Control of the Magnetic State of Massifs of Ferromagnetic Nanoparticles with the Aid of the Inhomogeneous Field of a Magnetic-Force-Microscope Probe 1

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1 ISSN X, The Physics of Metals and Metallography, 2010, Vol. 110, No. 7, pp Pleiades Publishing, Ltd., Control of the Magnetic State of Massifs of Ferromagnetic Nanoparticles with the Aid of the Inhomogeneous Field of a Magnetic-Force-Microscope Probe 1 V. L. Mironov, A. A. Fraerman, B. A. Gribkov, O. L. Ermolayeva, A. Yu. Klimov, S. A. Gusev, I. M. Nefedov, and I. A. Shereshevskii Institute of Physics of Microstructures, Russian Academy of Sciences, Nizhny Novgorod, Russia mironov@ipm.sci-nnov.ru Key words: magnetic-force microscopy, ferromagnetic nanoparticles, micromagnetism DOI: /S X X 1 1. INTRODUCTION An interest to the investigations of processes of the magnetization reversal of ferromagnetic nanoparticles under the action of a strongly inhomogeneous magnetic field of the probe of a magnetic-force microscope is caused by several reasons. First, the ordered massifs of nanodimensional ferromagnetic particles are universally recognized as promising media for the magnetic recording of information with a superhigh density [1 7]. In this connection, it is necessary to study the mechanisms of magnetization reversal of separate nanoparticles in local external magnetic fields [8, 9]. In this case, unique possibilities are provided by magnetic-force microscopy, since, on the one hand, the probe of a magnetic-force microscope (MFM) possesses significant stray fields, which cause effects of magnetization reversal, and, on the other hand, the MFM simultaneously makes it possible to control the results of this local action. Second, the ordered massifs of ferromagnetic nanoparticles are promising sources of embedded inhomogeneous magnetic fields in hybrid structures such as ferromagnet/superconductor, ferromagnet/metal, ferromagnet/semiconductor, which can influence the orbital and spin degrees of freedom of charge carriers in such systems. In this case, the magnitude and characteristic spatial scales of the inhomogeneity of the stray fields of the massifs of ferromagnetic nanoparticles are determined by the characteristics of the material, geometric dimensions and shape of the particles, and by the spatial periods of the structure [7, 10 14]. By changing the states of magnetization of single particles in a massif with the aid of the MFM probe, it is possible to 1 The work presents a survey of the results of studies of the processes of magnetization reversal of ferromagnetic nanoparticles under the action of the field of a magnetic-force-microscope probe. change the magnitude and the spatial configuration of the magnetic field induced by a given massif. One of the most informative methods of studying local states of magnetization in magnetic nanostructures is magnetic-force microscopy [15, 16]. Beginning from the pioneer works, many authors noted effects of local magnetization reversal of particles by the MFM probe [17 19] in the cases where the magnetic field of the MFM probe exceeded the coercivity of the particles to be investigated. Thus, for instance, experiments were carried out on the initiation, by the local field of the MFM probe, of transitions between the states with a homogeneous magnetization in elliptical Co particles in an external homogeneous remagnetizing field [19], and transitions were considered from a homogeneous to a vortex state of magnetization [17, 18] under the effect of the field of an MFM probe. However, these studies were quite irregular, and a whole series of problems connected with the influence of the field of the probe on the distribution of magnetization in the particles (such, as magnetization reversal mechanisms; controlling the sign of the vorticity of a vortex state; etc.) have not been touched at all. In this work, we give a survey of results of systematic studies of processes of the magnetization reversal of ferromagnetic nanoparticles of various shape induced by the magnetic field of the probe of a magnetic-force microscope [20 31]. 2. MAGNETIC FIELD OF THE MAGNETIC-FORCE-MICROSCOPE PROBE The modern technologies make it possible to prepare MFM probes of different configurations with variable parameters. The first MFM studies were conducted using probes that represented electrochemi- 1

2 2 MIRONOV et al. d y h z r (x, y, z) z = ax 2 + ay 2 Fig. 1. Model of a parabolic MFM probe with a magnetic coating: x, y, z are the coordinates of integration (r 2 = x 2 + y 2 ); h and ρ are the coordinates of the point of observation. cally sharpened wires from ferromagnetic materials [32 34]. Such probes made it possible to obtain a lateral resolution at a level of 10 nm [34]. At present, the probes from wires are mainly used in magnetic-force microscopes with systems of the registration of the power interaction of the probe and the sample on the basis of fiber optical interferometers [35, 36] and high- Q quartz resonators [37, 38]. A wider application in the magnetic-force microscopy obtained probe sensors in the form of a needle covered with a thin layer of a magnetic material and arranged on an elastic console (cantilever). Such sensors are fabricated using technologies of the microtreatment of silicon by methods of lithography and etching [39]. The registration of the force of interaction of the probe and the sample with the aid of such sensors is accomplished by an optical method according to the deviation of a laser beam upon the bend of the console [40, 41]. The technologies developed to date make it possible to form MFM probes in the form of pyramids and cones with different aspect ratios [42]. An increase in the spatial resolution of such probes is achieved due to the reduction in the area of the magnetic coating by the methods of lithography and ion etching [43 46]. In this case, on the tip of the probe there are formed single ferromagnetic particles with dimensions of nm, which leads to a considerable increase in spatial resolution. In recent years, there were intensely developed technologies of the fabrication of probes on the basis of carbon nanotubes. The magnetic probes are formed by coating nanotubes with thin ferromagnetic layers, ϕ x ρ by filling the internal space of nanotubes with a magnetic material, and also by the formation of a magnetic nanoparticle on the free tip of a nanotube [47 52]. The minimum sizes of such magnetic particles can be 10 nm, which is close to the physical limit caused by the superparamagnetism of small ferromagnetic particles [53]. Upon MFM studies of ultra-small particles, there appears a problem of the selection of an optimum shape and size of the probes. The matter is in that the value of the MFM response is determined by both the properties of the probe and by the spatial structure of the related stray fields [54 58]. Therefore, for various objects, depending on the configuration of stray fields, it is possible to select a probe with optimum parameters, which will ensure the highest sensitivity and high spatial resolution [59]. However, for a wide circle of problems of the interpretation of MFM images and the simulation of the processes of the influence of the probe on the magnetization of the objects to be investigated it is frequently sufficient to describe the probe within the framework of a simple model of a uniformly magnetized sphere [55 57, 60]. As an example, we examine a sufficiently simple, but realistic model of a probe in the form of a paraboloid covered with a layer of a magnetic material (Fig. 1). We will describe the field of the probe in a cylindrical coordinate system {ρ, ϕ, h}. Then, for a uniformly magnetized probe the components of the magnetic field H h and H ρ can be written as follows: H h ( ρ, h) M 2( z+ h) 2 ( ρ r) 2 = t dv, ( ( z+ h) 2 + ( ρ r) 2 ) 5/2 (1) H ρ ( ρ, h) = M t ( z+ h) ρ r dv. (2) ( ( z+ h) 2 + ( ρ r) 2 ) 5/2 where M t is the saturation magnetization of the material of the probe tip; and ρ = ρcosϕ i + ρsinϕ j is the polar vector in the plane x, y (ρ = ρ ). As an example, Fig. 2 shows the results of numerical calculations of H h and H ρ for an infinite parabolic probe with the parameters a = nm 1 (coefficient of the parabola) and d = 30 nm (thickness of the magnetic coating), which are close to the experimental parameters. For a comparison, the figure also presents spatial distributions of h and ρ, the components of the magnetic field of a spherical probe with close parameters. In Fig. 2 for convenience there are introduced normalized values H h = H h /H h (0, δ), H ρ = H ρ /H h (0, δ), and ρ = ρ/r t, where the thickness of the layer that covers the disk is δ = 20 nm, and the radius of the spherical tip of the probe is R t = 30 nm. Near the sur-

3 CONTROL OF THE MAGNETIC STATE OF MASSIFS 3 ~ H h (a) ~ H h ~ ρ ~ ρ Fig. 2. (a) Spatial distribution of H h (ρ) in the plane h = δ for (dashed line) parabolic and (solid line) spherical probes. Spatial distribution of H ρ (ρ) in the plane h = δ for (dashed line) parabolic and (solid line) spherical probes. face, the h component of the stray field of a parabolic probe (h = 0, ρ = 0) is written in the form H h ( 0, 0) 16παdM = t ( 1+ 4ad) (3) This quantity for the probes on the basis of Co (with the parameters a = nm 1, d = 30 nm) is approximately 13 koe and considerably exceeds the coercivity of submicron ferromagnetic particles. On the other hand, the MFM probe can be approximated by an effective uniformly magnetized spherical probe. In this case, the expressions for the components of the stray field are simpler: H h ( ρ, h) 3 4πM t R t ( h + R t) 2 ρ 2 = , 3 (( h+ ) 2 + ρ 2 ) 5/2 (4) 3 ( h+ R H ρ ( ρ, h) 4πM t R t )ρ = t (5) (( h+ R t ) 2 + ρ 2 ) 5/2 The spatial distributions of the h and ρ components of the field (4) and (5) for a spherical probe with R t = 30 nm are presented in Figs. 2a and 2b. It is seen from the figure that expressions (4) and (5) give a quite acceptable approximation of the distribution of the field of the parabolic probe, but they are much more convenient for the estimations of the field values and for the application in the micromagnetic simulation. 3. EFFECT OF THE MFM-PROBE FIELD ON THE DISTRIBUTION OF MAGNETIZATION IN THE SAMPLES INVESTIGATED Upon the study of samples by the methods of magnetic-force microscopy, it is usually assumed that the probe of the microscope does not change the structure of the magnetization of the object investigated. How- R t ever, in real MFM images there are frequently observed artifacts connected with the disturbing influence of the field of the probe on the magnetization of the sample; in particular, the different intensity of the bright and dark poles of MFM images of uniformly magnetized particles can be seen [61 63]. The effect of a local disturbance of the magnetization distribution in a thin uniformly magnetized ferromagnetic sample under the action of the field of an MFM probe was theoretically examined in [29, 30]. At the first stage, there was carried out a computer simulation of the processes of interaction of the uniformly magnetized sample with the MFM probe. The simulation was conducted with the aid of a special package of programs (SIMMAG) on the basis of a numerical solution to the Landau Lifshitz Gilbert (LLG) equations [64] for the magnetization of a sample in the field of the probe (the program was developed at the Institute of Metal Physics, Ural Division, Russian Academy of Sciences). In the calculations, the probe was represented as a sphere uniformly magnetized along the z axis (Fig. 3) with an effective magnetic moment m t = M t V t (M t is the saturation magnetization of the material of the probe tip; V t is the volume of the sphere, equal to the effective volume of the interacting part of the probe). As a sample, we selected a thin ferromagnetic particle of rectangular form with dimensions of nm. It was assumed that in the initial state the sample is uniformly magnetized, and the magnetization M 0 () r was directed along the x axis. Under a disturbance, the modulus of the vector of magnetization remains unchanged, and the deviation from the undisturbed state at each point of the sample is described by two angles polar (θ) and azimuthal (ϕ). The results of micromagnetic LLG calculations of the magnetization distribution for different magnitudes of the magnetic field of the probe are given in Fig. 4. The simula-

4 4 MIRONOV et al. m t x z M 0 χ ρ tion shows that in a weak field of the probe there is formed a region of disturbance, in which there appears a z component of magnetization perpendicular to the plane of the sample, and the magnetization vectors are partially turned along the field of the probe (Fig. 4). For finding parameters that characterize the excited region, we solved the problem of the magnetization redistribution in a thin uniformly magnetized ferromagnetic sample under the action of a weak field of the MFM probe, which was approximated by the field of a point magnetic dipole. The excited state of magnetization in the sample was described by a set of Euler equations for the polar and azimuthal angles, obtained as a result of the variation of the functional of energy, which takes into account the exchange, magnetostatic, and Zeeman energies, and also the anisotropy energy. In the first order of the perturbation theory, the set of equations for the polar (θ) and azimuthal (ϕ) angles, which characterize the direction of the local vector of magnetization, is written as follows: ( 2 ρ 2 ) 3ρθsinχ Δθ = ε ε ( 1 + ρ 2 ) 5/2 2 θ 3ρ( cosχ ϕsinχ) Δϕ ε k' 2 = + ϕ. ( ρ 2 + 1) 5/2 (6) where the normalized parameters entering into the equations are determined by the following expressions: ϕ M Fig. 3. Geometry of the problem. θ y Fig. 4. Disturbance of the magnetization in a uniformly magnetized sample under the effect of a weak field of the MFM probe. The position of the probe over the sample is shown by a dark dot. The region of the disturbance is outlined by a dashed line. Here, k is the anisotropy constant; V t is the volume of the probe; h is the height of the probe above the sample; l ex = A/M p is the exchange length; A is the 2 exchange constant; M p and M t are the saturation magnetizations of the materials of the sample and the probe tip, respectively; and r is the modulus of the radius vector drawn on the plane from the point that is located directly under the probe (Fig. 3). The solution for the θ component of the disturbance of the magnetization distribution can be represented in the integral form: s 2 exp( s) θρ ( ) = ε 1 ε J + s 2 0 ( sρ) ds. 0 ε 2 2 (8) Here, J 0 (sρ) is the zero-order Bessel function. The magnitude of the θ component of the magnetization disturbance in the maximum, under the condition ε 2 1, is equal to 2M θ( 0) 2ε t V = t 1 = M p l 2 ex h (9) This means that the magnitude of the θ component of the disturbance of the magnetization distribution depends, mainly, on the relationship between the volume of the probe, on the height at which the probe is located above the sample, and on the exchange length of the material of the film. At large distances from the probe, the θ component of the magnetization falls off proportional to the z component of the magnetic field of the probe and approximately it can be written in the form k' = k/ ( M p l 2 ex h); ε 1 = M t V t /M p l 2 ex h; ε 2 = 4π( h/l ex ) 2 ; ρ = r/h. (7) θρ ( ) = ε /2 ε 2 ( 2 ρ ). ( 1 + ρ 2 ) 5/2 (10)

5 CONTROL OF THE MAGNETIC STATE OF MASSIFS 5 The solution for the ϕ component of the disturbance also can be represented in the integral form: ϕρχ (, ) = ε 1 cosχ s2 exp( s) J k' 2 1 ( sρ) ds. + s 2 0 (11) Here, J 1 (sρ) is the first-order Bessel function. At short distances, the ϕ(ρ, χ) function can be described by the law ϕρ ( ) 3ε 1 ρcosχ = k' 2 (12) At large distances from the probe, the ϕ component of the disturbance falls off proportionally to the ρ component of the magnetic field of the probe: ϕρ ( ) 3ε 1 ρcosχ = k' 2 ( 1 + ρ 2 ) 3/2 (13) An analysis of the solutions shows that characteristic lateral dimensions of the excited region of the region of magnetization of the sample are determined only by the height of the location of the probe above the object: Δ θ, h. Δ ϕ (14) The solutions (8) and (11) obtained for the excited distribution of magnetization make it possible to estimate the distortions of MFM images induced by the probe. It is known that the phase contrast in the magnetic-force measurements is proportional to the derivative of the z component of the force of interaction of the probe with the sample with respect to the coordinate z: ΔΦ Q -- F 2 = k z (15) Then, the absolute value of the normalized additive to the MFM contrast, ΔΦ, can be calculated as follows: ΔΦ 2 = ( MH) dv. z 2 V t (16) The additive to the MFM contrast caused by the magnetization disturbance in the angle θ and, as a result, by the appearance of a surface magnetic charge, is equal to ΔΦ θ = 16π 2 M 2 p ε 2 1 ε 2 s3 exp( 2s ε 2 ) ( 1+ 4s ε 1 + s s 2 ε 2 ) ds. 0 (17) In the case where ε 2 1, the additional contrast ΔΦ θ (17) can be approximately represented as 57π 2 M 2 2 ΔΦ p ε 1 θ = ε 2 (18) The additive to the contrast ΔΦ ϕ caused by the magnetization redistribution in the plane of the sample and, as a result, by the appearance of a bulk charge, is equal to 8πM 2 2 ΔΦ p ε 1 ϕ = s exp( 2s) ( 3s 4 + 9s 2 k' k' 4 ) ds. ( k' 2 + s 2 ) 3 0 (19) In the case where k' 1, the additive to the MFM contrast caused by the ϕ component of the disturbance can approximately be represented in the form (20) The calculations show that at the characteristic parameters that correspond to the parameters of conducted MFM measurements, both k' and ε 2 are much greater than unity, and the additive ΔΦ ϕ caused by the ϕ component of the disturbance gives a smaller contribution to the contrast than the additive caused by the θ component. Indeed, the ratio of the contributions to the MFM contrast from different components in this case is equal to (21) Thus, the experimentally observed nonequivalence of the intensity of bright and dark poles in the MFM images of ferromagnetic nanoparticles is apparently caused precisely by the θ component of the magnetization disturbance. In a strong field of the probe, the magnetization directly under the probe is aligned along the force lines of the field (Fig. 5), while in the region where the Zeeman interaction energy is on the order of magnitude equal to the anisotropy energy there is formed a state, which is called in the literature magnetic antivortex [65 67] (shown in Fig. 5 by an arrow). Such strong magnetization disturbances due to the inhomogeneous field of the MFM probe can cause effects of magnetization reversal in ferromagnetic particles. ε 2 60πM 2 2 ΔΦ p ε 1 ϕ = ε 2 k' 2 ΔΦ ϕ ΔΦ θ πk' 2

6 6 MIRONOV et al. Fig. 5. Disturbance of the magnetization in a uniformly magnetized sample under the effect of a strong field of the MFM probe. The position of the probe over the sample is shown by a dark dot. The position of the antivortex structure of the magnetization is indicated by an arrow. 4. TRANSITIONS BETWEEN THE STATES WITH A HOMOGENEOUS MAGNETIZATION INDUCED BY THE MAGNETIC FIELD OF THE MFM PROBE IN ELLIPTIC FERROMAGNETIC NANOPARTICLES In the first experiments on the MFM magnetization reversal [27, 68], ordered massifs of submicron Fe Cr particles prepared by the method of interference laser annealing were used [69, 70]. Figure 6a displays an AFM image of a fragment of the sample. The lateral dimensions of the particles were nm and the thickness was on the order of 20 nm. The coercivity of the particles was on the order of G. After the magnetization of the massif in an external magnetic field, all particles were magnetized along the longer axes. Figure 6b shows a characteristic MFM image of part of a massif of FeCr particles. Upon MFM studies of the massif of Fe Cr particles with the use of a two-way method (tapping/lift mode), in the images obtained (Fig. 7a) there were revealed specific features associated with a change in the magnetic state of particles under the action of the MFM-probe field. The magnetization reversal of particles occurs during the first passage, when the MFM probe is located directly in the contact with the sample surface and the influence of its magnetic field on the state of the magnetization of particles is maximum. In the MFM images given, it is seen well that the magnetization reversal occurs at the moment when the MFM probe passes above the region of the particle that has a bright MFM contrast, which corresponds to the magnetic pole of the particle of the same type as the pole of the MFM probe. To configure stray fields created by the massifs of nanoparticles, there was developed a procedure of a selective magnetization reversal of separate nanoparticles in the massif. The experiments were conducted as follows. First, there was performed a scanning of a region of the massif of particles in a noncontact (constant height mode) regime. In the image obtained, there was selected some particle and a repeated scanning was performed only above this particle. In this case, in the process of scanning, the distance between the MFM probe and the upper face of the particle was gradually decreased until the particle changed its state of magnetization, which was detected by an abrupt change in the contrast in the line of scanning (see Fig. 7b). After this, the probe was moved away from the sample to a distance preventing repeated magnetization reversal. On the final stage of the procedure, a repeated scanning of the same region of the massif was performed at a high height for affirming the fact of the reversal of the magnetization of the particle. Figure 8 shows the sequential stages of the experiment on the magnetization reversal of single ferromagnetic Fe Cr particles. Directly before conducting this experiment, the sample was magnetized in a mag- µm (a) µm µm µm Fig. 6. (a) AFM image of a massif of Fe Cr particles. An MFM image of part of the massif of Fe Cr particles after magnetization in an external field.

7 CONTROL OF THE MAGNETIC STATE OF MASSIFS 7 μm (a) μM Fig. 7. (a) MFM image of a massif of FeCr particles that was obtained in the tapping/lift mode. MFM image of one of the particles in the process of magnetization reversal by the MFM probe. The moment of the change in the orientation of the magnetic moment is shown by an arrow. netic field of ~3 kg applied along the longer axes of the particles. Further, with the aid of the MFM probe there was produced a magnetization reversal of a single particle (Fig. 8a), then, of a second particle (Fig. 8b) and, finally, of a third one (Fig. 8c). In the final stage, the second particle was returned into the initial state (Fig. 8d). Thus, it was shown that with the aid of the MFM probe there is possible a controlled and reversible magnetization reversal of Fe Cr particles. To understand the mechanism of magnetization reversal of particles under the action of the inhomogeneous magnetic field of the MFM probe, we carried out a micromagnetic simulation of the process of interaction of the probe with the particle [68]. The simulation was carried out using a SIMMAG program package. The calculations showed that the magnetization reversal of elliptical particles during scanning is accompanied by a complex inhomogeneous rearrangement of the magnetization distribution inside the particles. The sequential stages of the process of magnetization reversal during the motion of the MFM probe across the particle are given in Fig. 9. The initial state of the particle corresponds to a state with a homogeneous magnetization (Fig. 9a). Then, the MFM probe, whose vector of magnetic moment is directed perpendicularly to the plane of the figure, is approached to the edge of the particle (Fig. 9b). It is clearly seen that the distribution of magnetization inside the particle experiences a disturbance under the action of the inhomogeneous field of the probe and that the directions of the magnetization vectors of the particle near the probe are aligned along the force lines of its magnetic field (Fig. 9b). During the further motion of the probe (Figs. 9c 9d) in the μm (a) μm μm μm (d) μm μm μm μm Fig. 8. Sequential stages of the experiment on the magnetization reversal of submicron Fe Cr particles. The particles that underwent magnetization reversal are outlined by dashed circles.

8 8 MIRONOV et al. (a) (d) (e) (f) Fig. 9. Sequential stages of the process of magnetization reversal of a uniformly magnetized Fe Cr particle under the effect of an inhomogeneous field of the MFM probe (the probe positions are indicated by gray dots). right-hand side of the particle there is formed a quasivortex distribution of magnetization. As the MFM probe goes out on the opposite side of the particle (Fig. 9e), the small disturbance of magnetization introduced by the probe is retained. At the same time, the direction of the magnetization vectors in the righthand side of the particle changes to opposite with respect to the initial state. Figure 9f demonstrate the final state, which corresponds to a uniform state with an opposite (with respect to the initial) direction of the magnetization. We applied the above algorithm of the magnetization reversal for creating built-in configurable sources of a strongly inhomogeneous field on the basis of chains of ferromagnetic particles. As an example, Fig. 10 displays the results of configuring a onedimensional array of eight elliptical Co particles by the MFM probe. The chain consisted of eight particles with lateral dimensions of nm and a height of 10 nm. The distance between the particles was equal to 150 nm. Preliminarily, the sample was in-plane magnetized in an external uniform field of 10 koe, so that all particles proved to be uniformly magnetized in the direction of the external field (Fig. 10a). Then, with the aid of the MFM probe in the chain of these particles there was configured a state with an antiferromagnetic ordering of magnetic moments (Fig. 10b). In this configuration, the average stray field from the chain is equal to zero. The configurable sources of magnetic field on the basis of such chains of particles were used by us for controlling critical current of edge-type Josephson junctions [71 73] and spin-dependent transport current in microbridges prepared from the magnetic semiconductor Ga 1 x Mn x As [14, 74]. (а) Fig. 10. Magnetization reversal of a chain of eight elliptic Co particles by the MFM probe: (a) MFM image of the chain in the initial uniformly magnetized state; MFM image of the same chain after the configuring of the state with antiferromagnetic ordered magnetic moments of the particles. The size of the MFM frames is μm. 5. TRANSITIONS BETWEEN THE HOMOGENEOUS AND VORTEX STATES OF MAGNETIZATION INDUCED BY THE MAGNETIC FIELD OF THE MFM PROBE IN ELLIPTIC NANOPARTICLES The state of magnetization of ferromagnetic nanodisks depends substantially on their geometric dimensions and shape [75 77]. Depending on the lateral sizes, thickness, and aspect ratio, the ground state of an elliptic disk can be quasi-uniform, weakly inhomogeneous (the so-called C or S states), or vortex. In a

9 CONTROL OF THE MAGNETIC STATE OF MASSIFS 9 (а) Fig. 11. MFM images of a massif of Co particles ( nm): (a) image taken in the tapping/lift mode; image of the same region taken in the constant-height mode. In the image (a), effects of the magnetization reversal by the field of the MFM probe are seen well. The scanning was performed in the direction parallel to the longer axis of particles (horizontal axis in the figure). The size of the MFM frames is 4 4 μm. number of works [17, 78, 79], it was experimentally shown that either states with a homogeneous magnetization or vortex states can be realized in elliptic disks of submicron size depending on their geometric parameters. However, the change in the type of the state of magnetization depending on the particle size does not have a nature of a sharp phase transition with a clear boundary that divides different states. There are ranges of geometric parameters, in which the free energy of a particle has several, rather than one, minima, which correspond to different configurations of magnetization, so that a ferromagnetic particle can possess several metastable states. This circumstance was apparently for the first time noted in our work [21]. The investigations conducted showed that in the ferromagnetic Co particles that have the shape of elliptic disks with lateral sizes of nm in the range of thicknesses of nm there are realized two metastable states either a uniform distribution of magnetization or a distribution in the form of a magnetic vortex. With an increase of the thickness, only a vortex state is realized in these particles, and for the particles with smaller thicknesses, a homogeneous state of magnetization is characteristic. The presence of this region of metastability served as a basis for the idea of controlling the direction of the vorticity of a magnetic vortex with the aid of an inhomogeneous field of the MFM probe [24, 26]. For this purpose, studies of transitions between the uniform and vortex states of magnetization induced by the MFM probe were carried out. In the experiments, elliptical Co particles with dimensions of nm and a thickness of 27 nm were used. The scanning of the massif of particles in the two-way (tapping/lift) regime showed that the formation of an MFM image is accompanied by the effects of magnetization reversal of particles by the field of the probe. Figure 11 presents MFM images of four particles obtained in the two-way (tapping/lift) mode (Fig. 11a) and in the single-pass (constant-height) regime (Fig. 11b). It is seen from Fig. 11a that in the tapping/lift mode in the process of scanning there occur changes in the magnetic state of particles accompanied by abrupt changes in the strength and nature of the MFM contrast. The undisturbed state of the same particles corresponded to a vortex state of magnetization (Fig. 11b). Figure 12 displays an increased MFM image of the particle obtained in the tapping/lift mode. The scanning was performed from the left to the right and from below upward. First, upon the scanning of the particle in the region A (Fig. 12), the MFM image has a low contrast. Then, on the boundary between the regions A and B, there occurs an abrupt change in the contrast, which attests to the fact that the particle converts into a magnetic state with a homogeneous magnetization (region B). Finally, at the boundary of the regions B and C, a sharp reduction in the contrast occurs. The process of the magnetization reversal in a particle can be interpreted as follows. At a certain time moment during the scanning above the central part of the particle (at the boundary AB), there occurs a transition from the initial vortex state (in Fig. 12, this corresponds to a counterclockwise magnetic vortex) into the state with a homogeneous magnetization. During a further scanning, the particle again converts (at the boundary BC) from the state with a homogeneous magnetization to a state (in Fig. 12, this corresponds to the transition into the state with an opposite vorticity of the magnetic vortex). The amplitude of the MFM response from the vortex state of magnetization is several times less than the value of the MFM response from the uniform state of the magnetization of the particle, which explains the jumps of magnitude of the MFM contrast upon the magnetization reversal. The transitions from the vortex to the homogeneous state and vice versa are accompanied by a com-

10 10 MIRONOV et al. μm C B 0.4 A μm Fig. 12. MFM image (tapping/lift mode) of an elliptic Co disk. Sharp changes in the MFM contrast are seen, which accompany the magnetization reversal of the particle. On the right-hand side, there are given MFM images of an elliptic particle corresponding to the tentative states of the magnetization in the regions A, B, and C. The direction of scanning coincides with the horizontal axis of the figure. (a) (d) Fig. 13. Sequential stages of the process of transition of a Co particle from a vortex state into a state with a homogeneous magnetization upon the movement of the MFM probe over the central region of the particle. The probe is designated by a gray dot; the direction of the probe motion is shown by arrows. plex rearrangement of magnetization inside the particles. Figure 13 gives the results of the LLG simulation of the transition between the vortex and uniform states of magnetization. As the initial state, the vortex distribution of magnetization was selected (Fig. 13a). In the simulation experiment, the MFM probe (the vector of the magnetic moment of the probe is directed upward, perpendicular to the plane of the figure) moved from the left to the right along the central region of the particle along its long axis. At the moment when the MFM probe approaches the boundary of the particle, the magnetization distribution undergoes a strong disturbance; in this case, the core of the vortex begins to be displaced from the center of the particle to its edge (Fig. 13b). During the further motion of the probe, the core is completely pushed out from the particle

11 CONTROL OF THE MAGNETIC STATE OF MASSIFS 11 (a) (d) (e) (f) Fig. 14. Sequential stages of the process of magnetization reversal of a particle upon the movement of the MFM probe over the edge of the particle. The probe is designated by a gray dot; the direction of the probe motion is shown by arrows. (Fig. 13c), and the magnetization inside the particle is mainly aligned along the force lines of the field of the probe. Finally, as a result of the passage of the probe above the particle, there is formed a homogeneous state of magnetization in it (Fig. 13d). The MFM-probe-induced transition from the state with a uniform magnetization into a vortex state occurs according to quite a different scenario. In this case, we simulated a situation in which the probe moved along the edge of the uniformly magnetized particle (Fig. 14) along the long axis of the ellipsis. At the initial moment, the magnetization of the particle has a uniform distribution (Fig. 14a). At the moment when the MFM probe (it is shown in the figure by a solid circle; the magnetic moment of the probe is directed upward, perpendicular to the plane of the figure) approaches the edge of the particle, the nearest vectors of magnetization are aligned along the force lines of the probe field (Fig. 14b), creating a vorticity in the nearest region of the particle. During the further motion of the probe, the magnetization in the particle undergoes a strong disturbance and becomes oriented along the edge of the particle, thereby forming a nucleus of a vortex state (Fig. 14c). At a certain time moment, in the fold of the magnetization near the probe there is formed a magnetic vortex (Fig. 14d), which subsequently is displaced to the center of the particle. As a result, at the end of the process of scanning there is realized a vortex state of magnetization in the particle. In the case examined in Fig. 14 the result of the passage of the MFM probe along the bottom edge of the uniformly magnetized particle is a vortex state with a clockwise direction of the vorticity. It is obvious that if the probe moved along the upper edge of the particle, then the result would be a vortex state with a counterclockwise vorticity. In practice, such MFM-probeinduced controlled transitions from the state with a uniform magnetization into the vortex state were used by us for controlling the sense of the vorticity of the shell of a magnetic vortex in elliptic ferromagnetic particles. The idea of experiments on changing the sign of the vorticity of the shell of a magnetic vortex with the aid of an MFM probe consisted in the following. The change of the sense of vorticity can be produced by means of a two-stage process, in which first there is produced a transition from the vortex state with a cer-

12 12 MIRONOV et al. y x 2 1 Defect particle (а) BC + OC BC Fig. 15. MFM images of one and the same region of a massif of Co particles taken in the process of magnetization reversal of a magnetic vortex by the MFM probe: (a) initial state (the central particle in the state BC + ); MFM images of the same particles taken in the process of scattering of the sample with a variable height; final state (particle in the state BC ). Size of MFM frames: 3 3 μm. tain direction of vorticity into the uniform state, and then, by means of an asymmetrical scanning by an MFM probe, there is initiated a transition from the uniform state to the vortex state with an opposite direction of the vorticity of the shell of the magnetic vortex. The experiments were conducted using Co particles with sizes of nm. The sequential stages of the process of changing the vorticity of the magnetic vortex under the action of the field of the MFM probe are presented in Fig. 15. In the initial state, the central particle of the massif (Fig. 15a) is in a state BC + (with the right-hand vorticity of the shell of the elliptic vortex). The image in Fig. 15a was obtained in the noncontact (constant-height mode) regime with the height of scanning h s = 50 nm. The second MFM image (Fig. 15b) was obtained as follows. At first, the scanning of the region of the massif was accomplished at a height h s = 50 nm. Then, as soon as the probe proved to be located above the central particle, the distance between the probe and the 50 nm 25 nm h s 1 2 Fig. 16. Variation of the height of scanning in the process of magnetization reversal of a magnetic vortex by the MFM probe (schematic). The changes in the height in regions 1 and 2 correspond to lines 1 and 2 in Fig. 15b. y sample h s was manually reduced to 15 nm (see scheme in Fig. 16). With the decrease in h s, there was detected a transition from the vortex state into the state with a homogeneous magnetization (BC + OC) accompanied by a sharp increase in the MFM contrast, after which the probe was raised by hand to h s = 50 nm. Then, upon scanning in the region between lines 1 and 2, there was observed an MFM contrast that corresponded to the state with a homogeneous magnetization (Fig. 15b). Then, on line 2, the probe was again lowered to the height H s = 15 nm and, after the transition OC BC was fixed, the probe again was raised to a height h s = 50 nm. After conducting this procedure, this region was repeatedly scanned in the noncontact (constant height mode) regime (Fig. 15c). As can be seen from Fig. 15, in the process of the magnetization reversal the central particle changed the direction of the vorticity of the vortex shell to the opposite. We used the above algorithm of the MFM-induced magnetization reversal OC BC for creating built-in configurable sources of a strongly inhomogeneous field on the basis of massifs of elliptic ferromagnetic particles. As an example, Fig. 17 shows the results of configuring a two-dimensional array of elliptic Co particles by the MFM probe. The massif consisted of elliptic Co disks with lateral sizes of nm and a height of 25 nm. Preliminarily, the sample was inplane magnetized in an external field of 10 koe, so that all the particles proved to be uniformly magnetized along the long axis (Fig. 17a). Then, with the aid of the MFM probe, part of the disks in the center of the massif was converted into the vortex state (Fig. 17b). Thus, in fact the field in the surrounding space on the modified part of the massif was turned off, since the stray fields from the particles in the vortex state are substantially less than from the uniformly magnetized particles. The configurable sources of an inhomogeneous magnetic field on the basis of massifs of particles were

13 used by us for controlling critical current in planar Josephson junctions [72]. CONTROL OF THE MAGNETIC STATE OF MASSIFS MFM-PROBE-INDUCED MAGNETIZATION REVERSAL OF CoPt NANODISKS WITH A PERPENDICULAR ANISOTROPY In [31], we studied some specific features of the local magnetization reversal of circular CoPt disks with a perpendicular anisotropy under the action of the field of an MFM probe. As was shown in Section 2, the characteristic spatial scale of stray fields of an MFM probe coincides on the order of magnitude with its effective diameter D t. Therefore, it was a priori expected that the processes of the magnetization reversal of disks with a diameter D p > D t and of disks with D p < D t must differ substantially. A comparative analysis of experimental and model MFM images and the simulation of the effects of magnetization reversal show that the characteristic effective diameter of MFM probes we used is D t = nm. Correspondingly, we used in the experiments two massifs of CoPt nanodisks, with a diameter D d = 200 nm (D d > D t ) and D d = 35 nm (D d < D t ). The massifs of nanodisks were fabricated by the methods of electronic lithography and ion etching of multilayer thin-film CoPt structures with a perpendicular anisotropy which were prepared at the University of Nebraska Lincoln, USA [31, 81, 82]. Two massifs of particles were prepared, which have the form of circular disks with a diameter of 35 and 200 nm. The spatial periods in the arrangement of particles in the massifs were 120 and 500 nm, respectively. In the first series of experiments, we investigated the magnetization reversal of circular disks with a diameter of 200 nm. Preliminarily, the sample was magnetized along the z axis (perpendicular to plane) in the external field of 15 koe, so that the magnetic moments of all particles in the initial state were directed oppositely with respect to the magnetic moment of the probe (in this case, the particles in the MFM image exhibit a bright contrast). The MFM images were recorded in the noncontact constant height regime; the height of scanning was 50 nm. The MFM-assisted recording of information was achieved during the scanning with a decrease in the spacing between the selected particle and the probe. In this case, the magnetization reversal of the particle was accompanied by a change in the contrast from bright to dark. Figure 18 presents MFM images of the sequential stages of the process of information recording in some region of the massif of CoPt particles. First, three particles located in one row (Fig. 18a) were subjected to magnetization reversal. Then, three following particles were inverted, arranged above the preceding ones (Fig. 18b). And finally, the magnetic (а) moments of the particles located along the edges were inverted (Figs. 18c, 18d), so that, as a result, a region in the form of a square was recorded on the massif of nanoparticles. However, as was shown in experiments, the magnetization reversal of these particles possesses some specific features. We did not succeed in carrying out the magnetization reversal of disks with a diameter of 200 nm by means of a single touch by the probe. The inversion of the magnetization occurred only during the traverse of the central part of the disk with the MFM probe. The details of this procedure of magnetization reversal are shown in Fig. 19. The characteristic moment of the magnetization reversal is shown in Fig. 19b. The distance between the probe and the particle was decreased during the scanning above the central part of the particle (line AB in Fig. 19b), and after a sharp inversion of the MFM contrast the probe was moved away from the surface of the sample to the initial position (50 nm). After this, the same part of the sample was scanned again to confirm the act of magnetization reversal (Fig. 19c). Such features of the process of the magnetization reversal of disks with R d > R c were also confirmed by the micromagnetic LLG simulation. First, it was shown that as the spherical probe (with a diameter of 100 nm) approaches the particle, a nucleation of microdomains with an opposite magnetization is observed inside the particle; however, the field of the probe is insufficient for the magnetization reversal via a single touchdown. The results of the LLG calculations of the process of touching a CoPt nanodisk by an MFM probe are presented in Fig. 20. During the first stage (Fig. 20a), there appears a radial component of magnetization caused by the radial component of the probe field. Then, in an annular region (Fig. 20b) there occurs a nucleation of a Fig. 17. Magnetization reversal of part of a massif of elliptic Co disks by the MFM probe: (a) MFM image of the massif in the initial uniformly magnetized state; MFM image of the same massif after OC BC magnetization reversal of a region in the center. Size of the MFM frames: μm.

14 14 MIRONOV et al. µm (a) µm µm µm µm µm (d) µm Fig. 18. (a) (d) Sequential MFM images of one and the same part of the massif of CoPt disks in the process of recording of information by the field of the MFM probe. Size of the MFM frames: 4 4 μm. A (а) Fig. 19. Magnetization reversal of a single CoPt disk by the MFM probe moving across the particle: (a) MFM image (constant-height mode) of the initial state of part of the massif with two noninverted particles; MFM image taken in the regime of scanning with a small height along the line AB (inversion of contrast is observed); final state of the same region after the MFM recording. Size of the MFM frames: 3 4 μm. B microdomain with an opposite orientation of the z component of the magnetization through a quasivortex state with a spiral distribution of magnetization. Then, the microdomain is enlarged and covers a region with a diameter on the order of 150 nm (Figs. 20c, 20d). This state is stable. After the removal of the probe, the magnetization returns to the initial state (Figs. 20e, 20f). The stability of the disk with a diameter of 200 nm is explained, apparently, by edge effects, since the propagation of a domain wall is accompanied by a significant increase in the magnetostatic energy at the edge of the disk. We also conducted an LLG simulation of the process of magnetization reversal in a 200-nm CoPt disk by moving the probe above the central region of the disk. The sequential stages of the model process are displayed in Fig. 21. The initial state (Fig. 21a) corresponds to a homogeneous magnetization directed along the z axis. During the first stage, a spherical MFM probe (with a diameter of 100 nm) is placed near the edge of the particle (Fig. 21b) and part of the magnetization changes its direction to the opposite. In fact, the strong field of the probe helps overcoming the barrier for the magnetization reversal connected with the edge of disk. Further, the probe moves through the particle and a wave of magnetization reversal propagates through the particle (Figs. 21c, 21d). At the final stage, when the probe passes over the middle of the particle, the process of magnetization reversal reaches the opposite edge of the disk (Fig. 21e) and then the

15 CONTROL OF THE MAGNETIC STATE OF MASSIFS 15 (a) (d) (e) (f) Fig. 20. Sequential stages of the process of magnetization reversal of CoPt particle with a diameter of 200 nm under the effect of the field of the MFM probe (a spherical probe (D t = 100 nm) is located over the center of the particle): (a d) magnetization distributions at various time moments in the course of magnetization reversal. The stable state that is realized in this system is shown in Fig. 20d. After the probe is raised upward, there occurs a relaxation of the magnetization distribution to the initial state (e f). The direction of the magnetization if the domains is shown by arrows. The boundaries of domains with opposite magnetizations are shown by dotted lines. disk completely changes the direction of its magnetization (Fig. 21f). In the second series of experiments, we investigated the magnetization reversal of CoPt particles with a diameter of 35 nm (D d < D t ). The period of the structure was 120 nm. In the initial state all particles were also magnetized perpendicular to the sample plane in the direction opposite to the magnetization direction of the probe. The magnetization reversal of the chosen disks was produced by touching their surface by the MFM probe, so that the record of information was achieved by means of a single contact. The sequential stages of the process of the magnetization reversal of the massif of particles are shown in Fig. 22. The magnetization reversal of particles was accompanied by the inversion of the MFM contrast. As a result, there was demonstrated a controlled selective magnetization reversal of single elements of the massif, which confirms the possibility of recording information by the MFM probe with a density at a level of 40 Gbit/in 2. The mechanism of magnetization reversal of particles with a perpendicular anisotropy in an external uniform field is discussed in many works. The most common models are the Stoner Wohlfarth model [83 85] (which assumes that the magnetization reversal occurs by means of a coherent rotation of the magnetization of the particle) and the model of the forma-

16 16 MIRONOV et al. (a) (d) (e) (f) Fig. 21. Sequential stages of the process of magnetization reversal upon the movement of the MFM probe over the central part of the CoPt disk with D d > D t (D d = 200 nm; D t = 100 nm). The position of the center of the spherical probe is shown by a dark circle. The directions of the magnetizations in the domains are shown by arrows. The boundaries of domains with opposite magnetizations are shown by dotted lines. tion of nuclei of a new phase (inverted domains) with an opposite magnetization [85 87]. However, the magnetization reversal of particles in a strongly inhomogeneous field of an MFM probe has a number of specific features. There was also carried out a computer simulation of the process of magnetization reversal of a nanodisk with a diameter of 35 nm under the action of the field of an MFM probe, which was represented in the form of a uniformly magnetized Co sphere with a diameter of 60 nm. The results of the simulation of the process of magnetization reversal of a disk in the field of an MFM probe are represented in Fig. 23. As can be seen from Fig. 23, the magnetization reversal of the nanodisk occurs through a substantially inhomogeneous vortex-like state. At first (Fig. 23a), the magnetization is directed upward along the z axis. In the first stage, a twisted state is formed (Fig. 23b). Then, a narrow annular region near the maximum of the radial component of the probe field changes the direction of the z component of magnetization (Fig. 23c). Finally, this nucleus with the opposite z component propagates to the entire particle (Figs. 23d 23f). Besides the studying of the dynamics of magnetization reversal, in the numerical LLG calculations we also determined the average magnetic energy of the disk in order to compare the barriers for the magnetization reversal in a homogeneous magnetic field (ΔE hom ) and in an inhomogeneous field of the tip of an

17 CONTROL OF THE MAGNETIC STATE OF MASSIFS 17 MFM probe (ΔE tip ). The LLG simulation showed that the magnetization reversal of CoPt disks with a diameter of 35 nm in a uniform field occurs by means of a coherent rotation of magnetization. In the calculations, we increased the amplitude of the external field until the process of the magnetization reversal began and at this moment determined the critical field and value of the magnetic energy in the critical state E hom *. The value of the barrier for the magnetization reversal was calculated as the difference between the energy in the critical state * and the energy in the initial state E hom (а) ΔE hom = E hom * E 0. (22) On the other hand, we also calculated the energy barrier for the magnetization reversal of CoPt nanodisks with a diameter of 35 nm in the inhomogeneous field of a spherical MFM probe with an effective radius of 30 nm. The radius of the probe tip R t was constant in these model calculations, and the value of the magnetic moment of the probe increased only due to an increase in the parameter of magnetization at saturation M t, so that the geometry of this numerical experiment did not change. At the moment of the beginning of the magnetization reversal, the value of the energy in the critical state E tip * was determined. The value of the barrier for the magnetization reversal in the inhomogeneous field of the probe was calculated as above: ΔE tip = E tip * E 0. (23) The numerical estimations based on LLG calculations showed that, first, the particle-averaged z component of the critical field of the probe, which causes the magnetization reversal of the disk, is less than an analogous critical homogeneous field and, second, the energy barrier for the magnetization reversal of a nanodisk in the critical field of an MFM probe is substantially lower than the barrier for its magnetization reversal in an external critical uniform field. Thus, for instance, for a 35-nm CoPt particle with an anisotropy constant K = erg/cm 3 the energy barrier for the magnetization reversal in the field of an MFM probe was ΔE = erg, while the barrier for the magnetization reversal in a homogeneous field was ΔE = erg. 7. MFM-PROBE-INDUCED MAGNETIZATION REVERSAL OF CROSS-SHAPED PARTICLES The interest in the magnetic states of cross-shaped ferromagnetic particles is caused by several aspects. First, the complex shape of particles leads to the formation of strongly inhomogeneous states. Second, in such particles there is realized a so-called antivortex distribution of magnetization [65 67, 88], which is manifested in an uncommon spin dynamics upon (d) Fig. 22. Sequential MFM images of one and the same part of a massif of particles in the process of a local magnetization reversal by the field of the probe: (a) MFM image of the first inverted particle; MFM image of two inverted particles; image of three inverted particles; (d) image of the region of the MFM recording in the form of regularly arranged particles with inverted magnetic moments. Size of the frame: nm. magnetization reversal [89 92] and can manifest itself in the topological Hall effect [93]. The massifs of cross-shaped Co particles of symmetrical and asymmetrical form were prepared by electronic lithography and ion (Ar + ) etching of Co layers on a silicon substrate. Characteristic SEM images of such massifs are shown in Fig. 24. The magnetic states were studied by the method of magnetic-force microscopy in a Solver HV vacuum microscope. A magnetic antivortex is a metastable state [4], which, however, can be stabilized via an optimum selection of geometric parameters, such as the length, width, and thickness of the branches of the cross. Depending on the aspect ratio (ratio of the length of a branch to its width), different states of magnetization can be realized in the particles. Thus, in the particles with a small aspect ratio g = a/b (g > 1, see Fig. 25a), quasi-vortex states of magnetization arise. Figure 25 displays a model distribution of magnetization and corresponding model and experimental MFM images of a magnetic vortex in a cross-shaped particle. With an increase in the aspect ratio, there are realized quasi-homogeneous distributions of the magnetization of two types, A and B in the particle. In the state A, in the branches of the cross there is realized a distribution with two incoming (outgoing) and two outgoing (incoming) directions of magnetization (Fig. 26a); in the state B, distributions with three incoming (outgoing) and one outgoing (incoming) directions of mag-

18 18 MIRONOV et al. (a) (d) (e) (f) Fig. 23. Calculated (LLG) magnetization distributions at various stages of the magnetization reversal of a CoPt nanodisk with a diameter of 35 nm under the effect of the magnetic field of the MFM probe. The probe is located directly over the center of the particle and touches its surface. The direction of the magnetization in the domains is shown by arrows. The boundaries of domains with opposite magnetizations are marked by a dotted line. (а) Fig. 24. Electron-microscopic images of massifs of crossshaped particles of (a) symmetrical and asymmetrical shape. netization arise (Fig. 27a). The model distributions of magnetization and corresponding model and experimental MFM images of symmetric crosses with lateral dimensions a = 600 nm and b = 100 nm (g = 6) are shown in Figs. 26 and 27. For the realization of an antivortex state, we prepared cross-shaped particles of asymmetrical form (Fig. 25b), in which two branches had sharpened ends, which increased the coercivity of the branches, and two other had thickenings, which reduced the coercivity of the branches. The lateral sizes were a = 1 μm, b = 100 nm (g = 10); the width of the part with the thickening was 150 nm; the thickness of the particle, 40 nm. The antivortex state was realized in such particles in the process of magnetization reversal in an external homogeneous field due to different coercivity of the

19 CONTROL OF THE MAGNETIC STATE OF MASSIFS 19 b a Fig. 25. Magnetic vortex in a cross-shaped particle: (a) model quasi-vortex distribution of magnetization; model distribution of MFM contrast from the magnetic vortex (Fig. 1a); experimental MFM image of a cross with lateral dimensions a = 600 nm and b = 200 nm (g = 3). Thickness of the particle: 40 nm. (а) Fig. 26. Quasi-homogeneous state A in a cross-shaped particle: (a) model distribution of magnetization; model distribution of MFM contrast; experimental MFM image of a cross with lateral dimensions a = 600 nm and b = 100 nm (g = 6). Thickness of the particle: 40 nm. (а) Fig. 27. Quasi-homogeneous state of type B in a cross-shaped particle: (a) model distribution of magnetization; model distribution of the MFM contrast; experimental MFM image of a cross with lateral dimensions a = 600 and 100 nm (g = 6). Thickness of the particle: 40 nm. branches of the cross [4]. The model distributions of magnetization and the corresponding model and experimental MFM images of a magnetic antivortex in the asymmetric Co crosses are presented in Fig. 28. We carried out experiments on the magnetization reversal of cross-shaped particles under the action of the field of the MFM probe. We examined both processes connected with the simple reorientation of a quasi-homogeneous state and processes of the transitions between different states with a quasi-homogeneous magnetization and processes of the formation of an antivortex state. The results of the experiments on the reorientation of a quasi-homogeneous state of type A are given in Fig. 29. At first, the sample was magnetized in an external magnetic field of ~1 kg applied along the diagonal of the cross, so that all the particles had a state with a quasi-homogeneous magnetization of type A (Fig. 29a). Then, by scanning at a low height

20 20 MIRONOV et al. (а) Fig. 28. Antivortex state in a cross-shaped particle: (a) model distribution of magnetization; model dist of the MFM contrast; experimental MFM image of an asymmetrical cross with lateral dimensions a = 1 μm and b = 100 nm (g = 10). Thickness of the particle: 40 nm. (а) (d) Fig. 29. Sequential stages of the magnetization reversal of a cross-shaped particle in a homogeneous state A by the MFM probe: (a) initial state; the central particle changed its magnetization direction by 90 ; the central particle changed the magnetization direction by 180 ; (d) the central particle changed the magnetization direction by 270 (clockwise). The regions of scanning upon magnetization reversal are shown by dotted lines. above the lower part of the cross, the particle was transformed into a quasi-homogeneous state with a magnetization rotated relative to the initial state by an angle of 90 (Fig. 29b). Then, by scanning along the vertical axis in the left-hand region of the cross (Fig. 29b), the particle was transformed into the state with the magnetization that was rotated by 90 more (Fig. 29c); and further, by scanning along the upper part of the cross, the magnetization was rotated once again by 90 (Fig. 29d). Thus, it was experimentally shown that with the aid of an MFM probe it is possible to perform a controlled reorientation of the magnetization distribution in cross-shaped particles. To clarify the mechanism of a reorientation of a quasi-homogeneous state under the action of the inhomogeneous magnetic field of the MFM probe, we carried out a micromagnetic simulation of the process of interaction of the probe with a cross-shaped particle. The simulation was carried out with the aid of a SIMMAG program package. The calculations showed that the magnetization reversal of particles during the scanning is accompanied by a complex inhomogeneous rearrangement of the magnetization distribution inside the particle. The sequential stages of the process of magnetization reversal during the motion of the MFM probe along the bottom edge of the particle (corresponding to the transition between the states shown in Figs. 29a and 29b) are given in Fig. 30. It is seen well that in the process of the probe passage the left-hand upper branch of the cross does not change the magnetization direction, whereas in all the other branches the magnetization direction changes to the opposite.

21 CONTROL OF THE MAGNETIC STATE OF MASSIFS 21 (a) (d) (e) (f) Fig. 30. Sequential model distributions of magnetization corresponding to the process of a reorientation of the quasi-homogeneous state of type A upon scanning by the MFM probe. The position of the probe is shown by a circle; the direction of the probe motion is shown by an arrow. In the second experiment, we studied the processes of the formation of an antivortex state under the action of the field of the MFM probe. We employed a twostage process, in which, first, the particle was transformed, by means of the local scanning, from the quasi-homogeneous state of type A (Fig. 31a) into a quasi-uniform state of type B (Fig. 31b) and then into an antivortex state (Fig. 31c). Figures 32 and 33 display the results of a computer simulation of the rearrangement of the magnetization distribution, which correspond to the processes of transitions from the quasi-homogeneous state of type A into the quasi-homogeneous state of type B (Fig. 32) and also from the quasi-homogeneous state of type B into the antivortex state (Fig. 33). In both cases the transitions of the particle from one state to another are the result of disturbing actions of the field of the probe during its motion along a certain trajectory above the particle. Using the procedure of a two-stage magnetization reversal, it is possible to form antivortex states of different symmetry in cross-shaped particles. To exemplify, Fig. 34 shows the results of experiments on the formation of magnetic antivortices whose magne-

22 22 MIRONOV et al. (а) (с) Fig. 31. MFM images of sequential stages of the formation of a magnetic antivortex: (a) initial homogeneous state of type A; resultant homogeneous state of type B; final antivortex state. The trajectory of the motion of the probe over the corresponding pole of the cross in the process of switching is shown by a dotted line. (a) (d) (e) (f) Fig. 32. Sequential model distributions of magnetization corresponding to the transition from the quasi-homogeneous state of type A into the quasi-homogeneous state of type B under the effect of the field of the MFM probe. The position of the probe is shown by a circle; the direction of the motion of the probe is shown by an arrow.

23 CONTROL OF THE MAGNETIC STATE OF MASSIFS 23 (a) (d) Fig. 33. Sequential model distributions of magnetization corresponding to the transition from the quasi-homogeneous state of type B into the antivortex state under the effect of the field of the MFM probe. The position of the probe is shown by a circle; the direction of the motion of the probe is shown by an arrow. tization distributions are rotated by an angle of 90 relative to each other. 8. CONCLUSIONS Thus, in this work we presented a survey of the results of studies of the processes of magnetization reversal in ferromagnetic nanoparticles under the action of the field of the probe of a magnetic-force microscope (MFM). Theoretically, in the approximation of weak field, we examined the effects of the disturbance of the magnetization distribution in objects induced by the magnetic field of the MFM probe, which influence the phase contrast in MFM measurements. It was shown that the basic contribution to the MFM contrast came from an additive connected with the θ component of the disturbance in the distribution of the magnetization of the sample. Procedures of a selective magnetization reversal of ferromagnetic nanoparticles in various massifs have been developed based on a redistribution of their magnetization in the local inhomogeneous field of the MFM probe. The possibilities were demonstrated of creating spatially inhomogeneous states in massifs of nanoparticles by means of probe-induced changes in the orientation of magnetic moments of single particles, and also by means of the MFM-probe-induced transformation of single particles into vortex states, which do not create stray fields. The techniques developed make it possible to realize configurable sources of strongly inhomogeneous magnetic fields. There are presented results of experiments on the MFM-assisted recording of information on massifs of CoPt nanodisks with a perpendicular magnetic anisot- (а) Fig. 34. Formation of antivortex states under the effect of the field of the MFM probe: (a) initial state; antivortex states formed by the MFM probe. In the central cross, there was formed a magnetic antivortex rotated by an angle of 90 relative to the other particles.