Fluxon mobility in an array of asymmetric superconducting quantum interference devices

Fluxon dynamics in the dc-biased array of asymmetric three-junction superconducting quantum interference devices (SQUIDs) is investigated. The array of SQUIDs is described by the discrete double sine-Gordon equation. It appears that this equation possesses a finite set of velocities at which the fluxon propagates with the constant shape and without radiation. The signatures of these velocities appear on the respective current-voltage characteristics of the array as inaccessible voltage intervals (gaps). The critical depinning current has a clear minimum as a function of the asymmetry parameter (the ratio of the critical currents of the left and right junctions of the SQUID), which coincides with the minimum of the Peierls-Nabarro potential.

Figure 1

Schematic view of the elementary cell of the SQUID array. The detailed equivalent scheme is given by Fig. 1 of Ref. [6]. Also see text for details.

Figure 2

Potential Eq. (4) for the different values of the anisotropy parameter: $\eta =0$ (curve 1), $\eta =0.1$ (curve 2), $\eta =0.4$ (curve 3), $\eta =0.6$ (curve 4), and $\eta =4$ (curve 5).

Figure 3

Sliding velocities as a function of $\eta$ for $\kappa =1$ (curve 1), $\kappa =0.5$ (curve 2), and $\kappa =0.25$ (curve 3). The solid lines are used as a guide for the eye.

Figure 4

Kink center of mass evolution in the Hamiltonian limit for $\kappa =0.5,\eta =0.6,v_1=0.399493$ with the initial boost velocity (a) $v_{*}=0.97v_1$, (b) $v_{*}=0.99v_1$, (c), (d) $v_{*}=v_1$, and (e) $v_{*}=1.25v_1$. The red line in (d) corresponds to the center of mass moving with the sliding velocity $X\left(t\right)\sim v_{1}t$.

Figure 5

Current-voltage curves for $\kappa =0.5,N=30,\alpha =0.05$ (black $⧫$), $\alpha =0.02$ (red $\circ$), $\alpha =0.01$ (blue $\diamond$), and (a) $\eta =0.1$, (b) $\eta =0.3$, (c) $\eta =0.6$, and (d) $\eta =1.5$. The blue solid lines correspond to the respective CVC in the continuum limit, Eq. (13). The red vertical lines in (b)–(d) are given by $4\pi v_1/N$, where $v_1$ is the sliding velocity for the respective value of $\eta$ (see Fig. 3). The inset in (a) shows the distribution of ${\dot{\varphi }}_n$ that correspond to the branches of the CVC pointed by the arrows at $\alpha =0.05$. The distribution corresponding to the branch on the right is given by $⧫$ while $\oplus$ corresponds to the branch on the left. The inset in (d) shows the details of CVCs in the neighbourhood of the sliding velocity for $\alpha =0.02$ ($\oplus$), $\alpha =0.01$ ($\diamond$), and $\alpha =0.005$ ($\Delta$).

Figure 6

Current-voltage curves for $\kappa =0.25,\alpha =0.05,\eta =0.6$ (black $⧫$), and $\eta =0.3$ (red $\circ$). The inset shows the details of CVCs in the neighborhood of the sliding velocity for $\eta =0.6,\alpha =0.05$ (black $⧫$), $\alpha =0.02$ (red $\circ$) $\alpha =0.01$ (blue $\times$), and $\alpha =0.0075$ (green $+$). The red vertical line marks the voltage drop that corresponds to the respective sliding velocity in the Hamiltonian limit.

Figure 7

Largest Lyapunov exponent as a function of bias for three branches of the CVC, given in Fig. 5. The parameters are: $\kappa =0.5,\alpha =0.05$, and $\eta =0.6$. Different lines correspond to the different branches of the CVC. See text for details.

Figure 8

Power spectra of the ${\dot{\varphi }}_{N/2}$ variable for the different values of bias at (a) $\gamma =0.11,\overline{V}=0.1789$; (b) $\gamma =0.08,\overline{V}=0.08241$; (c) $\gamma =0.075,\overline{V}=0.07585$; (d) $\gamma =0.045,\overline{V}=0.04825$.

Figure 9

Critical depinning current ($\circ$, left scale) and the PN barrier (solid black line, right scale) as a function of the asymmetry parameter $\eta$ for $\kappa =0.25$. The red line is used as a guide for the eye. The inset shows the distribution of the ${\varphi }_n$ for the pinned fluxon at $\eta =0.26,\gamma =0.07111$ ($⧫$) and $\eta =0.31,\gamma =0.079$ ($\circ$). The solid lines are used as guides for the eye.