Anderson-Higgs Mass of Magnons in Superconductor-Ferromagnet-Superconductor Systems

The Anderson-Higgs mechanism of mass generation is a generic concept in high-energy and condensed-matter physics. It shows up through the Meissner effect providing the expulsion of static and low-frequency magnetic fields from superconductors. However, it does not affect propagating electromagnetic waves with a spectrum gap determined by the plasma frequency, which is too large to be sensitive to the superconducting transition. Here we demonstrate the spectroscopic manifestation of the Anderson-Higgs mass, showing that it determines the spectrum gap of magnons in superconductor/ferromagnet/superconductor multilayers. Moreover, we show that this effect has been observed in recent experiments as a spontaneous ferromagnetic resonance frequency shift in such systems. Our theory explains many unusual experimental features and suggests effective controls over the magnon spectrum with tunable spectral gap and group-velocity reversal. These findings pave the way to a wide range of advanced functionalities for possible applications in magnonics.

Figure 1

(a) In the $F$ slab the in-plane magnetization rotation $\mathit{M}(t)$ is a zero-mass mode. (b) In the $S/F/S$ system the in-plane magnetization rotation $\mathit{M}(t)$ is a massive mode because of the the eddy supercurrents $\mathit{j}(t)$.

Figure 2

(a) Ferromagnetic resonance in $S/F/S$ system. Distributions of loop eddy currents ${\mathit{j}}_{\omega }$ and the magnetic field that they generate ${\mathit{H}}_{\omega }$ are shown schematically. (b) The magnetostatic mode in $S/F/S$ structure in Danone-Eschbach geometry with wave vector $\mathit{q}\perp {\mathit{M}}_0$.

Figure 3

(a),(b) Anderson-Higgs mass of magnons ${\mathrm{\Omega }}_{\mathrm{AH}}$ given by Eq. (16). (a) ${\mathrm{\Omega }}_{\mathrm{AH}}(T,d_F)$ normalized by ${\mathrm{\Omega }}_M=4\pi \gamma M_0$, showing saturation at $T\ll T_c$ and $d_F\gg \lambda$. (b) ${\mathrm{\Omega }}_{\mathrm{AH}}(T)$ for $d_F/{\lambda }_0=0.1,1,5$ shown by solid lines compared with ${\lambda }_0/\lambda (T)$ shown by the dashed line, where ${\lambda }_0=\lambda (T=0)$. (c) Comparison of $f_{\mathrm{FMR}}={\mathrm{\Omega }}_{\mathrm{FMR}}/2\pi$ given by Eq. (15) shown by solid lines with experimental data from Fig. 2(c) of Ref. [9] shown by dashed lines. Parameters are detailed in the text.

Figure 4

The evolution of dispersion ${\mathrm{\Omega }}_{\mathrm{mag}}(q)$ of surface spin waves in $S/F/S$ system at different temperatures at $H_0=0$. (a) $d_F=0.5{\lambda }_0$ and (b) $d_F=3{\lambda }_0$. In the latter case the dispersion changes from “forward” $v_g>0$ to “backward” $v_g<0$ with decreasing temperature. (c),(d) The set of curves ${\mathrm{\Omega }}_{\mathrm{mag}}(q)$ for different temperatures starting from $T=T_c$ (lower curve) to $T=0.5T_c$ (upper curve) with a step $0.01T_c$. The almost flat magnon spectrum shown by the red thick curve in (d) corresponds to the threshold at $T=0.98T_c$.