnetoresistance strongly. In particular, for out-of-plane magnetization, the negative MR increases with an increasing number of layers.

Acknowledgements This work was supported in part by the National Natural Science Foundation of China under grant No. 19774042, the Russian Foundation for the Basic Research under grant No. 99-02-39009, and the Research Grants Council of the Hong Kong SAR Government under Grant No. CUHK4129/98P.

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