Phase dynamics in graphene-based Josephson junctions in the presence of thermal and correlated fluctuations

We study by numerical methods the phase dynamics in ballistic graphene-based short Josephson junctions. A superconductor-graphene-superconductor system exhibits superconductive quantum metastable states similar to those present in normal current-biased Josephson junctions. We investigate the effects of thermal and correlated fluctuations on the escape time from these metastable states, when the system is driven by an oscillating bias current in the presence of Gaussian white and colored noise sources. Varying the intensity and the correlation time of the noise source, it is possible to analyze the behavior of the escape time, or switching time, from a superconductive metastable state in different temperature regimes. Moreover, we are able to clearly distinguish dynamical regimes characterized by the dynamic resonant activation effect, in the absence of noise source, and the stochastic resonant activation phenomenon induced by the noise. For low initial values of the bias current, the dynamic resonant activation shows double-minimum structures, strongly dependent on the value of the damping parameter. Noise-enhanced stability is also observed in the system investigated. We analyze the probability density function (PDF) of the switching times. The PDFs for frequencies within the dynamic resonant activation minima are characterized by single peaks with exponential tails. The PDFs for noise intensities around the maxima of the switching time, peculiarity of the noise-enhanced stability phenomenon, are composed of regular sequences of two peaks for each period of the driving current, with exponentially decaying envelopes.

Figure 1

Schematic view of a suspended SGS device. The electrons forming a Cooper pair, when they enter graphene, move into different K valleys, represented as orange cones. In the short-junction regime, $L\ll W$.

Figure 2

Washboard potential for SNS [see Eq. (3)] and SGS [see Eq. (9)] JJs (dashed and solid lines, respectively), for different values of the constant term of the bias current: $i_0=0.0$ (a), 0.5 (b), 0.9 (c). Also shown is the initial position (bottom of the potential well) of the “phase particle.” Blue dotted-dashed and pink dotted lines indicate the left and right absorbing barriers, respectively.

Figure 3

MST as a function of both $\omega$ and $\gamma$, for ${\tau }_c=0.0$ and different initial values of the bias current: (a) $i_0=0.0$ (no slope), (b) $i_0=0.1$ (small slope), (c) $i_0=0.5$ (intermediate slope), (d) $i_0=0.9$ (high slope). In all panels the values of the other parameters are $A=0.7,{\beta }_J=0.1$. The legend in panel (b) refers to all pictures.

Figure 4

(a) MST as a function of the driving frequency $\omega$, for $\gamma ={10}^{-4}$, and different initial values of the bias current, namely $i_0=\{0.0,0.1,0.5,0.9\}$. Solid and dotted lines represent results for SGS and SNS junctions, respectively. The dynamical resonant activation phenomenon is clearly evident. (b) Ensemble average trajectories of the phase particle for two values of the driving frequency, that is, ${\omega }_{dRA}^{0.1}\simeq 0.75$ and $0.95$, corresponding to the two minima of DRA observed at $i_0=0.1$. Blue dotted-dashed and pink dotted lines indicate the left and right absorbing barriers, respectively. (c) MST as a function of the driving frequency $\omega$, for $i_0=0.1$ and different values of noise intensity $\gamma$ ranging from $0.2$ to 2. The stochastic resonant activation phenomenon is clearly evident. In all panels the values of the other parameters are ${\tau }_c=0.0,A=0.7,{\beta }_J=0.1$.

Figure 5

MST as a function of the driving frequency $\omega$ and the damping parameter ${\beta }_J$, for $\gamma ={10}^{-4},{\tau }_c=0.0,i_0=0.1$, and $A=0.7$.

Figure 6

MST as a function of $\gamma$, for different values of $\omega ,i_0$, and ${\tau }_c$. Specifically, (a) $i_0=0.1,\omega \in [0.49,0.77]$ and ${\tau }_c=0.0$; (b) $i_0=0.1,\omega \in [0.9,1.02]$ and ${\tau }_c=0.0$; (c) $i_0=0,\omega =0.44$; (d) $i_0=0.1,\omega =0.44$; (e) $i_0=0.1,\omega =1.0$; (f) $i_0=0.5,\omega =0.6$; (g) $i_0=0.5,\omega =1.08$; (h) $i_0=0.9,\omega =1.18$. The values of the other parameters are $A=0.7,{\beta }_J=0.1$. The legend in panel (h) refers to all pictures except to panels (a) and (b).

Figure 7

MST as a function of $\gamma$, for $\omega =0.75,i_0=0.0,A=0.7,{\beta }_J=0.1$ and different values the noise correlation time: ${\tau }_c=0.0,1.0,5,10$. Symbols and lines represent results for SGS and SNS junctions, respectively.

Figure 8

Panels (a), (b), (c), and (d): PDFs as a function of the time $t$, varying $\omega$. Every picture is obtained fixing $\gamma ={10}^{-4},{\tau }_c=0,A=0.7$, and $i_0$ = (a) 0.0, (b) 0.1, (c) 0.5, (d) 0.9. The MST $\tau$ vs $\omega$ curves, corresponding to the dynamic RA effect [see solid lines in Fig. 4], are also shown. The PDF and $t$ axes are logarithmic. Panel (e): Semilog plot of the PDFs as a function of the time $t$, normalized to the driving period $T_p$, setting $\gamma ={10}^{-4},{\tau }_c=0,A=0.7,i_0=0.1$, and $\omega ={\omega }_{dRA}^{0.1}=0.75$, 0.95. The inset shows the PDF for $\omega =0.75$ of the bias currents $i_b\left(t\right)$ corresponding to escape or switching events of panel (e).

Figure 9

PDF as a function of the time $t$, normalized to the driving period $T_p$, varying $\gamma$. Every picture is obtained fixing ${\tau }_c=0,A=0.7$, and the values of $i_0$ and $\omega$. In detail, in each panel (a) $i_0=0,\omega =0.44$, (b) $i_0=0.1,\omega =0.44$, (c) $i_0=0.5,\omega =1.08$, and (d) $i_0=0.9,\omega =1.18$. Every picture shows also the MST versus $\gamma$ curve corresponding to the NES effect (see solid lines in Fig. 6), obtained using the same values for the other parameters. The insets in panels (a) and (b) show the detail of the peaks contained in one period of oscillation of the potential for $\gamma ={10}^{-2}$. The PDF and $\gamma$ axes are logarithmic. The legend refers to all panels.