[Paper Review] Magnetostatic Spin Waves

Kirk T. McDonald|ArXiv.org|Dec 3, 2003

Magnetic Properties and Applications5 references3 citations

TL;DR

This paper investigates magnetostatic spin waves in ferromagnetic and ferrimagnetic materials under the magnetostatic approximation, where time-varying magnetic fields are consistent with static electric fields. It derives that such waves are transverse, rotating magnetostatic modes with resonant frequencies dependent on external and internal fields, and shows the electric field is negligible, validating the magnetostatic assumption for low-frequency, long-wavelength spin waves.

ABSTRACT

In general, Maxwell's equations require that a wave of magnetic field be accompanied by a wave of electric field, and vice versa. In magnetic media it is possible to have waves of magnetization with negligible electric field. We discuss an example of this based on ferromagnetic spin waves.

Motivation & Objective

• To investigate the existence of time-varying magnetic fields (B and H) in a regime where electric fields are static, consistent with a broadened definition of magnetostatics.
• To examine two physical systems—ferromagnetic spin waves and waves in a ferrite cylinder—where magnetization perturbations are small compared to a static magnetization.
• To validate the magnetostatic approximation by showing that the induced electric fields are negligible compared to magnetic fields, justifying the use of magnetostatic equations.
• To derive the dispersion relation and mode structure of magnetostatic spin waves, particularly in cylindrical geometries with rotational symmetry.
• To demonstrate that the waves are transverse, rotating with a resonant angular velocity, and that the electric field remains subdominant under the condition $ a\omega/c \ll 1 $.

Proposed method

• Applies the magnetostatic approximation where $ \partial \mathbf{E}/\partial t = 0 $, leading to $ \partial^2 \mathbf{B}/\partial t^2 = 0 $, implying linear time dependence of B, which is excluded on physical grounds, thus enforcing static B and H.
• Uses the condition $ \nabla \times \mathbf{H} = 0 $ in the absence of free currents to describe magnetostatic waves, with $ \mathbf{B} = \mathbf{H} + 4\pi\mathbf{M} $, and $ \mathbf{M} = \mathbf{M}_0 + \mathbf{m} $, where $ \mathbf{m} \ll \mathbf{M}_0 $.
• For ferromagnetic spin waves, models the magnetization dynamics via $ d\mathbf{M}/dt = \vec{\Omega} \times \mathbf{M} $, with $ \vec{\Omega} = \Gamma \mathbf{B}_{\text{eff}} $, and $ \mathbf{B}_{\text{eff}} = \alpha \nabla^2 \mathbf{m} $, leading to a quadratic dispersion relation $ \omega = \alpha \Gamma M_0 k^2 $.
• For the ferrite cylinder, solves the magnetostatic equations in cylindrical coordinates using separation of variables, assuming a mode structure $ \phi_n \propto r^n e^{i(n\theta - \omega t)} $, and applies boundary conditions on $ b_r $ and $ h_\theta $ to determine allowed modes.
• Derives the resonant frequency $ \omega = \Gamma(H_0 + 2\pi M_0) $ for the lowest mode ($ n=1 $), showing it corresponds to a rotating transverse magnetic field.
• Evaluates the electric field via Faraday’s law, showing $ \mathbf{e} = (\omega/c) \phi \hat{\mathbf{z}} $, and confirms $ |e_z / b_r| \sim a\omega/(nc) \ll 1 $, validating the magnetostatic approximation.

Experimental results

Research questions

• RQ1 Can time-varying magnetic fields exist in a regime where electric fields are static, consistent with a broadened definition of magnetostatics?
• RQ2 What is the dispersion relation and mode structure of magnetostatic spin waves in a ferromagnet with weak spatially varying magnetization?
• RQ3 How do magnetostatic waves in a ferrite cylinder exhibit rotational symmetry and resonant frequencies under the magnetostatic approximation?
• RQ4 To what extent is the electric field induced by these waves negligible compared to the magnetic field, ensuring the validity of the magnetostatic approximation?
• RQ5 How does the motion of magnetization due to rotating waves lead to relativistic corrections in polarization and electric displacement, and are they negligible?

Key findings

• Magnetostatic spin waves in ferromagnets exhibit a quadratic dispersion relation $ \omega = \alpha \Gamma M_0 k^2 $, indicating low-frequency, long-wavelength behavior with $ \omega \ll ck $, justifying the magnetostatic approximation.
• In a ferrite cylinder, the lowest mode ($ n=1 $) corresponds to a transverse magnetic field that rotates about the $ z $-axis with angular velocity $ \omega $, and the magnetization also rotates with angular velocity $ \omega/n $.
• The resonant frequency of the mode is $ \omega = \Gamma(H_0 + 2\pi M_0) $, which depends on the external field $ H_0 $ and the internal demagnetizing field $ 2\pi M_0 $, with $ N_z = 0 $ for a cylinder.
• The electric field associated with the wave is $ \mathbf{e} = (\omega/c) \phi \hat{\mathbf{z}} $, and its magnitude is suppressed by a factor $ a\omega/(nc) \ll 1 $, confirming the magnetostatic approximation is valid.
• The electric displacement field $ \mathbf{d} $ is related to the electric field and relativistic polarization via $ \mathbf{d} = \mathbf{e} + 4\pi\mathbf{p} $, and the resulting violation of $ \nabla \times \mathbf{H} = 0 $ is second-order in $ a\omega/c $, confirming consistency.
• For $ n > 0 $, the scalar potential $ \phi_n $ rotates with angular velocity $ \omega/n $, and the condition $ v(r=a) = a\omega/n \ll c $ ensures relativistic consistency, which is satisfied in typical experiments.