Anomalous transport in biased ac-driven Josephson junctions: Negative conductances

We investigate classical anomalous electrical transport in a driven, resistively, and capacitively shunted Josephson junction device. Intriguing transport phenomena are identified in chaotic regimes when the junction is subjected to both a time-periodic (ac) and a constant biasing (dc) current. The dependence of the voltage across the junction on the dc exhibits a rich diversity of anomalous transport characteristics. In particular, depending on the chosen parameter regime, we can identify the so termed absolute negative conductance around zero dc bias, the occurrence of negative differential conductance, and, after crossing a zero conductance, the emergence of a negative nonlinear conductance in the nonequilibrium response regime remote from zero dc bias.

Figure 1

(Color online) The diagram depicts the dependence of the voltage versus the variation of the driving amplitude $i_1$ of the ac for a positive dc set at $i_0=0.0159$. Remaining parameters are chosen as $\sigma =0.143$, ${\Omega }_1=0.78$, and temperatures $D=0$ [dotted line (blue), left scale] and $D={10}^{-3}$ [solid line (black), right scale].

Figure 2

(Color online) The functional dependence of voltage $v$ on the dc current $i_0$ ($i\text{−}v$ characteristic) is compared for two sets of parameters, which either display negative conductance or Shapiro steps. Panel (a), which corresponds to ac frequency ${\Omega }_1=0.78$, ac-driving strength $i_1=0.73$, friction $\sigma =0.143$, and noise strength $D={10}^{-3}$, displays both absolute negative mobility in an interval containing $i_0=0$ and different regions with negative differential conductivity (cf. also the respective enlarged regions in the two insets). Panel (b) displays the $i\text{−}v$ characteristic for the parameter values of panel (a) (thin solid line) and the $i\text{−}v$ characteristic for ${\Omega }_1=7.8$, $i_1=73$, $\sigma =0.143$, and $D=0$ (thick solid line). Shapiro steps are clearly visible at $v=k{i}_0∕\sigma$ for $k=1∕2,1,2,3$ (dotted horizontal lines). Away from these steps, the $i\text{−}v$ charateristic follows the Ohmic law $v=i_0∕\sigma$ (dotted inclined line). Panel (c) presents the $i\text{−}v$ characteristic for the small driving frequency $({\Omega }_1=0.78)$ for positive bias currents $i_0$ up to $i_0=0.48$ (solid line) and the $i\text{−}v$ characteristic for the same set of junction parameters at zero noise $(D=0)$ (dotted line with wide distance of dots) and compares them with a rescaled version $v_c≔0.1v\left(10i_0\right)$ of the $i\text{−}v$ characteristic for the larger frequency such that the full range shown in panel (b) is displayed again. Whereas the linear overall behavior of the large frequency curve is interrupted only at a few Shapiro steps, the small frequency $i\text{−}v$ characteristic is dominated by the nonlinearities also away from the locking regions at $v={\Omega }_1∕2$ and $v=3{\Omega }_1$.

Figure 3

Voltage vs dc: The reentrant effect of the negative response $(v<0)$ is shown at a positive bias $(i_0>0)$ for the following set of parameters: $i_1=0.668$, $\sigma =0.143$, ${\Omega }_1=0.78$, and four values of temperature $D$. For small $i_0$, the absolute negative conductance is observed for nonzero temperature and it is solely induced by thermal fluctuations [cf. panel (a)]. For larger $i_0$, negative nonlinear conductance results as a deterministic effect. The corresponding long-time trajectory $\varphi \left(t^{\prime }\right)$ for $i_0=0.0796$ shown in panel (b) belongs to running states into the negative direction of $\varphi$ resulting in the large average voltage. The value of the amplitude of ac driving corresponds to the first negative-valued minimum of the mean voltage with respect to the ac (cf. in Fig. 1).

Figure 4

(Color online) The transport properties of the driven system in the parameter space $\{i_1,{\Omega }_1\}$ at a representative friction value of $\sigma =0.191$, dc bias $i_0=0.0159$, and zero noise strength $D=0$. All points in the parameter space, where a negative conductance occurs, are marked by symbols with different gray scales. The coding corresponds to different regimes of values assumed by the ratio $v∕{\Omega }_1$: dark gray (red) denotes the interval $(-0.94,0)$, light gray (green) $(-1.26,-0.94)$, and black (blue) corresponds to values less than $-1.26$. Most of the dark gray points correspond to chaotic trajectories. It turns out that the black regimes are most susceptible to thermal fluctuations, which means that the negative-valued conductance rapidly fades away toward positive values with increasing thermal noise intensity. In panel (a), one clearly can distinguish a large compact region exhibiting negative-valued conductance, being enlarged with the upper panel (b). In regions different from this one, we find that negative-valued conductance occurs in narrow “bands” only. For an experimental realization of negative conductance, the most promising parameter regimes are those regions marked by light gray. This holds, in particular, for the zoomed light gray regimes depicted in panel (b).

Figure 5

(Color online) This figure elucidates the transport features and the dynamical properties for parameter values in $\{i_1,{\Omega }_1\}$ space within the section obtained by fixing the angular driving frequency at ${\Omega }_1=1.1$ [cf. the dotted line in panel (b) of Fig. 4]. The blue (solid) line in panel (a) depicts the dependence of the average voltage on the amplitude of the ac for zero noise $D=0$. The influence of thermal fluctuations on the average voltage is shown for two temperature values $D={10}^{-3}$ (dashed line) and $D=3\times {10}^{-3}$ (dotted line). We do find that a negative-valued conductance indeed can survive at small nonzero temperatures. The transport properties of the system follow from the underlying complex dynamics. In panel (b), we show the bifurcation diagram: the Poincare section of the phase in the deterministic system. The attractors are color coded according to the sign of the assumed corresponding average voltage: blue yields a negative average voltage (also pointed by arrows with $v<0$), green gives zero (also pointed by arrows with $v=0$), and red amounts to a positive average voltage (also pointed by arrows with $v>0$).