Localized Spin-wave Modes Research Articles

Dipole-exchange spin wave spectrum in an anisotropic ferromagnetic waveguide with a rectangular cross section

A theory has been constructed that strictly describes the spectrum of dipole-exchange spin waves in an arbitrarily magnetized anisotropic ferrite waveguide with a rectangular cross section. The theory takes into account the spatial inhomogeneity of the internal magnetic field in the waveguide cross section. The influence of parameters of the ferrite waveguide on the distribution of the internal magnetic field in the waveguide cross section is analyzed. The dispersion characteristics of two waveguide types most widely used in practice are investigated. The dipole-exchange spin wave spectra calculated for a transversely magnetized waveguide are presented and the distributions of the dynamic magnetization in the waveguide cross section for several types of volume and localized spin-wave modes are constructed.

Intrinsic Localized Spin-Wave Modes in an Anisotropic Ferromagnet with Dzyaloshinsky—Moriya Interaction

The existence and properties of intrinsic localized spin-wave modes in a ferromagnetic XXZ spin chain with Dzyaloshinsky—Moriya interaction are investigated analytically in the semiclassical limit. The model Hamiltonian is quantized by introducing the Dyson—Maleev transformation and the coherent state representation is chosen as the basic representation of the system. By making use of the method of multiple scales combined with a quasidiscreteness approximation, the equation of motion for the coherent-state amplitude is reduced to the nonlinear Schrödinger equation. It is shown that a bright intrinsic localized spin-wave mode whose eigenfrequency lies below the bottom of the magnon frequency band can exist in the ferromagnetic system. We also show that the system can produce a dark intrinsic localized spin-wave mode, i.e., nonpropagating kink, whose eigenfrequency is below the upper of the magnon frequency band. In addition, we find that the introduction of the Dzyaloshinsky—Moriya interaction changes wave numbers in the Brillouin-zone corresponding to the appearance of intrinsic localized spin-wave modes.

Non-stationary excitation of two localized spin-wave modes in a nano-contact spin torque oscillator

We measure and simulate micromagnetically a framework based upon a nano-contact spin torque oscillator where two distinct localized evanescent spin-wave modes can be detected. The resulting frequency spectrum is composed by two peaks, corresponding to the excited modes, which lie below the ferromagnetic resonance frequency, and a low-frequency tail, which we attribute to the non-stationary switching between these modes. By using Fourier, wavelet, and Hilbert-Huang transforms, we investigate the properties of these modes in time and spatial domains, together with their spatial distribution. The existence of an additional localized mode (which was neither predicted by theory nor by previous numerical and experimental findings) has to be attributed to the large influence of the current-induced Oersted field strength which, in the present setup, is of the same order of magnitude as the external field. As a further consequence, the excited spin-waves, contrarily to what usually assumed, do not possess cylindrical symmetry: the Oersted field induces these modes to be excited at the two opposite sides of the region beneath the nano-contact.

INTRINSIC LOCALIZED MODES IN QUANTUM FERROMAGNETIC ISING–HEISENBERG CHAINS WITH SINGLE-ION UNIAXIAL ANISOTROPY

Based on the coherent-state method combined with the Dyson–Maleev representation of spin operators, the existence and properties of intrinsic localized spin-wave modes in quantum ferromagnetic Ising–Heisenberg chains with single-ion uniaxial anisotropy are investigated analytically in the semiclassical limit. With the help of the multiple-scale method combined with semidiscrete approximation, the equation of motion for the coherent-state amplitude is reduced to the nonlinear Schrödinger equation. It is found that, at the center of the Brillouin zone, a bright type intrinsic localized spin-wave mode can exist below the bottom of the linear spin-wave spectrum. Besides, we show that, at the boundary of the Brillouin zone, a dark type intrinsic localized spin-wave mode appears above the top of the linear spin-wave spectrum, which is different from the resonant nonpropagating kink mode.

Localized domain-wall excitations in patterned magnetic dots probed by broadband ferromagnetic resonance

We investigate the magnetization dynamics in circular Permalloy dots with spatially separated magnetic vortices interconnected by domain walls (double vortex state). We identify a novel type of quasi one-dimensional (1D) localized spin wave modes confined along the domain walls, connecting each of two vortex cores with two edge half-antivortices. Variation of the mode eigenfrequencies with the dot sizes is in quantitative agreement with the developed model, which considers a dipolar origin of the localized 1D spin waves or so-called Winter's magnons [J. M. Winter, Phys. Rev. 124, 452 (1961)]. These spin waves are analogous to the displacement waves of strings and could be excited in a wide class of patterned magnetic nanostructures possessing domain walls, namely in triangular, square, circular, or elliptic soft magnetic dots.

Dipolar spin waves of lateral magnetic superlattices

We investigate the high-frequency dynamics of dysprosium and cobalt gratings fabricated at the surface of a $\text{GaAs}/{\text{Al}}_{0.33}{\text{Ga}}_{0.67}\text{As}$ heterojunction. We detect the collective and localized spin-wave modes of the grating by measuring the photovoltage and the photoresistance induced in the two-dimensional electron gas (2DEG). The magnetic excitations couple to the 2DEG through their stray magnetic field. We perform a spectroscopy of dipolar-exchange spin waves as a function of microwave power, temperature, the tilt angle of the applied magnetic field, and by varying the structural and material parameters to change the strength of dipolar interactions. The data reveal two types of spin waves. Dipolar magnetization waves propagate across the grating through the magnetostatic interaction between the stripes. We derive an analytical expression of their dispersion curve and obtain a good fit of the ferromagnetic resonance broadening from first principles. The second type is dipolar edge spin waves which manifest through a series of sharp resonances at lower magnetic field. These waves are confined near the pole surfaces and interact very little with neighboring stripes. We calculate the eigenfrequencies of the quantized modes and obtain a qualitative explanation of the low-field resonances. The fit yields a value of the exchange stiffness constant of dysprosium, $A=1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}\text{ }\text{J}\text{ }{\text{m}}^{\ensuremath{-}1}$. Our experiments show that photovoltage measurements in hybrid semiconductor-ferromagnetic structures provide a sensitive and noninvasive tool for probing the spin waves of small magnets (10--500 nm).

Surface modes with biquadratic coupling and uniaxial anisotropy in semi-infinite Heisenberg antiferromagnet

We address the role and the influence of both biquadratic exchange and uniaxial anisotropy in the study of bulk and surface magnetic excitations for a two-sublattices semi-infinite Heisenberg antiferromagnet. The calculations apply to combined effect and are based on a matching technique within the framework of both non-interacting spin-wave theory and random phase approximation. Emphasis has been laid upon the advantage of handling bulk and surface spin precessional fields, characterized by their symmetry in particular. Analytic expressions for the bulk and surface equations of motion are also derived which enables us to analyze qualitatively the nature of the modes by observing the response of the energy branches to the variation of both biquadratic exchange and single-ion anisotropy field as well as exchange parameters occurring on the surface. Two numerical applications of the theoretical formalism developed in this work are described. First concerns the situation when the free surface model is assumed. The second application considers the existence of surface perturbations and treats the effects of the presence of biquadratic exchange term on the bulk and surface spin-wave spectra showing two kinds of surface propagating modes which depend on the presence or not of the uniaxial anisotropy contribution and also on the variations of the bulk–surface exchange parameters. A significant conclusion which this work reached relates to the checking of the results obtained in the literature by using our own theoretical formalism. We show that this formalism is very well adapted to calculations of localized bulk and surface spin-wave modes associated with the simultaneous presence of biquadratic exchange and uniaxial anisotropy. Also, the results obtained for the bulk spin fluctuations field as well as for the magnetic surface waves, are found to be in qualitative agreement with those obtained in the literature. The analytic formulation presented in this work provides an accurate and practical scheme for calculating the surface spin dynamics under the influence of surface exchange parameters, biquadratic coupling and uniaxial anisotropy terms.

Propagating volume and localized spin wave modes on a lattice of circular magnetic antidots

The spectrum of spin wave excitations on a nanometric two-dimensional periodical array of circular holes in a magnetic film was measured using the Brillouin light scattering technique. Two modes with positive group velocity in the frequency range between 4 and 7GHz were observed. Our calculations show that these correspond to the two lowest modes propagating along the edges of an effective stripe waveguide, perpendicular to the applied field, whose width is equal to the interhole distance. Moreover, a number of higher-frequency modes has been measured and identified as volume excitations of the same effective stripe.

Magnetization dynamics in microscopic spin-valve elements: Shortcomings of the macrospin picture

We have studied ultrafast magnetodynamics in micropatterned spin-valve structures using time-resolved x-ray photoemission electron microscopy combined with x-ray magnetic circular dichroism. Exciting the system with ultrafast field pulses of $250\phantom{\rule{0.3em}{0ex}}\mathrm{ps}$ width, we find the dynamic response of the free layer to fall into two distinctly different contributions. On the one hand, it exhibits localized spin wave modes that strongly depend on the shape of the micropattern. A field pulse applied perpendicular to the exchange bias field along the diagonal of a square pattern leads to the excitation of a standing spin wave mode with two nodes along the field direction. This mode is strongly suppressed for a pattern of elliptical shape. On the other hand, the integrated response of the free layer roughly follows a single-spin model with a damping constant of $\ensuremath{\alpha}=0.025$ independent of the shape and resembles the response of a critically damped forced oscillator.

Excitation of self-localized spin-wave bullets by spin-polarized current in in-plane magnetized magnetic nanocontacts: A micromagnetic study

It was shown by micromagnetic simulation that a current-driven in-plane magnetized magnetic nanocontact, besides a quasilinear propagating (``Slonczewski'') spin-wave mode, can also support a nonlinear self-localized spin-wave ``bullet'' mode that exists in a much wider range of bias currents. The frequency of the bullet mode lies below the spectrum of linear propagating spin waves, which makes this mode evanescent and determines its spatial localization. The threshold current for the excitation of the self-localized bullet is substantially lower than for the linear propagating mode, but finite-amplitude initial perturbations of magnetization are necessary to generate a bullet in our numerical simulations, where thermal fluctuations are neglected. Consequently, in these simulations the hysteretic switching between the propagating and localized spin-wave modes is found when the bias current is varied.

Large-amplitude coherent spin waves excited by spin-polarized current in nanoscale spin valves

We present spectral measurements of spin-wave excitations driven by direct spinpolarized current in the free layer of nanoscale Ir20Mn80/Ni80Fe20/Cu/Ni80Fe20 spin valves. The measurements reveal that large-amplitude coherent spin wave modes are excited over a wide range of bias current. The frequency of these excitations exhibits a series of jumps as a function of current due to transitions between different localized nonlinear spin wave modes of the Ni80Fe20 nanomagnet. We find that micromagnetic simulations employing the Landau-Lifshitz-Gilbert equation of motion augmented by the Slonczewski spin torque term (LLGS) accurately describe the frequency of the current-driven excitations including the mode transition behavior. However LLGS simulations give qualitatively incorrect predictions for the amplitude of excited spin waves as a function of current.

Localized magnetic excitations of coupled impurities in a transverse Ising ferromagnet

A Green's function formalism is used to calculate the spectrum of excitations of two neighboring impurities implanted in a semi-infinite ferromagnetic. The equations of motion for the Green's functions are determined in the framework of the Ising model in a transverse field and results are given for the effect of the exchange coupling, position and orientation of the impurities on the spectra of localized spin wave modes.

Magnetic field dependence of quantized and localized spin wave modes in thin rectangularmagnetic dots

The magnetic field dependences of the frequencies of standing spin-wavemodes in a tangentially magnetized array of thin rectangular permalloy dots(800 × 550 nm) were measured experimentally by a Brillouin light scattering technique and calculatedtheoretically using an approximate size-dependent quantization of the spin-wavevectorcomponents in the dipole-exchange dispersion equation for spin waves propagating in acontinuous magnetic film. It was found that the inhomogeneous internal bias magnetic fieldof the dot has a strong influence on the profiles of the lowest spin-wave standing modes. Inaddition, the dynamic magnetization distributions found for both longitudinally andtransversely magnetized long magnetic stripes gives a good approximation for modedistributions in a rectangular dot magnetized along one of its in-plane sides. Anapproximate analytic theory of exchange-dominated spin-wave modes, strongly localizedalong the dot edge that is perpendicular to the bias magnetic field, is developed. Agood quantitative agreement with the results of the BLS experiment is found.

Localized spin modes on the insulating antiferromagnetic stepped surface model

We present a numerical method to calculate the spin fluctuation dynamics on a stepped surface. The model discussed here consists of an extended antiferromagnetic surface step at the surface boundary of an insulating antiferromagnetic substrate. The stepped surface is formed by two straight steps dropped randomly and the spins moments of the steps and the substrate are considered as local with no electronic effects. The full magnetic problem arising from the absence of translational symmetry due to the presence of a magnetic surface and steps is considered and studied. The calculations concern in particular the energies of localized spin-wave modes near the surface steps and employ the matching procedure in the random-phase approximation and mean field approximation. Only the nearest-neighbor exchange interactions are considered between the spins in the model. The analytical formalism presented here is adapted from an earlier work on the vibrational spectra of two isolated steps, a structure that can be considered as a low dimensional system and solved for the three dimensional evanescent crystal spin field in the bulk and the surface domains around the steps. This spin field arises from the breakdown of the magnetic translation symmetry of the system. The results are used to calculate the spin mode energies associated with the steps and surface terraces. We show the presence of localized acoustic and optical spin wave modes propagating along the surface and the steps as well as the interface surface-steps, their fields are also described as evanescent in the plane normal to the surface step layers and depend on the nature of the exchange interaction near the steps.

Spin fluctuations dynamics on the insulating antiferromagnetic stepped surface model

We present a numerical method to calculate the spin fluctuations dynamics on the stepped surface. The model discussed here consists in an extended antiferromagnetic surface steps on the surface boundary of an insulating antiferromagnetic substrate. The stepped surface is formed by two straight steps dropped randomly and the spins moments of the steps and the substrate are considered as local with no electronic effects. The calculations concern in particular the energies of localized spin-wave modes near the stepped surface and employ the matching procedure in the random-phase approximation and mean field approximation. Only nearest neighbors exchange interactions are considered between the spins in the model. The theoretical formalism presented here solves for the three dimensional evanescent crystal spin field in the bulk and the surface domains around the steps, that arises owing to the breakdown of magnetic translation symmetry of the system. The results are used to calculate the spin modes energies associated with the steps and surface terraces. We show the presence of acoustic and optical localized spin-wave modes propagating along the surface and the steps, their spin fluctuations fields are also described as evanescent in the plane normal to the stepped surface and depend on the nature of the exchange parameters near the steps.

SPIN WAVE MODES AND MAGNON SCATTERING AT SURFACE NANOSTRUCTURE ON 2D HEISENBERG FERROMAGNETS

We investigate the spin fluctuation dynamics and the scattering phenomena at adsorbed magnetic nanostructure on the surface of two-dimensional Heisenberg ferromagnets. The surface is considered as an infinite slab of two coupled atomic layers, and the nanostructure as an isolated monatomic chain on the surface of a cubic lattice. No electronic effects are considered but local changes in exchange fields between the localized spins of the nanostructure and the magnetically ordered substrate are assumed to be dominant. The mathematical framework of the matching method is used to analyze both the localization and the scattering phenomena at the nanostructure boundaries. The energy of spin wave localized modes arising from the absence of magnetic translation symmetry is determined. The transmission and reflection probabilities as well as the spin wave average transmittance across the nanostructure are calculated and numerical results are presented in a large range of scattering energies. It is shown that there are several localized spin wave modes associated with the nanostructure which may be either optical or acoustical modes (depending on the system parameters), and the coherent coupling between these localized spin modes and the propagating spin modes of the system leads to Fano resonances in the scattering spectra.

Spatially resolved dynamics of localized spin-wave modes in ferromagnetic wires.

We have observed localized spin-wave modes in individual thin-film ferromagnetic wires using time-resolved Kerr microscopy as a micron-scale spectroscopic probe. The localization is due to the internal field profile present when an external field is applied in the plane of the film and perpendicular to the long axis of the wire. Spatially resolved spectra demonstrate the existence of distinct modes at the edges of a rectangular wire. Spectral images clearly show the crossover of the two edge modes into a single mode in low applied fields, in agreement with the results of micromagnetic simulations.

Surface and step dynamics of a semi-infinite insulating antiferromagnet system

We have carried out a theoretical study of the localized spin-wave modes near the surface step of the insulating Heisenberg antiferromagnet. In this work, we study the full magnetic problem arising from the absence of translational symmetry due to the presence of a magnetic surface and step. The calculation concerns in particular the spin fluctuation dynamics and employs the matching procedure in the random-phase approximation. Only the nearest neighbours exchange interactions are considered between the spins in the model. The analytical formalism presented here determines the bulk and evanescent spin fluctuation fields in the two-dimensional plane normal to the surface and step regions. The results are used to calculate the localized modes of magnons associated with the step and surface terraces. The present model may be generalized to treat the spin fluctuations dynamics of other extended surface imperfections or nanostructures, provided they preserve the translation symmetry of the ordered spins along a direction in the surface boundary. We show the presence of acoustic and optical localized magnon modes propagating along the surface and the step as well as the interface surface-step, their fields is also described as evanescent in the plane normal to the surface step layer and depends on the nature of the bulk-surface and bulk-step coupling exchanges.

Theory of localized spin wave modes at surface steps on ferromagnetic films

A Green function matrix method has been developed to investigate the spin dynamics in low-dimensional Heisenberg ferromagnetic nanostructures. The theory is based on a generalization of earlier work on the spin wave spectra of magnetic films and their interfaces, and in the present case applications are made to the surface step produced at a perpendicular interface between two films of the same material but different thickness. Analytic expressions are obtained for the dispersion relation and spectral intensities of the localized surface and edge spin waves and the quantized bulk spin waves near a step. The effects of varying the step height and modifying the exchange interactions near the step are studied. These factors influence the number of localized modes, as well as their frequency, intensity, and attenuation.

Magnetic evolution of the amorphous and nanocrystalline phases and interphase coupling during the crystallization ofFe87Zr6B6Cu: A ferromagnetic resonance study

We present results of ferromagnetic resonance measurements on the alloy ${\mathrm{Fe}}_{87}{\mathrm{Zr}}_{6}{\mathrm{B}}_{6}\mathrm{Cu}.$ These show the changes occurring in the sample, both magnetically and structurally, during the crystallization process. In this alloy the amorphous phase gradually crystallizes, forming nanocrystallites of \ensuremath{\alpha}-Fe in a remaining ferromagnetic amorphous matrix. Results show evidence of a disordered and Fe-rich interphase region surrounding the Fe crystallites. This interphase region initially grows along with the growth of the Fe nanocrystals. However, in the latter stages of the Fe nanocrystallization, this phase also crystallizes into the Fe phase and other crystalline phases, and disappears at the end of the Fe crystallization and the beginning of second crystallization. Evidence of an interface localized spin wave mode at the boundary between the amorphous ferromagnetic and Fe phases in the early stages of crystallization is also observed. At higher annealing temperatures, we observe another resonance absorption feature. This is due to the onset of the second stage of crystallization, where FeZr and iron-boride phases form. The results of the ferromagnetic resonance are analyzed in a multiphase regime, where the importance of the magnetic coupling between the various magnetic phases is evident. A minimum of the in-plane anisotropy is found to coincide with the softest magnetic properties of this alloy.