Dynamic transition of vortices into phase slips and generation of vortex-antivortex pairs in thin film Josephson junctions under dc and ac currents

We present theoretical and numerical investigations of vortices driven by strong dc and ac currents in long Josephson junctions described by a nonlinear integro-differential equation which takes into account nonlocal electrodynamics of films, vortex bremsstrahlung, and Cherenkov radiation amplified by the attraction of vortices to the edges of the junction. This work focuses on the dynamics of vortices in Josephson junctions in thin films where the effects of Josephson nonlocality are essential but London screening is negligible. We obtained an exact solution for a vortex driven by an arbitrary time-dependent current in an overdamped junction where the vortex turns into a phase slip if the length of the junction is shorter than a critical length which depends on current. Our analytical and numerical results show that the dynamic behavior of vortices depends crucially on the ohmic damping parameter. In overdamped junctions vortices expand as they move faster and turn into phase slips as current increases. In underdamped junctions vortices entering from the edges produce Cherenkov radiation generating cascades of expanding vortex-antivortex pairs, which ultimately drive the entire junction into a resistive phase slip state. Simulations revealed a variety of complex dynamic states of vortices under dc and ac currents which can manifest themselves in hysteretic current-voltage characteristics with jumps and regions with negative differential resistance resulting from transitions from oscillating to ballistic propagation of vortices, their interaction with pinning centers, and standing nonlinear waves in the junction.

Figure 1

(a) Wakes of Cherenkov radiation behind a moving vortex in an infinite junction calculated from Eq. (2) with $G\left(x\right)=K_0(x/\lambda )$ for ${\lambda }_J/\lambda =10,\eta =0.07$ and different $\beta$. (b) Initial stage of vortex instability at $\beta =0.76$.

Figure 2

Geometries of a Josephson junction in a thin film with the vortex parallel (a) and perpendicular (b) to the broad face of the film.

Figure 3

Current streamlines in the AJ vortex with $u=0.3$ and $l=0.2$ calculated from Eq. (22) and (23) at $\beta =0$. The red and blue dots show the positions of fictitious A vortices which produce the current streamlines in the upper and the lower half plane, respectively, as described in the text.

Figure 4

$\theta \left(x\right)$ in a static vortex calculated from Eq. (16) for different values of $\epsilon$ as described in the text.

Figure 5

Penetration of single vortices in an overdamped junction with $\eta =2$ and ${\beta }_0=1.05$ calculated for $k=0.02$ and $\epsilon =2\times {10}^{-3}$.

Figure 6

Snapshots of moving vortices in the middle of the junction calculated from Eq. (16) for different currents at $\eta =2,k=0.02$, and $\epsilon =2\times {10}^{-3}$.

Figure 7

Upper and lower limits between which $\theta (x,t)$ oscillates, calculated for $\eta =2$ and ${\beta }_0=7$. The red curve shows ${\theta }_s\left(x\right)$ described by Eq. (37).

Figure 8

The dc voltage $\overline{V}=\langle V\left(t\right)\rangle$ calculated from Eq. (38) for different values of $\eta$, where $\langle ...\rangle$ denotes time averaging, and $V_0={\varphi }_0{\omega }_J/2\pi c$.

Figure 9

A wake radiated behind the moving vortex at $\eta =1$ and ${\beta }_0=0.995$. Here the vortex exits from the right edge followed by penetration of antivortex.

Figure 10

Snapshots of dynamic patterns formed by counter-moving vortices and antivortices calculated for $\eta =1,\epsilon =2\times {10}^{-3},k=0.02,{\beta }_0=1.05$ (a) and ${\beta }_0=1.09$ (b). Different colors correspond to different times $t$ during the time period after which the phase structures repeat themselves periodically after shifting up in $\theta$. As current further increases, the patterns shown in Figs. 10 and 10 gradually turn into a phase slip profile similar to that is shown in Fig. 7. The asymmetry of $\theta (x,t)$ with respect to $x=0$ is due to the effect of the gradient in $\beta \left(x\right)$.

Figure 11

Initial state of generation of V-AV pairs calculated at $\eta =0.7$ and ${\beta }_0=0.995$. The Cherenkov splitting instability of a vortex occurs right after it enters the junction and ultimately results in the dynamic pattern similar to those shown in Fig. 10.

Figure 12

Dissipation power vs dc current calculated for different damping constants shows a quadratic behavior at currents well above the threshold of penetration of a vortex.

Figure 13

Fourier spectra of voltage $V\left(t\right)$ calculated for $\eta =0.8$ and different currents corresponding to different number of vortices in the junction (left panel). Right panel shows the dc $\overline{V}\left({\beta }_0\right)$, where the jumps result from the change of number of vortices: 1 (i); 4 (ii); 12 (iii); phase slip (iv).

Figure 14

A vortex depinned from the defect at the left edge of the junction accelerates and produces a V-AV pair at $x\approx 0.1$ after the next vortex enters the junction. Simulations were done for ${\delta }_0=0.5,\beta =0.8$, and $\eta =0.3$.

Figure 15

A vortex accelerates as it approaches the defect in the center and decelerates once it passes the defect in the case of $\zeta =0.01,{\delta }_0=0.15,{\beta }_0=0.98$, and $\eta =1$.

Figure 16

At $\eta =0.7$ even a weak defect can accelerate the approaching vortex so that it produces a critical radiation wake which generates a V-AV pair. Figure shows the dynamics of a vortex in the absence (left) and the presence (right) of a defect with ${\delta }_0=0.05,\zeta =0.05$, and ${\beta }_0=0.98$.

Figure 17

Interaction of a vortex penetrating from left with a V-AV pair produced simultaneously by a weak defect with ${\delta }_0=0.2$ and $\zeta =0.01,k=0.1$ and $\eta =1$ at the threshold current ${\beta }_0=0.98$. The vortex which entered from the left edge annihilates with the antivortex produced at the defect, and the remaining vortex exits from the right edge.

Figure 18

Oscillatory dynamics of vortices in an overdamped junction with $\eta =2$ at the penetration threshold ${\beta }_0=1.237$. The vortex enters the junction during the positive cycle of $\beta \left(t\right)$, stops midway when $\beta \left(t\right)=0$, turns around, and exits through the edge during the negative cycle of $\beta \left(t\right)$.

Figure 19

Ballistic penetration of vortices and antivortices into an overdamped junction with $\eta =2$ at ${\beta }_0=1.245$. Here vortices and antivortices traverse the junction and exit from the other end. Notice that the moving vortex extends nearly over the entire junction and produces no visible radiation.

Figure 20

Fourier spectrum of voltage at $\eta =2$ calculated for oscillatory vortex dynamics at ${\beta }_0=1.237$ and ballistic vortex penetration at ${\beta }_0=1.245$ represented in Figs. 18 and 19, respectively. The peaks in $V_{\omega }$ occur at the multiples of the ac frequency ${\omega }_n=n\omega$, where $\omega =\pi /30$ and $n=1,2,3,...$. Notice that voltage harmonics with even $n$ disappear as the vortex dynamics changes from oscillatory to ballistic.

Figure 21

AC power plots $\overline{P}\left({\beta }_0\right)$ for different damping constants. At large currents ${\beta }_0\gtrsim 3$, the curves $\overline{P}\left({\beta }_0\right)$ approach the ohmic limit $\overline{P}=P_0{\beta }_0^2/2\eta$.

Figure 22

Penetration of a second vortex during positive cycle on top of ballistic penetration of first vortex which results in the second step in $\overline{P}\left({\beta }_0\right)$ curve at $\eta =1$ in Fig. 21 calculated for ${\beta }_0=1.26$.

Figure 23

Penetration of radiating vortices and antivortices at $\eta =0.4$ and ${\beta }_0=1.034$.

Figure 24

Generation of V-AV pair by the accelerating antivortex exiting the junction at $\eta =0.7$ and ${\beta }_0=1.085$. Here the V-AV pair is produced inside the junction, unlike J vortices which only penetrate through the edges [46]. Inset shows instantaneous currents corresponding to different phase profiles at times: $2\pi /15$ (1); $13\pi /30$ (2); $17\pi /30$ (3); $39\pi /60$ (4); $7\pi /10$ (5); $23\pi /30$ (6).

Figure 25

Dynamic vortex patterns calculated at $\eta =0.7$ and: ${\beta }_0=1.092$ (a); ${\beta }_0=1.098$ (b); ${\beta }_0=1.149$ (c).

Figure 26

Dynamics of $\theta (-1/2,t)$ and $\theta (1/2,t)$ at the edges at $\eta =0.2$ and ${\beta }_0=1.1$. Here $\theta (x,t)$ remains nearly uniform along the junction, indicating a phase slip behavior.