Influence of surface losses and the self-pumping effect on current-voltage characteristics of a long Josephson junction

We have numerically investigated the dynamics of a long linear Josephson tunnel junction with overlap geometry. Biased by a direct current (dc) and an applied dc magnetic field, the junction has important applications as tunable high frequency oscillator [flux-flow oscillator (FFO)] in the millimeter and submillimeter range. The study is performed in the frame of a modified sine-Gordon model, which includes surface losses, self-pumping effect, and in an empirical way the superconducting gap. The electromagnetic coupling to the environment is modeled by a simple resistor-capacitor load ($RC$ load) placed at both ends of the FFO. In our model, the damping parameter depends both on the spatial coordinate and on the amplitude of the ac voltage. In order to find the dc current-voltage curves, the damping parameter has to be calculated self-consistently by successive approximations and time integration of the perturbed sine-Gordon equation. The modified model gives better qualitative agreement with experimental results than the conventional perturbed sine-Gordon model.

Figure 1

The distribution of overlap component of bias current $\eta \left(x\right)$. Short-dashed line: $\eta \left(x\right)=\frac{{\eta }_{0}L}{\pi \sqrt{x(L-x)}}$; solid curve: $x_0=11$, $x_1=25.5$, $p=0.13$, and $a=0.005$.

Figure 2

A series of dc current-voltage characteristics each obtained for incremented values (from left to right $\Gamma =1.85;2.0;2.2;\dots ;3.6;3.8$) of the external magnetic field for the following parameters: $L=40$, $\alpha =0.033$, $\beta =0.035$, $c_L=c_R=10$, $r_L=2$, and $r_R=100$. Thin lines: experimental measurements. Numerical simulations: triangles, crosses, circles, diamonds, etc., for the current profile depicted in Fig. 1 by the solid line.

Figure 3

Calculated current-voltage characteristics of the FFO. Numerical simulations for different values of the load; crosses, unloaded symmetrical case; thin lines, asymmetric load $r_L=2$, $r_R=100$; and circles, perfect matching $r_L=r_R=1$; all other parameters are the same as used for the IVCs in Fig. 2.

Figure 4

Current-voltage characteristics of the FFO. Thin line—experimental measurements. Numerical simulations including the self-pumping effect—circles. Inset: the damping factor $\alpha \left(x\right)$ at the middle of the junction (crosses) and spatially averaged over $x$ (circles).

Figure 5

Spatial dependence of the shunt damping $\alpha \left(x\right)$ for three currents ${\eta }_t$; $\Gamma =3.7$: curve 1, ${\eta }_t=0.3$; curve 2, ${\eta }_t=0.2$; and curve 3, ${\eta }_t=0.1$. The dashed line represents the distribution of $\eta \left(x\right)$.