$\delta$-biased Josephson tunnel junctions

The behavior of a long Josephson tunnel junction drastically depends on the distribution of the dc bias current. We investigate the case in which the bias current is fed in the central point of a one-dimensional junction. Such junction configuration has been recently used to detect the persistent currents circulating in a superconducting loop. Analytical and numerical results indicate that the presence of fractional vortices leads to remarkable differences from the conventional case of uniformly distributed dc bias current. The theoretical findings are supported by detailed measurements on a number of $\delta$-biased samples having different electrical and geometrical parameters.

Figure 1

(a) Gapped annular and (b) linear $\delta$-biased Josephson tunnel junctions. The base electrodes are in black, the top electrodes are in gray, and the junction areas are white. A gapped annular junction in a radially uniform magnetic field $H_r$ is topologically equivalent to a linear junction in an in- plane uniform field $H_{\parallel }$.

Figure 2

Three-dimensional sketch of a centrally injected $\delta$-biased linear planar Josephson tunnel junction.

Figure 3

(a) Phase profile ${\varphi }^e\left(x\right)$ for an infinitely long $\delta$-biased JTJ as in Eq. (10) with $\xi =0$ corresponding to $\gamma =4$, for $x∊\left[-10,10\right]$. (b), (c), and (d) show, respectively, ${\varphi }_x^e\left(x\right)$, ${\varphi }_{xx}^e\left(x\right)$, and $\text{sin}{\varphi }^e\left(x\right)$. It is $\text{cos}{\varphi }^e\left(x\right)=1-{\left[{\varphi }_x^e\left(x\right)\right]}^2/2$.

Figure 4

(a) Phase profile ${\varphi }^o\left(x\right)$ as in Eq. (15) for $\gamma =0$, $h=2$, and $l=20$, (b) its first derivative, (c) its second derivative, and (d) its cosines. Note that $\text{sin}{\varphi }^o\left(x\right)={\varphi }_{xx}^o\left(x\right)$ and $\text{cos}{\varphi }^o\left(x\right)=1-{\left[{\varphi }_x^o\left(x\right)\right]}^2/2$.

Figure 5

(a) $\varphi \left(x\right)$ as in Eq. (18) for $h=\gamma /2=2$ $\left(\zeta =\xi =0\right)$ and $l=20$; (b) $\text{sin}\varphi \left(x\right)$. Only the junction right side contributes to the Josephson current. It is $\text{sin}\varphi \left(x\right)={\varphi }_{xx}\left(x\right)$ and $\text{cos}\varphi =1-{\left[\varphi \left(x\right)\right]}^2/2$.

Figure 6

Numerically computed magnetic diffraction patterns for different junction normalized lengths. From (a) to (d) $i_c$ vs $h$, respectively, for $l=2,4,8,16$. The full dots refer to point-injected bias current, while, for the sake of comparison, the open dots correspond to the well-known case of a uniformly biased junction.

Figure 7

Numerically computed phase profiles for higher-order modes with $l=16$ and $h=1.99$ and different $\gamma$ values: (a) 2 vortex mode for $\gamma =4.15$ and (b) 3 vortex mode for $\gamma =4.6$.

Figure 8

Log-log plot of the measured zero-field critical currents $i_c\left(0\right)$ of many $\delta$-biased JTJs versus their reduced length $L/{\lambda }_j$. The critical current has been normalized to the small junction theoretical value calculated as the 70% of the current jump at the junction gap voltage. The solid line is obtained from Eq. (11) with ${\gamma }_c$ as in the empirical Eq. (12).

Figure 9

Experimental magnetic diffraction patterns taken for linear samples in a uniform in-plane field with increasing normalized lengths $l=L/{\lambda }_j$. (a) $L=250\mu \text{m}$ and ${\lambda }_j\simeq 80\mu \text{m}$, (b) $L=300\mu \text{m}$ and ${\lambda }_j\simeq 80\mu \text{m}$, (c) $L=150\mu \text{m}$ and ${\lambda }_j\simeq 12\mu \text{m}$, and (d) $L=300\mu \text{m}$ and ${\lambda }_j\simeq 12\mu \text{m}$.

Figure 10

Experimental magnetic diffraction patterns in a radially uniform magnetic field $H_r$ for two gapped annular samples having the same critical current density $J_c=3\text{kA}/{\text{cm}}^2$ (corresponding to ${\lambda }_j\simeq 12\mu \text{m}$) and different lengths $L$. (a) $L=82\mu \text{m}$ and (b) $L=200\mu \text{m}$.

Figure 11

Profiles of the first zero-field step for overlap junctions with $L=150\mu \text{m}$ and $J_c=3\text{kA}/{\text{cm}}^2$ (corresponding to ${\lambda }_j\simeq 12\mu \text{m}$). (a) $\delta$ bias (solid line) and (b) uniform bias (dashed line). The data were taken a $T\approx 6\text{K}$. The arrows indicate the switching direction at the top of the step.