Theory of macroscopic quantum tunneling in high-$T_c$ $c$-axis Josephson junctions

We study macroscopic quantum tunneling (MQT) in $c$-axis twist Josephson junctions made of high-$T_c$ superconductors in order to clarify the influence of the anisotropic order parameter symmetry (OPS) on MQT. The dependence of the MQT rate on the twist angle $\gamma$ about the $c$ axis is calculated by using the functional integral and the bounce method. Due to the $d$-wave OPS, the $\gamma$ dependence of standard deviation of the switching current distribution and the crossover temperature from thermal activation to MQT are found to be given by $\cos 2\gamma$ and $\sqrt{\cos 2\gamma }$, respectively. We also show that a dissipative effect resulting from the nodal quasiparticle excitation on MQT is negligibly small, which is consistent with recent MQT experiments using ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ intrinsic junctions. These results indicate that MQT in $c$-axis twist junctions becomes a useful experimental tool for testing the OPS of high-$T_c$ materials at low temperature, and suggest high potential of such junctions for qubit applications.

Figure 1

(Color online) Schematic picture of the $c$-axis twist Josephson junction. $\gamma$ is the twist angle around the $c$ axis.

Figure 2

The twist angle $\gamma$ dependence of the critical current $I_C$. Broken line is proportional to $\cos 2\gamma$.

Figure 3

(Color online) The twist angle $\gamma$ dependences of (a) the switching current distribution $P\left(\eta \right)$ and (b) the mass increment $\delta M$ due to the quasiparticle dissipation.

Figure 4

(Color online) The standard deviation ${\sigma }_n$ of the switching current distribution $P\left(\eta \right)$ for various $\gamma$. The dotted line is proportional to $T^{2∕3}$. The arrows point to the crossover temperature $T^{*}$ calculated from the approximate formula (53).

Figure 5

The twist angle $\gamma$ dependences of (a) the standard deviation ${\sigma }_n$ of the switching current distribution $P\left(\eta \right)$ and (b) the crossover temperature $T^{*}$. The broken lines are proportional to $\cos 2\gamma$ and $\sqrt{\cos 2\gamma }$ in (a) and (b), respectively.

Figure 6

(Color online) The dependences of the crossover temperature $T^{*}$ on (a) the sweep rate $v$, (b) the critical current $I_C\left(0\right)$ which is proportional to ${\Delta }_0$ [we set $I_C\left(0\right)=48.54\mu \mathrm{A}$ at ${\Delta }_0=30\mathrm{meV}$], and (c) the capacitance $C$ for $c$-axis junction with $\gamma =0$.

Figure 7

Numerically obtained bounce solution for the action with the Ohmic nonlocal $\beta$ term for ${\omega }_0{\tau }_0=1$ is shown by the solid line. The usual bounce solution for the action with local approximation, which is given by ${\psi }_0\left(\tilde{\tau }\right)=3∕{\cosh }^2(\tilde{\tau }∕2)$, is also shown by the broken line.

Figure 8

The normalized bounce action $\overline{S}$ as a function of the cutoff parameter ${\tau }_0$ obtained by numerical calculation (solid line) and perturbative method (dotted line). In the limit ${\omega }_0{\tau }_0\to 0$, the bounce action becomes 4.8, which corresponds to the local approximation.