Evaluation of Doppler shifts in $d$-wave superconductor tunneling junctions

The energy levels of quasiparticles in superconductors experience Doppler shifts due to the influence of a finite current flow. The influence of the Doppler shift is particularly pronounced in unconventional superconductors with gap nodes. In the case of a normal/insulator/superconductor junction of $d$-wave superconductors, a zero-bias conductance peak exhibits a peak shift in an applied magnetic field. Although the shifts have been detected in experiments, the shifts showed different field responses depending on each experiment. In this paper, we evaluate the magnitude of the shift based on realistic junction shapes to elucidate the origin of the variability in experimental Doppler shifts. We find that the peak shift depends on the junction shape; the splitting is significantly suppressed when the junction is formed at a protruding position. Also, the shift is suppressed by the penetration of magnetic flux quanta, resulting in a nonlinear response to the magnetic field. The present results explain the origin of the variability in experimental Doppler shifts.