Conditions for observing Shapiro steps in a Bi₂Sr₂CaCu₂O_8+δ high-T_c superconductor intrinsic Josephson junction: Numerical calculations

Conditions to observe Shapiro steps clearly and stably are studied for an intrinsic Josephson junction (IJJ) in Bi₂Sr₂CaCu₂O_8+δ 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-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 SR_shunt of the junction cross section S and the shunt resistance R_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 μ(i₀) as a function of the normalized external dc current i₀. Furthermore, the effects of the noise, the normalized CP tunneling currents, the SR_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 SR_shunt, and (3) the response for the high J_c junction is much better than that for the low one.