Hybridization of Spin and Plasma Waves in Josephson Tunnel Junctions Containing a Ferromagnetic Layer

We study dynamics of tunnel Josephson junctions with a thin ferromagnetic layer $F$ [superconductor-insulator-ferromagnet-superconductor ($SIFS$) junctions]. On the basis of derived equations relating the superconducting phase and magnetic moment to each other we analyze collective excitations in the system and find a new mode which is a hybrid of plasmalike and spin waves. The latter are coupled together in a broad range of parameters characterizing the system. Using the solution describing the collective modes we demonstrate that besides the Fiske steps new peaks appear on the $I\mathrm{\text{−}}V$ characteristics due to oscillations of the magnetic moment $M$ in the ferromagnetic layer. Thus, by measuring the $I\mathrm{\text{−}}V$ curve of the $SIFS$ junctions, one can extract information about the spectrum of spin excitations in the ferromagnet $F$.

Figure 1

(a) Schematic picture of the system under consideration. The thick line between the $S$ and $F$ layers means an insulating layer; (b) Spectrum of coupled spin and plasmalike modes. On the vertical and horizontal axes we plot the quantity $(\omega /{\Omega }_J{)}^2$, and $(k{l}_J{)}^2$, respectively. The following parameters are chosen: $({\Omega }_J/{\Omega }_M{)}^2=2$, $(l_J/l_M)=1$, and $s=0.05$.

Figure 2

Correction to the $I\mathrm{\text{−}}V$ characteristics due to interaction of Josephson oscillations and collective modes. The normalized voltage is defined as $V_{\mathrm{norm}}={\Omega }_V/{\Omega }_J$. Solid curves correspond to a small parameter $s\equiv 2({\tilde{d}}_F/\beta {\lambda }_L)=0.05$ (negligible effect of the $F$ layer), point curves correspond to $s=0.4$. The plots are presented for the following parameters: (a) ${\Omega }_M/{\Omega }_J=2$, ${\kappa }_H{l}_J=2$; (b) ${\Omega }_M/{\Omega }_J=1$, ${\kappa }_H{l}_J=4$. The damping coefficients are ${\gamma }_J/{\Omega }_J=0.2$, ${\gamma }_M/{\Omega }_J=0.1$.