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

Abstract

We find that a conventional s-wave superconductor in proximity to a three-dimensional Dirac material (3DDM), to all orders of perturbation in tunneling, induces a combination of s- and p-wave pairing only. We show that the Lorentz invariance of the superconducting pairing prevents the formation of Cooper pairs with higher orbital angular momenta in the 3DDM. This no-go theorem acquires stronger form when the probability of tunneling from the conventional superconductor to positive and negative energy states of 3DDM are equal. In this case, all the p-wave contribution except for the lowest order, identically vanish and hence we obtain an exact result for the induced p-wave superconductivity in 3DDM. Fierz decomposing the superconducting matrix we find that the temporal component of the vector superconducting order and the spatial components of the pseudovector order have odd-frequency pairing symmetry. We find that the latter is odd with respect to exchange of position and chirality of the electrons in the Cooper pair and is a spin-triplet, which is necessary for NMR detection of such an exotic pseudovector pairing. Moreover, we show that the tensorial order breaks into a polar vector and an axial vector and both of them have conventional pairing symmetry except for being a spin triplet. According to our study, for gapless 3DDM, the tensorial superconducting order will be the only order that is odd with respect to the chemical potential μ. Therefore we predict that a transverse p−n junction binds Majorana fermions. This effect can be used to control the neutral Majorana fermions with electric fields.

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