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A spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying
collective excitation In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum. For exa ...
s occur in magnetic lattices with
continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to ano ...
. From the equivalent quasiparticle point of view, spin waves are known as
magnon A magnon is a quasiparticle, a collective excitation of the electrons' spin structure in a crystal lattice. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. Magnons carry a fixed amount of en ...
s, which are bosonic modes of the spin lattice that correspond roughly to the
phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechani ...
excitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces a
ferromagnet Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
's
spontaneous magnetization Spontaneous magnetization is the appearance of an ordered spin state ( magnetization) at zero applied magnetic field in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or . Overview Heated to temper ...
. The energies of spin waves are typically only in keeping with typical Curie points at room temperature and below.


Theory

The simplest way of understanding spin waves is to consider the Hamiltonian \mathcal for the Heisenberg ferromagnet: :\mathcal = -\frac J \sum_ \mathbf_i \cdot \mathbf_j - g \mu_ \sum_i \mathbf \cdot \mathbf_i where is the exchange energy, the operators represent the
spins The spins (as in having "the spins")Diane Marie Leiva. ''The Florida State University College of Education''Women's Voices on College Drinking: The First-Year College Experience"/ref> is an adverse reaction of intoxication that causes a state of v ...
at
Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
points, is the Landé -factor, is the
Bohr magneton In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum. The Bohr magneton, in SI units is defined as \mu_\m ...
and is the internal field which includes the external field plus any "molecular" field. Note that in the classical continuum case and in dimensions
Heisenberg ferromagnet A spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as ...
equation has the form :\mathbf_t=\mathbf \times \mathbf_. In and dimensions this equation admits several integrable and non-integrable extensions like the Landau-Lifshitz equation, the Ishimori equation and so on. For a ferromagnet and the ground state of the Hamiltonian , 0\rangle is that in which all spins are aligned parallel with the field . That , 0\rangle is an eigenstate of \mathcal can be verified by rewriting it in terms of the spin-raising and spin-lowering operators given by: :S^\pm = S^x \pm i S^y resulting in :\mathcal = -\frac J \sum_ S^z_i S^z_j - g \mu_ H \sum_i S^z_i - \frac J \sum_ (S^+_i S^-_j + S^-_i S^+_j) where has been taken as the direction of the magnetic field. The spin-lowering operator annihilates the state with minimum projection of spin along the -axis, while the spin-raising operator annihilates the ground state with maximum spin projection along the -axis. Since : S^z_i, 0\rangle = s, 0\rangle for the maximally aligned state, we find :\mathcal , 0\rangle = \left (-Js^2 -g \mu_ H s \right )N, 0\rangle where N is the total number of Bravais lattice sites. The proposition that the ground state is an eigenstate of the Hamiltonian is confirmed. One might guess that the first excited state of the Hamiltonian has one randomly selected spin at position rotated so that :S^z_i , 1\rangle = (s-1), 1\rangle, but in fact this arrangement of spins is not an eigenstate. The reason is that such a state is transformed by the spin raising and lowering operators. The operator S^+_i will increase the -projection of the spin at position back to its low-energy orientation, but the operator S^_j will lower the -projection of the spin at position . The combined effect of the two operators is therefore to propagate the rotated spin to a new position, which is a hint that the correct eigenstate is a spin wave, namely a superposition of states with one reduced spin. The exchange energy penalty associated with changing the orientation of one spin is reduced by spreading the disturbance over a long wavelength. The degree of misorientation of any two near-neighbor spins is thereby minimized. From this explanation one can see why the
Ising model The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
magnet with discrete symmetry has no spin waves: the notion of spreading a disturbance in the spin lattice over a long wavelength makes no sense when spins have only two possible orientations. The existence of low-energy excitations is related to the fact that in the absence of an external field, the spin system has an infinite number of degenerate ground states with infinitesimally different spin orientations. The existence of these ground states can be seen from the fact that the state , 0\rangle does not have the full rotational symmetry of the Hamiltonian \mathcal, a phenomenon which is called
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
.


Magnetization

In this model the magnetization :M = \frac where is the volume. The propagation of spin waves is described by the Landau-Lifshitz equation of motion: :\frac = -\gamma\mathbf M \times\mathbf H - \frac where is the gyromagnetic ratio and is the damping constant. The cross-products in this forbidding-looking equation show that the propagation of spin waves is governed by the torques generated by internal and external fields. (An equivalent form is the Landau-Lifshitz-Gilbert equation, which replaces the final term by a more "simply looking" equivalent one.) The first term on the right hand side of the equation describes the precession of the magnetization under the influence of the applied field, while the above-mentioned final term describes how the magnetization vector "spirals in" towards the field direction as time progresses. In metals the damping forces described by the constant are in many cases dominated by the eddy currents. One important difference between phonons and magnons lies in their
dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
s. The dispersion relation for phonons is to first order linear in wavevector , namely , where is frequency, and is the velocity of sound. Magnons have a parabolic dispersion relation: where the parameter represents a "
spin stiffness The spin stiffness or spin rigidity or helicity modulus or the "superfluid density" (for bosons the superfluid density is proportional to the spin stiffness) is a constant which represents the change in the ground state energy of a spin system as a ...
." The form is the third term of a Taylor expansion of a cosine term in the energy expression originating from the dot-product. The underlying reason for the difference in dispersion relation is that the order parameter (magnetization) for the ground-state in ferromagnets violates
time-reversal symmetry T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that entropy increases as time flows toward the futur ...
. Two adjacent spins in a solid with lattice constant that participate in a mode with wavevector have an angle between them equal to .


Experimental observation

Spin waves are observed through four experimental methods:
inelastic neutron scattering Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...
, inelastic
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
scattering (
Brillouin scattering Brillouin scattering (also known as Brillouin light scattering or BLS), named after Léon Brillouin, refers to the interaction of light with the material waves in a medium (e.g. electrostriction and magnetostriction). It is mediated by the refra ...
,
Raman scattering Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrational energy being gained by ...
and inelastic
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10  nanometers, corresponding to frequencies in the range 30&nb ...
scattering), inelastic electron scattering (spin-resolved electron energy loss spectroscopy), and spin-wave resonance (
ferromagnetic resonance Ferromagnetic resonance, or FMR, is coupling between an electromagnetic wave and the magnetization of a medium through which it passes. This coupling induces a significant loss of power of the wave. The power is absorbed by the precessing magneti ...
). In the first method the energy loss of a beam of neutrons that excite a magnon is measured, typically as a function of scattering vector (or equivalently momentum transfer), temperature and external magnetic field. Inelastic neutron scattering measurements can determine the dispersion curve for magnons just as they can for
phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechani ...
s. Important inelastic neutron scattering facilities are present at the ISIS neutron source in Oxfordshire, UK, the Institut Laue-Langevin in
Grenoble lat, Gratianopolis , commune status = Prefecture and commune , image = Panorama grenoble.png , image size = , caption = From upper left: Panorama of the city, Grenoble’s cable cars, place Saint- ...
, France, the
High Flux Isotope Reactor The High Flux Isotope Reactor (HFIR) is a nuclear research reactor at Oak Ridge National Laboratory (ORNL) in Oak Ridge, Tennessee, United States. Operating at 85 MW, HFIR is one of the highest flux reactor-based sources of neutrons for cond ...
at
Oak Ridge National Laboratory Oak Ridge National Laboratory (ORNL) is a U.S. multiprogram science and technology national laboratory sponsored by the U.S. Department of Energy (DOE) and administered, managed, and operated by UT–Battelle as a federally funded research an ...
in Tennessee, USA, and at the
National Institute of Standards and Technology The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical s ...
in Maryland, USA. Brillouin scattering similarly measures the energy loss of
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s (usually at a convenient visible wavelength) reflected from or transmitted through a magnetic material. Brillouin spectroscopy is similar to the more widely known
Raman scattering Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrational energy being gained by ...
, but probes a lower energy and has a superior energy resolution in order to be able to detect the meV energy of magnons. Ferromagnetic (or antiferromagnetic) resonance instead measures the absorption of
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
s, incident on a magnetic material, by spin waves, typically as a function of angle, temperature and applied field. Ferromagnetic resonance is a convenient laboratory method for determining the effect of
magnetocrystalline anisotropy In physics, a ferromagnetic material is said to have magnetocrystalline anisotropy if it takes more energy to magnetize it in certain directions than in others. These directions are usually related to the principal axes of its crystal lattice. I ...
on the dispersion of spin waves. One group at the Max Planck Institute of Microstructure Physics in Halle, Germany proved that by using spin polarized electron energy loss spectroscopy (SPEELS), very high energy surface magnons can be excited. This technique allows one to probe the dispersion of magnons in the ultrathin ferromagnetic films. The first experiment was performed for a 5 ML Fe film. With momentum resolution, the magnon dispersion was explored for an 8 ML fcc Co film on Cu(001) and an 8 ML hcp Co on W(110), respectively. The maximum magnon energy at the border of the surface Brillouin zone was 240 meV.


Practical significance

When magnetoelectronic devices are operated at high frequencies, the generation of spin waves can be an important energy loss mechanism. Spin wave generation limits the linewidths and therefore the
quality factor In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy ...
s ''Q'' of ferrite components used in
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
devices. The reciprocal of the lowest frequency of the characteristic spin waves of a magnetic material gives a time scale for the switching of a device based on that material.


See also

*
Magnonics Magnonics is an emerging field of modern magnetism, which can be considered a sub-field of modern solid state physics. Magnonics combines the study of waves and magnetism. Its main aim is to investigate the behaviour of spin waves in nano-structure ...
*
Holstein–Primakoff transformation The Holstein–Primakoff transformation in quantum mechanics is a mapping to the spin operators from boson creation and annihilation operators, effectively truncating their infinite-dimensional Fock space to finite-dimensional subspaces. One impo ...
*
Spin engineering Spin engineering describes the control and manipulation of quantum spin systems to develop devices and materials. This includes the use of the spin degrees of freedom as a probe for spin based phenomena. Because of the basic importance of quant ...


References

* * * * {{Refend


External links


Spin Waves
Biennial International Symposium for discussion of the latest advances in fundamental studies of dynamic properties of various magnetically ordered materials.

performing Brillouin scattering measurements. Magnetic ordering Waves de:Spinwelle pl:Fale spinowe