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The response of the magnetic nanoparticles lattice to a Gaussian magnetic field pulse
Superlattices and Microstructures 132 (2019) 106158
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The response of the magnetic nanoparticles lattice to a Gaussian magnetic field pulse Anatolij M. Shutyi, Svetlana V. Eliseeva ∗, Dmitrij I. Sementsov Department of High Technology Physics and Engineering, Ulyanovsk State University, Lev Tolstoy 42, 432700, Ulyanovsk, Russian Federation
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Keywords: Lattices of magnetic dipoles Magnetic moment Uniaxial anisotropy Magnetization reversal Magnetic field pulse System response Homogeneous and binary lattices
ABSTRACT The paper investigates the response of homogeneous and binary planar lattices of magnetic nanoparticles with uniaxial anisotropy to the action of a short Gaussian magnetic field pulse. It is shown that the influence of the dipole–dipole interaction and the presence of two types of nanoparticles leads to modulation of the response amplitude and its decrease. A complex periodic dependence of the amplitude and duration of the system response on the duration of the pulse has been revealed. The research shows the possibility of a pulse-induced magnetization reversal of the entire lattice or its individual parts upon defining the lattice composition and the parameters of the bias field and the effective pulse.
1. Introduction The attention of researchers to nanoparticles and ordered structures on their basis is due to both their unusual physical and chemical properties, and their fast-paced introduction into modern nanotechnologies [1]. Magnetic regular structures of various dimensions formed on basis of ferromagnetic metal nanoparticles [2,3] represent a particular interest. Due to their small size, magnetic nanoparticles are generally single-domain, which makes it possible to simplify the description of structures based on them. However, the description of the dynamic properties of magnetic nanostructures is very difficult, since the interaction between the particles of the system is determined not only by their local ordering of the particles, but also by their own magnetic moment, magnetic anisotropy, coercivity, scattering fields [4–6]. Nevertheless, in recent decades a considerable progress has been made in understanding the properties and processes in magnetic nanoparticles systems [7–11]. In magnetic lattices created artificially by nanotechnologies the main contribution to the interaction of the magnetic moments of nanoparticles is made by the dipole–dipole interaction. It was shown in [12–14] that the discreteness of the structures leads to essential differences of the equilibrium states and the dynamic regimes arising in the process of magnetization reversal of linear chains and square lattices of nanoparticles from the properties of macroscopic single-domain objects. These differences, for example, may include bistable states of the lattices due to the presence of various orientational configurations with unequal total magnetic moment. Regular ensembles of magnetic nanoparticles of various dimensions can serve as a medium for super-dense recording and storage of information. The possibility of recording information on a magnetic dipole lattice is based on a change in the equilibrium configuration of magnetic moments due to radio pulses of a magnetic field, and the reading is provided by excitation of the resulting configuration by a low-power radio pulse at the ferromagnetic resonance frequency and by scanning the response frequency of the dipole system [15,16].
∗ Corresponding author. E-mail address: [email protected] (S.V. Eliseeva).
https://doi.org/10.1016/j.spmi.2019.106158 Received 6 August 2018; Received in revised form 28 March 2019; Accepted 11 June 2019 Available online 12 June 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 132 (2019) 106158
A.M. Shutyi et al.
The increase in the functional efficiency of magnetic nanostructures requires systematic studies both to optimize their synthesis and to clarify the specific features of magnetism in systems of magnetic nanoparticles. In this paper, the response of a planar lattice 6 × 6 of magnetic nanoparticles to a magnetic field pulse in the presence of uniaxial anisotropy in nanoparticles is studied on the basis of a numerical solution of dynamic equations. This type of anisotropy is the most common for nanoparticles, and also important for information recording systems. Both lattices with one kind of nanoparticles (i.e., homogeneous lattices) and lattices with two alternating types of nanoparticles differing in the magnitude of the magnetic moment (binary lattices) are being studied. Previously unknown features of the dependence of oscillations amplitude of the magnetic moments and the duration of the lattice response from the Gaussian pulse duration have been revealed. The processes of complete and partial magnetization reversal of homogeneous and binary lattices under the magnetic field pulse are studied. 2. Initial equations Let us consider a plane dipole lattice 6 × 6 consisting of single-domain spherical nanoparticles with a magnetic moment |𝐦𝑖 | = 𝑚 and a uniaxial magnetic anisotropy with a constant 𝐾𝑢 . We assume that the lattice is square with the distance between the centers of the nearest nanoparticles (lattice parameter) 𝑟0 . The dynamics of each magnetic moment of the particles is described by the Hilbert equation, which has the following form for the dimensionless parameters [14,17]: 𝜕𝝁𝑖 𝜕𝝁𝑖 = −𝝁𝑖 × 𝐡𝑒𝑓 (1) 𝑖 − 𝛼𝝁𝑖 × 𝜕𝜏 , 𝜕𝜏 where 𝝁𝑖 = 𝐦𝑖 ∕𝑚, 𝛼 is a dimensionless dissipation parameter, 𝜏 = (𝐽 𝛾)𝑡 is dimensionless time, 𝛾 is a gyromagnetic ratio, 𝐽 = 𝑚∕𝑉0 is the magnetization of the material, 𝑅 and 𝑉0 = (4𝜋∕3)𝑅3 are the radius and volume of the nanoparticle. The effective magnetic field acting on a particle includes an external field, a uniaxial anisotropy field, a dipole–dipole interaction field, and a scattering field of a spherical particle: [ ] ∑ 3(𝝁𝑛 𝐞𝑖𝑛 )𝐞𝑖𝑛 − 𝝁𝑛 𝑒𝑓 𝐡𝑖 = 𝐡 + 𝑘𝑢 𝐧(𝝁𝐢 𝐧) + − 𝝁𝑖 . (2) 3 𝑙𝑖𝑛 𝑛≠𝑖 Here dimensionless parameters are also introduced 𝐞𝑖𝑛 = 𝐫𝑖𝑛 ∕𝑟𝑖𝑛 where 𝐫𝑖𝑛 and 𝑟𝑖𝑛 are the radius vector and distance between the 3 = 𝑟3 ∕𝑅, and the <> external field is 𝐡 = 𝐇∕𝐽 and the uniaxial anisotropy constant is given 𝑖th and 𝑛th dipoles, 𝑙𝑖𝑛 𝑖𝑛 by the expression 𝑘𝑢 = 𝐾𝑢 ∕𝐽 2 . The dimensionless lattice parameter used in numerical calculations is denoted as 𝑙0 = 𝑟0 ∕(𝑉0 )1∕3 . The vector equation (1) in the further analysis is represented by three scalar equations. Thus, the projection of this equation on the 𝑋 axis has the form: ( ) 𝑒𝑓 ( ) 𝑒𝑓 ) ( ) 𝜕𝜇𝑖𝑥 ( 2 (3) = 𝜇𝑖𝑧 + 𝛼𝜇𝑖𝑥 𝜇𝑖𝑦 ℎ𝑒𝑓 1 + 𝛼2 𝑖𝑦 − 𝜇𝑖𝑦 + 𝛼𝜇𝑖𝑧 𝜇𝑖𝑥 ℎ𝑖𝑧 − 𝛼 1 − 𝜇𝑖𝑥 ℎ𝑖𝑥 . 𝜕𝜏 The equations for the remaining projections have a similar form and can be obtained from (3) by cyclic permutation of the components. Let us consider the transition from dimensionless quantities to dimensional lattice of nanoparticles consisting of the 𝑁 atoms of iron: the magnetic moment of nanoparticles 𝑚 ≈ 2.2𝜇𝐵 𝑁, where 𝜇𝐵 is a Bohr magneton. Thus, for a stable spherical configuration of 𝑁 = 561 atoms, the radius of the nanoparticle is 𝑅 = 1.364 ⋅ 10−7 cm, and 𝑚 ≈ 1.145 ⋅ 10−17 erg/Oe (or 𝑚 ≈ 1.145 ⋅ 10−20 A m2 ). With these parameters, the magnetization of the nanoparticle material is 𝐽 ≃ 1078 G (or 𝐽 ≃ 8.58 ⋅ 104 A/m). Taking into consideration 𝛾 = 1.76 ⋅ 107 (Oe s)−1 we obtain the following numerical estimates: for time 𝑡 = (1∕𝛾𝐽 )𝜏 ≈ 5.27 ⋅ 10−11 𝜏 s, for frequencies 𝜔 = (𝛾𝐽 ) Ω ≃ 1.9 ⋅ 1010 Ω s−1 , for external field 𝐻 = 𝐽 ℎ ≃ 1.078 h kOe (or 𝐻 ≃ 85.81 h kA/m), for anisotropy constants 𝐾𝑢 = 𝐽 2 𝑘𝑢 ≈ 1.162 ⋅ 106 𝑘𝑢 erg/cm3 (or 𝐾𝑢 ≃ 1.162 ⋅ 105 𝑘𝑢 J/m3 ). Next, we will study the dynamics of the lattice 6 × 6 whose nanoparticles are either identical (homogeneous lattices) or represent two subsystems that are equal in number to the nanoparticles, but differ in their magnetic moment (binary lattices). In the latter case, nanoparticles of the same type alternate with nanoparticles of a different type. The coordinate system is chosen so that the 𝑋 axis is perpendicular to the plane of the lattice, and the other two axes are parallel to the sides of the lattice. The direction of the easy axis of the magnetic anisotropy coincides with the 𝑌 axis, the anisotropy constant is assumed to be equal to 𝑘𝑢 = 1 (or 𝐾𝑢 = 1.162 ⋅ 106 erg/cm3 which corresponds to the real value of the constant induced anisotropy in iron). The dissipation parameter is assumed to be equal to 𝛼 = 0.01. The connection of this parameter with the line width 𝛥𝐻 and the frequency of ferromagnetic resonance 𝐻𝑟𝑒𝑠 is given by the expression 𝛥𝐻 = 2𝛼𝐻𝑟𝑒𝑠 . For a spherical particle 𝐻𝑟𝑒𝑠 = 𝜔∕𝛾 and at 𝜔 = 2𝜋 ⋅ 1010 s−1 (10 GHz) equals 𝐻𝑟𝑒𝑠 ≃ 3.6 kOe, the width of the line can be 𝛥𝐻 ≈ (50−100) Oe, which determines that the selected value of the dissipation parameter. Equilibrium orientations and precessional dynamic regimes of the total magnetic moment of the entire lattice are determined on the basis of numerical analysis, which is carried out using the fourth-order Runge–Kutta method. 3. The magnetic moments response to the magnetic field pulse Consider the response of the lattice to the Gaussian pulse of the magnetic field: [ ] ℎ(𝜏) = ℎ0 exp −(𝜏 − 𝜏𝑖 )2 ∕2𝜏02 ,
(4)
where ℎ0 , 𝜏𝑖 𝜏0 are the peak value of the field, the time shift of the maximum of the pulse and its duration. For a homogeneous lattice, the value |𝜇𝑖 | is always equal to unity. Fig. 1 shows the time dependence of the 𝑥-component of the total magnetic moment 2
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∑ 𝑀 = 𝜇𝑖 of a homogeneous lattice under the action of a pulse with parameters ℎ0 = 10, 𝜏𝑖 = 200 and 𝜏0 = 1. The magnetic field of the pulse is linearly polarized in the case (a, b) along the 𝑋 axis and in the case (c) along the 𝑍 axis. The system in the direction of the 𝑌 axis is also affected by a constant magnetic field ℎ𝑦 = 0.1 (the effective magnetic field acting on the dipoles will be close to the value 0.1). The distance between the centers of the nearest nanoparticles is assumed to be equal to 𝑙0 = 𝑟0 ∕𝑅 = 10, 7 (a – thin and bold lines, respectively), 𝑙0 = 5, 2 (b, c – thin and bold lines, respectively). It can be seen that the effect of the dipole–dipole interaction, which is enhanced by the approach of nanoparticles, leads to modulation and a decrease in the response amplitude of the system. With the field pulse polarized along a perpendicular axis 𝑋, the influence of the dipole–dipole interaction to the dynamics of the magnetic moments becomes stronger. Next, we will only consider a lattice with a weak dipole–dipole interaction with 𝑙0 = 10. Fig. 2 shows the time dependence of the response of a binary lattice under the action of a magnetic field pulse, polarized along the 𝑋 axis and having the parameters indicated above (the pulse profile is shown in the inset of the figure). In the direction of the 𝑌 axis, the magnetic moments of the lattice are acted upon by a constant magnetic field of the value ℎ𝑦 = 1 (a–c) and ℎ𝑦 = 0.5 (d–f). Cases (a) and (d) correspond to a homogeneous lattice with |𝜇𝑖 | = 1. For a binary lattice the magnetic moments of nanoparticles in sublattices are different: |𝜇𝑖 | = 𝜇01 , 𝜇02 . In cases (b) and (e) is 𝜇01 = 1.1, 𝜇02 = 0.9, in cases (c) and (f) 𝜇01 = 1.2, 𝜇02 = 0.8. It can be seen that the presence of two types of nanoparticles with close but different magnetic moments leads to the appearance in the response of the system of an additional modulating frequency, the value of which increases with increasing difference between the magnetic moments of the two subsystems of dipoles. In this case, a decrease in the effective field acting on the magnetic dipoles leads to a decrease in the fundamental frequency of the system response and to an increase in the relaxation time of the precession dynamics of the magnetic moments. The dynamics of the precession motion of the magnetic moment after the action of the field pulse on the lattice can be represented by the projections of the trajectory of the magnetic moment on the coordinate planes. For the cases (a, b, d, e) considered in Figs. 2 and 3 (a, b, c, d, respectively) shows the projections of the trajectories of the total magnetic moment of the lattice on the plane 𝑋𝑍. It can be seen that for a given peak value of acting of the effective magnetic pulse, there is a short time interval of precession of the magnetic moments around the axis along which the field momentum is directed, after which occurs a transition to precession around the direction of the effective field. 4. Dependence of the response on the pulse duration The shape of the trajectory and the amplitude of the oscillations of the total magnetic moment of the lattice after the action of the field pulse depends in a complex manner on the parameters of both the lattice and the pulse, in particular, on its duration. Fig. 4 shows a diagram that determines the dependence on the pulse duration 𝜏0 of the extreme values of the 𝑥-component of the precessing magnetic moment of the system on the time interval 250 ≤ 𝜏 ≤ 280 after the action of the field pulse with parameters ℎ0 = 10, 𝜏𝑖 = 200. The number of points corresponding to each value 𝜏0 is approximately equal to the number of half-periods of damped oscillations of the magnetic moment, which fit in the taken time interval for a given value of the parameter 𝜏0 . The system is assumed to be homogeneous with |𝜇𝑖 | = 1, to which a constant magnetic field ℎ𝑦 = 1 is applied in the direction of the 𝑌 axis. It can be seen from the diagram that this dependence is of a complex character: alternation of the maximum and minimum oscillation amplitudes takes place. The maxima of the diagram are modified with increasing pulse duration. For 𝜏0 ≤ 0.6 they are separated by a drop in maxima. With a further increase in the pulse duration, the maxima of the diagram decrease. Fig. 5 shows the time dependence of the oscillation of the magnetic moment of the system (with the above lattice parameters and peak value field) under the action of a pulse length 𝜏0 = 0.1, 0.12 (a, b), which corresponds to the region of the drop in the diagram maxima. The oscillations are of a complex character with a frequency that is small at the beginning of the process and which increases strongly with time. The amplitude variation is nonmonotonic — modulation and time intervals of its sharp increase and decrease are observed. Fig. 6 shows the time dependence 𝑀𝑥 (𝜏) and the projection to the 𝑋𝑍 plane of the trajectory of the magnetic moment of the lattice after the pulse of duration 𝜏0 = 0.65, 0.8, 0.9 (a–c). These values 𝜏0 correspond to approximately to two maxima and one minimum in the diagram in Fig. 4. For values of the pulse duration corresponding to the maximums of the diagram, the response of the magnetic moments reaches the largest amplitudes and duration by time. In the case of the pulse duration, which corresponds to a minimum in the diagram, the trajectory of the oscillations of the magnetic moments turns out to be rapidly approaching the equilibrium state, as a result of which the amplitude of the oscillations does not reach large values, and the oscillations decay rapidly enough. 5. Lattice magnetization reversal under the action of the pulse Under the action of the magnetic field pulse, a partial or complete reversal of the dipole lattice can be performed. For its realization, it is necessary to approximate the equilibrium orientations of the magnetic moments by means of an external magnetizing field. As a result of the action of the magnetic anisotropy and the field oriented along the 𝑍 axis, the angle between the equilibrium orientations is in the interval 𝜋 < 𝜑 < 0 (depending on the magnitude of the field and the uniaxial anisotropy constant). First we consider a homogeneous lattice with magnetic moments of nanoparticles |𝜇𝑖 | = 1 to which a magnetizing field ℎ𝑧 = 0.5 is applied. Fig. 7 shows the time dependence of the two components of the total magnetic moment of the lattice upon its magnetization reversal under the action of a pulse with parameters 𝜏0 = 0.65, ℎ0 = 10, 𝜏𝑖 = 200. Three remagnetization processes are shown in the figure: the first (a, b) corresponds to the initial configuration (a) and final configuration (b); the second (c, d) corresponds to 3
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Fig. 1. The time dependence of the 𝑥-component of the magnetic moments of homogeneous lattice at a pulse action with ℎ0 = 10, 𝜏𝑖 = 200 and 𝜏0 = 1, polarized along the axes 𝑋 (a, b) and 𝑍 (c); a magnetizing field ℎ𝑦 = 0.1; the distance between the centers of the nearest nanoparticles 𝑙0 = 10, 7 (a – thin and bold lines), 𝑙0 = 5, 2 (b, c – thin and bold lines); dissipation parameter 𝛼 = 0.01.
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Fig. 2. The time dependence of the lattice response of the lattice with 𝑙0 = 10 under the action of the pulse (the profile of a pulse polarized along the 𝑋 axis is shown in the inset) and a magnetizing fields ℎ𝑦 = 1 (a–c) and ℎ𝑦 = 0.5 (d–f); the magnetic moments |𝜇𝑖 | = 1 for (a, d) and |𝜇𝑖 | = 𝜇01 , 𝜇02 , 𝜇01 = 1.1, 𝜇02 = 0.9 for (b, e), 𝜇01 = 1.2, 𝜇02 = 0.8 for (c, f).
the initial configuration (b) and the final configuration (d); the third (e, f) corresponds to the initial configuration (d) and the final configuration (f). The magnetization reversal of each individual dipole with these parameters has a random character, but after each 5
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Fig. 3. Projections of the dynamics trajectories of the magnetic moment of the lattice under the field ℎ𝑦 = 1 (a, b) and ℎ𝑦 = 0.5 (c, d) and magnetic moments |𝜇𝑖 | = 1 (𝑎, 𝑐), |𝜇𝑖 | = 𝜇01 , 𝜇02 , 𝜇01 = 1.1, 𝜇02 = 0.9 for (b, d).
Fig. 4. The diagram of the dependence on the pulse duration (with ℎ0 = 10, 𝜏𝑖 = 200) of the extreme values of the 𝑥-component of the precessing magnetic moment of the system (on the interval 250 ≤ 𝜏 ≤ 280); |𝜇𝑖 | = 1, a magnetizing field ℎ𝑦 = 1.
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Fig. 5. The time dependence on of the lattice response time for of the lattice per pulse with ℎ0 = 10, 𝜏𝑖 = 200 and 𝜏0 = 0.1, 0.12 (a, b); |𝜇𝑖 | = 1, ℎ𝑦 = 1.
pulse action the lattice comes to a new value of the total magnetic moment. At the initial stage, a sharp change in the orientation of the magnetic moments of the dipoles is observed. After this, first, oscillations of magnetic moments with a large amplitude arise, then (through 𝛥𝜏 ≈ 170) the amplitude of the oscillations sharply decreases, and the system approaches the next equilibrium state. We note that the distance between the nearest nanoparticles here, as before, is assumed to be equal 𝑙0 = 10, as a result of which the dipole–dipole interaction is very weak, but in this case it can affect the process of magnetization reversal of lattice dipoles. Fig. 8 shows the time dependence of the 𝑦-component of the magnetic moment of the lattice during magnetization reversal under the action of the pulse with 𝜏0 = 1 (the other parameters are taken as before), and the final configurations of the magnetization reversal are also presented. For case (a), the initial configuration is in Fig. 7a; for cases (b) and (c) the initial configurations are in (a) and (b), respectively. At these pulse parameters, the time interval corresponding to the large precession amplitude of the magnetic moments turns out to be short, and most of the lattice nanoparticles undergo a magnetization reversal. Next, we consider the magnetization reversal of a binary lattice consisting of alternating dipoles with 𝜇01 = 1, 𝜇02 = 2. Fig. 9 shows the time dependence of the 𝑦-component of the magnetic moment of the lattice, which is magnetized by the field ℎ𝑧 = 0.2, under the action of a magnetic pulse with parameters 𝜏0 = 1, 𝜏𝑖 = 200 and ℎ0 = 10, 20 (a, b). Dependence 1 (a) corresponds to the magnetization reversal of the lattice from configuration I to configuration II, i.e. the magnetization reversal of only one type of magnetic moments is realized. Dependence 2 (a) is inverse: the action of the same pulse on the configuration II returns the lattice to the configuration I. By increasing the pulse amplitude remagnetization of all (or nearly all) of the lattice can be achieved. Thus, the 1 (b) dependence corresponds to the magnetization reversal of the lattice from the configuration I to the configuration III, in which only two angular dipoles remained unmagnetized (which may be explained by the action of a weak dipole–dipole interaction). Dependence 2 (b) corresponds to magnetization reversal of the lattice from configuration II to the configuration IV, at which all dipoles with 𝜇02 = 2 and most of the dipoles with 𝜇01 = 1 are reversed. Thus, by choosing the parameters of the acting pulse, as well as the magnitude and direction of the magnetizing field, it is possible to achieve magnetization reversal of various parts inhomogeneous in composition lattice of the magnetic nanoparticles. 7
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Fig. 6. The time dependence on of the lattice response time of the lattice per for the pulse with ℎ0 = 10, 𝜏𝑖 = 200, 𝜏0 = 0.65, 0.8, 0.9 (a–c) and the corresponding projections of the trajectories of the total magnetic moment; |𝜇𝑖 | = 1, ℎ𝑦 = 1.
6. Conclusion The investigation of the response dynamics of a magnetic nanoparticles lattice with uniaxial anisotropy to the action of a short Gaussian pulse showed that the influence of the dipole–dipole interaction leads to modulation and a decrease in the amplitude of 8
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Fig. 7. The time dependence of the components of the magnetic moment of a homogeneous lattice (|𝜇𝑖 | = 1) with aupon magnetization reversal by a pulse with 𝜏0 = 0.65, ℎ0 = 10, 𝜏𝑖 = 200; a magnetizing field ℎ𝑧 = 0.5; for (a, b) initial configuration – (a), final configuration – (b), for (c, d) initial configuration – (b), final configuration – (d), for (e, f) initial configuration – (d), final configuration – (f).
the lattice response. However, even at the distance between the nearest dipoles 𝑟 ≥ 10𝑅 the dipole–dipole interaction in most cases can be neglected. The presence in the lattice of two different, types of dipoles, but with similar magnetic moments, leads to the 9
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Fig. 8. The time dependence of the 𝑦-component of the magnetic moment of the lattice (|𝜇𝑖 | = 1) upon magnetization reversal by a pulse with 𝜏0 = 1, ℎ0 = 10, 𝜏𝑖 = 200 and finite magnetization reversal configurations; the initial configuration for (a) is shown in Fig. 7a; for (b) and (c) configuration is shown in (a) end (b), respectively.
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Fig. 9. The time dependence of the y-component of the magnetic moment of the lattice with |𝜇𝑖 | = 𝜇01 , 𝜇02 , where 𝜇01 = 1, 𝜇02 = 2; with magnetizing field ℎ𝑧 = 0.2 and pulse with 𝜏0 = 1, 𝜏𝑖 = 200 and ℎ0 = 10, 20 (a, b). The case 1 (a) corresponds to the magnetization reversal from configuration I to configuration II, 2 (a) inverse (from II to I), 1 (b) from I to III, 2 (b) from II to IV.
appearance in the response of the system of two close frequencies and to the corresponding modulation of the dynamics of the total magnetic moment of the lattice. There is a strong dependence of the amplitude and duration of the response of magnetic moments on the duration of the acting pulse. Moreover, this dependence has a nonmonotonic character: the amplitude of the response increases periodically and, having reached one of the maxima (which decrease with increasing pulse duration), begins to decrease to the minimum values. At pulses corresponding to the indicated minima, the trajectory of the precessing magnetic moment rapidly approaches the equilibrium state of the dipole, as a result of which the amplitude of the response is small. In the case of sufficiently short pulses, there are regions of values of its duration that correspond to an increase in the response (by several times) of the magnetic moments. With the aid of a magnetizing field lying in the plane of the system and directed perpendicularly to the easy axis of the magnetic anisotropy, a convergence of the equilibrium orientations of the magnetic dipoles is achieved. The result can be realized a pulse reversal of the lattice or a part of it. In this case, the magnetization reversible part of the system can be specified by introducing several types of dipoles into the lattice, differing value of the magnetic moment, and also by selecting the parameters of the acting pulse and the value of the magnetizing field. Acknowledgments The reported study was partially supported by the Ministry of Education and Science of the Russian Federation (Contract No. 14.Z50.31.0015 and No. 3.6825.2017/BCh), and was partially funded by RFBR, Russian Federation and the government of Ulyanovsk, Russian Federation region according to the research project No. 18-42-730001/18. References [1] P. Sheng, Z. Tang, Nano Science and Technology: Novel Structures and Phenomena, CRC Press, 2003. [2] S. Gusev, Y.N. Nozdrin, M. Sapozhnikov, A. Fraerman, Collective effects in man-made two-dimensional lattices of ferromagnetic particles, Usp. Fiz. Nauk 170 (3) (2000) 331–333. 11
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[3] G. Srajer, L. Lewis, S. Bader, A. Epstein, C. Fadley, E. Fullerton, A. Hoffmann, J. Kortright, K.M. Krishnan, S. Majetich, et al., Advances in nanomagnetism via X-ray techniques, J. Magn. Magn. Mater. 307 (1) (2006) 1–31. [4] S. Gudoshnikov, B.Y. Liubimov, A. Popova, N. Usov, The influence of a demagnetizing field on hysteresis losses in a dense assembly of superparamagnetic nanoparticles, J. Magn. Magn. Mater. 324 (22) (2012) 3690–3694. [5] T. Kiseleva, S. Zholudev, A. Novakova, T. Grigoryeva, The enhanced magnetodeformational effect in Galfenol/polyurethane nanocomposites by the arrangement of particle chains, Compos. Struct. 138 (2016) 12–16. [6] M.F. Hansen, P.E. Jönsson, P. Nordblad, P. Svedlindh, Critical dynamics of an interacting magnetic nanoparticle system, J. Phys.: Condens. Matter 14 (19) (2002) 4901. [7] R. Skomski, Nanomagnetics, J. Phys.: Condens. Matter 15 (20) (2003) R841. [8] E. Meilikhov, R. Farzetdinova, Magnetic properties of two-dimensional random systems of ising dipoles, J. Magn. Magn. Mater. 268 (1–2) (2004) 237–250. [9] P. Bondarenko, A.Y. Galkin, B. Ivanov, Phase diagram of a two-dimensional square lattice of magnetic particles with perpendicular anisotropy, J. Exp. Theor. Phys. 112 (6) (2011) 986–1003. [10] A.A. Fraerman, Magnetic states and transport properties of ferromagnetic nanostructures, Phys.-Usp. 55 (12) (2012) 1255. [11] S. Dzian, B. Ivanov, Collective oscillations of the magnetic moments of a chain of spherical magnetic nanoparticles with uniaxial magnetic anisotropy, J. Exp. Theor. Phys. 116 (6) (2013) 975–979. [12] A.M. Shutyi, D.I. Sementsov, Vortex structures in planar lattices of magnetic dipoles in the presence of exchange coupling, JETP Lett. 99 (12) (2014) 695–701. [13] A.M. Shutyi, S.V. Eliseeva, D.I. Sementsov, Equilibrium state of planar arrays of magnetic dipoles in the presence of exchange interaction, Phys. Rev. B 91 (2) (2015) 024421. [14] A.M. Shutyı, D.I. Sementsov, Dynamics of the magnetic moments for chain of dipoles in domain wall, J. Magn. Magn. Mater. 401 (2016) 1033–1038. [15] N. Eibagi, J.J. Kan, F.E. Spada, E.E. Fullerton, Role of dipolar interactions on the thermal stability of high-density bit-patterned media, IEEE Magnet. Lett. 3 (2012) 4500204. [16] E. Meilikhov, R. Farzetdinova, Effective field theory for disordered magnetic alloys, Phys. Solid State 56 (4) (2014) 707–714. [17] A.G. Gurevich, G.A. Melkov, Magnetization Oscillations and Waves, CRC Press, 1996.
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/yspmi
The response of the magnetic nanoparticles lattice to a Gaussian magnetic field pulse Anatolij M. Shutyi, Svetlana V. Eliseeva ∗, Dmitrij I. Sementsov Department of High Technology Physics and Engineering, Ulyanovsk State University, Lev Tolstoy 42, 432700, Ulyanovsk, Russian Federation
ARTICLE
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Keywords: Lattices of magnetic dipoles Magnetic moment Uniaxial anisotropy Magnetization reversal Magnetic field pulse System response Homogeneous and binary lattices
ABSTRACT The paper investigates the response of homogeneous and binary planar lattices of magnetic nanoparticles with uniaxial anisotropy to the action of a short Gaussian magnetic field pulse. It is shown that the influence of the dipole–dipole interaction and the presence of two types of nanoparticles leads to modulation of the response amplitude and its decrease. A complex periodic dependence of the amplitude and duration of the system response on the duration of the pulse has been revealed. The research shows the possibility of a pulse-induced magnetization reversal of the entire lattice or its individual parts upon defining the lattice composition and the parameters of the bias field and the effective pulse.
1. Introduction The attention of researchers to nanoparticles and ordered structures on their basis is due to both their unusual physical and chemical properties, and their fast-paced introduction into modern nanotechnologies [1]. Magnetic regular structures of various dimensions formed on basis of ferromagnetic metal nanoparticles [2,3] represent a particular interest. Due to their small size, magnetic nanoparticles are generally single-domain, which makes it possible to simplify the description of structures based on them. However, the description of the dynamic properties of magnetic nanostructures is very difficult, since the interaction between the particles of the system is determined not only by their local ordering of the particles, but also by their own magnetic moment, magnetic anisotropy, coercivity, scattering fields [4–6]. Nevertheless, in recent decades a considerable progress has been made in understanding the properties and processes in magnetic nanoparticles systems [7–11]. In magnetic lattices created artificially by nanotechnologies the main contribution to the interaction of the magnetic moments of nanoparticles is made by the dipole–dipole interaction. It was shown in [12–14] that the discreteness of the structures leads to essential differences of the equilibrium states and the dynamic regimes arising in the process of magnetization reversal of linear chains and square lattices of nanoparticles from the properties of macroscopic single-domain objects. These differences, for example, may include bistable states of the lattices due to the presence of various orientational configurations with unequal total magnetic moment. Regular ensembles of magnetic nanoparticles of various dimensions can serve as a medium for super-dense recording and storage of information. The possibility of recording information on a magnetic dipole lattice is based on a change in the equilibrium configuration of magnetic moments due to radio pulses of a magnetic field, and the reading is provided by excitation of the resulting configuration by a low-power radio pulse at the ferromagnetic resonance frequency and by scanning the response frequency of the dipole system [15,16].
∗ Corresponding author. E-mail address: [email protected] (S.V. Eliseeva).
https://doi.org/10.1016/j.spmi.2019.106158 Received 6 August 2018; Received in revised form 28 March 2019; Accepted 11 June 2019 Available online 12 June 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 132 (2019) 106158
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The increase in the functional efficiency of magnetic nanostructures requires systematic studies both to optimize their synthesis and to clarify the specific features of magnetism in systems of magnetic nanoparticles. In this paper, the response of a planar lattice 6 × 6 of magnetic nanoparticles to a magnetic field pulse in the presence of uniaxial anisotropy in nanoparticles is studied on the basis of a numerical solution of dynamic equations. This type of anisotropy is the most common for nanoparticles, and also important for information recording systems. Both lattices with one kind of nanoparticles (i.e., homogeneous lattices) and lattices with two alternating types of nanoparticles differing in the magnitude of the magnetic moment (binary lattices) are being studied. Previously unknown features of the dependence of oscillations amplitude of the magnetic moments and the duration of the lattice response from the Gaussian pulse duration have been revealed. The processes of complete and partial magnetization reversal of homogeneous and binary lattices under the magnetic field pulse are studied. 2. Initial equations Let us consider a plane dipole lattice 6 × 6 consisting of single-domain spherical nanoparticles with a magnetic moment |𝐦𝑖 | = 𝑚 and a uniaxial magnetic anisotropy with a constant 𝐾𝑢 . We assume that the lattice is square with the distance between the centers of the nearest nanoparticles (lattice parameter) 𝑟0 . The dynamics of each magnetic moment of the particles is described by the Hilbert equation, which has the following form for the dimensionless parameters [14,17]: 𝜕𝝁𝑖 𝜕𝝁𝑖 = −𝝁𝑖 × 𝐡𝑒𝑓 (1) 𝑖 − 𝛼𝝁𝑖 × 𝜕𝜏 , 𝜕𝜏 where 𝝁𝑖 = 𝐦𝑖 ∕𝑚, 𝛼 is a dimensionless dissipation parameter, 𝜏 = (𝐽 𝛾)𝑡 is dimensionless time, 𝛾 is a gyromagnetic ratio, 𝐽 = 𝑚∕𝑉0 is the magnetization of the material, 𝑅 and 𝑉0 = (4𝜋∕3)𝑅3 are the radius and volume of the nanoparticle. The effective magnetic field acting on a particle includes an external field, a uniaxial anisotropy field, a dipole–dipole interaction field, and a scattering field of a spherical particle: [ ] ∑ 3(𝝁𝑛 𝐞𝑖𝑛 )𝐞𝑖𝑛 − 𝝁𝑛 𝑒𝑓 𝐡𝑖 = 𝐡 + 𝑘𝑢 𝐧(𝝁𝐢 𝐧) + − 𝝁𝑖 . (2) 3 𝑙𝑖𝑛 𝑛≠𝑖 Here dimensionless parameters are also introduced 𝐞𝑖𝑛 = 𝐫𝑖𝑛 ∕𝑟𝑖𝑛 where 𝐫𝑖𝑛 and 𝑟𝑖𝑛 are the radius vector and distance between the 3 = 𝑟3 ∕𝑅, and the <
(4)
where ℎ0 , 𝜏𝑖 𝜏0 are the peak value of the field, the time shift of the maximum of the pulse and its duration. For a homogeneous lattice, the value |𝜇𝑖 | is always equal to unity. Fig. 1 shows the time dependence of the 𝑥-component of the total magnetic moment 2
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∑ 𝑀 = 𝜇𝑖 of a homogeneous lattice under the action of a pulse with parameters ℎ0 = 10, 𝜏𝑖 = 200 and 𝜏0 = 1. The magnetic field of the pulse is linearly polarized in the case (a, b) along the 𝑋 axis and in the case (c) along the 𝑍 axis. The system in the direction of the 𝑌 axis is also affected by a constant magnetic field ℎ𝑦 = 0.1 (the effective magnetic field acting on the dipoles will be close to the value 0.1). The distance between the centers of the nearest nanoparticles is assumed to be equal to 𝑙0 = 𝑟0 ∕𝑅 = 10, 7 (a – thin and bold lines, respectively), 𝑙0 = 5, 2 (b, c – thin and bold lines, respectively). It can be seen that the effect of the dipole–dipole interaction, which is enhanced by the approach of nanoparticles, leads to modulation and a decrease in the response amplitude of the system. With the field pulse polarized along a perpendicular axis 𝑋, the influence of the dipole–dipole interaction to the dynamics of the magnetic moments becomes stronger. Next, we will only consider a lattice with a weak dipole–dipole interaction with 𝑙0 = 10. Fig. 2 shows the time dependence of the response of a binary lattice under the action of a magnetic field pulse, polarized along the 𝑋 axis and having the parameters indicated above (the pulse profile is shown in the inset of the figure). In the direction of the 𝑌 axis, the magnetic moments of the lattice are acted upon by a constant magnetic field of the value ℎ𝑦 = 1 (a–c) and ℎ𝑦 = 0.5 (d–f). Cases (a) and (d) correspond to a homogeneous lattice with |𝜇𝑖 | = 1. For a binary lattice the magnetic moments of nanoparticles in sublattices are different: |𝜇𝑖 | = 𝜇01 , 𝜇02 . In cases (b) and (e) is 𝜇01 = 1.1, 𝜇02 = 0.9, in cases (c) and (f) 𝜇01 = 1.2, 𝜇02 = 0.8. It can be seen that the presence of two types of nanoparticles with close but different magnetic moments leads to the appearance in the response of the system of an additional modulating frequency, the value of which increases with increasing difference between the magnetic moments of the two subsystems of dipoles. In this case, a decrease in the effective field acting on the magnetic dipoles leads to a decrease in the fundamental frequency of the system response and to an increase in the relaxation time of the precession dynamics of the magnetic moments. The dynamics of the precession motion of the magnetic moment after the action of the field pulse on the lattice can be represented by the projections of the trajectory of the magnetic moment on the coordinate planes. For the cases (a, b, d, e) considered in Figs. 2 and 3 (a, b, c, d, respectively) shows the projections of the trajectories of the total magnetic moment of the lattice on the plane 𝑋𝑍. It can be seen that for a given peak value of acting of the effective magnetic pulse, there is a short time interval of precession of the magnetic moments around the axis along which the field momentum is directed, after which occurs a transition to precession around the direction of the effective field. 4. Dependence of the response on the pulse duration The shape of the trajectory and the amplitude of the oscillations of the total magnetic moment of the lattice after the action of the field pulse depends in a complex manner on the parameters of both the lattice and the pulse, in particular, on its duration. Fig. 4 shows a diagram that determines the dependence on the pulse duration 𝜏0 of the extreme values of the 𝑥-component of the precessing magnetic moment of the system on the time interval 250 ≤ 𝜏 ≤ 280 after the action of the field pulse with parameters ℎ0 = 10, 𝜏𝑖 = 200. The number of points corresponding to each value 𝜏0 is approximately equal to the number of half-periods of damped oscillations of the magnetic moment, which fit in the taken time interval for a given value of the parameter 𝜏0 . The system is assumed to be homogeneous with |𝜇𝑖 | = 1, to which a constant magnetic field ℎ𝑦 = 1 is applied in the direction of the 𝑌 axis. It can be seen from the diagram that this dependence is of a complex character: alternation of the maximum and minimum oscillation amplitudes takes place. The maxima of the diagram are modified with increasing pulse duration. For 𝜏0 ≤ 0.6 they are separated by a drop in maxima. With a further increase in the pulse duration, the maxima of the diagram decrease. Fig. 5 shows the time dependence of the oscillation of the magnetic moment of the system (with the above lattice parameters and peak value field) under the action of a pulse length 𝜏0 = 0.1, 0.12 (a, b), which corresponds to the region of the drop in the diagram maxima. The oscillations are of a complex character with a frequency that is small at the beginning of the process and which increases strongly with time. The amplitude variation is nonmonotonic — modulation and time intervals of its sharp increase and decrease are observed. Fig. 6 shows the time dependence 𝑀𝑥 (𝜏) and the projection to the 𝑋𝑍 plane of the trajectory of the magnetic moment of the lattice after the pulse of duration 𝜏0 = 0.65, 0.8, 0.9 (a–c). These values 𝜏0 correspond to approximately to two maxima and one minimum in the diagram in Fig. 4. For values of the pulse duration corresponding to the maximums of the diagram, the response of the magnetic moments reaches the largest amplitudes and duration by time. In the case of the pulse duration, which corresponds to a minimum in the diagram, the trajectory of the oscillations of the magnetic moments turns out to be rapidly approaching the equilibrium state, as a result of which the amplitude of the oscillations does not reach large values, and the oscillations decay rapidly enough. 5. Lattice magnetization reversal under the action of the pulse Under the action of the magnetic field pulse, a partial or complete reversal of the dipole lattice can be performed. For its realization, it is necessary to approximate the equilibrium orientations of the magnetic moments by means of an external magnetizing field. As a result of the action of the magnetic anisotropy and the field oriented along the 𝑍 axis, the angle between the equilibrium orientations is in the interval 𝜋 < 𝜑 < 0 (depending on the magnitude of the field and the uniaxial anisotropy constant). First we consider a homogeneous lattice with magnetic moments of nanoparticles |𝜇𝑖 | = 1 to which a magnetizing field ℎ𝑧 = 0.5 is applied. Fig. 7 shows the time dependence of the two components of the total magnetic moment of the lattice upon its magnetization reversal under the action of a pulse with parameters 𝜏0 = 0.65, ℎ0 = 10, 𝜏𝑖 = 200. Three remagnetization processes are shown in the figure: the first (a, b) corresponds to the initial configuration (a) and final configuration (b); the second (c, d) corresponds to 3
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Fig. 1. The time dependence of the 𝑥-component of the magnetic moments of homogeneous lattice at a pulse action with ℎ0 = 10, 𝜏𝑖 = 200 and 𝜏0 = 1, polarized along the axes 𝑋 (a, b) and 𝑍 (c); a magnetizing field ℎ𝑦 = 0.1; the distance between the centers of the nearest nanoparticles 𝑙0 = 10, 7 (a – thin and bold lines), 𝑙0 = 5, 2 (b, c – thin and bold lines); dissipation parameter 𝛼 = 0.01.
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Fig. 2. The time dependence of the lattice response of the lattice with 𝑙0 = 10 under the action of the pulse (the profile of a pulse polarized along the 𝑋 axis is shown in the inset) and a magnetizing fields ℎ𝑦 = 1 (a–c) and ℎ𝑦 = 0.5 (d–f); the magnetic moments |𝜇𝑖 | = 1 for (a, d) and |𝜇𝑖 | = 𝜇01 , 𝜇02 , 𝜇01 = 1.1, 𝜇02 = 0.9 for (b, e), 𝜇01 = 1.2, 𝜇02 = 0.8 for (c, f).
the initial configuration (b) and the final configuration (d); the third (e, f) corresponds to the initial configuration (d) and the final configuration (f). The magnetization reversal of each individual dipole with these parameters has a random character, but after each 5
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Fig. 3. Projections of the dynamics trajectories of the magnetic moment of the lattice under the field ℎ𝑦 = 1 (a, b) and ℎ𝑦 = 0.5 (c, d) and magnetic moments |𝜇𝑖 | = 1 (𝑎, 𝑐), |𝜇𝑖 | = 𝜇01 , 𝜇02 , 𝜇01 = 1.1, 𝜇02 = 0.9 for (b, d).
Fig. 4. The diagram of the dependence on the pulse duration (with ℎ0 = 10, 𝜏𝑖 = 200) of the extreme values of the 𝑥-component of the precessing magnetic moment of the system (on the interval 250 ≤ 𝜏 ≤ 280); |𝜇𝑖 | = 1, a magnetizing field ℎ𝑦 = 1.
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Fig. 5. The time dependence on of the lattice response time for of the lattice per pulse with ℎ0 = 10, 𝜏𝑖 = 200 and 𝜏0 = 0.1, 0.12 (a, b); |𝜇𝑖 | = 1, ℎ𝑦 = 1.
pulse action the lattice comes to a new value of the total magnetic moment. At the initial stage, a sharp change in the orientation of the magnetic moments of the dipoles is observed. After this, first, oscillations of magnetic moments with a large amplitude arise, then (through 𝛥𝜏 ≈ 170) the amplitude of the oscillations sharply decreases, and the system approaches the next equilibrium state. We note that the distance between the nearest nanoparticles here, as before, is assumed to be equal 𝑙0 = 10, as a result of which the dipole–dipole interaction is very weak, but in this case it can affect the process of magnetization reversal of lattice dipoles. Fig. 8 shows the time dependence of the 𝑦-component of the magnetic moment of the lattice during magnetization reversal under the action of the pulse with 𝜏0 = 1 (the other parameters are taken as before), and the final configurations of the magnetization reversal are also presented. For case (a), the initial configuration is in Fig. 7a; for cases (b) and (c) the initial configurations are in (a) and (b), respectively. At these pulse parameters, the time interval corresponding to the large precession amplitude of the magnetic moments turns out to be short, and most of the lattice nanoparticles undergo a magnetization reversal. Next, we consider the magnetization reversal of a binary lattice consisting of alternating dipoles with 𝜇01 = 1, 𝜇02 = 2. Fig. 9 shows the time dependence of the 𝑦-component of the magnetic moment of the lattice, which is magnetized by the field ℎ𝑧 = 0.2, under the action of a magnetic pulse with parameters 𝜏0 = 1, 𝜏𝑖 = 200 and ℎ0 = 10, 20 (a, b). Dependence 1 (a) corresponds to the magnetization reversal of the lattice from configuration I to configuration II, i.e. the magnetization reversal of only one type of magnetic moments is realized. Dependence 2 (a) is inverse: the action of the same pulse on the configuration II returns the lattice to the configuration I. By increasing the pulse amplitude remagnetization of all (or nearly all) of the lattice can be achieved. Thus, the 1 (b) dependence corresponds to the magnetization reversal of the lattice from the configuration I to the configuration III, in which only two angular dipoles remained unmagnetized (which may be explained by the action of a weak dipole–dipole interaction). Dependence 2 (b) corresponds to magnetization reversal of the lattice from configuration II to the configuration IV, at which all dipoles with 𝜇02 = 2 and most of the dipoles with 𝜇01 = 1 are reversed. Thus, by choosing the parameters of the acting pulse, as well as the magnitude and direction of the magnetizing field, it is possible to achieve magnetization reversal of various parts inhomogeneous in composition lattice of the magnetic nanoparticles. 7
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Fig. 6. The time dependence on of the lattice response time of the lattice per for the pulse with ℎ0 = 10, 𝜏𝑖 = 200, 𝜏0 = 0.65, 0.8, 0.9 (a–c) and the corresponding projections of the trajectories of the total magnetic moment; |𝜇𝑖 | = 1, ℎ𝑦 = 1.
6. Conclusion The investigation of the response dynamics of a magnetic nanoparticles lattice with uniaxial anisotropy to the action of a short Gaussian pulse showed that the influence of the dipole–dipole interaction leads to modulation and a decrease in the amplitude of 8
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Fig. 7. The time dependence of the components of the magnetic moment of a homogeneous lattice (|𝜇𝑖 | = 1) with aupon magnetization reversal by a pulse with 𝜏0 = 0.65, ℎ0 = 10, 𝜏𝑖 = 200; a magnetizing field ℎ𝑧 = 0.5; for (a, b) initial configuration – (a), final configuration – (b), for (c, d) initial configuration – (b), final configuration – (d), for (e, f) initial configuration – (d), final configuration – (f).
the lattice response. However, even at the distance between the nearest dipoles 𝑟 ≥ 10𝑅 the dipole–dipole interaction in most cases can be neglected. The presence in the lattice of two different, types of dipoles, but with similar magnetic moments, leads to the 9
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Fig. 8. The time dependence of the 𝑦-component of the magnetic moment of the lattice (|𝜇𝑖 | = 1) upon magnetization reversal by a pulse with 𝜏0 = 1, ℎ0 = 10, 𝜏𝑖 = 200 and finite magnetization reversal configurations; the initial configuration for (a) is shown in Fig. 7a; for (b) and (c) configuration is shown in (a) end (b), respectively.
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Fig. 9. The time dependence of the y-component of the magnetic moment of the lattice with |𝜇𝑖 | = 𝜇01 , 𝜇02 , where 𝜇01 = 1, 𝜇02 = 2; with magnetizing field ℎ𝑧 = 0.2 and pulse with 𝜏0 = 1, 𝜏𝑖 = 200 and ℎ0 = 10, 20 (a, b). The case 1 (a) corresponds to the magnetization reversal from configuration I to configuration II, 2 (a) inverse (from II to I), 1 (b) from I to III, 2 (b) from II to IV.
appearance in the response of the system of two close frequencies and to the corresponding modulation of the dynamics of the total magnetic moment of the lattice. There is a strong dependence of the amplitude and duration of the response of magnetic moments on the duration of the acting pulse. Moreover, this dependence has a nonmonotonic character: the amplitude of the response increases periodically and, having reached one of the maxima (which decrease with increasing pulse duration), begins to decrease to the minimum values. At pulses corresponding to the indicated minima, the trajectory of the precessing magnetic moment rapidly approaches the equilibrium state of the dipole, as a result of which the amplitude of the response is small. In the case of sufficiently short pulses, there are regions of values of its duration that correspond to an increase in the response (by several times) of the magnetic moments. With the aid of a magnetizing field lying in the plane of the system and directed perpendicularly to the easy axis of the magnetic anisotropy, a convergence of the equilibrium orientations of the magnetic dipoles is achieved. The result can be realized a pulse reversal of the lattice or a part of it. In this case, the magnetization reversible part of the system can be specified by introducing several types of dipoles into the lattice, differing value of the magnetic moment, and also by selecting the parameters of the acting pulse and the value of the magnetizing field. Acknowledgments The reported study was partially supported by the Ministry of Education and Science of the Russian Federation (Contract No. 14.Z50.31.0015 and No. 3.6825.2017/BCh), and was partially funded by RFBR, Russian Federation and the government of Ulyanovsk, Russian Federation region according to the research project No. 18-42-730001/18. References [1] P. Sheng, Z. Tang, Nano Science and Technology: Novel Structures and Phenomena, CRC Press, 2003. [2] S. Gusev, Y.N. Nozdrin, M. Sapozhnikov, A. Fraerman, Collective effects in man-made two-dimensional lattices of ferromagnetic particles, Usp. Fiz. Nauk 170 (3) (2000) 331–333. 11
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[3] G. Srajer, L. Lewis, S. Bader, A. Epstein, C. Fadley, E. Fullerton, A. Hoffmann, J. Kortright, K.M. Krishnan, S. Majetich, et al., Advances in nanomagnetism via X-ray techniques, J. Magn. Magn. Mater. 307 (1) (2006) 1–31. [4] S. Gudoshnikov, B.Y. Liubimov, A. Popova, N. Usov, The influence of a demagnetizing field on hysteresis losses in a dense assembly of superparamagnetic nanoparticles, J. Magn. Magn. Mater. 324 (22) (2012) 3690–3694. [5] T. Kiseleva, S. Zholudev, A. Novakova, T. Grigoryeva, The enhanced magnetodeformational effect in Galfenol/polyurethane nanocomposites by the arrangement of particle chains, Compos. Struct. 138 (2016) 12–16. [6] M.F. Hansen, P.E. Jönsson, P. Nordblad, P. Svedlindh, Critical dynamics of an interacting magnetic nanoparticle system, J. Phys.: Condens. Matter 14 (19) (2002) 4901. [7] R. Skomski, Nanomagnetics, J. Phys.: Condens. Matter 15 (20) (2003) R841. [8] E. Meilikhov, R. Farzetdinova, Magnetic properties of two-dimensional random systems of ising dipoles, J. Magn. Magn. Mater. 268 (1–2) (2004) 237–250. [9] P. Bondarenko, A.Y. Galkin, B. Ivanov, Phase diagram of a two-dimensional square lattice of magnetic particles with perpendicular anisotropy, J. Exp. Theor. Phys. 112 (6) (2011) 986–1003. [10] A.A. Fraerman, Magnetic states and transport properties of ferromagnetic nanostructures, Phys.-Usp. 55 (12) (2012) 1255. [11] S. Dzian, B. Ivanov, Collective oscillations of the magnetic moments of a chain of spherical magnetic nanoparticles with uniaxial magnetic anisotropy, J. Exp. Theor. Phys. 116 (6) (2013) 975–979. [12] A.M. Shutyi, D.I. Sementsov, Vortex structures in planar lattices of magnetic dipoles in the presence of exchange coupling, JETP Lett. 99 (12) (2014) 695–701. [13] A.M. Shutyi, S.V. Eliseeva, D.I. Sementsov, Equilibrium state of planar arrays of magnetic dipoles in the presence of exchange interaction, Phys. Rev. B 91 (2) (2015) 024421. [14] A.M. Shutyı, D.I. Sementsov, Dynamics of the magnetic moments for chain of dipoles in domain wall, J. Magn. Magn. Mater. 401 (2016) 1033–1038. [15] N. Eibagi, J.J. Kan, F.E. Spada, E.E. Fullerton, Role of dipolar interactions on the thermal stability of high-density bit-patterned media, IEEE Magnet. Lett. 3 (2012) 4500204. [16] E. Meilikhov, R. Farzetdinova, Effective field theory for disordered magnetic alloys, Phys. Solid State 56 (4) (2014) 707–714. [17] A.G. Gurevich, G.A. Melkov, Magnetization Oscillations and Waves, CRC Press, 1996.
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