In this chapter, we will restrict our attention to the ferrites and a few other closely related materials. The great interest in ferrites stems from their unique combination of a spontaneous magnetization and a high electrical resistivity. The observed magnetization results from the difference in the magnetizations of two non-equivalent sub-lattices of the magnetic ions in the crystal structure. Materials of this type should strictly be designated as “ferrimagnetic” and in some respects are more closely related to anti-ferromagnetic substances than they are to ferromagnetics in which the magnetization results from the parallel alignment of all the magnetic moments present. We shall not adhere to this special nomenclature except to emphasize effects, which are due to the existence of the sub-lattices.

Ferromagnetic materials

We develop a generalized Langevin spin dynamics (GLSD) algorithm where both the longitudinal and transverse (rotational) degrees of freedom of atomic magnetic moments are treated as dynamic variables. This removes the fundamental limitation associated with the use of stochastic Landau-Lifshitz (sLL) equations, in which the magnitude of magnetic moments is assumed constant. A generalized Langevin spin equation of motion is shown to be equivalent to the sLL equation if the dynamics of an atomic moment vector is constrained to the surface of a sphere. A fluctuation-dissipation relation for GLSD and an expression for the dynamic spin temperature are derived using the Fokker-Planck equation. Numerical simulations, performed using ferromagnetic iron as an example, illustrate the fundamental difference between the twoand three-dimensional dynamic evolution of interacting moments, where the three-dimensional GLSD includes the treatment of both transverse and longitudinal magnetic excitations.

Longitudinal magnetic fluctuations in Langevin spin dynamics

The magnetic properties of materials only receive a satisfactory microscopic explanation by reference to the quantum structure of the constituent atoms and molecules. Quantum states of electrons are always dynamic in nature and they can involve surprisingly large internal electric currents. Bohr’s theory of the hydrogen atom, for example, describes an electron in its lowest energy state as orbiting the atomic nucleus with a period, T, of about 10−16 s. Transport of electronic charge, of magnitude e, around a circuit in this time corresponds to an average electric current of I = e/T ~ 1 mA. Taking into account the very small radius of this orbit (r ~ 1/2 A = 0.5 × 10−10 m) gives a much more modest figure for the magnetic dipole moment, m, of the current loop, namely m = I πr2 ~ 10−23 A m2.

Magnetic properties of materials

Using composites of polyvinylidene fluoride (PVDF) and carbon nanostructures embedded with Co-nanoparticles we demonstrate that electromagnetic shielding effectiveness depends strongly on the graphitic carbon concentration and the magnetic properties of Co-particles. Cobalt nanoparticles encapsulated by graphitic carbon embedded in an amorphous carbon-matrix were synthesized by a one-pot pyrolysis method at two different synthesis temperatures, TS = 800 °C (Co-800) and 1000 °C (Co-1000). We demonstrate that TS plays an important role in determining the structure, morphology and magnetic properties of the carbonaceous matrix, the graphite layer and the Co nanoparticles. Higher amounts of graphitic carbon and high saturation magnetization were observed for the Co-1000 sample than that for the Co-800 sample. We observed that the electromagnetic interference (EMI) shielding behavior of the PVDF-Co-1000 nanocomposite shows higher shielding effectiveness than that of the PVDF-Co-800 specimen. A more inhomogeneous dielectric medium in the PVDF-Co-1000 composite results in higher dielectric loss and impedance mismatch. A direct correlation between the shielding effectiveness with dielectric permittivity and magnetic permeability is demonstrated. The synergy between the multiple reflections at the interfaces and absorption of the microwave radiation in the conducting species confirms that a higher degree of graphitization and highly magnetic particles in nanocomposites are effectively superior for EMI shielding of microwave radiation.

Enhancing absorption dominated microwave shielding in Co@C-PVDF nanocomposites through improved magnetization and graphitization of the Co@C-nanoparticles.

A dynamic spin-fluctuation theory that directly takes into account nonlocality of thermal spin fluctuations and their mode-mode interactions is developed. The Gaussian approximation in the theory is improved by a self-consistent renormalization of the mean field and spin susceptibility due to the third-and fourth-order terms of the free energy, respectively. This eliminates the fictitious first-order phase transition, which is typical for the Gaussian approximation, and yields a proper second-order phase transition. The effect of nonlocal spin correlations is enhanced by taking into account uniform fluctuations in the single-site mean Green function. Explicit computational formulae for basic magnetic characteristics are obtained. The extended theory is applied to the calculation of magnetic properties of Fe-Ni Invar. Almost full agreement with experiment is achieved for the magnetization, Curie temperature, and local and effective magnetic moments.

Extended dynamic spin-fluctuation theory of metallic magnetism.

The method for calculation of the mean and local magnetic characteristics of ferromagnetic metals at finite temperatures, developed by the author and co-workers in previous papers and based on using the real band structure and spin fluctuations, is applied to Fe, Co and Ni. The initial parameters of the calculation are the electron density of states and the magnetic moment at T = 0 for a specific metal. The magnetization, the Curie temperature, the local magnetic moment and the uniform and local susceptibilities are calculated for ferromagnetic and paramagnetic states. The results are in agreement with experimental data over a wide range of temperatures.

https://iopscience.iop.org/article/10.1088/0953-8984/11/25/307/pdf

Calculation of the magnetic properties of Fe, Co and Ni with account taken of the real band structure and spin fluctuations

Temperature dependence of the magnetic characteristics of Fe and Fe–Ni Invar is considered in the dynamic nonlocal approximation of the spin-fluctuation theory. Calculations by several numerical methods show that the magnetic characteristics can have a discontinuous jump at high temperatures, well below the Curie temperature. Using the methods of catastrophe theory, we investigate the effect of small changes in the initial data on the results of the calculation. It is demonstrated that the discontinuous jump can only be smoothed but cannot be eliminated entirely without a significant change in the system of equations of the spin-fluctuation theory. Possible variants of such changes are discussed.

B. I. Reser

Papers

Extended dynamic spin-fluctuation theory of metallic magnetism.

Calculation of the magnetic properties of Fe, Co and Ni with account taken of the real band structure and spin fluctuations

Problem of temperature dependence in the dynamic spin-fluctuation theory for strong ferromagnets

Short-range order above the Curie temperature in the dynamic spin-fluctuation theory

Dynamic Spin-Fluctuation Theory of Metallic Magnetism