Tunable $\pm \phi ,{\phi }_0$, and ${\phi }_0\pm \phi$ Josephson junction

We study a 0-$\pi$ dc superconducting quantum interference device (SQUID) with asymmetric inductances and critical currents of the two Josephson junctions (JJs). By considering such a dc SQUID as a black box with two terminals, we calculate its effective current-phase relation $I_s\left(\psi \right)$ and the Josephson energy $U\left(\psi \right)$, where $\psi$ is the Josephson phase across the terminals. We show that there is a domain of parameters where the black box has the properties of a $\phi$ JJ with degenerate ground state phases $\psi =\pm \phi$. The $\phi$ domain is rather large, so one can easily construct a $\phi$ JJ experimentally. We derive the current phase relation and show that it can be tuned in situ by applying an external magnetic flux resulting in a continuous transition between the systems with static solutions $\psi =\pm \phi ,\psi ={\phi }_0 \left({\phi }_0\ne 0,\pi \right)$ and even $\psi ={\phi }_0\pm \phi$. The dependence of ${\phi }_0$ on applied magnetic flux is not $2\pi$ (one flux quantum) periodic.

Figure 1

The energy $U\left(\psi \right)$ of the system given by Eq. (5) for different values of the asymmetry parameter $\alpha$ and for ${\beta }_1={\beta }_2=0.7 \left({\alpha }_c\approx -0.417\right)$. Inset shows the schematics of the circuit considered.

Figure 2

The domain of the $\phi$ state (pink/gray) as a function of parameters $\alpha ,{\beta }_1$, and ${\beta }_2$.

Figure 3

Josephson energy $U\left(\psi \right)$ for ${\beta }_1=0.4,{\beta }_2=0.6,r_1=0.4$ and different $f=0...1$ specified next to each curve. (a) $\alpha =0.7>|{\alpha }_c|=0.5$; (b) $\alpha =0.4<|{\alpha }_c|=0.5$.

Figure 4

The ground state phase $\pm \phi \left(\alpha \right)$. The horizontal dashed line shows the unstable static solution $\psi =0$ in the region $\alpha <{\alpha }_c$. Thin line shows the approximation given by Eq. (A11).

Figure 5

The phase diagram of 0-$\pi$ dc SQUID [ground state phase $\phi ({\beta }_1,{\beta }_2)]$ for (a) $\alpha =-0.6$ and (b) $\alpha =-0.8$. The dashed line shows the boundary between trivial $\psi =0$ and $\psi =\pm \phi$ ground states given by expression (15). Continuous lines are the lines of the constant ground state phase $\phi$. Its value is given next to each line.

Figure 6

Circulating current as a function of $\alpha$. Tilted dashed line shows the value of the critical current of the $\alpha$ junction. The horizontal dashed-dotted line shows the maximum value of persistent current reached at $\alpha =-1$ and calculated using Eq. (17). Vertical dashed-dotted line shows the value ${\alpha }_{\mathrm{circ}}^{\max }$ calculated using Eq. (18).

Figure 7

The amplitudes of the spontaneous circulating current $I_{\mathrm{circ}}\left({\beta }_{\mathrm{\Sigma }}\right)$ and spontaneous flux $\mathrm{\Phi }\left({\beta }_{\mathrm{\Sigma }}\right)/{\mathrm{\Phi }}_0$ for fixed $\alpha =-0.7$.

Figure 8

Numerically calculated dependencies ${\gamma }_{c\pm }\left(\alpha \right)$ for (a) ${\beta }_1={\beta }_2=0.7$ and (b) ${\beta }_1=0.1$ and ${\beta }_2=0.9$.

Figure 9

The dependence of the critical current ${\gamma }_c$ of the device on the normalized applied magnetic field (frustration) $f$ for ${\beta }_1={\beta }_2=0.7 (\pi {\beta }_L=1.4,{\alpha }_c\approx -0.417)$ and different $\alpha$.