Conductance of a superconducting Coulomb-blockaded Majorana nanowire
AbstractIn the presence of an applied magnetic field introducing Zeeman spin splitting, a superconducting (SC) proximitized one-dimensional (1D) nanowire with spin-orbit coupling can pass through a topological quantum phase transition developing zero-energy topological Majorana bound states (MBSs) on the wire ends. One of the promising experimental platforms in this context is a Coulomb-blockaded island, where by measuring the two-terminal conductance one can in principle investigate the MBS properties. Here, we theoretically study the tunneling transport of a single electron across the superconducting Coulomb-blockaded nanowire at finite temperature in order to obtain the generic conductance equation. By considering all possible scenarios where only MBSs are present at the ends of the nanowire, we compute the nanowire conductance as a function of the magnetic field, the temperature, and the gate voltage. In the simplest 1D topological SC model, the oscillations of the conductance peak spacings (OCPSs) arising from the Majorana overlap from the two wire ends manifest an increasing oscillation amplitude with increasing magnetic field (in disagreement with a recent experimental observation). We develop a generalized finite-temperature master-equation theory including not only multiple subbands in the nanowire, but also the possibility of ordinary Andreev bound states in the nontopological regime. Inclusion of all four effects (temperature, multiple subbands, Andreev bound states, and MBSs) provides a complete picture of the tunneling transport properties of the Coulomb-blockaded nanowire. Based on this complete theory, we indeed obtain OCPSs whose amplitudes decrease with increasing magnetic field in qualitative agreement with recent experimental results, but this happens only for rather high temperatures with multisubband occupancy and the simultaneous presence of both Andreev bound states and MBSs in the system. Thus, the experimentally observed OCPSs manifesting decreasing amplitude with increasing magnetic field can be explained in our theory only if the experimental magnetic field range encompasses both the nontopological and the topological regimes so that both Andreev bound states and Majorana bound states are contributing to these oscillations as well as the applicable electron temperature in the nanowire is rather high. A particularly significant aspect of our theory is that in such a high-temperature Coulomb-blockaded nanowire, the OCPSs no longer have a one-to-one correspondence with the nanowire quasiparticle energy spectrum as is generic in the low-temperature unblockaded situation. This implies that the OCPSs cannot be used to conclude about the low-energy spectrum so that no statement can be made about the so-called “topological protection” based on such OCPSs. In particular, the length dependence of the oscillation peak in such a situation is nongeneric and does not directly contain useful information about the Majorana splitting energy, reflecting only the physics of Andreev bound states in the finite-size nanowires used in the experiment.