Zeeman Energy Term Research Articles

Influence of magnetic external field and particle size on the formation of a single domain state

Magnetic particles are widely employed in applications ranging from medical to industrial. Their magnetic state, a single-domain versus a multi-domain configuration, depends on their size as well as the strength and frequency of an applied magnetic field. The critical size of a single-domain ferromagnetic/ferrimagnetic particle can be determined by minimizing the total free energy, which should include the Zeeman energy term when an external field is applied. To study the effect of an applied field on the critical size of single-domain particles as a function of their size, we performed numerical simulations using MuMax3 software for Fe, Co, Fe3O4, MgZn ferrites, and BaFe12O19. These materials exhibit a wide range of values for magnetic properties such as saturation magnetization and anisotropy. We determined the critical field at which a given sized particle becomes a single-domain magnetic particle. Our MuMax3 calculations for an iron spherical particle were compared with Kondorskii’s analytical theory.

Magnetostatic dipolar energy of large periodic Ni fcc nanowires, slabs and spheres

Abstract The computational effort to calculate the magnetostatic dipolar energy, MDE, of a periodic cell of N magnetic moments is an O(N2) task. Compared with the calculation of the Exchange and Zeeman energy terms, this is the most computationally expensive part of the atomistic simulations of the magnetic properties of large periodic magnetic systems. Two strategies to reduce the computational effort have been studied: An analysis of the traditional Ewald method to calculate the MDE of periodic systems and parallel calculations. The detailed analysis reveals that, for certain types of periodic systems, there are many matrix elements of the Ewald method identical to other elements, due to some symmetry properties of the periodic systems. Computation timing experiments of the MDE of large periodic Ni fcc nanowires, slabs and spheres, up to 32,000 magnetic moments in the periodic cell, have been carried out and they show that the number of matrix elements that should be calculated is approximately equal to N, instead of N2/2, if these symmetries are used, and that the computation time decreases in an important amount. The time complexity of the analysis of the symmetries is O(N3), increasing the time complexity of the traditional Ewald method. MDE is a very small energy and therefore, the usual required precision of the calculation of the MDE is so high, about 10−6 eV/cell, that the calculations of large periodic magnetic systems are very expensive and the use of the symmetries reduces, in practical terms, the computation time of the MDE in a significant amount, in spite of the increase of the time complexity. The second strategy consists of parallel calculations of the MDE without using the symmetries of the periodic systems. The parallel calculations have been compared with serial calculations that use the symmetries.

Effects of a magnetic field on the optoelectronic properties of mono- and bi-layer transition metal dichalcogenides

We report a theoretical study on the effects of an external magnetic field on the electronic and optical properties of mono- and bi-layer MoS2, and bilayer MoS2/WS2 heterostructures. The direction of an applied magnetic field determines (i) the strength of the coupling between the atomic orbital moments of the structure and the field, and (ii) Zeeman contribution to the splitting of the VB and CB levels with the amplitude of the applied field. Whereas for a magnetic field applied perpendicular to the structure, calculated real part of optical conductivity reveals optical transitions red-shifted compared to zero magnetic field case, conductivity is unaffected by the the amplitude of an in-plane applied magnetic field. We show that through modifying atomic d-orbitals conduction and valence band states by an applied electric field, we can determine and control the impact of a magnetic field on the optical response of these materials. We employ our parametrized tight-binding model with non-orthogonal sp3d5 orbitals and spin–orbit coupling, which was advanced to include the effects of an external magnetic field through Peierls substitution in the wave vector and the Zeeman energy term.

Creation and Annihilation of Skyrmions in the Frustrated Magnets with Competing Exchange Interactions

In triangular-lattice magnets, the coexistence of third-neighbor antiferromagnetic and nearest-neighbor ferromagnetic exchange interactions can induce rich magnetic phases including noncoplanar skyrmion crystals. Based on Monte Carlo simulation, we studied the dependence of magnetic phase transition on exchange interaction strength. Under the consideration of uniaxial anisotropy and magnetic field both perpendicular to the film plane, a large antiferromagnetic exchange interaction induces a high frustration. When the value of antiferromagnetic exchange interaction is one and a half times larger than the ferromagnetic one, a magnetic phase composed of canting spin stripes, never observed in the chiral magnets, forms. Interestingly, different canting spin stripes along three 120 degree propagation directions may coexist randomly in a magnetic phase, attesting that the canting spin stripes are three-fold degenerate states akin to helices and the multiple state of canting spin stripes is a circular configuration with zero skyrmion charge number. Moreover, skyrmions and antiskyrmions can be observed simultaneously in the configuration at the low temperature nearly close to 0 K, and their configuration and diameter properties are discussed. Finally, the mechanisms of skyrmion creation and annihilation are properly interpreted by comparing exchange and Zeeman energy terms.

Correlation of exchange bias with magneto-structural effects across the compensation temperature of Co(Cr1–xFex)2O4 (x = 0.05 and 0.075)

A small amount of Fe (5% and 7.5%) substitution in the Cr-site of the multiferroic compound CoCr2O4 leads to a magnetization reversal. In these compounds, we report a sign change in the exchange bias across the compensation temperature, accompanied by a non-monotonic change in the local moments across the compensation temperature. Such non-monotonic change in the magnetic moments is triggered by a similar change in the lattice structure. We relate here the sign change of exchange bias with that of the crystalline energy of the lattice and the Zeeman energy term arising from the anti-site disorder.

Chiral phase transition in a planar four-Fermi model in a tilted magnetic field

We study a planar four-Fermi Gross-Neveu model in the presence of a tilted magnetic field, with components parallel and perpendicular to the system's plane. We determine how this combination of magnetic field components, when applied simultaneously, affects the phase diagram of the model. It is shown that each component of the magnetic field causes a competing effect on the chiral symmetry in these fermionic systems. While the perpendicular component of the magnetic field tends to make the chiral symmetry breaking stronger, the effect of the parallel component of the field in these planar systems is to weaken the chiral symmetry through the enhancement of the Zeeman energy term. We show that this competing effect, when combined also with temperature and chemical potential, can lead to a rich phase diagram, with the emergence of multiple critical points and reentrant phase transitions. We also study how the presence of these multiple critical points and reentrant phases can manifest in the quantum Hall effect. Our results provide a possible way to probe experimentally chiral symmetry breaking and the corresponding emergence of a gap (i.e., the presence of a nonvanishing chiral vacuum expectation value) in planar condensed matter systems of current interest.

Numerical methods for computing the ground state of spin-1 Bose-Einstein condensates in a uniform magnetic field

We propose efficient and accurate numerical methods for computing the ground-state solution of spin-1 Bose-Einstein condensates subjected to a uniform magnetic field. The key idea in designing the numerical method is based on the normalized gradient flow with the introduction of a third normalization condition, together with two physical constraints on the conservation of total mass and conservation of total magnetization. Different treatments of the Zeeman energy terms are found to yield different numerical accuracies and stabilities. Numerical comparison between different numerical schemes is made, and the best scheme is identified. The numerical scheme is then applied to compute the condensate ground state in a harmonic plus optical lattice potential, and the effect of the periodic potential, in particular to the relative population of each hyperfine component, is investigated through comparison to the condensate ground state in a pure harmonic trap.

Dynamics of Spin-2 Bose–EinsteinCondensates

We numerically simulate the dynamics of a spin-2 Bose–Einsteincondensate. We find that the initial phase plays an important rolein the spin component oscillations. The spin mixing processes canfully cancel out due to quantum interference when taking someinitial special phase. In all the spin mixing processes, the totalspin is conversed. When the initial population is mainly occupiedby a component with the maximal or minimal magnetic quantumnumber, the oscillations of spin components cannot happen due tothe total spin conversation. The presence of quadratic Zeemanenergy terms suppresses some spin mixing processes so that theoscillations of spin components are suppressed in some initialspin configuration. However, the linear Zeeman energy terms haveno effects on the spin mixing processes.

Bragg-Williams model of Fe-Co order-disorder phase transformations in a strong magnetic field

A modified Bragg-Williams statistical model with nearest-neighbor magnetic exchange and chemical interactions and an additional Zeeman energy term has been used to predict an increase in the critical ordering temperature (Torder) of Fe-Co alloys in the presence of a large magnetic field (ΔTorder∼20K near equiatomic composition for H=100T). The magnitude of the predicted ΔTorder increases with increasing field and is only predicted to be significant (ΔTorder≳2−3K near equiatomic) for H≳10T. The predicted shift in the ordering temperature is due to the higher average exchange interaction per atom in the ordered phase resulting in a larger theoretical Curie temperature as compared to the disordered phase. The dependence of the average magnetic moment of Fe on the degree of order was neglected in this work, however it is expected to contribute to an additional shift in Torder. Theoretical magnetization versus temperature curves are also presented for equiatomic FeCo subject to uniform applied magnetic fields.

The perturbative construction of the effective soft‐spin Hamiltonian of the system of magnetized nano‐loops

AbstractThe effective large‐scale Hamiltonian of a planar system of nano‐loops in a weakly excited flux‐closed magnetized state has been constructed by means of a perturbative technique based on micromagnetic theory. The Hamiltonian is written by means of two classes of collective variables: the continuous soft spins and discrete vorticity charges. Analytical and numerical calculations of the inter‐loop magnetostatic energy are compared for a pair of magnetic nano‐loops. The transformation from small‐scale to collective variables is performed for intra‐loop exchange‐coupling, magnetostatic and Zeeman energy terms. Evidence of correlations of uniform vortex charges in low‐energy configurations is uncovered numerically. The generalization of the perturbative method that deals with more realistic out‐of‐plane excitations is also considered. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Monte Carlo model of the MCD behavior of hematite colloids

We made Monte Carlo simulations of the energy distribution and magnetic moment contributions of surface and core sites in a model particle of spherical shape, intended as a simple representation of hematite colloids. We included exchange interactions between nearest neighbors, plus anisotropy and Zeeman energy terms. Model particles built with origin between lattice sites show a temperature dependence of magnetization consistent with previous measurements and explain the puzzling results of magnetic circular dichroism in samples of nanostructured hematite recently obtained with synchrotron radiation.

Quantum phase interference for quantum tunneling in spin systems

The point-particle-like Hamiltonian of a biaxial spin particle with external magnetic field along the hard axis is obtained in terms of the potential field description of spin systems with exact spin-coordinate correspondence. The Zeeman energy term turns out to be an effective gauge potential which leads to a nonintegrable pha se of the Euclidean Feynman propagator. The phase interference between clockwise and anticlockwise under barrier propagations is recognized explicitly as the Aharonov-Bohm effect. An additional phase which is significant for quantum phase interference is discovered with the quantum theory of spin systems besides the known phase obtained with the semiclassical treatment of spin. We also show the energ y dependence of the effect and obtain the tunneling splitting at excited states with the help of periodic instantons.

Dependence of magnetization reversal on the crystallite size in MnBi thin films: Experiment, theory, and computer simulation

A micromagnetic model is proposed to explain the magnetization reversal of thermally evaporated MnBi thin films. The model assumes that the films consist of grains on a square lattice with perpendicular magnetic anisotropy. In order to describe the magnetic reversal of a particular grain, dipole coupling, wall energy, Zeeman energy, and an energy barrier for the reversal of a single grain are taken into account. Disorder is included by random fluctuations of the lateral grain size in the Zeeman-energy term. The simulation is carried out by using a Monte Carlo method. The experimentally observed decrease in coercivity with increasing film thickness is accurately described by the model and can be explained by an increase in lateral grain size. The asymmetry of the slope of the hysteresis curve of thicker MnBi films can be understood in terms of disorder. In the presence of fluctuations in grain size, large grains reverse more easily than small ones leading to an asymmetry of the slope of the hysteresis curve. \textcopyright{} 1996 The American Physical Society.

Zeeman energy, interference and Neutron Spin Echo: A minimal theory

In some polarized neutron beam experiments, such as Neutron Spin Echo spectroscopy, it is essential to follow exactly the minute changes of the neutron kinetic energy due to interaction with the magnetic fields, e.g. in a spin flip process. It is emphasized that in the Amperian current loop model of microscopic magnetism, shown to be valid by the experiments, the Zeeman energy term can be treated as a bona fide potential energy. This allows us to describe both the spin and the spatial motion of the neutron by using simple notions of wave propagation across potential steps. In this picture Larmor precessions are represented as interference between two Stern-Gerlach states ↑ and ↓. The approach proves to be sufficient for the interpretation of all known polarized neutron beam phenomena, including crystal interferometry. It also leads to the surprising prediction that, as a pure quantum effect, under special circumstances the Larmor precession frequency can be different from its classical value.