Advances in Condensed Matter Physics

Volume 2017 (2017), Article ID 6209206, 15 pages

https://doi.org/10.1155/2017/6209206

## Excitation Spectrum of the Néel Ensemble of Antiferromagnetic Nanoparticles as Revealed in Mössbauer Spectroscopy

Institute of Physics and Technology, Russian Academy of Sciences, Nakhimovskii pr. 36-1, Moscow 117218, Russia

Correspondence should be addressed to Mikhail A. Chuev

Received 13 November 2016; Accepted 8 March 2017; Published 11 June 2017

Academic Editor: Jan A. Jung

Copyright © 2017 Mikhail A. Chuev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The excitation spectrum of the Néel ensemble of antiferromagnetic nanoparticles with uncompensated magnetic moment is deduced in the two-sublattice approximation following the exact solution of equations of motion for magnetizations of sublattices. This excitation spectrum represents four excitation branches corresponding to the normal modes of self-consistent regular precession of magnetizations of sublattices and the continuous spectrum of nutations of magnetizations accompanying these normal modes. Nontrivial shape of the excitation spectrum as a function of the value of uncompensated magnetic moment corresponds completely to the quantum-mechanical calculations earlier performed. This approach allows one to describe also Mössbauer absorption spectra of slowly relaxing antiferromagnetic and ferrimagnetic nanoparticles and, in particular, to give a phenomenological interpretation of macroscopic quantum effects observed earlier in experimental absorption spectra and described within the quantum-mechanical representation.

#### 1. Introduction

A wide application of materials containing nanosized antiferromagnetic particles in different branches of nanotechnology is primarily due to a number of specific structural, magnetic, and thermodynamic properties of these materials found within long-term fundamental studies. However, these real materials are still characterized by different experimental techniques on the basis of phenomenological Néel approach describing a superposition of antiferromagnetism and superparamagnetism of uncompensated magnetic moments on two magnetic sublattices [1, 2]. Practical models for analyzing experimental data taken on antiferromagnetic nanoparticles are based mainly on the representation of uncompensated magnetic moment and a rather simplified treatment of antiferromagnetism in terms of linear magnetic susceptibility introduced also by Néel [3, 4]. However, the ground state for antiferromagnetic nanoparticles should be much more complicated as compared to that for a bulk sample, which is evidenced from the atomic-scale magnetic modeling [5, 6] that in its turn is hardly possible to be used for analyzing experimental data in practice due to computational expenses followed by an uncontrollable accuracy of calculations.

Meanwhile, a quantum-mechanical model for describing thermodynamic properties of an ensemble of ideal (compensated) antiferromagnetic nanoparticles is recently developed [7, 8]. This model clarifies principally the difference in thermodynamic behavior of ferromagnetic and antiferromagnetic particles revealed in spectroscopic measurements without the assistance of uncompensated magnetic moment and describes qualitatively macroscopic quantum effects earlier observed repeatedly in experimental Mössbauer absorption spectra of antiferromagnetic and even ferrimagnetic nanoparticles [9–13]. It was also shown that taking the uncompensated spin in the account does not change the qualitative pattern of these effects but is reduced to small numerical corrections of the shape of the absorption spectrum of the ensemble of antiferromagnetic particles [14]. Note that the earlier studies of macroscopic quantum phenomena in small antiferromagnetic particles have also explored the same Néel idea of the uncompensated magnetic moment, but again within a simplified treatment of the ground state and the lowest energy levels [15, 16].

The quantum-mechanical model [7, 8, 14] can be easily realized in numerical calculations and efficiently used for analyzing experimental data, in particular, for analyzing a large array of Mössbauer spectra taken at different temperatures [17]. However, for solving the last task in a complete manner one should take into consideration relaxation processes that is the corresponding model of magnetic dynamics to be developed. On the other hand, the macroscopic quantum effects observed in the Mössbauer spectra of antiferromagnetic nanoparticles are treated within the quantum-mechanical model [8, 14] only in terms of wave functions and mean values of macrospins of sublattices for different energy levels with no phenomenological explanation. The latter can be found only in the macroscopic limit. The corresponding continuum model of the magnetic dynamics of an ensemble of ideal antiferromagnetic nanoparticles in the two-sublattice approximation is recently proposed in solving the equations of motion for magnetizations of sublattices [18]. This model has demonstrated the nontrivial character of the excitation spectrum of particles in the form of four excitation branches corresponding to four normal modes of self-consistent uniform precession of vectors of magnetizations of sublattices around the easy axis. Two of these modes are known well from the classical theory of antiferromagnetic resonance in the zero external magnetic field [19, 20]. Two other modes have the ferromagnetic character and were completely out of the focus of interest of researchers.

However, the normal modes of the uniform precession are only partial solutions of equations of motion of uniform magnetizations of sublattices, while the general solution of these equations should contain nutations at the background of the uniform precession in analogy with problems of a sphere pendulum and a heavy gyroscope [21]. The continuous energy spectrum of nutations of magnetizations of sublattices accompanying the normal modes of their regular precession for ideal antiferromagnetic nanoparticles has been recently described in [22]. In fact, the presence of the excitation branch corresponding to one of the ferromagnetic normal modes with the local energy minimum for the vectors of sublattice magnetizations precessing in the equatorial plane and nutations of magnetizations accompanying this normal mode gives the phenomenological explanation of macroscopic quantum effects observed in Mössbauer absorption spectra [9–13] and described in the quantum-mechanical model of antiferromagnetic nanoparticles [7, 8, 14].

The continuous models described in [18, 21] are the basis for further development of the theory of nanoparticles with different magnetic natures which can be implemented as the method for quantitative analysis of experimental data, in particular, a large array of the Mössbauer spectra of nanoparticles measured for the last half-century [23]. Taking into account the general validity of the Néel idea, the next stage on the way to solve this problem is a generalization of these continuous models developed for ideal antiferromagnetic nanoparticles for the case of the presence of an uncompensated magnetic moment, in particular, for studying the effect of the latter on the excitation spectrum and the shape of Mössbauer spectra of an ensemble of “uncompensated” antiferromagnetic nanoparticles and its temperature evolution. This study was aimed at solving these problems.

#### 2. Excitation Spectrum of Antiferromagnetic Nanoparticles

In analogy with [18, 22], let us start the analysis with the simplest expression for the energy density of an antiferromagnetic particle with the exchange interaction constant , the constant of the axial magnetic anisotropy in the approximation of two sublattices with magnetizations and :Here, and are angles between vectors and and the easy axis. The only difference from the case of ideal antiferromagnetic particles [18, 22] is that the absolute values of magnetizations and in (1) can be not equal. We have neglected distinction of magnetic anisotropy for two sublattices in (1) because of small values of uncompensated magnetic moment.

In accordance with the classical theory of the antiferromagnetic resonance [19, 20] and ferromagnetic resonance [24], the phenomenological consideration can be performed within the assumption that the magnetic moment of each th sublattice precesses in the internal effective field:and equations of motion for the sublattice magnetizations can be presented in the following form: Here, is the magnetomechanical ratio for th sublattice. In our case effective magnetic fields acting on each of sublattices are determined by expressionsHere, we have introduced effective values of the exchange field, , and the anisotropy field, , in accordance with [19, 20, 24] as well as normalized values of magnetizations of sublattices and their projections on the easy axis with the unit vector for th sublattice.

It is convenient to look for axially symmetric solutions of equations of motion (3) in the form of the self-consistent and uniform precession of vectors and around the easy axis:Substituting these expressions into equations of motion (3), one comes to the combined equations for the components of the sublattice magnetizationsHere, we have introduced the ratio of the sublattice spinsnormalized values of the magnetization precession frequency and the anisotropy constantdefined by the exchange interaction energy densityFurther without loss of a community, we will consider that ≤ 1 and the normalized value of uncompensated magnetic moment in the case of the Néel ensemble of antiferromagnetic nanoparticles.

The solutions of (6a) and (6b) are four normal modes of self-consistent and homogeneous precession of the vectors and around the easy axis (see Figure 1). These modes are defined by parametric relations between the components of the sublattice magnetizations, and , which are rather complicated for perception so that we represent only the equation for the normalized precession frequency Analysis of equations like (11) in physically meaningful limiting cases of small anisotropy energy as compared to the exchange energy () and small deviations of the vectors and from the easy axis allows one to describe the antiferromagnetic resonance [19, 20] and ferromagnetic resonance [24] on a phenomenological level. However, solution of (6a) and (6b) in a general form allows one to find new qualitative features of the thermodynamics of an ensemble of antiferromagnetic nanoparticles [18, 22] and to develop a technique for quantitative treatment of the experimental data [17].