Superconducting proximity in three dimensional Dirac materials: odd-frequency, pseudoscalar, pseudovector and tensor-valued superconducting orders

摘要

We find that a conventional s-wave superconductor in proximity to threedimensional Dirac material (3DDM), to all orders of perturbation in tunneling,induces a combination of s and p-wave pairing only. We show that the Lorentzinvariance of the superconducting pairing prevents the formation of Cooperpairs with higher orbital angular momenta in the 3DDM. This no-go theoremacquires stronger form when the probability of tunneling from the conventionalsuperconductor to positive and negative energy states of 3DDM are equal. Inthis case all the p-wave contribution except for the lowest order, identicallyvanish and hence we obtain an exact result for the induced p-wavesuperconductivity in 3DDM. Fierz decomposing the superconducting matrix we findthat temporal component of the vector superconducting order and spatialcomponents of the pseudo-vector order are odd-frequency pairing. We find thatthe latter is odd with respect to exchange of position and chirality of theelectrons in the Cooper pair and is spin-triplet which is necessary for NMRdetection of such an exotic pseudo- vector pairing. Moreover, we show that thetensorial order breaks into a polar vector and an axial vector and both of themare conventional pairing except for being spin-triplet. According to our study,for gapless 3DDM the tensorial superconducting order will be the only orderwhich is odd with respect to the chemical potential {\mu}. Therefore we predictthat a transverse p-n junction binds Majorana fermions. This effect can be usedto control the neutral Majorana fermions with electric fields.