Kittel's Theory Research Articles

Magnetoresistance of Antiphase Boundaries in Sr2FeMoO6−δ

Antiphase boundaries (APB) in strontium ferromolybdate induce antiferromagnetically coupled spins of the Fe ions at the boundary affecting both the magnetic properties and the electron transport across the boundary. Herein, first, 3D presentation of the APB in strontium ferromolybdate is presented. For the first time, the weighted antiferromagnetic (AF) modulus, the bulk ferrimagnetic exchange stiffness constant, and the AF exchange stiffness constant across the APB of strontium ferromolybdate are estimated. This is done by analyzing carefully selected experimental data of the magnetoresistance (MR). The boundary width and boundary energy are calculated by means of Kittel's theory of the structure of ferromagnetic domains in films and small particles. The evaluation of these parameters deepens the understanding of MR in strontium ferromolybdate ceramic and thin‐film samples possessing a high density of APBs.

Permeability spectra of planar M‐type barium hexaferrites with high Snoek’s product by two‐step sintering

AbstractHigh‐frequency ferrite ceramics exhibiting high permeability and low magnetic loss have been recently attracting more and more attention owing to the rapid development of modern communication technologies. It is known that the magnetic performance of ferrites is strongly associated with the morphology of their polycrystalline structure. In this work, we focus on the control of grain growth and densification process for planar Co‐Ti doped M‐type barium hexaferrites, BaCoxTixFe12‐xO19, by one‐step sintering (OSS) and two‐step sintering (TSS) techniques. Experimental results indicate that a uniform and fine‐grained microstructure is achieved by the TSS method. More importantly, the ferrites prepared by TSS demonstrate high permeability (µ' = 18‐20), low magnetic loss (tan δm = 0.1‐0.3), and high Snoek's product (>25 GHz) in the frequency range of 100‐400 MHz. By fitting experimental data, we have determined that low magnetic loss is derived from the small damping coefficient of spin rotation in terms of Kittel's theory. Therefore, the TSS technique provides an effective and efficient approach to prepare planar BaM hexaferrite materials for ultra‐high frequency (UHF) communication devices requiring low loss and high Snoek's product.

Magnetic and electrical properties of ε-Fe3N on c-plane GaN

ε-Fe3N films on c-plane GaN are fabricated through nitridation of amorphous Fe films. From x-ray diffraction, the structures of these ε-Fe3N films are found to consist of two crystal orientations of ε-Fe3N, i.e. (0 0 2) and (1 1 1). The binding energies of Fe ions in the samples are analysed by x-ray photoelectron spectroscopy, both Fe–N and Fe–O–N peaks are observed. Scanning electron microscopy shows that the surface morphology of the ε-Fe3N films is island-like. In particular, these films are ferromagnetic at room temperature, with a coercivity of about 200 Oe according to the measured magnetic hysteresis loop. The ferromagnetism is further studied by ferromagnetic resonance spectra. With increasing angle between the incident field and the sample plane, the resonance magnetic fields of Fe and ε-Fe3N films exhibit different behaviours; the difference can be interpreted by Kittel's theory. In addition, the I–V curves and Hall measurement indicate that the ε-Fe3N films are good conductors at room temperature. On the whole, these magnetic and electrical properties demonstrate that the ε-Fe3N films are appropriate for GaN-based spintronic devices.

Control of gigahertz permeability and permittivity dispersion by means of nanocrystallization in FeCo based nanocrystalline alloy

An amorphous (FeCo) based alloy has been prepared by a rapid quench method. Subsequent annealing on the amorphous samples gives rise to the coexistence of two magnetic phases: amorphous matrix and nanocrystalline grains (α′-FeCo) with an average size of 9.8 nm. Permeability dispersion behaviors have been studied by Kittel theory [C. Kittel, J. Phys. Radium 12, 332 (1951)]. The results show that these two magnetic phases contribute to the permeability dispersion. The Cole-Cole dispersion law [K. S. Cole and R. H. Cole, J. Chem. Phys. 9, 341 (1941)] has been employed to explain the permittivity dispersion within microwave region based on the assumption that multiple dielectric relaxation processes existing. Our results indicate the possibility of tuning the high frequency permeability and permittivity values of (FeCo) based alloy by controlling the magnetic microstructure, which suggests an alternative method to develop smart electromagnetic materials.

Universal Properties of Ferroelectric Domains

Based on the Ginzburg-Landau approach, we generalize the Kittel theory and derive the interpolation formula for the temperature evolution of a multidomain polarization profile P(x,z). We resolve the long-standing problem of the near-surface polarization behavior in ferroelectric domains and demonstrate polarization vanishing instead of the usually assumed fractal domain branching. We propose an effective scaling approach to compare the properties of different domain-containing ferroelectric plates and films.

Influence of grain growth on the high-frequency behaviour of ferromagnetic Fe–Co–(M)–N (M=Ta, Hf) nanocomposite films

The CMOS compatible ferromagnetic Fe–Co–(M)–N (M=Ta, Hf) films were investigated with regard to their grain size-dependent frequency behaviour. Predominantly Fe 33Co 40Ta 10N 17 films were deposited by reactive r.f. magnetron sputtering. These films were compared to Fe 36Co 44Hf 9N 11 films. In order to induce an in-plane uniaxial anisotropy H u as well as to investigate the grain growth behaviour, the films were annealed in a static magnetic field. The in-plane uniaxial anisotropy field of around 4 mT as well as a good soft magnetic behaviour with a saturation polarisation of approximately 1.2–1.4 T could be observed after heat treatment. Ferromagnetic resonance frequencies (FMR) of approximately up to 2.4 GHz could be achieved according to the Kittel theory. Depending on the heat treatment, high-frequency losses through energy dissipation was made conspicuous by means of the full-width at half-maximum (FWHM) Δ f eff of the imaginary part of the frequency-dependent permeability which was between 0.4 and 1 GHz. This FWHM was basically discussed in terms of two-magnon scattering theories, in combination with the Herzer random anisotropy model. In order to correlate the resonance line broadening with a phenomenological damping parameter α eff, which ranged from about 0.0125 to 0.028, the modified Landau–Lifschitz–Gilbert equation was used to fit and describe the permeability spectra of the ferromagnetic films.

High-frequency behavior of magnetic composites

The paper reviews recent progress in the field of microwave magnetic properties of composites. The problem under discussion is developing composites with high microwave permeability that are needed in many applications. The theory of magnetic composites is briefly sketched with the attention paid to the laws governing the magnetic frequency dispersion in magnetic materials and basic mixing rules for composites. Recent experimental reports on the microwave performance of magnetic composites, as well as data on the agreement of the mixing rules with the measured permeability of composites that are available from the literature are discussed. From the data, a conclusion is made that the validity of a mixing rule is determined by the permeability contrast in the composite, i.e., the difference between permeability of inclusions and that of the host matrix. When the contrast is low, the Maxwell Garnet mixing rule is frequently valid. When the contrast is high, which is of the most interest for obtaining high microwave permeability of a composite, no conventionally accepted theory is capable of accurately predicting the permeability of the composites. Therefore, the mixing rules do not allow the microwave properties of magnetic composites to be predicted when the permeability of inclusions is high, that is the case of the most interest. Because of that, general limitations to the microwave performance of composites are of importance. In particular, an important relation constraining the microwave permeability of composites follows from Kittel's theory of ferromagnetic resonance and analytical properties of frequency dependence of permeability. Another constraint concerning the bandwidth of electromagnetic wave absorbers follows from the Kramers–Kronig relations for the reflection coefficient. The constraints are of importance in design and analysis of electromagnetic wave absorbers and other devices that employ the microwave magnetic properties of composites, such as magnetic substrates for microwave antennas, microwave inductors, etc.

The effect of perturbative interactions among localized electrons on the Verwey transition in magnetite

We modify the current Straessler–Kittel theory of phase transitions by allowing for weak perturbative interactions among electrons localized on adjacent lattice sites. The solutions obtained in the absence of such interactions are stable against the perturbative influences. However, the detailed quantitative modeling of the Verwey transition in magnetite requires that such interactions be taken into account.

Use of the Straessler–Kittel theory of phase transitions: thermodynamic properties and application to the Verwey transition

Abstract Thermodynamic quantities derived from the Straessler–Kittel theory of phase transitions are derived and evaluated. The applicability of these results to the Verwey transition of magnetite is briefly discussed.

Paleointensity determination, magnetic domain structure, and selection criteria

This special section is focused on the impact of Kittel's theory of magnetic domains on rock magnetism, and this paper is focused on the implications for paleointensity determinations. Absolute paleointensity determination is based on specific properties of thermoremanent magnetizations (TRMs), mainly the linear dependence of a TRM's intensity on the inducing field and the additivity and independence of partial TRMs. These properties are exhibited by populations of noninteracting single‐domain (SD) grains but not by populations of multidomain grains. The situation is more complex in the pseudo‐single‐domain population, the most common one in natural rocks usable for paleointensity studies, which sometimes exhibit SD‐like behavior but not always. If the experiments were restricted to SD rocks, it would not be possible to find enough outcrops to have the spatial and temporal resolution needed for interpretation of the data in terms of geodynamo. However, as paleointensity determination is very time consuming, a preselection of suitable samples is necessary. A new procedure will be proposed using a direct test of partial TRM's properties with a fully automated experiment, fast to implement.

Introduction to Special Section: Frontiers in Rock Magnetism and Domain Theory—Basic and Applied

Kittel [1946], then of the erstwhile Bell Laboratories, predicted that magnetization reversal in small uniformly magnetized, or single‐domain particles will occur only if a large negative magnetic field (∼2×102 mT) was applied. In other words, once magnetically saturated and left in the ambient Earth's magnetic field, the single‐domain (SD) particles would exhibit high values of magnetic remanence, as well as high coercive forces. The terrestrial geomagnetic “recording media,” sediments and rocks with strongly ferrimagnetic particles in them, owe their magnetic stability or “memory” over geological timescales to such high coercive forces. Stacey (1963) applied Kittel's theory to calculate SD size thresholds for magnetite and hematite, the two most common iron oxide minerals in the Earth's crust.

The internal energy of ferroelectric domain structures characterized by pre-fractals of the pentad Cantor sets

The laminated ferroelectric 180 degrees domain structure optically observed in uniaxial ferroelectric KDP is periodic along the normal to the domain walls. Each half-period of the domain structure is characterized by the first four pre-fractals of the pentad Cantor sets. In order to investigate the thermodynamic stability of these pre-fractal domain structures, the electrostatic energy UE(n) and the wall energy UW(n) of the nth pre-fractal domain structure have been formulated as functions of the period in the framework of the Kittel theory. It is shown that under a constant period the electrostatic energy UE(3) of the third pre-fractal domain structure takes the least value while the wall energy UW(n) increases monotonically with increasing ordinal number n of the pre-fractal domain structure. The equilibrium half-period minimizing the internal energy including UE(n) and UW(n) is proportional to the square root of the thickness of a crystal plate. The equilibrium half-period increases monotonically with increasing ordinal number n of the pre-fractal domain structure. The equilibrium internal energy takes the least value in the zeroth pre-fractal domain structure and the second-least value in the second prefractal domain structure. The ratio of the latter to the former is nearly equal to unity, with the value of 1.10.

Peculiarities of the flucutational aftereffect following first order magnetostructural phase transition in (Fe,Ni)Rh alloy

Behaviours of the parameters S T and S H as well as irreversible “susceptibilities” k irr and d irr, which characterize the magnetic aftereffect of strain, are studied for the (Fe 0.965Ni 0.035) 0.49Rh 0.51 alloy in the region of temperature and magnetic field induced first order magnetostructural phase transitions. The universal constants kT/ r and kT/ q (which are independent of the shape of specimen and kinds of dependences of strain versus temperature and field) as well as activation volumes ι' T and ι' H are estimated from the experimental data. Inapplicability of the Kittel's theory for the description of the investigated phase transition in FeRh based alloys is discussed. The results obtained in the work are compared with known ones for high-coercivity materials.

Dipolar forces and stripe domain structure in ferromagnetic plates near the phase transition

The authors discuss the domain structure in a ferromagnetic plate with strong uniaxial anisotropy in the perpendicular direction. The inclusion of dipolar interactions produces anisotropic magnetisation correlations above the ordering temperature T0. They calculate the dependence of the width D of incipient domains on plate thickness L in the temperature region from slightly above to just below the transition temperature. They find that D approximately L1/3 in this region, in contrast to the Kittel relation D approximately L1/2 for stripe domains at temperatures well below T0. At low temperatures the width of the walls is very small compared with the distance between them and so they do not interact with each other. They determine the range of validity of the Kittel theory and they use this to suggest substances upon which it might be possible to make appropriate observations.

Structural phase transitions in lead zirconate

The structural phase transitions occurring in PbZrO3 have been investigated using X-ray diffraction and dielectric constant measurements. The lattice parameters, Pb ion displacements and ZrO6 octahedral tilts in the antiferroelectric, orthorhombic phase have been determined as functions of temperature between 20 and 228 degrees C using continuously recording X-ray diffraction photography. The temperature dependence of the Pb ion displacement is discussed with reference to the Kittel theory of antiferroelectricity. The lattice parameters and structural data derived from these results are used to describe the temperature dependence of the birefringence in PbZrO3.

Relaxation of rare earth and cobalt ions in polycrystalline yttrium iron garnet

Relaxation at room temperature of rare earths and cobalt ions in ferrimagnetic garnets has been studied by measuring the three linewidths : ΔH, resonance linewidth, Δ H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">eff</inf> , effective linewidth, ΔH <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</inf> , spinwave linewidth. These linewidths have been measured at 9 GHz Δ H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">eff</inf> , far from resonance, Δ H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> in perpendicular and in parallel pumps. From experimental results, relations between ΔH, Δ H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">eff</inf> and Δ H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> have been found which are in good agreement with the theory. The broadening δ (ΔH), due to the introduction of rare earths ions, allows the relaxation time τ (about 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-13</sup> sec) of some rare earths ions to be deduced from Kittel's theory. At room temperature, this theory explains the gyromagnetic properties of Sm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3+</sup> , Tb <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3+</sup> , Dy <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3+</sup> , Ho <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3+</sup> and Er <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3+</sup> ions. An adaptation of this theory allows us to explain also the behaviour of Gd <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3+</sup> ions. Relation between δ (ΔH) and the saturation magnetization 4π M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</inf> are explained for the Sm, Dy, Ho and Er ions by an extension of Kittel's theory, but this explanation is not unique.

Indirect nuclear interaction coupling constants for metallic copper

Ruderman and Kittel's theory is used to derive values for the indirect nuclear interaction coupling constants for copper. The ratio of the coupling constants for copper and silver is found to be 8.5 for all neighbours, in excellent agreement with experiment.

The antiferromagnetic-ferromagnetic transition in iron-rhodium alloys

The critical field required to induce this transition in Fe-Rh alloys close to the equiatomic composition has been measured using pulsed magnetic fields up to 280 kOe. Measurements of the thermal expansion and high field magnetization have also been carried out. It is found that application of Kittel's theory of exchange inversion to the results leads to serious numerical discrepancies. A four sublattice model is developed which indicates the importance of the excess entropy of the ferromagnetic over the antiferromagnetic phase in exciting the antiferromagnetic-ferromagnetic transition. The exchange iteration β between iron sublattices is small and negative and, although its lattice parameter dependence is large and positive, there is no suggestion of exchange inversion corresponding to a change in sign of β at the antiferromagnetic-ferromagnetic transition. Thus the ferromagnetic state is stabilized by the excess entropy even in the presence of antiferromagnetic interactions and this stability is maintained right up to the Curie temperature

Pulsed nuclear magnetic resonance in a tin single crystal

Pulsed nuclear magnetic resonance studies of an isotopically pure 119Sn single crystal were made at 78 °K. An attempt to observe anisotropy in the spin–lattice relaxation time was inconclusive, but gave T1T = (34 ± 2) ms deg as the average value. Spin–spin relaxation was dominated by the pseudo-exchange interaction. The numerical value for this, (2.0 ± 0.5) kc/s, is in agreement with the value calculated from the Rudermann–Kittel theory. A strong pseudodipolar interaction was also observed.

Entropy Changes of Ferromagnetic-Antiferromagnetic Transitions from Magnetic Measurements

The thermodynamics of magnetic processes in several compounds undergoing ferro- or ferrimagnetic to antiferromagnetic transitions are studied using critical field-transition temperature data obtained with a miniature-coil, pulsed-field technique. The zero-field transition entropy change in each material is calculated by means of the magnetic Clapeyron equation. The results are discussed in terms of the Kittel theory of exchange inversion. The variation of exchange constant α with lattice parameter c is found to vary over only a small range, and is roughly proportional to the reciprocal of the sublattice magnetization.