Josephson flux-flow oscillator: The microscopic tunneling approach

We elaborate a theoretical description of large Josephson junctions which is based on Werthamer's microscopic tunneling theory. The model naturally incorporates coupling of electromagnetic radiation to the tunnel currents and, therefore, is particularly suitable for description of the self-coupling effect in Josephson junction. In our numerical calculations we treat the arising integro-differential equation, which describes temporal evolution of the superconducting phase difference coupled to the electromagnetic field, by the Odintsov-Semenov-Zorin algorithm. This allows us to avoid evaluation of the time integrals at each time step while taking into account all the memory effects. To validate the obtained microscopic model of large Josephson junction we focus our attention on the Josephson flux-flow oscillator. The proposed microscopic model of flux-flow oscillator does not involve the phenomenological damping parameter, rather the damping is taken into account naturally in the tunnel current amplitudes calculated at a given temperature. The theoretically calculated current-voltage characteristics is compared to our experimental results obtained for a set of fabricated flux-flow oscillators of different lengths.

Figure 1

Amplitudes of the pair (a) and quasiparticle (b) tunnel currents. Solid red and blue lines represent fit to the real and imaginary parts of the pair and quasiparticle currents in the form of a sum of exponents (18) with $N=8$ terms. The exact theoretical tunnel current amplitudes based on which the fitting was done are shown by dashed lines for comparison. To illustrate the behavior of the tunnel current amplitudes in the subgap region, $20\times$ zoom of the imaginary parts of the tunnel current amplitudes is shown in both figures. Relative difference of the fitted and exact amplitudes defined by Eq. (20) is shown in (c). Tunnel currents amplitudes in this figure are presented without the account of the pair current suppression (${\alpha }_{\mathrm{supp}}=1$).

Figure 2

Experimental (a),(b) and theoretical (c),(d) IVCs of FFO. In experimental IVCs the color scale of its branches corresponds to the rise in the SIS mixer dc current from 0 to 25% of the current step $I_g$ at the gap voltage (the more precise definition of $I_g$ is given in Ref. [107]). The data for the SIS mixer where dc current rises above the 25% threshold is painted by the same (red) color as the 25% rise. Power output in the numerical IVCs is expressed in units $V_g^2/Z_c{k}^2$ and is cut at the 0.12 threshold. Panels (a) and (b) show experimental IVCs of the FFOs with length $80\mu \mathrm{m}$ and $400\mu \mathrm{m}$, respectively. The corresponding numerical IVCs calculated with the use of the MTT are presented in (c) for $80\mu \mathrm{m}$ and (d) for $400\mu \mathrm{m}$. Values of the normalized external magnetic field $h_{\mathrm{ext}}$ vary with the step 0.07 from 1.20 to 4.28. In both numerical calculations Josephson penetration length is taken to be 5.5 $\mu \mathrm{m}$, normalized gap frequency $k=3.3$, surface damping $\beta =0.02$, and pair current suppression ${\alpha }_{\mathrm{supp}}=0.7$.