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# Magnetic microwires as macrospins in a long-range dipole-dipole interaction

(Parte **1** de 3)

Magnetic microwires as macrospins in a long-range dipole-dipole interaction

L. C. Sampaio,* E. H. C. P. Sinnecker, and G. R. C. Cernicchiaro Centro Brasileiro de Pesquisas Fısicas/CNPq, Rua Dr. Xavier Sigaud, 150, URCA, 22290-180, Rio de Janeiro, RJ, Brazil

M. Knobel Instituto de Fısica ‘‘Gleb Wataghin,’’ Universidade Estadual de Campinas (UNICAMP), Caixa Postal 6165,

Campinas 13083-970, SP, Brazil

M. Vazquez and J. Velazquez† Instituto de Magnetismo Aplicado (UCM/RENFE) and Instituto de Ciencia de Materiales (CSIC), PO Box 155,

28230 Las Rozas, Madrid, Spain ~Received 15 October 1999!

The long-range dipole-dipole interaction in an array of ferromagnetic microwires is studied through magnetic hysteresis measurements and Monte Carlo simulation. The experimental study has been performed on glass-coated amorphous Fe77.5Si7.5B15 microwire with diameter of 5 m and lengths from 5 to 60 m. Hysteresis loops performed at room temperature for an array of N microwires (N52, 3, 4, and 5! exhibit jumps and plateaux on the demagnetization, each step correspondent to the magnetization reversal of an individual wire. A model has been constructed taking into account the fact that the magnetization reversal is nucleated at the ends of each wire, under the influence of a dipolar field due to all other wires. Measurements for two wires allowed us to conclude that the dipolar field ~or constant coupling! is independent of distance, at least for an array of a few wires. With the exception of three wires, where frustration seems to be present, the predicted reversal fields of our model are in good agreement with measurements. In order to study the role played by the number of wires on the demagnetization process, we calculate hysteresis loops for a large number of wires through the Monte Carlo method.

Long-range interactions are very common in nature in a wide range of sizes, varying from astrophysical to atomic scales. In condensed matter physics, one of the more remarkable manifestations of long-range interactions is in magnetism. Interactions among magnetic entities are the core of basic and applied studies of modern magnetic materials. For instance, in magnetic materials the long-range dipolar interactions can play a fundamental role in the magnetic properties, being responsible for the formation of certain domain structures and the dynamics of magnetization reversal processes. In addition, advances in fabrication techniques ~including lithography! have given rise to the possibility of producing nanostructured solids with especially interesting physical properties. In particular, it is possible to obtain controlled arrays of magnetic wires with diameters of a few nanometers, which are of practical interest in the design and optimization of magnetoresistive heads for ultrahigh-density data storage applications. In such systems the contribution of the dipole-dipole interaction on the magnetic properties becomes yet more relevant, because long-range interactions can strongly modify the magnetic response of the system to an external excitation.

Although an array composed of a few ferromagnetic wires could in principle seem a quite simple problem to study and model, it is striking to notice how complex this problem can turn out to be. Some recent investigations have dealt with the dynamics of magnetization processes, which can include localized excitations and/or collective modes, independent of the parity of the system,1 weak chaotic behavior,2 and even the possibility to tailor the value of the coercivity, in the case of nanoscopic flat wires.3 The complication in the study of dipolar interactions is that the magnetic fields resulting from the interaction depend on the magnetization state of each entity, which, in turn, depend on the effective field of neighboring elements. In spin systems the long-range character of a dipole-dipole interaction is an inherent difficulty to solve the Hamiltonians due to the large number of neighbors that one has to take into account in calculations. Several works have been made using Monte Carlo simulations calculating the magnetic domain structure and magnetic hysteresis including the dipolar interaction term in the Ising or Heisenberg Hamiltonian.4,5

An intrinsic difficulty in the study of magnetic interactions is the fact that it is extremely difficult to characterize a single magnetic element using most conventional magnetometry techniques. Also, the predictions of numerical simulations are intricate to compare with real systems, owing to the necessity to introduce several approximations in the modeled problem. However, in this work we make use of a convenient macroscopic configuration, placing together several ferromagnetic microwires covered with glass. Such microwires exhibit a strong magnetic anisotropy with an easy magnetization direction along the axis of the wire with a main single domain practically extending along the whole wire. This fact will allow us to consider each one of such microwires as a elemental magnetic moment. The stray fields created by the microwires couple the magnetization of the neighboring wires, affecting the magnetic state of each single

PHYSICAL REVIEW B 1 APRIL 2000-IVOLUME 61, NUMBER 13 wire. This system is relatively easy to study both experimentally and theoretically, and, in the case of few wires it is possible to obtain analytical solutions. Some interesting aspects of the long-range character of dipole-dipole interaction and their influences on the magnetic properties appear clearly in the obtained results. The exact solutions and experimental data can be compared with Monte Carlo simulations, which are necessary to employ when the array is formed by a large number of wires. As mentioned before, although this arrangement seems to be rather simple, it displays a variety of interesting aspects which certainly would apply in other physical systems.

Concerning the experimental measurements, they have been performed in glass-coated amorphous microwires with nominal composition Fe77.5Si7.5B15, diameter of 5 m, and the thickness of the glass coating of 7.5 m. Glass-coated amorphous microwires are presently attracting an increasing interest from both basic and applied points of view ~for reviews see Ref. 6!. Their metallic core, being structurally amorphous and with typical diameter from 1 to 30 m, is covered by an insulating Pyrex-like coating with thickness between 1 and 20 m. They are fabricated by means of Taylor-Ulitovsky technique by which the molten metallic alloy and its glassy coating are rapidly quenched and drawn to a kind of composite microwire typically a few kilometers long. This family of microwires displays quite remarkable magnetic properties, that together with their tiny dimensions and the protective coating make them potential candidates for many sensor applications.7

Owing to the amorphous nature of such microwires, their unique magnetic behavior depends on the strength and the distribution of magnetoelastic anisotropy. That is first determined by the magnetostriction constant, which is mainly a function of composition.8 For the present alloy composition the saturation magnetostriction takes a value of 231025.I n turn, the internal stresses ~as strong as 103 MPa! depend on the ratio cover thickness to core diameter, which is controlled by the fabrication parameters, and also on particular processing as thermal treatments and chemical etching of the coating.8 When axially magnetized these wires exhibit lowfield square hysteresis loops with a single and large Barkhausen jump.

We have measured magnetic hysteresis loops in arrays of

N microwires (1<N<5) placed side by side, all parallel, each one touching its nearest neighbors. Their lengths vary from 5 to 60 m, cut from a single long microwire. We performed the measurements by using either a superconducting quantum interference device ~SQUID! magnetometer ~Quantum Design, MPMS XL model! or a very sensitive magnetic-flux integrator. Essentially, the difference in these two systems is the sensitivity and the time of measurement. Although the hysteresis loops measured in the SQUID do not exhibit either noise or drift, which appear in the fluxintegrator, a single hysteresis measurement can take a few hours in the SQUID, while the same loop in the fluxintegrator takes about 1 min. The flux-integrator was used for rapid measurements, for instance, to measure the distribution of reversal field values. We will focus our attention on measurements performed at room temperature, but also several loops were performed at low temperatures @4<T(K)

<300#, in several attempts to strengthen the interactions among the wires. However, at low temperatures there is a change in the domain structure of the microwires, probably owing to the increasing internal stresses induced by the different thermal expansion coefficients of the ferromagnetic alloy and the covering glass. Therefore, the loops which are rather square at room temperature turn out to lose this property at low temperatures. This characteristic has been also reported for Co-based microwires.9

In the case of one 5 m long wire (N51, see Fig. 1! the hysteresis curve exhibits a typical square loop, with characteristic large Barkhausen jumps. The observation of such square loops, labeled as magnetic bistability, has been interpreted as in the case of in-water-quenched wires, considering the remagnetization processes of the inner core between two stable remanence states.10 That internal core mainly consists of a single axial domain, but at the ends of the wire a closure domain structure appears at finite applied fields to reduce the otherwise quite high magnetostatic energy. Of course, for very short microwires closure structures coming from both ends overlap at the middle of the sample, destroying the magnetic bistability.1 The critical length to observe bistability in the microwires of the present study is less than 5 m. Nevertheless, in spite of the existence of these closure structures some stray field is generated in the surroundings of the microwire. Upon application of reversed field a domain wall depins from one end of the wire and propagates along the wire resulting in the observed magnetization jump.

Small differences in the measured coercive fields (HC) were detected when the magnetic field was applied along positive or negative directions ~in Fig. 1 the HC values correspond to 20.89 and 0.79 Oe, respectively!. There can be several origins for the fluctuation of HC values. When dif- ferent samples are investigated, the fluctuation in HC is probably due to different magnetoelastic anisotropies induced in the wire ends during the cutting process, which can generate different levels of mechanical stresses. Since the nucleation of a domain wall starts in defects located at the extremities of the wires, and the number and strength of these defects depend on the cutting, the switching field of one wire can be slightly different for magnetic fields applied in opposite directions, and it can also vary in different samples. We have

FIG. 1. Hysteresis loop for one microwire at room temperature.

The reversal field is 20.89 Oe for negative reversal field and 0.79 Oe for positive reversal field. The coercive field is defined as the mean value, uHCu50.85 Oe.

investigated this fluctuation in several samples, and we have found a maximum variation in HC of about 0.10 Oe. Another distribution in the HC values arises from thermal fluctuations, and it occurs even when the same sample is measured several times. To determine this distribution, we measured

HC several times in the same sample and with the field applied in the same direction, and it was found that the width of this distribution is around 0.03 Oe. However, we cannot exclude that this intrinsic fluctuation of the reversal field can arise simply from the fact that after each reversal the closure domain structures cannot be exactly the same, thus introducing a fluctuation in the next magnetization reversal. From the above discussions we consider the absolute value of HC of this microwire as the mean value obtained from many differ- ent measurements, HC50.85 Oe. Let us now consider two wires ~5 m long! placed side by side with their axes parallel. In this case the distance between their axes is twice the thickness of the coating plus twice the radius of the ferromagnetic core, i.e., around 20 m. The corresponding hysteresis loops exhibit two clear Barkhausen jumps steps and a plateau nearly at zero magnetization ~see Fig. 2!. This plateau corresponds to the configuration of two wires with opposite magnetization directions. It is worth noting that the first jump occurs at magnetic fields lower than HC , while the second one occurs for fields larger than HC . These reversal fields will be named H2i and H2ii , and their values are 0.38 and 1.10 Oe, respectively, for positive demagnetizing field. For negative demagnetizing field the values of H2i and H2ii were found to be 20.49 and 21.12 Oe, respectively. As will be shown below, the splitting of the

HC in two reversal fields has its origin in the dipole-dipole interaction between the wires. Varying the number of wires, the hysteresis loops exhibit several steps on the demagnetization ~see Figs. 2–5!, each one corresponding to the reversal of the magnetization of a single wire. As observed, the number of jumps equals the number of wires with the exception of the particular case of 3 wires.

In order to understand the existence of jumps and plateaux in the demagnetization curves, let us initially consider the simplest arrangement: two parallel magnetic wires in the presence of a positive and saturating magnetic field applied parallel to the axis of wires. Such a situation yields the magnetization of both wires and the applied field to stay parallel, pointing in the same direction. Notice that beyond the applied magnetic field ~H! each wire feels the influence of a dipolar field (Hi,j) due to the presence of the other wire, where Hi,j is the field of the wire i over the wire j. This dipolar field Hi,j is given by2

FIG. 3. Hysteresis loop for three microwires at room temperature. Note that the second step does not appear. The magnetization is normalized to the saturation value. The arrows represent the magnetic configuration of the wires.

FIG. 4. Hysteresis loop for four microwires at room temperature. The magnetization is normalized to the saturation value. The arrows represent the magnetic configuration of the wires.

FIG. 2. Hysteresis loop for two microwires at room temperature.

The mean reversal fields on the demagnetization are H2 i

520.43 Oe and H2ii521.1 Oe. The magnetization is normalized to the saturation value. The arrows represent the magnetic configu- ration of the wires.

FIG. 5. Hysteresis loop for five microwires at room temperature.

The magnetization is normalized to the saturation value. The arrows represent the magnetic configuration of the wires.

8978 PRB 61L. C. SAMPAIO et al.

where Kn is a geometric factor and Mi is the magnetization of the ith wire. As the coefficient Kn depends in principle on the distance between interacting wires, the subscipt n denotes the distance between the wires, K1 corresponds to nearest neighbors, K2 to second neighbors, and so on. Thus, for two wires one can easily write down the mutual dependence pro-

duced by the dipole-dipole interaction through the functions

2KnMi. Applying now a reversal magnetic field, one notices that both applied field and dipolar fields act in the same direction, antiparallel to the magnetization of both wires. Let us simplify the analysis by considering two quasi-identical wires, i.e., both wires have the same magnetization M and coercive field HC . It is important to emphasize that on the ideal demagnetization process the field necessary to reverse the mag- netization of an individual wire has always the same value

~the coercive field, HC! and this value, but for the abovementioned fluctuations, is characteristic of the internal mag- netic properties ~anisotropies! of that particular wire. Hence, at the first jump (H2i ), in spite of the fact that the applied field is H2i , the effective field is equal to HC . Therefore, the condition to the first wire to reverse its magnetization is

Once the first wire reverses its magnetization, the configuration of the system is given by two wires with compensated magnetization (↑#), which is more stable than the previous configuration since the dipolar fields now act parallel to the magnetization of both wires. Now, to reverse the magnetization of the second wire a stronger external field is required because this field has to compensate the dipolar contribution.

Actually, the dipolar interaction acts on the wires as a bias field with opposite direction, decreasing and increasing the reversal field of the first step and the second jumps, respectively. Note that the plateau or difference between the rever- sal fields H2i and H2ii corresponds to dipole-dipole interaction between the wires, and it is given by 2K1M. Before proceeding with the discussion and extending the reasoning to the case of several wires, let us discuss several important points regarding the dipole-dipole interactions of two wires. First, it is worth noticing that although we have considered, for the sake of simplicity, two wires with the same magnetization and coercive field, the fluctuations in these values are essential to observe the effects of dipolar interaction.2 If both wires would have exactly the same magnetic properties they would feel exactly the same effective field, and would reverse at exactly the same field value, H2i .

However, in real situations the wires are not identical, and they can display fluctuations in both magnetization and coercive fields ~see above!, and therefore one of the wires reverses the magnetization before the other one, leading to an intermediate, more stable structure, and to the splitting of the reversal field.

Second, it is important to further discuss some details about the coupling constant Kn . We consider that the magnetization reversal is nucleated at the ends of the wires when the effective magnetic field acting on the wire is equal to

HC . Calculating the dipolar field nearby one of the wires it is easy to show that the component of its dipolar field along the axis of the second wire at its ends is given by where p is the strength of the point poles, located at the extremeties of the wire of length L and rij is the distance between the wires i and j.12 Note that in our configuration the magnetizations of the wires are either parallel or antiparallel, and the distances among wires are well known. Since the magnetization is pL, one identifies Kn as given by tems are composed of a few wires with distances never exceeding a few tenths of millimeters, with length of the wires of about 5 m we have always the condition rij!L fulfilled.

Therefore, the constants of coupling Kn become independent of distance, at least in the range of a few wires.

In order to better understand the role of the constant of coupling and its dependence on rij and L we have evaluated the dipole-dipole interaction 2K1M for various lengths of the wires. The distance between the wires is fixed since the wires are placed side by side adjacent to each other as mentioned above. The experimental value of 2K1M was obtained from the width of the plateau of the hysteresis loops measured for two wires using the flux-integrator setup. The results are shown in Fig. 6. Note that the coupling rapidly increases for the shortest wires and seems to become negligible for wires longer than around 40 m, at least within our experimental sensitivity. For the range of lengths larger than 5 m the hysteresis loop for one wire exhibits bistability ~square loop! and the hysteresis loop for two wires are all identical differing only on the width of the plateau. However, for wires with

FIG. 6. Width of the plateau in compensated magnetic configuration ~zero magnetization! measured from the demagnetization curve of two wires. It represents the strength of dipole-dipole inter- action ~see text!. The line is a fitting following the power law L21.2 .

length of about 1 m the hysteresis loop for one wire does not exhibit bistability anymore, preventing equivalent analysis.

According to our model the quantity 2K1M is propor-

early on L following L/rij3 . The conjunction of the two limits

should exhibit a maximum in the plot 2K1M vs L, which was clearly not experimentally observed. Thus, from Fig. 6

we conclude that in our samples we are always working in the limit rij!L. In this range of length, the data were fitted by the power law, L21.2, which, however, is not in good

agreement with the expected dependence (L22).

(Parte **1** de 3)