When the magnetization of a ferromagnet tilts away from its equilibrium direction, it precesses around the effective magnetic field. This motion can be observed as ferromagnetic resonance (FMR). Similar to a spinning top, the precession of magnetization decays with time owing to friction. This phenomenon, known as magnetic damping, occurs because the spin system transfers energy and angular momentum to its surroundings.

What is the cause of this damping? Spin–orbit coupling (SOC) plays an essential role in metallic ferromagnets. It couples the spin of an electron to its orbital motion, allowing the spin angular momentum to be transferred to other degrees of freedom, such as lattice vibrations. Consequently, the precession of the magnetization gradually relaxes toward the direction of the effective magnetic field.

An important question is how the magnetic damping changes with temperature. At low temperatures, the damping can be very strong and decreases rapidly as the temperature increases, reflecting the effects of electron scattering. This behavior is understood in terms of the electronic transitions associated with the SOC. However, the situation becomes more complicated near the Curie temperature (T_c), where the spontaneous magnetization disappears and strong spin fluctuations develop. How can the damping behavior at low temperatures be connected to that near and above T_c?

In this study, this question was examined using the general theory of spin dynamics. The key idea is that the relaxation rate (1/τ) that quantifies how rapidly the precession decays, is determined by the ratio of two quantities: the torque correlation function (F(T)), representing microscopic spin relaxation, and the transverse magnetic susceptibility (χ_₊) (T)), describing how easily the magnetization responds to a transverse magnetic field. This framework clarifies how the conventional microscopic theory of magnetic damping at low temperatures relates to a more general description of spin relaxation.

At low temperatures, the relaxation rate decreases rapidly with increasing temperature because electron scattering suppresses the contribution of electronic transitions within the same energy band. Below T_c, where the temperature dependence of F(T) is weak, the relaxation rate is approximately governed by 1/τ ∝ M^α-2(2 < α < 3). Near T_c, qualitative analysis suggests that SOC alone may not produce a pronounced critical peak, whereas magnetic dipole–dipole interactions can enhance damping through critical spin fluctuations. Above T_c , a Korringa-type behavior, whereby 1/τ ∝ T –T_c, is expected theoreticaally.

Thus, the present study provides a unified physical outlook of magnetic damping over a broad temperature range (low temperatures to T_c to the paramagnetic region). Clarifying the relation between microscopic electronic relaxation and the collective magnetic response explains how apparently different damping behaviors can be understood using a common theoretical framework.

(Written by Akimasa Sakuma)

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