Expert opinion: Magnetic nanoparticles: theory and modern technological applications

    As we already wrote earlier, the NITU MISiS annually hosts the Christmas Lectures event . As part of this event, our leading scientists give a lecture on their research areas and major achievements. We have already published the expert opinion of D.V. Holberg about the lecture.

    Today we would like to publish the expert opinion of our leading scientist, project manager "Development and application of amorphous ferromagnetic microwires for the creation of new sensors, composite materials and devices based on them" Dr. Sci. Professor Usov Nikolay Alexandrovich .
    His lecture "Modern magnetism in applications: magnetic recording, biomedicine, microelectronics"took place on December 3 and made a vivid impression on all students, and many researchers noted his pedantry in the preparation of the material. Our directing team edited a very good video for that lecture, by the way, it lasted longer than everyone else. We could not help asking the professor to write an expert opinion for us. As always, Nikolai Alexandrovich very responsibly approached the task and two months later he sent us the material. Of course, this is not a popular science format, and to understand it will require a university level of knowledge of physics.
    In his expert opinion, the professor will talk about Magnetic Nanoparticles and their modern technological applications.


    The micrograph shows magnetic needle-like nanoparticles of chromium dioxide (CrO2) obtained by hydrothermal synthesis in the presence of small modifying additives of tin (Sn) and antimony (Sb).
    Particles are collected in the shape of an egg, due to their high magnetic characteristics. This material can be used in magnetic recording devices and spin electronics.

    Ensembles of magnetic nanoparticles are very widely used in modern nanotechnology. It is enough to mention such important applications of magnetic nanoparticles as superdense magnetic recording of information, magnetic fluids with unique rheological properties, highly coercive permanent magnets, etc. Recently, very promising biomedical applications of magnetic nanoparticles, such as magnetic resonance imaging, targeted drug delivery, magnetic hyperthermia, deep cleaning of biological media from toxins and impurities, etc. d.

    Chemists, physicists, engineers and technologists have been working with various ensembles of magnetic nanoparticles for many years, seeking to optimize the physicochemical properties of ensembles for a variety of technical applications. This work is far from complete. This is due, firstly, to the fact that the phenomenon of magnetism itself is quite difficult to study. And secondly, it is very difficult to work with nano-objects, which can be observed only with the help of advanced electron microscopes.

    A ferromagnetic substance has a special magnetic order, which is absent in ordinary substances. Namely, at each point of the ferromagnetic body there is a magnetization vector M®, the length of which is constant, and is equal to a physical quantity called the saturation magnetization of the substance, | M® | = Ms. Saturation magnetization is the number of elementary magnetic moments per unit volume of a ferromagnetic substance, the behavior of which is correlated by quantum-mechanical exchange interaction [1,2]. The main subject of study in ferromagnetism is the analysis of the possible types of distribution of the M® vector over the volume of a ferromagnetic body, depending on the applied magnetic field and other factors. It turns out that the vector M® cannot change abruptly, abruptly, but can only smoothly rotate in a magnetized body from point to point, keeping its length. Thus, in essence, magnetism is a three-dimensional vector field.

    Note that we live surrounded by various physical fields. For example, the inhomogeneous temperature distribution inside and around us is a three-dimensional scalar temperature field. This field is described by a single function T (r, t), which can depend not only on the position of a point in the space r, but also on time t. To describe a vector field, three functions are needed - the projections of this vector on the axis of the Cartesian coordinate system, {Mx (r, t), My (r, t), Mz (r, t)}. An essential property of a magnetic vector that radically distinguishes the field of this vector from other physical fields is the constancy of the length of the magnetic vector, Mx2 (r, t) + My2 (r, t) + Mz2 (r, t) = Ms2, which is dictated by the laws of quantum mechanics [ 1,2]. This relationship is non-linear because it connects the squares of quantities. Therefore, the study of ferromagnetism requires the use of special nonlinear mathematics, which is much more complicated than ordinary mathematical analysis. In addition, any magnetized body creates inside and around itself a distribution of the magnetic field, H®, which itself affects the distribution of magnetization in the ferromagnetic body. When moving away from the magnetized body, the field H® decreases in space slowly, in proportion to ~ 1 / r3, that is, it is long-range. This means that even sufficiently distant portions of the magnetized body are connected by magnetic interaction, that is, their behavior is consistent. that is, it is a long-range. This means that even sufficiently distant portions of the magnetized body are connected by magnetic interaction, that is, their behavior is consistent. that is, it is a long-range. This means that even sufficiently distant portions of the magnetized body are connected by magnetic interaction, that is, their behavior is consistent.

    These two circumstances — the nonlinearity of the equations that describe the distribution of the M® vector in space, and the long-range nature of the magnetic interaction, extremely complicate the theoretical analysis of the properties of ferromagnetic materials. Although the basic equations of phenomenological ferromagnetism were formulated by Landau and Lifshitz a long time ago, in their famous work of 1935 [3], significant progress in the development of the theory of ferromagnetism occurred only in the 90s of the last century, in connection with the development of powerful computer simulation methods. Until now, magnetic nanoparticles remain one of the central objects of the theory of ferromagnetism, and are still an important area of ​​experimental research. The fact is that an extended ferromagnetic body has a large number of magnetic degrees of freedom. Really, in a macroscopically large body, the M® vector can expand in space in a huge number of ways. This phenomenon is spoken of as the presence of a large number of stable magnetization distributions, which can also be easily transformed into each other. Therefore, the properties of an extended ferromagnet are difficult to control, since it is difficult to fix the magnetic state of such a body.

    However, it is clear that the number of magnetic degrees of freedom decreases sharply with decreasing body volume. Indeed, quantum-mechanical exchange interaction allows only fairly smooth changes in the vector M® in space, at characteristic lengths exceeding the so-called exchange length Lex. In good ferromagnets, such as iron, cobalt, nickel and their alloys, the exchange length is in the order of magnitude of 20-30 nanometers. If the characteristic size of the nanoparticle D is less than or of the order of the exchange length, D <Lex, then the reversal of the magnetization vector within such a particle is energetically unfavorable. It is more rigorous to speak of the characteristic size of a single domain, Dc, [4] which in the so-called magnetically soft ferromagnets is close to the exchange length, Dc ~ Lex. Particles with sizes smaller than the diameter of the single domain, D <Dc, are magnetized uniformly, that is, their magnetization vector is independent of coordinates, M = const. Such particles, with the simplest magnetic structure, are called single domain. A single-domain particle is a small permanent natural magnet that is almost impossible to demagnetize. If the particle size exceeds the size of the single domain, D> Dc, then inhomogeneous magnetization distributions, usually of the vortex type, can develop in such a particle.

    Fig. 1. The state of homogeneous magnetization in a spherical cobalt nanoparticle with a diameter of D = 36 nm (left) and an inhomogeneous vortex state in the same nanoparticle with a diameter of D = 56 nm (right), obtained using three-dimensional computer simulation.

    In Fig. Figure 1 shows the homogeneous and vortex distributions of magnetization calculated by modern numerical methods [5] in spherical cobalt nanoparticles of different diameters. The three-dimensional distributions of the magnetization vector in these particles are represented in these figures by arrows of a fixed length. To determine the single-domain diameter of a spherical cobalt nanoparticle, one needs to calculate the energy diagram of these states, schematically shown in Fig. 2.

    Fig. 2. Schematic energy diagram of stable magnetic states of a nanoparticle depending on its radius.

    As can be seen from Fig. 2, the total specific energy of the uniformly magnetized state of the particle (black line) does not depend on the radius of the particle, while the total energy of the vortex state (red curve) rapidly decreases with increasing radius. The intersection point of these curves determines the radius of a single domain of a particle ac = Dc / 2. If the particle radius r <ac, then the homogeneous state has the lowest total energy, while the vortex state in a certain range of radii can exist as metastable, that is, be stable, but have more energy. If r> ac, then the vortex state will be the lowest in energy. In this case, a homogeneous state can exist as metastable in the range of radii ac <r <Rc, where Rc is the stability radius of the homogeneous state.

    For most technical applications, it is convenient to work with an ensemble of single-domain nanoparticles, whose properties can be relatively accurately predicted and controlled. Under all circumstances, a single-domain nanoparticle retains its full magnetic moment, while the average magnetic moment of a particle in a vortex state can be small, since the magnetic vectors in this state close themselves. For a particle in a vortex state, the average value of the magnetic moment substantially depends on the magnitude of the external magnetic field acting on the particle. From the diagram Fig. Figure 2 shows that if the ensemble has a size dispersion of nanoparticles near the radius of a single domain, then in fact such an ensemble is an uncontrolled mixture of particles with different properties, single domain and not single domain. If you still consider

    We see that an ensemble of magnetic nanoparticles is a rather complex physical system whose properties are determined by many different factors. In most cases (although not always) the ensemble consists of nanoparticles of the same chemical composition. Therefore, the particles of the ensemble can be characterized by a single set of material magnetic parameters, i.e., saturation magnetization Ms, the type of magnetic anisotropy, and the value of the magnetic anisotropy constant K. Magnetic anisotropy determines the distinguished directions (the so-called light axes) in space with respect to the axes of symmetry of the crystal. In the absence of an external magnetic field, the particle magnetization vector spontaneously orientates along the light axes of the magnetic crystal. But specifying magnetic parameters alone is completely insufficient for a complete characterization of an ensemble of nanoparticles. It is necessary to know the distribution of nanoparticles in size and shape; the number and orientation of the light axes of anisotropy of nanoparticles (oriented or non-oriented ensemble); distribution of centers of nanoparticles in space. For example, the centers of nanoparticles can be located periodically, forming a certain spatial lattice, or occupy random positions, with some average distance between the particles.

    Note that in real experiments, as a rule, fairly dense ensembles of particles are studied, the properties of which differ significantly from the properties of rare ensembles. A theoretical study of the properties of a dense ensemble of particles is hindered by the long-range nature of the magnetic dipole interaction between the particles of the ensemble. Because of this, for an ensemble in which there are Np particles, it is necessary to take into account Np2 pair interactions of particles, so that the computational complexity increases rapidly with increasing number of particles in the ensemble.

    Further, the environment in which the ensemble is located has a significant effect on the properties of the ensemble. We should distinguish between relatively low viscosity media in which the particles of the ensemble under the influence of an external magnetic field or magnetic dipole interaction of neighboring particles can rotate as a whole, and media such as a solid matrix in which the rotation of the nanoparticles as a whole is impossible. Finally, the ambient temperature can significantly affect the properties of an ensemble of particles of sufficiently small sizes (superparamagnetic nanoparticles). If the temperature of the medium exceeds the so-called temperature of blocking the magnetic moments of nanoparticles, then the temperature fluctuations of the magnetic moments of individual particles significantly reduce the average magnetic moment of the ensemble.

    Let's move on to technical applications. The magnetic moment of a single-domain magnetic nanoparticle with uniaxial magnetic anisotropy has two distinguished directions in space. At a sufficiently low (for example, room) temperature, a particle can remain indefinitely in each of these two magnetic states. Thus, it retains the memory of the acquired magnetic state, which means that it can store information without loss for a sufficiently long time. If we arbitrarily assign the value “0” to the direction of the magnetic moment of the particle up, and the value “1” to the direction of the magnetic moment down, as shown in Figure 3 left, then some binary text from a sequence of zeros and ones can be stored in a specially prepared magnetic state of an ensemble of nanoparticles. Currently, in the process of magnetic recording, one bit of information is recorded not on one, and for a whole collection of 20-40 closely spaced magnetic nanoparticles. The transition to recording on the principle of “one bit - one particle” would significantly increase the density of magnetic recording of information.
    However, the essential technical difficulties that must be overcome in order to realize this interesting idea are quite obvious. First, the particles of the ensemble should be substantially identical and periodically located in the plane, with a lattice period of the order of the size of the nanoparticle. It is even more surprising that chemists recently learned to create similar, almost ideally periodic structures of magnetic nanoparticles [6] using self-assembly processes, that is, self-organization of ensemble particles during their growth during a chemical reaction (see Fig. 3, right).


    Fig. 3. The principle of super-dense magnetic recording of information on individual magnetic nanoparticles with two magnetic states (left) and the periodic structure of FePt magnetic nanoparticles obtained by the chemical self-assembly method.

    However, in order to implement the idea of ​​super-dense recording of information, in addition to implementing the correct geometric structure of the ensemble, it is necessary to ensure a sufficiently large value of the magnetic anisotropy constant of the synthesized nanoparticles. In principle, FePt particles with the correct crystalline structure, in which the planes of iron atoms regularly alternate with the planes of platinum atoms, have a record high value of the magnetic anisotropy constant, K = 5 * 107 erg / cm3, [7] This allows maintaining the stability of the magnetic state of the particle during time and for particles of sufficiently small diameter.

    Indeed, in order to transfer the magnetic moment of a particle between two directions of easy magnetization, it is necessary to overcome the energy barrier of heightwhere V is the volume of the nanoparticle. To prevent spontaneous torque transfer due to thermal fluctuations for a sufficiently long time (10 years), it is necessary to fulfill the strict condition KV> (50 - 70) kBT, [7] where T is the ambient temperature, kB is the Boltzmann constant. With a decrease in the particle diameter, the energy barrier rapidly falls, but the large value of the magnetic anisotropy constant of the particle allows one to maintain the indicated inequality for particles of nanometer sizes. Unfortunately, the FePt particles grown by self-assembly [6] are in a misoriented magnetic state, when iron and platinum atoms occupy arbitrary positions in the crystal lattice. And in this case, the constant of the magnetic anisotropy of the substance is small, several orders of magnitude less than the specified passport value. Despite considerable efforts to transfer FePt particles to an ordered magnetic state using annealing and other methods, this fundamental problem has not yet been solved. In general, the problem of the influence of small fluctuations in the magnetic state of temperature fluctuations of their magnetic moments is known as the superparamagnetic limit [7], and is still waiting to be resolved.

    Thus, temperature fluctuations of the magnetic moments of particles lead to significant difficulties in creating magnetic carriers of superdense information recording. At the same time, they prove to be very useful for the development of one of the interesting biomedical applications of magnetic nanoparticles, namely, the method of magnetic hyperthermia, intended for the treatment of dangerous oncological diseases. It was experimentally proved [8] that maintaining the temperature of the affected organ at about 42 ° C for 20 to 30 minutes. leads to necrosis of cancer cells, more susceptible to high temperature than normal tissues. Many ferromagnetic materials are able to absorb the energy of an external alternating magnetic field and thereby heat the surrounding tissue. However, magnetic nanoparticles have significant advantages for magnetic hyperthermia, because: a) ensembles of superparamagnetic nanoparticles are able to provide extremely large values ​​of specific energy absorption, of the order of 1 kW per gram of substance; b) due to their small size, nanoparticles can penetrate deep into biological materials; c) nanoparticles of iron oxides are non-toxic or slightly toxic for a living organism; d) they have short periods of elimination from the body.

    As we saw above, a magnetic nanoparticle is a very strong natural magnet, since the characteristic field of magnetization reversal even of a magnetically soft iron oxide particle at room temperature is quite large, Hc (0) ~ 2K / Ms ~ 400 E. It is important, however, that the energy of the barrier separating the magnetic potential wells decreases with decreasing particle volume, V ~ R3, and can be compared with the characteristic thermal energy kBT. In this case, due to thermal fluctuations of the magnetic moment, the particle loses its average constant magnetization and becomes superparamagnetic. The characteristic residence time of the magnetic moment in a given potential well (Neel relaxation time) is estimated as , where the constant[9]. The relaxation time decreases exponentially with decreasing particle diameter. As soon as it becomes on the order of or less than the characteristic time of measuring the magnetic moment , the time -average magnetic moment of the particle is equal to zero.

    But the phenomenon of superparamagnetism has a positive side. Thermal fluctuations, swinging the magnetic moment of the particles in the potential well, effectively lower the energy barrier and significantly reduce the magnitude of the magnetization reversal field of the particle. Therefore, an ensemble of superparamagnetic particles is able to magnetize in an external variable magnetic field of moderate amplitude, H0 ~ 100 - 200 Oe, which is extremely important for magnetic hyperthermia, since this simplifies the conditions for creating an alternating magnetic field and reduces the cost of the necessary equipment.

    As is known from thermodynamics [1, 2], the intensity of absorption of energy of an alternating magnetic field is proportional to the area of ​​the hysteresis loop of an ensemble of magnetic nanoparticles. In our group, theoretical calculations of low-frequency hysteresis loops of rarefied ensembles of magnetic nanoparticles of various types were performed [10, 11], as well as the corresponding experimental measurements performed by the original method [12, 13]. Theoretical calculations showed a significant dependence of the hysteresis loops on the frequency of an alternating magnetic field as shown in Fig. 4.


    Fig. 4. Magnetic nanoparticles are able to effectively remotely absorb the energy of an alternating external magnetic field, and thereby heat the environment. However, this ability significantly depends on the frequency of exposure, and a number of other physical parameters.

    A theoretical analysis shows [10] (see Fig. 5) that the hysteresis loops of a superparamagnetic ensemble also very strongly depend on the average particle diameter if the particles are fixedly fixed in the surrounding nonmagnetic medium. This important fact was confirmed in a number of recent experiments, although at the same time a significant dependence of the specific energy absorption on a number of other factors was demonstrated, such as the influence of the magnetic dipole interaction in dense ensembles of magnetic nanoparticles [12, 13].


    Fig. 5. Theoretical calculation [] of specific absorption of energy of an alternating magnetic field by sparse ensembles of cobalt nanoparticles (f = 500 kHz, H = 200 Oe) and magnetite (f = 400 kHz, H = 120 Oe) depending on the particle diameter.

    Magnetic hyperthermia, being a local and remote exposure, apparently does not have such serious side effects as chemo- or radiotherapy [8]. It seems that the successful development of magnetic hyperthermia will depend on the successful solution of several problems. First of all, it is necessary to improve the methods for preparing ensembles of nanoparticles with a sufficiently large specific energy absorption in an alternating magnetic field of moderate amplitude. This will reduce the dose of nanoparticles, sufficient to achieve a positive therapeutic effect. Ideally, it would be desirable to learn how to locally heat small volumes of tissue in order to suppress small, very dangerous neoplasms at an early stage. Further, it is necessary to ensure the creation of an alternating magnetic field of sufficient amplitude, with the necessary spatial distribution in a given area of ​​the body, at reasonable cost for energy, guaranteed safety from electric shock, moderate cost. Finally, it is necessary to learn how to control the effect itself, choosing the amplitude and frequency of the magnetic field, the magnetic and geometric parameters of the nanoparticles, the time and frequency of the exposure, taking into account the electrodynamic and thermal parameters of the medium. It is also highly desirable to control the spatial and temporal distribution of temperature in the affected area. Currently, these problems are in the focus of attention of researchers of various profiles. magnetic and geometric parameters of nanoparticles, time and frequency of exposure, taking into account the electrodynamic and thermal parameters of the medium. It is also highly desirable to control the spatial and temporal distribution of temperature in the affected area. Currently, these problems are in the focus of attention of researchers of various profiles. magnetic and geometric parameters of nanoparticles, time and frequency of exposure, taking into account the electrodynamic and thermal parameters of the medium. It is also highly desirable to control the spatial and temporal distribution of temperature in the affected area. Currently, these problems are in the focus of attention of researchers of various profiles.

    Cited literature
    [1] G. S. Krinchik, Physics of Magnetic Phenomena (Moscow, Moscow State University, 1985).
    [2] C.V. Vonsovsky, Magnetism (Moscow, Science, 1972).
    [3] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935).
    [4] WF Brown, Jr., Micromagnetics (Wiley-Interscience, New York - London, 1963).
    [5] NA Usov and JW Tucker. Material Science Forum 373-376, 429 (2001).
    [6] S. Sun, CB Murray, D. Weller, L. Folks, and A. Moser, Science 287, 1989 (2000).
    [7] D. Weller and A. Moser, IEEE Trans. Magn. 35, 4423 (1999).
    [8] QA Pankhurst, NKT Thanh, SK Jones, J. Dobson, J. Phys. D: Appl. Phys. 42, 224001 (2009).
    [9] WF Brown, Jr., Phys. Rev. 130, 1677 (1963).
    [10] NA Usov, J. Appl. Phys. 107, 123909 (2010).
    [11] NA Usov, B.Ya. Liubimov, J. Appl. Phys. 112, 023901 (2012).
    [12] SA Gudoshnikov, B. Ya. Liubimov, and NA Usov, AIP Advances 2, 012143 (2012)
    [13] SA Gudoshnikov, B.Ya. Liubimov, AV Popova, NA Usov. J. Magn. Magn. Mater. 324, 3690 (2012)

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