Spin wave

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In

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

${\displaystyle {\mathcal {H}}=-{\frac {1}{2}}J\sum _{i,j}\mathbf {S} _{i}\cdot \mathbf {S} _{j}-g\mu _{\rm {B}}\sum _{i}\mathbf {H} \cdot \mathbf {S} _{i}}$

where J is the

${\displaystyle \mathbf {S} _{t}=\mathbf {S} \times \mathbf {S} _{xx}.}$

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 ${\displaystyle |0\rangle }$ is that in which all spins are aligned parallel with the field H. That ${\displaystyle |0\rangle }$ is an eigenstate of ${\displaystyle {\mathcal {H}}}$ can be verified by rewriting it in terms of the spin-raising and spin-lowering operators given by:

${\displaystyle S^{\pm }=S^{x}\pm iS^{y}}$

resulting in

${\displaystyle {\mathcal {H}}=-{\frac {1}{2}}J\sum _{i,j}S_{i}^{z}S_{j}^{z}-g\mu _{\rm {B}}H\sum _{i}S_{i}^{z}-{\frac {1}{4}}J\sum _{i,j}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})}$

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

${\displaystyle S_{i}^{z}|0\rangle =s|0\rangle }$

for the maximally aligned state, we find

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

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

${\displaystyle S_{i}^{z}|1\rangle =(s-1)|1\rangle ,}$

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 ${\displaystyle S_{i}^{+}}$ will increase the z-projection of the spin at position i back to its low-energy orientation, but the operator ${\displaystyle S_{j}^{-}}$ 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 ${\displaystyle |0\rangle }$ does not have the full rotational symmetry of the Hamiltonian ${\displaystyle {\mathcal {H}}}$, a phenomenon which is called spontaneous symmetry breaking.

In this model the magnetization

${\displaystyle M={\frac {N\mu _{\rm {B}}gs}{V}}}$

where V is the volume. The propagation of spin waves is described by the Landau-Lifshitz equation of motion:

${\displaystyle {\frac {d\mathbf {M} }{dt}}=-\gamma \mathbf {M} \times \mathbf {H} -{\frac {\lambda \mathbf {M} \times (\mathbf {M} \times \mathbf {H} )}{M^{2}}}}$

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

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

Spin waves are observed through four experimental methods:

• In
  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
  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.^[1] 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.^[2]
  The maximum magnon energy at the border of the surface Brillouin zone was 240 meV.

• Magnonics
• Holstein–Primakoff transformation
• Spin engineering