Conditions for observing Shapiro steps in aBi2Sr2CaCu2O8+δhigh-Tcsuperconductor intrinsic Josephson junction: Numerical calculations

Abstract

Conditions to observe Shapiro steps clearly and stably are studied for an intrinsic Josephson junction (IJJ) in ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ high-${T}_{c}$ superconductors. The current equation normalized by the critical current ${I}_{c}(T)$ is solved fully numerically. In the calculations, the quasiparticle tunneling current is evaluated by using the normalized $I\text{\ensuremath{-}}V$ characteristics obtained within the $d$-wave symmetry superconducting gap, while the Cooper-pair (CP) one is calculated on the basis of the general way in which the coherent and incoherent CP tunneling currents can be correctly calculated within the $d$-wave treatment and the current due to thermal noises is also simulated by using normal random numbers. It is found that the product $S{R}_{\mathit{\text{shunt}}}$ of the junction cross section $S$ and the shunt resistance ${R}_{\mathit{\text{shunt}}}$, and the critical current density ${J}_{c}$ are important junction parameters, and moreover, that the current equation of the IJJ with no shunt resistance depends on only a universal curve $\ensuremath{\mu}({i}_{0})$ as a function of the normalized external dc current ${i}_{0}$. Furthermore, the effects of the noise, the normalized CP tunneling currents, the $S{R}_{\mathit{\text{shunt}}}$ product, the normalized amplitude ${i}_{r}$ of external ac modulation, and the ${J}_{c}$ on observing the Shapiro steps are studied. When the IJJ is operated under the condition that the shunt resistance is added and the external ac modulation frequency ${f}_{r}$ is higher than the plasma frequency ${f}_{p}$, it is found that (1) clear and stable Shapiro steps with good responses are obtained within the wide range of ${i}_{r}$, (2) the response does not so largely depend on the value of $S{R}_{\mathit{\text{shunt}}}$, and (3) the response for the high ${J}_{c}$ junction is much better than that for the low one.