Devil's staircases in the IV characteristics of superconductor/ferromagnet/superconductor Josephson junctions

We study the effect of coupling between the superconducting current and magnetization in the superconductor/ferromagnet/superconductor Josephson junction under an applied circularly polarized magnetic field. Manifestation of ferromagnetic resonance in the frequency dependence of the amplitude of the magnetization and the average critical current density is demonstrated. The IV characteristics show subharmonic steps that form devil's staircases, following a continued fraction algorithm. The origin of the found steps is related to the effect of the magnetization dynamics on the phase difference in the Josephson junction. The dynamics of our system is described by a generalized RCSJ model coupled to the Landau-Lifshitz-Gilbert equation. We justify analytically the appearance of the fractional steps in IV characteristics of the superconductor/ferromagnet/superconductor Josephson junction.

Figure 1

Geometry of the S/F/S Josephson junction with cross-sectional area $L_y{L}_z$ in uniform magnetic field $H_0$ and circularly polarized magnetic field $H_{ac}$.

Figure 2

(a) Manifestation of the FMR in the frequency dependence of the maximum of magnetization component $m_y$ and the average critical current density at bias current $I=1.16$. Lines added to guide the eye. (b) Frequency dependence of the maximum of magnetization component $m_y$ at different damping $\alpha$ and amplitude of circularly polarized magnetic field $h_{ac}$. Other parameters are the same as in (a). (c) Comparison with a linearized case at ${\epsilon }_J=0.02$.

Figure 3

(a) IV characteristic of S/F/S junction at ferromagnetic resonance. The case in Ref. [13] is shown by the dashed line for comparison, shifted by 0.8 to the right for clarity; (b) and (c) enlarge the parts of IV characteristic marked by rectangles in (a).

Figure 4

IV characteristics of the S/F/S junction at ferromagnetic resonance without oscillating electric field ($A=0$) and two characteristics at amplitudes $A=0.3$ and $A=1$. Here ${\mathrm{\Omega }}_r={\omega }_r/{\omega }_c$, other parameters are the same as in Fig. 3. For clarity, the curves at $A=0.3$ and $A=1$ have been shifted to the right, by $\mathrm{\Delta }I=0.6$ and $\mathrm{\Delta }I=1.2$, respectively, relative to the IV characteristic at $A=0$.

Figure 5

Different possibilities of the frequency locking excluding (red) and including (blue) the Josephson energy in the effective magnetic field.