# Spin wave

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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 magnons, which are bosonic modes of the spin lattice that correspond roughly to the phonon excitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces a ferromagnet's spontaneous magnetization. The energies of spin waves are typically only in keeping with typical Curie points at room temperature and below.

## Theory An illustration of the precession of a spin wave with a wavelength that is eleven times the lattice constant about an applied magnetic field. The projection of the magnetization of the same spin wave along the chain direction as a function of distance along the spin chain.

The simplest way of understanding spin waves is to consider the Hamiltonian ${\mathcal {H}}$ for the Heisenberg ferromagnet:

${\mathcal {H}}=-{\frac {1}{2}}J\sum _{i,j}\mathbf {S} _{i}\cdot \mathbf {S} _{j}-g\mu _{\rm {B}}\sum _{i}\mathbf {H} \cdot \mathbf {S} _{i}$ where J is the exchange energy, the operators S represent the spins at Bravais lattice points, g is the Landé g-factor, μB is the Bohr magneton and H is the internal field which includes the external field plus any "molecular" field. Note that in the classical continuum case and in 1 + 1 dimensions Heisenberg ferromagnet equation has the form

$\mathbf {S} _{t}=\mathbf {S} \times \mathbf {S} _{xx}.$ In 1 + 1, 2 + 1 and 3 + 1 dimensions this equation admits several integrable and non-integrable extensions like the Landau-Lifshitz equation, the Ishimori equation and so on. For a ferromagnet J > 0 and the ground state of the Hamiltonian $|0\rangle$ is that in which all spins are aligned parallel with the field H. That $|0\rangle$ is an eigenstate of ${\mathcal {H}}$ can be verified by rewriting it in terms of the spin-raising and spin-lowering operators given by:

$S^{\pm }=S^{x}\pm iS^{y}$ resulting in

${\mathcal {H}}=-{\frac {1}{2}}J\sum _{i,j}S_{i}^{z}S_{j}^{z}-g\mu _{\rm {B}}H\sum _{i}S_{i}^{z}-{\frac {1}{4}}J\sum _{i,j}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})$ where z has been taken as the direction of the magnetic field. The spin-lowering operator S annihilates the state with minimum projection of spin along the z-axis, while the spin-raising operator S+ annihilates the ground state with maximum spin projection along the z-axis. Since

$S_{i}^{z}|0\rangle =s|0\rangle$ for the maximally aligned state, we find

${\mathcal {H}}|0\rangle =\left(-Js^{2}-g\mu _{\rm {B}}Hs\right)N|0\rangle$ 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 i rotated so that

$S_{i}^{z}|1\rangle =(s-1)|1\rangle ,$ 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 z-projection of the spin at position i back to its low-energy orientation, but the operator $S_{j}^{-}$ will lower the z-projection of the spin at position j. 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 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 ${\mathcal {H}}$ , a phenomenon which is called spontaneous symmetry breaking.

In this model the magnetization

$M={\frac {N\mu _{\rm {B}}gs}{V}}$ where V is the volume. The propagation of spin waves is described by the Landau-Lifshitz equation of motion:

${\frac {d\mathbf {M} }{dt}}=-\gamma \mathbf {M} \times \mathbf {H} -{\frac {\lambda \mathbf {M} \times (\mathbf {M} \times \mathbf {H} )}{M^{2}}}$ where γ is the gyromagnetic ratio and λ 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 λ are in many cases dominated by the eddy currents.

One important difference between phonons and magnons lies in their dispersion relations. The dispersion relation for phonons is to first order linear in wavevector k, namely ώ = ck, where ω is frequency, and c is the velocity of sound. Magnons have a parabolic dispersion relation: ώ = Ak2 where the parameter A represents a "spin stiffness." The k2 form is the third term of a Taylor expansion of a cosine term in the energy expression originating from the SiSj 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. Two adjacent spins in a solid with lattice constant a that participate in a mode with wavevector k have an angle between them equal to ka.

## Experimental observation

Spin waves are observed through four experimental methods: inelastic neutron scattering, inelastic light scattering (Brillouin scattering, Raman scattering and inelastic X-ray scattering), inelastic electron scattering (spin-resolved electron energy loss spectroscopy), and spin-wave resonance (ferromagnetic resonance). 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 phonons. Important inelastic neutron scattering facilities are present at the ISIS neutron source in Oxfordshire, UK, the Institut Laue-Langevin in Grenoble, France, the High Flux Isotope Reactor at Oak Ridge National Laboratory in Tennessee, USA, and at the National Institute of Standards and Technology in Maryland, USA. Brillouin scattering similarly measures the energy loss of photons (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, 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 microwaves, 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 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 factors Q of ferrite components used in microwave 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.

## Related Research Articles

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The classical XY model is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's n-vector model for n = 2.

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In quantum mechanics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates : In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In quantum physics, the spin–orbit interaction is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.

In chemistry and physics, the exchange interaction is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical force, it is not a true force as it lacks a force carrier.

In quantum mechanics, Landau quantization refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, called Landau levels. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau.

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This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

In quantum mechanics, orbital magnetization, Morb, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, Mspin, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non-magnetic material by an applied magnetic field.

Magnetic resonance is a quantum mechanical resonant effect that can appear when a magnetic dipole is exposed to a static magnetic field and perturbed with another, oscillating electromagnetic field. Due to the static field, the dipole can assume a number of discrete energy eigenstates, depending on the value of its angular momentum quantum number. The oscillating field can then make the dipole transit between its energy states with a certain probability and at a certain rate. The overall transition probability will depend on the field's frequency and the rate will depend on its amplitude. When the frequency of that field leads to the maximum possible transition probability between two states, a magnetic resonance has been achieved. In that case, the energy of the photons composing the oscillating field matches the energy difference between said states. If the dipole is tickled with a field oscillating far from resonance, it is unlikely to transition. That is analogous to other resonant effects, such as with the forced harmonic oscillator. The periodic transition between the different states is called Rabi cycle and the rate at which that happens is called Rabi frequency. The Rabi frequency should not be confused with the field's own frequency. Since many atomic nuclei species can behave as a magnetic dipole, this resonance technique is the basis of nuclear magnetic resonance, including nuclear magnetic resonance imaging and nuclear magnetic resonance spectroscopy.

In condensed matter and atomic physics, Van Vleck paramagnetism refers to a positive and temperature-independent contribution to the magnetic susceptibility of a material, derived from second order corrections to the Zeeman interaction. The quantum mechanical theory was developed by John Hasbrouck Van Vleck between the 1920s and the 1930s to explain the magnetic response of gaseous nitric oxide and of rare-earth salts. Alongside other magnetic effects like Paul Langevin's formulas for paramagnetism and diamagnetism, Van Vleck discovered an additional paramagnetic contribution of the same order as Langevin's diamagnetism. Van Vleck contribution is usually important for systems with one electron short of being half filled and this contribution vanishes for elements with closed shells.

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