Model for missing Shapiro steps due to bias-dependent resistance

Abstract

Majorana zero modes are predicted in several solid state systems such as hybrid superconductor-semiconductor structures and topological insulators coupled to superconductors. One of the expected signatures of Majorana modes is the fractional 4$\pi$ Josephson effect. Evidence in favor of this effect often comes from a.c. Josephson effect measurements and focuses on the observation of missing first or higher odd-numbered Shapiro steps. However, the disappearance of the odd Shapiro steps has also been reported in conventional Josephson junctions where no Majorana modes are expected. In this paper, we present a phenomenological model that displays suppression of the odd Shapiro steps. We perform resistively-shunted junction model calculations and introduce peaks in differential resistance as function of the bias current. In the presence of only the standard 2$\pi$ Josephson current, for chosen values of peak positions and amplitudes, we can suppress the odd Shapiro steps, or any steps, thus providing a possible explanation for the observation of missing Shapiro steps.