Noise rectification by a superconducting loop with two weak links

We consider a superconducting loop with two weak links that encloses a magnetic flux. The weak links are unequal and are treated as Josephson junctions with non-sinusoidal phase dependence. We devise a model that takes into account the fluctuation of the critical currents, due to the fluctuations of the order parameter in the weak links. These fluctuations are important near the onset of superconductivity; in this regime they may significantly weaken and eventually disconnect the superconducting loop. As a consequence of these fluctuations and of the resistive noise in the junctions, the average dc voltage does not vanish. Our model can be easily extended to provide a qualitative description of a recent experiment.

Figure 1

Schematic representation of the superconducting loop. The vertical thick lines represent segments which are “strongly” superconducting and the horizontal branches are weak links, which are modeled by Josephson junctions. These junctions are modeled by “pure” junctions in parallel with a resistor and a capacitor.

Figure 2

Current through a Josephson junction as a function of the phase difference for the case considered in Ref. 14. Here the unit of current is $wc{\xi }_A{H}_c^2∕{\Phi }_0$, where $w$ is the cross section of the wire, ${\xi }_A$ the coherence length in the strongly superconducting wire and $H_c$ the thermodynamic transition field. The length of the weak link is $2d{\xi }_A=0.002{\xi }_A$ and the curves are for several values of $\Gamma$, a parameter that defines the ratio between the strengths of superconductivity in both materials. The curves are marked with the number ${\log }_{10}(\Gamma ∕d)$. Only the curve for $\Gamma ={10}^{-1}d$ is nearly sinusoidal. In order to make the curve visible, the current values for the case $\Gamma ={10}^{-3}d$ were multiplied by 400.

Figure 3

The values of ${\gamma }_i$ behave as the coordinates of a particle in two dimensions (with anisotropic mass) which feels the potential in this graph, in addition to an anisotropic viscosity and a Langevin force. In this graph the darker areas represent lower potentials. This potential is periodic in ${\gamma }_1+{\gamma }_2$, with period $4\pi$. The parameters used to evaluate this graph are $L=I_{\mathrm{c}1}=I_{\mathrm{c}2}=1$, ${\beta }_1=-{\beta }_2=0.7$, ${\phi }_x=\pi ∕2$.

Figure 4

Voltage as a function of the magnetic flux ${\phi }_x∕\pi =2\Phi ∕{\Phi }_0$. (This function is odd and periodic in $\Phi$ with period ${\Phi }_0$.) The parameters used here were $\alpha =1$, $C=0$, $L=1$, $\beta =-0.7$, ${\delta }_0=0.02$, ${\delta }_1=0$, $\tau =5\times {10}^4$, $\eta =0.45$. For the dependence of the voltage on the parameters, see the text. The curve is a guide for the eye.

Figure 5

Voltage as a function of the strength of the fluctuations of the maximal supercurrent of the junctions. The flux is fixed at ${\phi }_x=0.5\pi$ and all the other parameters (except for ${\delta }_0$) are the same as in Fig. 4. The curve is a guide for the eye.

Figure 6

Average voltage as a function of the amplitude of the external ac current. The upper curve was evaluated for a period $2\pi ∕\omega =5\times {10}^4$, but we also searched for periods in the range between 50 and $5\times {10}^6$, and no appreciable difference was found; the lower curve is for $2\pi ∕\omega =10$. The other parameters used here were $\alpha =2$, $C=0$, $L=0.03$, $\beta =0.3$, ${\delta }_0=0.02$, ${\delta }_1=0$, $\tau =800$, $\eta =0.45$ and ${\phi }_x=\pi ∕2$.