Interaction of Josephson and magnetic oscillations in Josephson tunnel junctions with a ferromagnetic layer

We study the dynamics of Josephson junctions with a thin ferromagnetic layer F [superconductor-ferromagnet-insulator-ferromagnet-superconductor (SFIFS) junctions]. In such junctions, the phase difference $\varphi$ of the superconductors and magnetization $M$ in the F layer are two dynamic parameters coupled to each other. We derive equations describing the dynamics of these two parameters and formulate the conditions of validity. The coupled Josephson plasma waves and oscillations of the magnetization $M$ affect the form of the current-voltage (I-V) characteristics in the presence of a weak magnetic field (Fiske steps). We calculate the modified Fiske steps and show that the magnetic degree of freedom not only changes the form of the Fiske steps but also the overall view of the I-V curve (new peaks related to the magnetic resonance appear). The I-V characteristics are shown for different lengths of the junction including those which correspond to the current experimental situation. We also calculate the power $P$ absorbed in the system if a microwave radiation with an ac in-plane magnetic field is applied (magnetic resonance). The derived formula for the power $P$ essentially differs from the one which describes the power absorption in an isolated ferromagnetic film. In particular, this formula describes the peaks related to the excitation of standing plasma waves as well as the peak associated with the magnetic resonance.

Figure 1

Schematic construction of a SFIFS junction of the “overlap” geometry.

Figure 2

Correction to the I-V characteristics of an SFIFS JJ in a weak external magnetic field due to interaction of Josephson oscillations and spin-wave modes. The correction is plotted as a function of normalized voltage $V_{\mathrm{norm}}={\Omega }_V/{\Omega }_J$ for different values of the parameter $s$. The figures are presented for the following parameters: (a) ${\Omega }_M/{\Omega }_J={\kappa }_H{l}_J=8\text{and}L/l_J=0.75$, (b) ${\Omega }_M/{\Omega }_J=8,{\kappa }_H{l}_J=12\text{and}L/l_J=0.75$, (c) ${\Omega }_M/{\Omega }_J={\kappa }_H{l}_J=8\text{,}\text{and}L/l_J=2$, and (d) ${\Omega }_M/{\Omega }_J=8,{\kappa }_H{l}_J=12,\text{and}L/l_J=2$. The damping coefficients are ${\gamma }_R/{\Omega }_J=0.4\text{and}{\gamma }_M/{\Omega }_J=0.3$.

Figure 3

Current correction $\delta \eta$ as a function of normalized voltage $V_{\mathrm{norm}}={\Omega }_V/{\Omega }_J$ for long JJ with normalized junction length $L/l_J=10$. The function $\delta \eta$ is displayed for (a) ${\Omega }_M/{\Omega }_J={\kappa }_H{l}_J=8$, (b) ${\Omega }_M/{\Omega }_J=8,{\kappa }_H{l}_J=12$, and for different values of the parameter $s$. The damping coefficients are ${\gamma }_R/{\Omega }_J=0.4\text{and}{\gamma }_M/{\Omega }_J=0.3$.

Figure 4

Spectrum of coupled spin and plasmalike modes. (a) The function $Z\left(q\right)={[\omega \left(q\right)/{\Omega }_J]}^2$ determined by Eq. (40) is shown for finite values of $s$ and in the inset, for the case $s=0$. (b) The dependence $\delta Z\left(\delta q\right)$ that represents the spectrum close to the crossing point is plotted according to Eq. (39). The following parameters are chosen: $Z_M={({\Omega }_M/{\Omega }_J)}^2=4\text{and}{l}_M/l_J=0.1$ .

Figure 5

Frequency dependence of the normalized absorbed power ${\mathcal{P}}_n$ in the JJ as a function of the normalized frequency $\omega /{\Omega }_J$ for different values of the parameter $s$ and for different damping coefficients ${\gamma }_R/{\Omega }_J\text{and}{\gamma }_M/{\Omega }_J$. The figures (a) and (b) are shown for the parameters $L_n=3,{\gamma }_M/{\Omega }_J=0.1,\text{and}{({\Omega }_M/{\Omega }_J)}^2=5$ and, respectively, (a) ${\gamma }_R/{\Omega }_J=0.1$ and (b) ${\gamma }_R/{\Omega }_J=1$. With regard to (c) and (d), we have chosen $L_n=1,{({\Omega }_M/{\Omega }_J)}^2=5$ and in (c) ${\gamma }_M/{\Omega }_J=0.1$.

Figure 6

Frequency dependence of the normalized absorbed power ${\mathcal{P}}_n$ as a function of the normalized frequency $\omega /{\Omega }_J$ for really short junctions. Here, too, the figures are displayed for different values of the parameter $s$ and for the choice ${\gamma }_R/{\Omega }_J=1,{\gamma }_M/{\Omega }_J=0.1,\text{and}{({\Omega }_M/{\Omega }_J)}^2=5$. In figure (a), $L_n=0.2,$ whereas in figure (b), $L_n=0.6$.