Fiske steps and Abrikosov vortices in Josephson tunnel junctions

We present a theoretical and experimental study of the Fiske resonances in the current-voltage characteristics of “small” Josephson junctions with randomly distributed misaligned Abrikosov vortices. We obtained that in the presence of Abrikosov vortices, the resonant interaction of electromagnetic waves, excited inside a junction, with the ac Josephson current manifests itself by Fiske steps in a current-voltage characteristics even in the absence of external magnetic field. We found that the voltage positions of the Fiske steps are determined by a junction size but the Fiske-step magnitudes depend both on the density of trapped Abrikosov vortices and on their misalignment parameter. We measured the magnetic field dependence of both the amplitude of the first Fiske step and the Josephson critical current of low-dissipative small Nb-based Josephson tunnel junctions with artificially introduced Abrikosov vortices. A strong decay of the Josephson critical current and a weak nonmonotonic decrease in the first Fiske-step amplitude on the Abrikosov vortex density were observed. The experimentally observed dependencies are well described by the developed theory.

Figure 1

Schematic cross section of a Josephson tunnel junction with a misaligned Abrikosov vortex. $\delta$ is the misalignment length. ${\lambda }_{L_1}$ and ${\lambda }_{L_2}$ are the London penetration depths of the superconductors $S_1$ and $S_2$ separated by insulating layer $I$ forming the tunnel barrier. The orientation of the two magnetic field, $B_{\perp }$ and $B_{\parallel }$, is provided.

Figure 2

$I\text{−}V$ curves recorded at $B_{\parallel }=2.2\text{G}$ when the first Fiske step is maximized (open circles) and at $B_{\parallel }=12.7\text{G}$ when the Fiske step is suppressed (closed circles). The method allowing to extract both the amplitude of the first Fiske step, $I_F^{\left(1\right)}$, and its voltage position, $V_1$, is shown. The experimental first Fiske-step branch is fitted by a linear function in the current interval between $I_c^F/2$ and $I_c^F$ (dashed line). The experimental quasiparticle branch is fitted by a polynomial function (solid line). The voltage position $V_1$ is obtained from the point of the intersection of the dashed line with the solid line. The amplitude of the first Fiske step is $I_F^{\left(1\right)}=I_c^F-I_q$, where $I_q$ is the quasiparticle current at voltage $V_1$. $V^{I_c}$ is the threshold voltage to define the Josephson critical current.

Figure 3

The experimental $I_c\left(B_{\parallel }\right)$ curve (open circles) and theoretical fit by Eq. (10) (solid line). The trapped Abrikosov vortices are absent.

Figure 4

Experimental $I_F^{\left(1\right)}\left(B_{\parallel }\right)$ curve for the junction without trapped Abrikosov (open circles). Solid line represents the theoretical dependence calculated by Eqs. (4, 5) for $\gamma =1.1\times {10}^{10}{\text{s}}^{-1}$. Dashed line is the result of calculation without $J_1^2\left(\frac{a}{2}\right)$ term at $\gamma =4.3\times {10}^9{\text{s}}^{-1}$ according to the original theoretical analysis of Ref. 27.

Figure 5

$I_c\left(B_{\parallel }\right)$ curves measured after cooling of the junction in perpendicular magnetic field $B_{\perp }$ of various values.

Figure 6

$I_F^{\left(1\right)}\left(B_{\parallel }\right)$ curves measured after the junction was cooled in perpendicular magnetic field $B_{\perp }$ of various values.

Figure 7

Open squares: experimental dependence of the maximum Josephson critical current $I_c^{max}$ on the density of trapped Abrikosov vortices $n_A$; open circles: experimental dependence of the maximum amplitude of the first Fiske step $I_F^{\left(1\right),max}$ on the density of trapped Abrikosov vortices $n_A$. Solid line is the theoretical $I_F^{\left(1\right),max}\left(n_A\right)$ curve, calculated by Eqs. (7, 11). The values of $\gamma =2.2\times {10}^{10}{\text{s}}^{-1}$ and a mean misalignment parameter over all vortex densities $\overline{\delta }=0.8\mu \text{m}$ were used; full circles represent the theoretical $I_F^{\left(1\right),max}\left(n_A\right)$ dependence obtained by Eqs. (8, 11).