Nonlinear Viscous Vortex Motion in Two-Dimensional Josephson-Junction Arrays

When a vortex in a two-dimensional Josephson junction array is driven by a constant external current it may move as a particle in a viscous medium. Here we study the nature of this viscous motion. We model the junctions in a square array as resistively and capacitively shunted Josephson junctions and carry out numerical calculations of the current-voltage characteristics. We find that the current-voltage characteristics in the damped regime are well described by a model with a {\bf nonlinear} viscous force of the form $F_D=\eta(\dot y)\dot y={{A}\over {1+B\dot y}}\dot y$, where $\dot y$ is the vortex velocity, $\eta(\dot y)$ is the velocity dependent viscosity and $A$ and $B$ are constants for a fixed value of the Stewart-McCumber parameter. This result is found to apply also for triangular lattices in the overdamped regime. Further qualitative understanding of the nature of the nonlinear friction on the vortex motion is obtained from a graphic analysis of the microscopic vortex dynamics in the array. The consequences of having this type of nonlinear friction law are discussed and compared to previous theoretical and experimental studies.Comment: 14 pages RevTex, 9 Postscript figure

Giant Shapiro steps for two-dimensional Josephson-junction arrays with time-dependent Ginzburg-Landau dynamics

Two-dimensional Josephson junction arrays at zero temperature are investigated numerically within the resistively shunted junction (RSJ) model and the time-dependent Ginzburg-Landau (TDGL) model with global conservation of current implemented through the fluctuating twist boundary condition (FTBC). Fractional giant Shapiro steps are found for {\em both} the RSJ and TDGL cases. This implies that the local current conservation, on which the RSJ model is based, can be relaxed to the TDGL dynamics with only global current conservation, without changing the sequence of Shapiro steps. However, when the maximum widths of the steps are compared for the two models some qualitative differences are found at higher frequencies. The critical current is also calculated and comparisons with earlier results are made. It is found that the FTBC is a more adequate boundary condition than the conventional uniform current injection method because it minimizes the influence of the boundary.Comment: 6 pages including 4 figures in two columns, final versio

Orientational pinning and transverse voltage: Simulations and experiments in square Josephson junction arrays

We study the dependence of the transport properties of square Josephson Junctions arrays with the direction of the applied dc current, both experimentally and numerically. We present computational simulations of current-voltage curves at finite temperatures for a single vortex in the array ($Ha^2/\Phi_0=f=1/L^2$), and experimental measurements in $100\times1000$ arrays under a low magnetic field corresponding to $f\approx0.02$. We find that the transverse voltage vanishes only in the directions of maximum symmetry of the square lattice: the [10] and [01] direction (parallel bias) and the [11] direction (diagonal bias). For orientations different than the symmetry directions, we find a finite transverse voltage which depends strongly on the angle $\phi$ of the current. We find that vortex motion is pinned in the [10] direction ($\phi=0$), meaning that the voltage response is insensitive to small changes in the orientation of the current near $\phi=0$. We call this phenomenon orientational pinning. This leads to a finite transverse critical current for a bias at $\phi=0$ and to a transverse voltage for a bias at $\phi\not=0$. On the other hand, for diagonal bias in the [11] direction the behavior is highly unstable against small variations of $\phi$, leading to a rapid change from zero transverse voltage to a large transverse voltage within a few degrees. This last behavior is in good agreement with our measurements in arrays with a quasi-diagonal current drive.Comment: 9 pages, 9 figure

Vortex reflection at boundaries of Josephson-junction arrays

We study the propagation properties of a single vortex in square Josephson-junction arrays (JJA) with free boundaries and subject to an applied dc current. We model the dynamics of the JJA by the resistively and capacitively shunted junction (RCSJ) equations. For zero Stewart-McCumber parameter $\beta_c$ we find that the vortex always escapes from the array when it gets to the boundary. For $\beta_c\geq 2.5$ and for low currents we find that the vortex escapes, while for larger currents the vortex is reflected as an antivortex at one edge and the antivortex as a vortex at the other, leading to a stationary oscillatory state and to a non-zero time-averaged voltage. The escape and the reflection of a vortex at the array edges are qualitatively explained in terms of a coarse-grained model of a vortex interacting logarithmically with its image. We also discuss the case when the free boundaries are at $45$ degrees with respect to the direction of the vortex motion. Finally, we discuss the effect of self-induced magnetic fields by taking into account the full-range inductance matrix of the array, and find qualitatively equivalent results.Comment: 14 pages RevTex, 9 Postscript figure

Transverse phase-locking in fully frustrated Josephson junction arrays: a new type of fractional giant steps

We study, analytically and numerically, phase locking of driven vortex lattices in fully-frustrated Josephson junction arrays at zero temperature. We consider the case when an ac current is applied {\it perpendicular} to a dc current. We observe phase locking, steps in the current-voltage characteristics, with a dependence on external ac-drive amplitude and frequency qualitatively different from the Shapiro steps, observed when the ac and dc currents are applied in parallel. Further, the critical current increases with increasing transverse ac-drive amplitude, while it decreases for longitudinal ac-drive. The critical current and the phase-locked current step width, increase quadratically with (small) amplitudes of the ac-drive. For larger amplitudes of the transverse ac-signal, we find windows where the critical current is hysteretic, and windows where phase locking is suppressed due to dynamical instabilities. We characterize the dynamical states around the phase-locking interference condition in the $IV$ curve with voltage noise, Lyapunov exponents and Poincar\'e sections. We find that zero temperature phase-locking behavior in large fully frustrated arrays is well described by an effective four plaquette model.Comment: 12 pages, 11 figure