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Using surface plasmons to detect spin inertia

H. Y. Yuan

TL;DR

Spin dynamics in magnets has been described by the Landau-Lifshitz-Gilbert (LLG) equation, but recent experiments indicate an inertial nutation term, leading to nutation spin waves at THz frequencies. The authors propose to detect and quantify spin inertia by coupling THz nutation spin waves to graphene surface plasmons in a graphene$|$ferromagnet heterostructure, using an Otto geometry to excite and measure hybrid modes. They derive a complete dispersion relation for the hybrid magnon–plasmon modes from the inertial LLG and Maxwell equations, and show that the reflection spectrum of THz light exhibits a dip at the nutation frequency whose position directly determines the nutation time $\eta$. The approach is universal for magnetic insulators and may generalize to metals and antiferromagnets, enabling ultrafast, nanoscale spintronics and nanophotonics via spin-inertia effects.

Abstract

Recent experiments demonstrate that spin dynamics may acquire an inertial effect in a few metallic magnets, deviating from the traditional inertia-free dynamics. It remains an open question to ascertain the physical mechanisms and universality of the spin inertia across diverse magnetic systems. Here, we show that spin inertia generates nutation spin waves in the terahertz regime, which can hybridize with the surface plasmons in two-dimensional (2D) conducting materials such as graphene. By exciting hybrid spin wave-plasmon modes and analyzing the reflection spectrum of a 2D material$|$magnet heterostructure, we propose a method to quantitatively determine the strength of spin inertia in magnetic layers. Our approach is universally applicable to all types of magnetic insulators and could advance the future exploration of the magnitude and physical mechanism of spin inertia.

Using surface plasmons to detect spin inertia

TL;DR

Spin dynamics in magnets has been described by the Landau-Lifshitz-Gilbert (LLG) equation, but recent experiments indicate an inertial nutation term, leading to nutation spin waves at THz frequencies. The authors propose to detect and quantify spin inertia by coupling THz nutation spin waves to graphene surface plasmons in a grapheneferromagnet heterostructure, using an Otto geometry to excite and measure hybrid modes. They derive a complete dispersion relation for the hybrid magnon–plasmon modes from the inertial LLG and Maxwell equations, and show that the reflection spectrum of THz light exhibits a dip at the nutation frequency whose position directly determines the nutation time . The approach is universal for magnetic insulators and may generalize to metals and antiferromagnets, enabling ultrafast, nanoscale spintronics and nanophotonics via spin-inertia effects.

Abstract

Recent experiments demonstrate that spin dynamics may acquire an inertial effect in a few metallic magnets, deviating from the traditional inertia-free dynamics. It remains an open question to ascertain the physical mechanisms and universality of the spin inertia across diverse magnetic systems. Here, we show that spin inertia generates nutation spin waves in the terahertz regime, which can hybridize with the surface plasmons in two-dimensional (2D) conducting materials such as graphene. By exciting hybrid spin wave-plasmon modes and analyzing the reflection spectrum of a 2D materialmagnet heterostructure, we propose a method to quantitatively determine the strength of spin inertia in magnetic layers. Our approach is universally applicable to all types of magnetic insulators and could advance the future exploration of the magnitude and physical mechanism of spin inertia.

Paper Structure

This paper contains 1 section, 25 equations, 2 figures.

Table of Contents

  1. Appendix: Derivation of the dispersion relation

Figures (2)

  • Figure 1: (a) Cross-sectional view of a DE$|$GRA$|$FM$|$DE hybrid structure with $z$-axis being the thickness direction. The nutation spin waves and surface plasmons are coupled at the interface. The right panel sketches the nutation motion (blue curve) superimposed above the normal precession of magnetization on a sphere with magnetization length as the radius. The red and purple arrows sketch the directions of precessional and nutational torques, respectively. (b) Dispersion relation of the Damon-Eshbach mode (red line) and nutation surface mode (blue line). Parameters of yttrium iron garnet (YIG) are used: $\mu_0M_s=0.175~\mathrm{T},~\mu_0H_e=0.3~\mathrm{T},~d=1~\mu m$, $\eta=0.5~\mathrm{ps}$. (c) Dispersion relation of the hybrid plasmon-spin wave modes. $E_F=0.1~\mathrm{eV}$. The symbols are numerical solutions of Eq. \ref{['dispersion_hybrid_infite_fm']} while the lines represent the analytical results calculated based on Eq. \ref{['analytical_hybrid_excitation']}.
  • Figure 2: (a) Scheme of a DE$|$DE$|$GRA$|$FM hybrid structure. An electromagnetic wave inputs from medium 4 and penetrates into medium 3 as an evanescent wave, exciting the surface plasmons and spin waves at the interface of media 3 and 2. (b) Reflection spectrum of the system as a function of incident wave frequency. The orange line plots the amplitude of spin waves. (c) Local minimum position of the reflection spectrum as a function of nutation time $\eta$. The red line plots the theoretical nutation frequency according to Eq. (11a). $\theta=-67.58^\circ,~\eta=0.5~\mathrm{ps},d_{DE}=5~\mu m, \epsilon_4=14,\epsilon_3=2,\epsilon_2=10.8,~E_F=0.1\mathrm{eV},~\alpha=10^{-4},~\mu_0H_e=0.3~\mathrm{T}$. Both the dissipation of plasmons and magnons are included as $\Gamma=0.1~\mathrm{meV},~\alpha=10^{-4}$.