Induced ferro-ferromagnetic exchange bias in nanocrystalline systems

Journal of Magnetism and Magnetic Materials 377 (2015) 424–429

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Induced ferro-ferromagnetic exchange bias in nanocrystalline systems J.C. Martínez-García, M. Rivas n, J.A. García Departamento de Física, Campus de Viesques, Universidad de Oviedo, 33203 Gijón, Spain

art ic l e i nf o

a b s t r a c t

Article history: Received 17 June 2014 Received in revised form 27 October 2014 Available online 30 October 2014

An unusual magnetic hysteresis consisting of horizontally shifted and distorted loops appears in some Co-based nanocrystalline systems in which soft and hard ferromagnetic phases coexist. The bias field can be tuned at room temperature by premagnetising treatments. Several works attributed the origin of this effect to the dipolar interaction, while little attention has been paid to the exchange interaction contribution due to its short-range nature. In this paper the relative importance of the dipolar and exchange interactions is investigated by means of micromagnetic simulations. It is demonstrated that the exchange coupling, though a nearest-neighbour interaction, has far-reaching repercussions in the magnetic configuration, and substantially prevails over the magnetostatic interaction as the cause of the asymmetrical magnetisation reversal. The straightforward conclusion is that we are dealing with a ferro-ferromagnetic exchange bias effect. & 2014 Elsevier B.V. All rights reserved.

Keywords: Nanocrystalline magnetic material Magnetic interaction Micromagnetic simulation Exchange bias

1. Introduction The magnetic properties of nanocrystalline materials are mainly dictated by their grain size and intergrain distance. Densely compacted agglomerations of nanoparticles with sizes smaller than the exchange correlation length produce extremely soft magnetic behaviour [1,2], while larger sizes (typically above 20 nm) give rise to magnetic hardening with respect to the amorphous counterpart [3]. An intermediate kind of nanocrystalline systems may be achieved from an amorphous precursor by an adequate devitrification resulting in a low concentration of nonagglomerated nanoparticles embedded in the residual amorphous matrix. Depending on the nature of the crystalline phase and the magnetic character of the matrix, the material can be a combination of ferromagnetic hard and soft phases presenting biased hysteresis loops (HL) [4–6]. Biased HL are typical of systems in which ferro/antiferromagnetic interfaces are exchange coupled [7–10] and which are either field cooled or submitted to large pulsed fields [11]. Other studies can be found in the literature in which the exchange bias (EB) is produced by coupling of ferri/antiferro, ferri/ferri and antiferro/antiferro interfaces (see [12–14] and references there in). Nonetheless, antiferro or ferrimagnetic phases are absent in the material studied in this work, in spite of which a clear biasing is observed without need of field cooling. This anomalous hysteresis has often been attributed to the dipolar interaction between the magnetically hard crystallites precipitated during the annealing n

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http://dx.doi.org/10.1016/j.jmmm.2014.10.123 0304-8853/& 2014 Elsevier B.V. All rights reserved.

and the still soft residual matrix [4,5,15], neglecting the exchange coupling between the two phases. Other hard/soft ferro/ferro multiphase composite systems with biased HL have been reported in which the effect has been attributed to magnetostatic interactions [16–20]. Contrariwise, this paper concerns the hard/soft ferro/ferro exchange interaction and demonstrates that, except for samples with very specific geometries, it is the main responsible for the shift of the HL and other features of the magnetisation reversal in this kind of nanocrystallised material. A simple model is presented in which the dipolar and exchange interactions between both hard and soft phases are taken into account giving rise to twisted magnetisation configurations as in exchange-spring multilayers [21–23], the large differences in the magnetisation behaviour coming from the extensiveness of the soft phase. From the experimental point of view, the work focuses on partially devitrified samples of Co66Fe4Mo2Si16B12 as representative of a group of Co-based nanocrystallised materials presenting biased HL with a common phenomenology [6,24,25]. HL shifts as large as five times the coercive field have been previously reported by the authors [15,26] in this system which, together with the persistence of the effect [27], points in the right direction for technical application in devices in which the hysteresis loops of their soft magnetic cores must be displaced, like switching power supplies, pulse transformers, magnetic sensors or giant magnetoimpedance-based sensors. But besides its potential technological relevance, this material constitutes a very interesting system to go further in the understanding of the influence of internal magnetic interactions on the hysteretic features.

J.C. Martínez-García et al. / Journal of Magnetism and Magnetic Materials 377 (2015) 424–429

2. Experimental procedure and results

425

0.6 Hp = 0 kA/m Hp = 200 kA/m

0.4 0.2 0

| Hb| (A/m)

µ0 M (T)

The experiments have been carried out on partially devitrified samples obtained by isothermal annealing of amorphous ribbons of Co66Fe4Mo2Si16B12. In previous works [15,26] it was already reported that annealing this alloy at temperatures slightly below the crystallisation temperature (Tcr = 558 °C ) give rise to samples with very low volume fractions (between 2 × 10−4 and 5 × 10−3) of nanocrystals. X-Ray Diffraction and Selected Area Electron Diffraction results revealed that Co3B and hcp Co, both ferromagnetic, are the most probable crystalline phases. Transmission Electron Microscopy (TEM) micrographs allowed the observation of particles with sizes between 15 and 80 nm. Fig. 1 shows two TEM images taken from a sample annealed at 530 °C for 12 min in which the typical morphology of this kind of hard-soft system can be appreciated: some agglomerates of only 1–6 nanoparticles completely surrounded by amorphous material. The magnetic hysteresis curves were obtained at room temperature in an inductive computer-controlled hysteresismeter working at 3 mHz [28]. The loops of the as-quenched samples are typical of this soft alloy with very small coercive field (Hc = 2.4 A/m ) and a saturation field that is mainly dependent on the shape demagnetising field. These are conventional odd-symmetrical M–H loops, in the sense that M(H) in the descending branch is equal to −M ( − H) in the ascending one. After annealing, the hysteretic behaviour changes dramatically in several aspects: the coercive field increases remarkably and the loop becomes asymmetrical, appearing horizontally shifted and distorted. Moreover, the loop displacement and asymmetry can be modified at room temperature by subjecting the sample to a relatively strong premagnetising field Hp (20–200 kA/m) before measuring.

−0.2

400 200

−0.4

0 0

−2

−1

0 H (kA/m)

100 Hp (kA/m)

1

200

2

Fig. 2. Loops of a sample annealed at 530 °C for 12 min and measured before and after applying a premagnetising field Hp ¼200 kA/m. Inset: Bias field as a function of the premagnetizating field.

It was previously stated that (i) the shift and distortion of the resulting loop are strongly dependent on the component of Hp along the measurement direction, (ii) the effect is saturated at Hp ≃ 200 kA/m , and (iii) opposite premagnetising fields give rise to opposite displacements and distortions. The dependence of the shift with the premagnetising field value is shown in Fig. 2 characterised by the bias field Hb defined as the applied field value at which the central point of the loop is placed.

3. Model and discussion

Fig. 1. TEM micrographs of the particles embedded in the amorphous residual matrix taken on a sample which was devitrified at 530 °C for 12 min.

Based on the fact that the magnetic system consists of a very low concentration of ferromagnetic hard crystallites embedded in a ferromagnetic soft amorphous matrix, the asymmetry of the magnetisation reversal can be explained in terms of the effect that the hard particles produce in their surroundings provided that (i) their magnetisation will remain practically unchanged while the external field sweeps from positive to negative values to trace the loop of the soft phase, and (ii) the ferromagnetic exchange coupling at the grain boundaries is strong enough to pin the magnetic moments of the amorphous matrix. Applying a premagnetising field produces then an orientation of the magnetisations of the crystalline grains whose effect is equivalent to an unidirectional anisotropy. Larger premagnetising fields induce larger biasing (see inset in Fig. 2); the fact that there is a little biasing even with no premagnetisation indicates that there is a certain degree of order in the orientation of the magnetisations of the precipitated grains just after annealing. It should be then remarked that, although they may seem saturated, all the biased HL are in fact minor loops. The strong exchange interaction between a crystallite and the surrounding material will push the magnetic moments of the latter at the grain boundary to be parallel to the magnetisation of the former. As a consequence, both the magnetic poles due to the crystallite magnetisation M1 and to the matrix magnetisation M2 coexist on the grain boundary. In order to evaluate this effect, the particles will be approximated by uniformly magnetised spheres of radius R with an effective magnetisation of M1 − M2. As a first approximation to the problem, we will analyse solely the effect of the magnetostatic interaction. With this purpose we have implemented numerically a simple model in which the magnetic moments of the amorphous material align with the magnetic field resulting from the superposition of the externally

426

J.C. Martínez-García et al. / Journal of Magnetism and Magnetic Materials 377 (2015) 424–429

have used a variational method to minimise the magnetic energy. At this stage, we neglect the effect of the dipolar interaction and the magnetic anisotropy (take into account that the anisotropy constant of the amorphous precursor is very small [29,30]), so only two energy terms are required: the Zeeman and the exchange energies. The x-axis is taken along the line joining the particles, a stands for the distance between their centres, and R for their radii. The magnetic energy of the amorphous material is then given by

1 H

M/Ms

0.5

0

E=

-0.5

∫R

a−R

(εex + ε Z ) dx

(1)

where

-1 -40

1

-20

0 H (A/m)

40

H

→→ ε Z = − μ 0 M2 H ·m

H

(2)

(3)

is the linear density of Zeeman energy, corresponding to the in→ teraction between the external exciting field H and the magnetic moments of the soft phase. The magnetisation configuration which minimises the energy is obtained solving the Euler–Lagrange equations for the Lagrangian

0

-0.5

-1 -400

⎤ ⎡ ⎛ dm x ⎞2 ⎛ dm y ⎞2 ⎛ dm z ⎞2⎥ ⎟ +⎜ εex = A ⎢ ⎜ ⎟ ⎟ +⎜ ⎢ ⎝ dx ⎠ ⎝ dz ⎠ ⎥⎦ ⎝ dy ⎠ ⎣

is the linear density of exchange energy, mx, my, and mz being the Cartesian components, in normalised units, of the magnetisation → → of the amorphous phase, m = M2/M2, and

0.5 M/Ms

20

Wire Disk -200

0 H (A/m)

200

400

Fig. 3. Magnetisation curve simulated for (a) a cubic and (b) a wire/disk-shaped portion of soft material with a hard particle in its centre, taking as only magnetic interaction the magnetostatic one.

⎛ dm x dm y dm z ⎞ ⎟ = εex + ε Z 3 ⎜m x , m y , m z , , , dx dy dz ⎠ ⎝

d2m x dx2

applied field and the magnetostatic one produced by the presence of the crystal. If we consider the crystal placed in the centre of a cube of amorphous material, the magnetisation of the crystal and the edge of the cube parallel to the direction of the external field, the resulting HL is that of Fig. 3 (a) (for this simulation the following parameters have been chosen: μ0 M1 = 1.75 T and μ0 M2 = 0.55 T , according to the composition of crystallites and matrix; grain radius R ¼15 nm and side length of the cube L ¼3 μm, in agreement with the observed morphology). As can be appreciated in the figure, there is no shift effect of the magnetostatic field on the curve. But things are strongly dependent on the geometry: if the crystal is surrounded by a cylinder of amorphous material the resulting HL depends on the relative dimensions. Fig. 3(b) shows the results for two extreme cases: wire geometry (h ≫ r , with r ≥ R ), and disk geometry (h ≪ r , with h ≥ 2R ). For the wire shape, the average dipolar field is positive which leads to a shift of the curve to the left, while for the disk shape, the effect is opposite (in both cases the magnetisation of the crystal, the cylinder axis and the applied field have been taken parallel). It can be concluded at this point that the magnetostatic interaction between hard and soft phases will have a negligible effect as far as the distribution of the former around the later is compensated. Let us analyse now the effect of the exchange coupling. In the simplest simulated model, a line of amorphous material placed between two spherical particles has been considered. In order to numerically compute the effect of the exchange interaction, we

(4)

→ which can be expressed, for H along the z-direction, in the following way:

d2m y dx2 d2m z dx2

=0

(5)

=0

=−

(6) μ 0 M2 2A

H

(7)

with the additional condition

m x2 + m y2 + m z2 = 1

(8)

From a mathematical point of view, the simulation is reduced to solving three independent boundary problems for mx, my and mz at each point of the line, coupled by condition (8). The boundary conditions are imposed by the exchange pinning of the magnetic moments of the soft phase at the crystal boundaries. As we have assumed the hypothesis of the crystals being so hard magnetically that their magnetisations are independent of the external field (this assumption will be revised on a subsequent step of the modelling), the boundary conditions will remain constant,

→ →i m = M1 /M1i → →f m = M1 /M1f

x=R x=a−R

(9) (10)

→f →i M1 and M1 standing for the magnetisation vectors of the initial and final particles (which is equivalent to considering an infinite exchange strength between grains and matrix).

J.C. Martínez-García et al. / Journal of Magnetism and Magnetic Materials 377 (2015) 424–429

427

1

1

0.5 M/Ms

0.5

mz

H = −500 A/m H = −531 A/m

0

0

H = −734 A/m H= = −8000 A/m

H >0 d H =0 d H <0 d

-0.5

-0.5 -1 -8

-1 0

100

200

300

400

-4

4

8

H (kA/m)

500

x (nm)

0

Fig. 5. Simulated magnetisation curves taking into account the exchange and the magnetostatic interactions for different relative orientations of the material line. In red, for null magnetostatic interaction. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

magnetic moments of the amorphous phase,

→ → , d = − μ 0 H d· m

(11)

From the new lagrangian

⎛ dm x dm y dm z ⎞ ⎟ = , ex + , Z + , d 3 ⎜m x , m y , m z , , , dx dx dx ⎠ ⎝

(12)

the following Euler–Lagrange equations are deduced

d2m x dx2 d2m y Fig. 4. Distribution of the magnetisation in the direction of the applied field along the line between two positively magnetised crystals. Below, a schematic view of the magnetisation twist along the line between the two crystals.

dx2 d2m z dx2

Fig. 4 shows the result of the simulation for several values of the applied field. The magnetisation of both crystals has been taken parallel to the z-axis, while the other parameters were R¼ 15 nm, μ0 M1 = 0.75 T , μ0 M2 = 0.55 T (according to the observed crystallisation) and A = 0.4 × 10−11 J/m (see Ref. [31]). Three magnetic configurations are schematised, from the case in which the system is just starting to go out from positive saturation to a case close to negative saturation. It is noticeable that to get to negative saturation a large negative field, capable to switch the crystal magnetisation, would be required. Hence, these are actually minor HL although the majority soft matrix is quasi-saturated. It is remarkable that, in spite of its short range, involving mainly nearest-neighbours, the exchange interaction propagates along the material as a mechanical wave, restricting abrupt changes in the orientation of the magnetic moments and conditioning the overall magnetisation [32]. The simulated demagnetisation curve for a field of amplitude Hm = 8 kA/m is shown in Fig. 5 (red dashed–dotted line). The obvious effects of the exchange interaction are the deformation of the curve and its shift to negative values of the applied field. As a last step of our analysis of the relative importance of the exchange and dipolar interactions, we will consider both terms in the total energy. We need only to modify equation (1) to include the linear density of energy of the magnetostatic interaction be→d tween the dipolar field H produced by the crystals and the

=−

=−

=−

μ 0 M2 2A μ 0 M2 2A μ 0 M2 2A

(Hx + Hxd )

(H y + H yd )

(Hz + Hzd )

(13)

(14)

(15)

which would be uncoupled except for the additional condition (8). Fig. 5 shows the results of the simulation for two cases (in addition to the already described non-dipolar one): both the applied field and the magnetisation of the crystals oriented along (i) the x-direction and (ii) the z-direction. The difference between both configurations resides on the dipolar field, which in case (i) is positive, while in (ii) is negative. The comparison of the three curves confirms the following idea: the way in which the soft phase is distributed around the hard particles will determine the relative importance of the dipolar interaction with respect to the exchange coupling; in the case of a compensated distribution of particles within the soft matrix, the exchange effect will largely prevail over the dipolar one. Once the predominance of the exchange influence has been established, we will drop the dipolar energy term for the subsequent analyses. The influence of the relative orientation of the particle magnetisation with respect to the applied field is remarkable both on the displacement and on the shape of the magnetisation curve. Fig. 6 shows the results of the simulation for different orientations; it can be appreciated that when the crystal magnetisation is perpendicular to the applied field direction, θ = 90°, no shift is produced and the curve bends smoothly, while for θ = 0 the shift is maximum and the approach to positive saturation is abrupt. The described computational model does not produce any hysteresis as far as it does not contain any non-conservative term

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modelling, that the exchange interaction largely prevails upon the magnetostatic one as the most plausible cause for the biasing effect. In fact, the magnetostatic interaction itself could only justify the biasing for very specific crystalline morphologies. Noteworthily, being a short-range interaction, the exchange coupling between adjacent magnetic moments propagates in the soft matrix as a mechanical wave, conditioning the overall magnetic configuration.

1

M/Ms

0.5

0

-0.5

-1 -8

Acknowledgements

0oo 45o 90 -6

-4

-2

0 2 H (kA/m)

4

6

8

Fig. 6. Magnetisation curves for different angles between the magnetisation of the crystals and the applied field.

1

M/Ms

0.5

0

-0.5

-1 -20

-10

0

10

20

H (kA/m) Fig. 7. Simulated HL assuming Stoner–Wohlfarth behaviour for the crystalline particles.

in the energy Lagrangian (4). Nevertheless, the experimental HL present an appreciable hysteresis which, given the softness of amorphous matrix, must come mainly from the presence of the crystals. In the following lines we will drop the condition of the magnetisation of the crystals being immutable and instead will assume it follows the Stoner–Wohlfarth coherent rotation model. For the simulation, this assumption makes the boundary conditions (9) and (10) vary with the applied field according to such model. An illustrating result of this procedure is shown in Fig. 7: for this computation two monodomain particles with anisotropy field 50 kA/m and easy axes making 45° and 60° respectively with the applied magnetic field direction have been considered. For a magnetic field amplitude of 25 kA/m, only the magnetisation of one of these particles switches which produces all the effects observed in the experimental HL: shift to the left and distortion.

4. Conclusions A biasing of the HL has been induced in nanocrystalline Co66Fe4Mo2Si16B12 by simple magnetic treatments at room temperature. All the aspects of the phenomenon are consistent with an explanation in terms of the magnetic interactions between the crystalline particles, which are magnetically hard, and the much softer amorphous matrix. On the basis of a strong pinning of the matrix at the interface of the crystalline grains, it has been proved by simple computational

This work was supported in part by the Spanish government under project MAT2012-33405 and the Principality of Asturias under project SV-PA-13-ECOEMP-34.

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