Topologically nontrivial and trivial zero modes in chiral molecules

Recently, electron transport along chiral molecules has been attracting extensive interest and a number of intriguing phenomena have been reported in recent experiments, such as the emergence of zero-bias conductance peaks in the transmission spectrum upon the adsorption of single-helical protein on superconducting films. Here, we present a theoretical study of electron transport through a two-terminal single-helical protein sandwiched between a superconducting electrode and a normal-metal one in the presence of a perpendicular magnetic field. As the proximity-induced superconductivity attenuates with the distance from superconducting media, the pairing potential along the helix axis of the single-helical protein is expected to decrease exponentially, which is characterized by the decay exponent $\ensuremath{\lambda}$ and closely related to the experiments. Our results indicate that (i) a zero-bias conductance peak of $2{e}^{2}/h$ appears at zero temperature and the peak height (width) decreases (broadens) with increasing temperature, and (ii) this zero-bias peak can split into two peaks, which are in agreement with the experiments [see, e.g., H. Alpern et al. Nano Lett. 19, 5167 (2019)]. Remarkably, Majorana zero modes are observed in this protein-superconductor setup in a wide range of model parameters, as manifested by the ${Z}_{2}$ topological invariant and the Majorana oscillation. Interestingly, a specific region is demonstrated for decaying superconductivity, where topologically nontrivial and trivial zero modes coexist and the bandgap remains constant. With increasing the pairing potential, the topologically nontrivial zero modes will transform to the trivial ones without any bandgap closing-reopening, and the critical pairing potential of the phase transition attenuates exponentially with $\ensuremath{\lambda}$. Additionally, one of the two zero modes can be continuously shifted from one end of the protein toward the other end contacted by the normal-metal electrode. The underlying physics of the topologically nontrivial and trivial zero modes is discussed.